George N Frantziskonis
 Professor, Civil EngineeringEngineering Mechanics
 Professor, Materials Science and Engineering
 Member of the Graduate Faculty
 (520) 6214347
 Civil Engineering, Rm. 206A
 Tucson, AZ 85721
 frantzis@email.arizona.edu
Biography
George Frantziskonis received his civil engineering degree from the Aristotle University in Greece, in 1982 and his doctorate in engineering mechanics from the University of Arizona in 1986. He joined the University of Arizona faculty in 1988 after a visiting position at Aristotle University and consulting for the industry. He is a registered professional civil engineer in the State of Arizona and in the European Union state members. His teaching interests and activities include contemporary, collaborativestyle (flipped classroom), multimedia course delivery methods tailored to classroombased, distance learning, and classes of large enrollment. He has taught numerous courses in the broad areas of civil, mechanical, aerospace, engineering mechanics, and engineering design. He has directed graduate students in civil engineering, engineering materials, and mechanical, aerospace engineering. Frantziskonis’s areas of research interests and activities include multiand interdisciplinary multiscale modeling, simulation, and experimentation, material characterization and applications, probabilistic and multiscale problem formulation and description and applications to safety and reliability, behavior of materials at nanoscale, reactiondiffusiontransport and reactive flow problems, smart materials at a multiplicity of scales. Applications include material characterization and modeling, response of materials to rare stochastic events, and reactiondiffusiontransport phenomena in energy production processes. He has published extensively in journals in the areas of (some journals where his work has been published are shown in parentheses), the diversity coming primarily from extending multiscale methods to several areas of science and engineering:
 Mechanics/Materials ( J. Solids & Structures, Acta Mechanica, Comp. Structures)
 Physics ( Rev. B, Phys. Rev. E, J. Phys. A, J. Phys. Cond. Matter)
 Civil Engineering (ASCE journals, & Geotechnics, Structural Control and Health Monitoring)
 Mechanical Engineering ( Mech. Rev., J. Comp.aided Design)
 Materials Science and Engineering (Smart Mater. & Struct., Intelligent Mat. Systems, Mod. Simul. Mater. Sci. Eng., Comput. Matl. Sci.)
 Chemical Engineering ( J. Chem. React. Engr., Ultras. Sonochemistry, Industrial & Engineering Chemistry Research, Chemical Engineering Science)
 Computational Physics ( Comp. Phys., Chaos, Solitons and Fractals)
 Energy (Energy Conv. & Management)
He holds a tenured professor appointment with the Civil and Architectural Engineering Mechanics Department and a courtesy appointment with the Material Science and Engineering Department at the University of Arizona. He has worked as visiting professor in France, Norway, and Greece, and has taught courses in France and Germany. He has also worked at Department of Defense laboratories for three summers. Awards he has received include the NSF Presidential Young Investigator award and the Fulbright Scholar award.
Degrees
 Ph.D. Engineering Mechanics
 University of Arizona, Tucson, Arizona, United States
 Progressive Damage and Constitutive Behavior of Materials
 B.S. Civil Engineering
 Aristotle University, Thessaloniki, Greece
 An Algorithm for Twoway Traffic Light Synchronization
Work Experience
 Aristotle University (2015)
 National University of Science and Technology (2001  2002)
 Université Bordeaux 1 (2001)
 Laboratoire de Mecanique et Technologie (LMT) (1994  1995)
 Institute National Polytechnique de Grenoble, Laboratoire 3S (1994)
 Air Force Office of Scientific Research (1993  1997)
 Petroleum and Geological Engineering, University of Oklahoma (1991)
 University of Arizona, Tucson, Arizona (1988  Ongoing)
 Institute for Steel Structures, Aristotle University (1986  1988)
Awards
 Accreditation Review Panel Member (fivemember panel), University of Crete
 NSF, Fall 1991
 Fulbright
 Fulbright scholarship for sixmonth visit to the Department of Theoretical Physics, Trondheim, Norway, 2001, Fall 2001
Licensure & Certification
 Professional Engineer, Civil, European Union State Members (1993)
 Professional Engineer, Civil, State of Arizona (2006)
Interests
Research
 Multiscale Material and Structural Characterization and Applications. Behavior of Materials at Nanoscale  Surface Effects and Insensitivity to Defects. Stochastic and Multiscale Material Description and Applications to Material and Structural Performance and Reliability. Multiscale Interpretation of Experimental Results. Multiscale and Multiphysics Problems. Scatter in Material Behavior.  Compressed Air Energy Storage.
Teaching
Undergraduate Education:Course Experience and Relevant Activities: Fundamentals of Engineering Design—Handson team based experience for freshmen. Engineering Mechanics, Statics—Core Engineering course. Computer programming for Civil Engineers. Strength of Materials—This core course is currently being modified to incorporate learning via the internet, particularly with respect to interactive homework completion. It has also been taught in Grenoble, France, during a sabbatical leave. Engineering Design Graphics—Computer based drafting. Junior Field Trip—Practicum experience for Juniors. Materials Laboratory—experimental property identification of various engineering materials. Engineering Design—Senior design project, a major, team based, Engineering Project. Introductory Finite Element Analysis—Technical elective course for seniors Aerospace and Mechanical Engineering Senior Design Project (team mentoring).Graduate Education:Course Experience and Relevant Activities: Theory of Elasticity and Applications—Research as well as application oriented basic elasticity theory; students majoring in various Engineering disciplines (Mechanics, Aerospace, Mechanical, Mining, Applied Math, etc.) take this course. Plasticity Theory and Applications—Computationally and research oriented plasticity for “traditional” and modern materials. Energy Methods in Mechanics—This “classical” course has been extended to include energy based stability, and chaotic behavior of structures. Continuum Mechanics—Basic principles of solid mechanics. Fracture Mechanics—This includes traditional and nontraditional themes, i.e. statistical fracture, scaling phenomena, numerical analysis, etc.Advanced tools, i.e. symbolic computations by computer, and numerical computations using the program Mathematica have been incorporated in some graduate courses.Multimedia technologies are progressively incorporated into undergraduate and graduate education.
Courses
202122 Courses

Dissertation
CE 920 (Fall 2021) 
Mechanics of Solids
CE 215 (Fall 2021) 
Probabil+Stat Civil Engr
CE 310 (Fall 2021)
202021 Courses

Dissertation
CE 920 (Spring 2021) 
Mechanics of Solids
CE 215 (Spring 2021) 
Dissertation
CE 920 (Fall 2020) 
Dissertation
EM 920 (Fall 2020) 
Mechanics of Solids
CE 215 (Fall 2020) 
Research Topics
CE 596A (Fall 2020) 
Statics
CE 214 (Fall 2020)
201920 Courses

Dissertation
CE 920 (Spring 2020) 
Dissertation
EM 920 (Spring 2020) 
Mechanics of Solids
CE 215 (Spring 2020) 
Statics
CE 214 (Spring 2020) 
Dissertation
CE 920 (Fall 2019) 
Dissertation
EM 920 (Fall 2019) 
Mechanics of Solids
CE 215 (Fall 2019) 
Probabil+Stat Civil Engr
CE 310 (Fall 2019)
201819 Courses

Dissertation
EM 920 (Spring 2019) 
Independent Study
CE 599 (Spring 2019) 
Mechanics of Solids
CE 215 (Spring 2019) 
Statics
CE 214 (Spring 2019) 
Dissertation
EM 920 (Fall 2018) 
Independent Study
CE 599 (Fall 2018) 
Mechanics of Solids
CE 215 (Fall 2018) 
Probabil+Stat Civil Engr
CE 310 (Fall 2018)
201718 Courses

Dissertation
CE 920 (Spring 2018) 
Mechanics of Solids
CE 215 (Spring 2018) 
Research Topics
CE 596A (Spring 2018) 
Statics
CE 214 (Spring 2018) 
Mechanics of Solids
CE 215 (Fall 2017) 
Statics
CE 214 (Fall 2017)
201617 Courses

Dissertation
EM 920 (Spring 2017) 
Independent Study
EM 599 (Spring 2017) 
Mechanics of Solids
CE 215 (Spring 2017) 
Research Topics
CE 596A (Spring 2017) 
Research Topics
EM 596A (Spring 2017) 
Dissertation
EM 920 (Fall 2016) 
Independent Study
EM 599 (Fall 2016) 
Mechanics of Solids
CE 215 (Fall 2016) 
Statics
CE 214 (Fall 2016)
201516 Courses

Dissertation
CE 920 (Spring 2016) 
Elasticity Thry+Applictn
EM 504 (Spring 2016) 
Mechanics of Solids
CE 215 (Spring 2016) 
Statics
CE 214 (Spring 2016)
Scholarly Contributions
Books
 Frantziskonis, G. N. (2017). Essentials of Mechanics of Materials, 3rd edition. Lancaster, PA: DESTech Publications.
 Frantziskonis, G. N. (2013). Essentials of the Mechanics of Materials. DEStech Publications, Inc.
 Frantziskonis, G. N. (1998). Probamat21st Century: Probabilities and Materials: Tests, Models, and Applications for the 21st Century. Kluwer Academic Pub.
 Desai, C. S., Krempl, E., & Frantziskonis, H. (1991). Constitutive Laws for Engineering Materials. ASME Press, New York.
Chapters
 Pannala, S., Simunovic, S., & Frantziskonis, G. (2010). Multiscale/multiphysics modeling of biomass thermochemical processes. In Computational Modeling in Lignocellulosic Biofuel Production(pp 245272). ACS publishers.
 Frantziskonis, G. N., & Blodgett, M. P. (1998). Multlscale Material Characterization and Applications. In PROBAMAT21st Century: Probabilities and Materials(pp 367378). Springer.
 Frantziskonis, G., Matikas, T. E., Karpur, P., Krishnamurthy, S., & Shaw, L. (1995). Lattice Analysis to Assess Fibermatrix Interface Behavior under Various Experimental Configurations. In Computational Mechanics’ 95(pp 25632568). Springer Berlin Heidelberg.
 Frantziskonis, G. (1994). Crack pattern related universal constants. In Probabilities and Materials(pp 361375). Springer.
Journals/Publications
 Sadat, M. R., Muralidharan, K., Frantziskonis, G. N., & Zhang, L. (2021). From atomicscale to mesoscale: A characterization of geopolymer composites using molecular dynamics and peridynamics simulations. Computational Materials Science, 186(110038). doi:https://doi.org/10.1016/j.commatsci.2020.110038
 Gur, S., & Frantziskonis, G. N. (2020). Design of Porous and Functionally Graded NiTi Smart Energy Absorbers Considering Synthetic Parametric Uncertainty. Journal of Intelligent Material Systems and Structures, 122. doi:0.1177/1045389X20977908
 Gur, S., Sadat, M. R., Frantziskonis, G. N., Bringuier, S., Zhang, L., & Muralidharan, K. (2019). The effect of grainsize on fracture of polycrystalline silicon carbide: A multiscale analysis using a molecular dynamicsperidynamics framework. COMPUTATIONAL MATERIALS SCIENCE, 159, 341348.
 Gur, S., Frantziskonis, G. N., & Muralidharan, K. (2018). Atomistic simulation of shape memory effect (SME) and superelasticity (SE) in nanoporous NiTi shape memory alloy (SMA). COMPUTATIONAL MATERIALS SCIENCE, 152, 2837.
 Marmarokopos, K., Doukakis, D., Frantziskonis, G., & Avlonitis, M. (2018). Leak Detection in Plastic Water Supply Pipes with a High SignaltoNoise Ratio Accelerometer. MEASUREMENT & CONTROL, 51(12), 2737.
 Sadat, M. R., Bringuier, S., Muralidharan, K., Frantziskonis, G., & Zhang, L. (2018). Atomicscale dynamics and mechanical response of geopolymer binder under nanoindentation. COMPUTATIONAL MATERIALS SCIENCE, 142, 227236.
 Frantziskonis, G. N., & Gur, S. (2017). Computational simulations for the development of novel solidstate smart NiTiAl thermal diodes. JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, 28(15), 20822094.
 Frantziskonis, G. N., & Gur, S. (2017). Length scale effects and multiscale modeling of thermally induced phase transformation kinetics in NiTi SMA. MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING, 25(4).
 Gur, S., Frantziskonis, G. N., & Aifantis, K. E. (2017). A computation study on the interplay between surface morphology and electrochemical performance of patterned thin film electrodes for Liion batteries. JOURNAL OF POWER SOURCES, 360, 504515.
 Gur, S., Frantziskonis, G. N., & Mishra, S. K. (2017). Thermally modulated shape memory alloy friction pendulum (tmSMAFP) for substantial nearfault earthquake structure protection. STRUCTURAL CONTROL & HEALTH MONITORING, 24(11).
 Gur, S., Frantziskonis, G. N., Pannala, S., & Daw, C. S. (2017). Application of WaveletBased Methods for Accelerating MultiTimeScale Simulation of Bistable Heterogeneous Catalysis. INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH, 56(9), 23932406.
 Gur, S., Manga, V. R., Bringuier, S., Muralidharan, K., & Frantziskonis, G. N. (2017). Evolution of internal strain in austenite phase during thermally induced martensitic phase transformation in NiTi shape memory alloys. COMPUTATIONAL MATERIALS SCIENCE, 133, 5259.
 Gur, S., & Frantziskonis, G. N. (2016). Atomistic study on size effects in thermally induced martensitic phase transformation of NiTi. Smart Materials Research, 2016(7512642), 12. doi:http://dx.doi.org/10.1155/2016/7512642
 Gur, S., & Frantziskonis, G. N. (2016). Linking simulations and experiments for the multiscale tracking of thermally induced martensitic phase transformation in NiTi SMA. MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING, 24(7).
 Gur, S., & Frantziskonis, G. N. (2016). Thermally Modulated Superelastic SMA Dampers for Vibration Control. ATINER'S Paper Series.
 Gur, S., Danielson, T., Xiong, Q., Hin, C., Pannala, S., Frantziskonis, G., Savara, A., & Daw, C. S. (2016). Waveletbased surrogate time series for multiscale simulation of heterogeneous catalysis. CHEMICAL ENGINEERING SCIENCE, 144, 165175.
 Gur, S., Mishra, S. K., & Frantziskonis, G. N. (2016). Thermomechanical strain ratedependent behavior of shape memory alloys as vibration dampers and comparison to conventional dampers. JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, 27(9), 12501264.
 Konstantinidis, A. A., Frantziskonis, G. N., Askes, H., & Aifantis, E. C. (2016). The use of nanoindentation for determining internal lengths and the constitutive response of monument materials: models and experiments. Journal of the Mechanical Behavior of Materials, 25, 5760. doi:10.1515/jmbm20160003
 Gur, S., Mishra, S. K., & Frantziskonis, G. N. (2015). Thermomechanical strain rate dependent behavior of SMAs as vibration dampers and comparison to conventional dampers. Journal of Intelligent Material Systems and Structures, 2015, 115. doi:10.1177/1045389X15588628
 Frantziskonis, G. (2014). On the strength reliability of statistically heterogeneous materials with microstructure at diverse scales. International Journal for Multiscale Computational Engineering, 12(3).
 Frantziskonis, G. N. (2014). Erratum: Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids [Phys. Rev. E 83, 066703 (2011)]. Physical Review E, 89(5), 059901.
 Frantziskonis, G., & Denis, A. (2003). Complementary entropy and wavelet analysis of drillingability data. MATHEMATICAL GEOLOGY, 35(1), 89103.More infoThe paper combines wavelet and entropy analysis of nonstationary drillingability data in order to obtain optimum information on the mechanical behavior of different geological formations. Both methods are multiscale in nature, and while entropy analysis provides information on stationary subdomains, wavelet analysis identifies dominant scales or range of scales at which the entropy analysis is most useful. The combination of the techniques yields relatively simple results that can aid the drilling process by providing information on delays to be expected.
 Buban, D. M., & Frantziskonis, G. N. (2013). Shape memory alloy fracture as a deployment actuator. Smart Materials and Structures, 22(11).More infoAbstract: The paper reports an experimental investigation into using shape memory alloy (SMA) as a deployment actuator. SMA specimens were heated and pulled to failure or pulled and heated to failure, thus developing an environmental and structural operating envelope for application as deployment mechanisms. The experimental results strongly suggest that SMAs can be implemented as deployment actuators. Recorded data shows that SMA fracture is possible over a wide range of temperatures and strains, filling a material performance gap not found in the literature. The obtained information allows design engineers to appropriately size SMAs given the design requirements for achieving the desired deployment effects. The major conclusion of the reported work is that SMAs as actuators are strong competitors to typical existing deployment efforts that use explosive or nonexplosive actuators having implementation drawbacks such as the expense associated with special handling and the volume encountered in mounting the devices. © 2013 IOP Publishing Ltd.
 Frantziskonis, G. N. (2013). Multiscale unified prediction of size/scale and HallPetch effects in the mechanics of polycrystalline materials. Journal of the Mechanical Behavior of Materials, 22(12), 6771.
 Frantziskonis, G. N. (2013). Temporal scaling in fatigue life of materials and incorporation of temporal events in Paris’s law. Journal of the Mechanical Behaviour of Materials, 21(56), 175180.
 Muralidharan, K., Deymier, P. A., & Frantziskonis, G. (2013). CompressedAir Energy Storage Systems for StandAlone OffGrid Photovoltaic Modules.
 Pannala, S., Simunovic, S., Frantziskonis, G., Nimlos, M., & Crowley, M. (2013). Multiscale/Multiphysics Modeling of Biomass Thermochemical Processes. COMPUTATIONAL MODELING IN LIGNOCELLULOSIC BIOFUEL PRODUCTION, 1052, 245271.
 Proczka, ., j., j., Muralidharan, K., Vilela, D., Simmons, J., & Frantziskonis, G. (2013). Guidelines for the Efficient Sizing of Pressure Vessels for Compressed Air Energy Storage. Energy Conversion and Management, 65, 597605.
 Frantziskonis, G., & Frantziskonis, G. N. (2011). Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids. Physical review. E, Statistical, nonlinear, and soft matter physics, 83(6 Pt 2).More infoThis paper reports the lattice Boltzmann method (LBM) based formulation for viscoelastic fluids with both volumetric and shear viscoelasticity. The relaxation limit of the viscoelastic fluid formulation yields the LBM for elastic solids with both volumetric or pressure (p) and shear (s) wave propagation modes. The reflection of a twodimensional p wave from an obstacle (wedge) inclined to the propagation direction of the p wave is studied together with the convergence and stability behavior of the LBM as the lattice size and lattice time step decrease. The model is capable of accurately predicting the mode change (p to s) due to the reflection. The model provides a unique unified approach capable of simulating fluids, viscoelastic fluids, and solids within a single LBM framework, thus avoiding interface problems between different simulation methods. The paper concentrates on the wave propagation part of the model, in the quasielastic regime.
 Frantziskonis, G., Muralidharan, K., Deymier, P., Simunovic, S., Nukala, P., & Pannala, S. (2011). Timeparallel multiscale/multiphysics framework. JOURNAL OF COMPUTATIONAL PHYSICS, 228(21), 80858092.More infoWe introduce the timeparallel compound wavelet matrix method (tpCWM) for modeling the temporal evolution of multiscale and multiphysics systems. The method couples time parallel (TP) and CWM methods operating at different spatial and temporal scales. We demonstrate the efficiency of our approach on two examples: a chemical reaction kinetic system and a nonlinear predatorprey system. Our results indicate that the tpCWM technique is capable of accelerating timetosolution by 23orders of magnitude and is amenable to efficient parallel implementation. (C) 2009 Elsevier Inc. All rights reserved.
 FRANTZISKONIS, G., & DESAI, C. (1987). ANALYSIS OF A STRAIN SOFTENING CONSTITUTIVE MODEL. INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, 23(6), 751767.
 FRANTZISKONIS, G., & LORET, B. (1995). SCALEDEPENDENT CONSTITUTIVE RELATIONS  INFORMATION FROM WAVELET ANALYSIS AND APPLICATION TO LOCALIZATION PROBLEMS. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 14(6), 873892.More infoThis study explores wavelet analysis as a tool to include the influence of scale on material behavior. The wavelet representation of the failure state of a onedimensional problem provides the relevant scale as a very important parameter in the material law. The consequences are paramount since the local scale involved, and its influence on material response, is decided from the global deformation field, rather then being defined as a fixed quantity independent of deformation. The scale dependent constitutive relations are examined in detail with respect to localization problems in the context of viscoplasticity.
 Mishra, S. K., Chaudhuri, S. R., Chakraborty, S., & Frantziskonis, G. (2010). Spectral characterization of the stochastically simulated vehicle queue on bridges. ENGINEERING WITH COMPUTERS, 25(4), 367378.More infoMonte Carlo simulation in conjunction with Fourier transform based spectral windowing is used to model the live load on bridges. Vehicles are classified into a few groups and the probability distributions of axle weight and length associated with each group are estimated. The vehicle arriving at an instant is determined through Monte Carlo simulation, which uses a vehicle group density function derived from measurement data on the relative contribution of each group in total vehicles. The weight and length of the arriving vehicle is also simulated by Monte Carlo using the distribution function for the corresponding group. Vehicle arrivals are modeled by the Poisson distribution. The vehicle velocities are realized through spectral simulation based on decaying power spectra of the velocity time series. The simulations are performed for a sufficient time interval in several lanes, thus the ensemble sampling of load is obtained. Fourier transform based windowing is used to characterize the power spectra of mechanical load on the bridge. The study shows the white noise nature of the load spectral density, which is in agreement with the assumptions of previous investigators. Parametric sensitivity of the spectra is also performed and recommendations are made to include sitespecific parameters in the model. Finally, applications are illustrated for frequency domain random vibration analysis of a simple model of bridge structures.
 Mishra, S. K., Deymier, P., Muralidharan, K., Frantziskonis, G., Pannala, S., & Simunovic, S. (2010). Modeling the coupling of reaction kinetics and hydrodynamics in a collapsing cavity. Ultrasonics sonochemistry, 17(1), 258265.
 Villela, D., Kasinathan, V. V., De Valle, S., Alvarez, M., Frantziskonis, G., Deymier, P., & Muralidharan, K. (2010). Compressedair energy storage systems for standalone offgrid photovoltaic modules. PVSC, IEEE, 962967.
 DANIEWICZ, D., & FRANTZISKONIS, G. (2009). EDGE DELAMINATION IN LAMINATED COMPOSITES. COMPOSITE STRUCTURES, 21(3), 141153.More infoA damage evolution theory for the effects of the rather complicated edge delamination phenomena on composite laminate response is developed. The theory is a mechanicsbased formulation which quantifies the damage development of a laminate under general loading, and incorporates it directly into the laminate constitutive equations.
 FRANTZISKONIS, G., KARPUR, P., MATIKAS, T., KRISHNAMURTHY, S., & JERO, P. (1994). FIBERMATRIX INTERFACE  INFORMATION FROM EXPERIMENTS VIA SIMULATION. COMPOSITE STRUCTURES, 29(3), 231247.More infoThis study explores a novel procedure for obtaining quantitative information on the mechanical properties of the fibermatrix interface in composite materials. The method, based on lattice discretization of a medium, simulates actual experiments in detail, including fiber breakage, matrix yield and/or cracking, and interface failure. The paper concentrates on two experiments performed commonly, the socalled fragmentation test for metal matrix, acid the pushout/pullout test for metal as well as ceramic matrix composites. Based on the documented capability of the method to simulate actual experimental data, reliable values of (homogenized) interface properties can be obtained. In addition, the simulations provide further understanding of the mechanisms involved during the relevant testing. Although this study presents results from basic problems, the method is general enough to include effects of residual stress, of high temperature environment, and of dynamic crack propagation, as well as threedimensional details of the interface failure process. The potential exists for simulating nondestructive wavebased techniques aimed at evaluating interface properties.
 Frantziskonis, G. N., & Matikas, T. E. (2009). Multiscale WaveletBased Analysis and Characterization of Fretting Fatigue Damage in Titanium Alloys. Materials transactions, 50(7), 17581767.
 Frantziskonis, G. N., & Matikas, T. E. (2009). Multiscale waveletbased analysis and characterization of fretting fatigue damage in titanium alloys. Materials Transactions, 50(7), 17581767.More infoAbstract: Wavelet analysis is used to rationalize information at various scales in several branches of science, including particle physics, biology, electrical engineering, fluid mechanics, and medicine. However, this powerful technique has not been applied extensively to characterize structures of materials, fretting damage for the present case, even though many critical questions could be addressed. In particular, the following unsolved problems are considered in this paper: (a) The first problem deals with the quantitative characterization of fretted surfaces in a Ti6A14V alloy. This is investigated by analyzing profilometric digital images of fretted surfaces obtained in a range of magnifications. Wavelet analysis of the data is able to identify, by examining the wavelet coefficients, dominant length scales as those regions in scalespace where the energy of the wavelet transform and/or peaks of local concentration dominate. For the range of magnifications examined, i.e. from 1.25× to 100×, the ̃20× magnification is identified as the one with the most useful information. (b) An alternative procedure is employed for the second use of wavelets which deals with the nonuniformity of the contact regions. Wavelet analysis is employed to identify partially slipping regions, which result in the "pattern" of the fretted surface morphology. ©2009 The Thermoelectrics Society of Japan.
 Frantziskonis, G., Muralidharan, K., Deymier, P., Simunovic, S., Nukala, P., & Pannala, S. (2009). Timeparallel multiscale/multiphysics framework. Journal of Computational Physics, 228(21), 80858092.
 Mishra, S. K., Chaudhuri, S. R., Chakraborty, S., & Frantziskonis, G. (2009). Spectral characterization of the stochastically simulated vehicle queue on bridges. Engineering with Computers, 25(4), 367378.
 Mishra, S. K., Chaudhuri, S. R., Chakraborty, S., & Frantziskonis, G. (2009). Spectral characterization of the stochastically simulated vehicle queue on bridges. Engineering with Computers, 25(4), 367378.More infoAbstract: Monte Carlo simulation in conjunction with Fourier transform based spectral windowing is used to model the live load on bridges. Vehicles are classified into a few groups and the probability distributions of axle weight and length associated with each group are estimated. The vehicle arriving at an instant is determined through Monte Carlo simulation, which uses a vehicle group density function derived from measurement data on the relative contribution of each group in total vehicles. The weight and length of the arriving vehicle is also simulated by Monte Carlo using the distribution function for the corresponding group. Vehicle arrivals are modeled by the Poisson distribution. The vehicle velocities are realized through spectral simulation based on decaying power spectra of the velocity time series. The simulations are performed for a sufficient time interval in several lanes, thus the ensemble sampling of load is obtained. Fourier transform based windowing is used to characterize the power spectra of mechanical load on the bridge. The study shows the white noise nature of the load spectral density, which is in agreement with the assumptions of previous investigators. Parametric sensitivity of the spectra is also performed and recommendations are made to include sitespecific parameters in the model. Finally, applications are illustrated for frequency domain random vibration analysis of a simple model of bridge structures. © 2009 SpringerVerlag London Limited.
 Mishra, S. K., Muralidharan, K., Deymier, P. A., Frantziskonis, G., Pannala, S., & Simunovic, S. (2009). WaveletBased Spatial Scaling of Coupled ReactionDiffusion Fields. International Journal for Multiscale Computational Engineering, 6(4), 281297.More infoMultiscale schemes for transferring information from fine to coarse scales are typically based on homogenization techniques. Such schemes smooth the fine scale features of the underlying fields, often resulting in the inability to accurately retain the fine scale correlations. In addition, higherorder statistical moments (beyond mean) of the relevant field variables are not necessarily preserved. As a superior alternative to averaging homogenization methods, a waveletbased scheme for the exchange of information between a reactive and diffusive field in the context of multiscale reactiondiffusion problems is proposed and analyzed. The scheme is shown to be efficient in passing information along scales,from fine to coarse, i.e., upscaling as well as from coarse to fine, i.e., downscaling. It incorporates fine scale statistics (higherorder moments beyond mean), mainly due to the capability of wavelets to represent fields hierarchically. Critical to the success of the scheme is the identification of dominant scales containing the majority of the useful information. The dominant scales in effect specify the coarsest resolution possible. The scheme is applied in detail to the analysis of a diffusive system with a chemically reacting boundary. Reactions are simulated using kinetic Monte Carlo (kMC) and diffusion is solved by finite differences (FDs). Spatial scale differences are present at the interface of the kMC sites and the diffusion grid. The computational efficiency of the scheme is compared to results obtained by averaging homogenization, and to results from a benchmark scheme that ensures spatial scale parity between kMC and FD.
 FRANTZISKONIS, G., & DESAI, C. (2008). CONSTITUTIVE MODEL WITH STRAIN SOFTENING. INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, 23(6), 733750.
 Frantziskonis, G., Muralidharan, K., Deymier, P., Simunovic, S., & Pannala, S. (2008). Time Parallel Scalable Multiphysics/Multiscale Framework. arXiv preprint arXiv:0804.0017.
 Frantziskonis, G., Renaudin, P., & Breysse, D. (2008). Heterogeneous solids .1. Analytical and numerical 1D results on boundary effects. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 16(3), 409423.More infoThis twopart study explores the consequences of material heterogeneity on the response of an externally loaded structure/specimen near its boundaries. The single deviation from the framework of classical (homogeneous) linear elasticity is that the elastic properties of the material form a stationary random field. It is shown that the consequences near boundaries can be quite adverse. Specifically, in the first part, investigation of a bar subjected to uniaxial load demonstrates that the strain field near the two ends is significantly and unfavorably different than in the bulk. Here, ''unfavorably' concurs to fracture initiation. Further, it is shown that for brittle materials, both size effects and data scatter of peak strength are closely related to boundary phenomena. The (approximate) analytical solution of the problem provides significant insight, and allows examination of the efficiency of numerical solution schemes for problems of the type studied herein.
 Mishra, S. K., Muralidharan, K., Deymier, P. A., Frantziskonis, G., Pannala, S., & Simunovic, S. (2008). Waveletbased spatial scaling of coupled reactiondiffusion fields. International Journal for Multiscale Computational Engineering, 6(4).
 Mishra, S. K., Muralidharan, K., Pannala, S., Simunovic, S., Daw, C. S., Nukala, P., Fox, R., Deymier, P. A., & Frantziskonis, G. N. (2008). Spatiotemporal compound wavelet matrix framework for multiscale/multiphysics reactor simulation: Case study of a heterogeneous reaction/diffusion system. International Journal of Chemical Reactor Engineering, 6(1).
 Muralidharan, K., Mishra, S. K., Frantziskonis, G., Deymier, P. A., Nukala, P., Simunovic, S., & Pannala, S. (2008). Dynamic compound wavelet matrix method for multiphysics and multiscale problems. PHYSICAL REVIEW E, 77(2).
 Muralidharan, K., Mishra, S. K., Frantziskonis, G., Deymier, P., Nukala, P., Simunovic, S., & Pannala, S. (2008). Dynamic compound wavelet matrix method for multiphysics and multiscale problems. Physical Review E, 77(2), 026714.
 Deymier, P., Oh, K., Muralidharan, K., Frantzikonis, G., & Runge, K. (2006). Selection of domains for coarse and fine levels of description in mixedpotential simulations. Journal of computeraided materials design, 13(13), 1744.
 FRANTZISKONIS, G., DESAI, C., TANG, F., & DANIEWICZ, D. (2006). DEGRADATION MECHANISMS IN BRITTLE MATERIALS INVESTIGATED BY ULTRASONICSCANNING. ENGINEERING FRACTURE MECHANICS, 42(2), 347369.More infoAn experimental study of degradation mechanisms and patterning in brittle materials is reported. Specimens of a castin brittle material were subjected to external mechanical load. At various prepeak load levels, through transmission ultrasonic measurements were taken at several locations of the sample. The damage evolution and its patterning were studied through analysis of the attenuation of the transmitted wave. For monotonically increasing load significant attenuation was observed at low load levels, while for unloading the attenuation changes were relatively small. The test results indicate random and nonsymmetric distribution of attenuation within the sample. However, in general, higher attenuation was observed close to the loadfree surfaces than in the interior of the sample. This observation signifies the socalled skin effect. Occasionally, at a few locations of various samples the energy of the received wave increased with increasing external load. This may indicate unloading of partially formed cracks. Measured surface strains are compared to the overall ones and the degradation patterning within the samples. The ultrasonic measurements are examined with respect to estimating the internal material length. The relation between ultrasonic wave attenuation and mechanical dissipated energy is examined.
 Frantziskonis, G., & Breysse, D. (2006). DIFFERENTIAL SETTLEMENTS OF STRUCTURES FOUNDED ON HETEROGENEOUS SOILS.
 Frantziskonis, G., & Deymier, P. (2006). Surface effects at the nanoscale significantly reduce the effects of stress concentrators. Probabilistic Engineering Mechanics, 21(3), 277286.More infoAbstract: At nanoscales, the large surface over volume ratio is shown to be instrumental in eliminating or significantly reducing the adverse effects of nanoscale stress concentrators (NSCs) such as impurities, inclusions, pores, and cracks. Using molecular dynamics (MD) simulations, Cu crystals with and without NSCs are strained in tension and in shear, at two strain rates, one being an order of magnitude larger than the other. Cubeshape crystals with periodic boundary conditions show sensitivity to NSCs similar to macrosize samples where fracture mechanics works well. For such crystals, atomistic defects cluster near the loaded surfaces, the clustering being stimulated significantly by the NSCs. Crystals with nonperiodic boundary conditions, however, show insensitivity to NSCs, for the sample sizes examined herein, i.e., cubes up to about 30 nm side length. Atomistic defects do not cluster near the loading surfaces but rather distribute over the entire sample. Even though the spatial distribution of atomistic defects depends on the presence of NSCs, the total number of such defects is found to be independent of the presence of NSCs for the cubic crystals. The reason for this is the presence of a "vast" amount of surfaces, for the nonperiodic boundary conditions case, where numerous atomistic defects initiate, making the number of defects initiating from NSCs insignificant. Provided that the average energy in creating these defects is constant, a robust explanation of the insignificance of NSCs emerges. © 2005 Elsevier Ltd. All rights reserved.
 Frantziskonis, G., & Deymier, P. (2006). Surface effects at the nanoscale significantly reduce the effects of stress concentrators. Probabilistic engineering mechanics, 21(3), 277286.
 Frantziskonis, G., & Deymier, P. (2006). The effects of stress concentrators on strength of materials at nanoscale: A molecular dynamics study. Mechanics Research Communications, 33(3), 352358.
 Frantziskonis, G., & Deymier, P. (2006). The effects of stress concentrators on strength of materials at nanoscale: A molecular dynamics study. Mechanics Research Communications, 33(3), 352358.More infoAbstract: There is evidence that when at least one spatial dimension of a material component is in the nanometer range, the effects of nanosize stress concentrators (NSCs) such as impurities, inclusions, pores, and cracks are either eliminated or significantly reduced. The aim of the paper is to examine such evidence using atomistic simulation techniques for a crystalline metal and identify the critical dimensions below which the effects of NSCs are minimal or even nonexistent. The preliminary results reported herein show that for Cu single crystals subjected to constant external strain rate, such critical dimensions are larger than about 30 nm. Since atomistic details are crucial in understanding material behavior at such scales, the paper points to the need for multiscale simulations techniques, presently being developed, for identifying critical dimensions and for examining slow strain rates. Based on the results, the paper presents simulationbased explanations why NSCs may be insignificant at nanoscales. © 2005 Elsevier Ltd. All rights reserved.
 Frantziskonis, G., Mishra, S. K., Pannala, S., Simunovic, S., Daw, C. S., Nukala, P., Fox, R. O., & Deymier, P. A. (2006). Waveletbased spatiotemporal multiscaling in diffusion problems with chemically reactive boundary. International Journal for Multiscale Computational Engineering, 4(56), 755770.More infoAbstract: Chemically reacting flows over catalytic and noncatalytic surfaces are one of the elementary operations in chemical processing plants. The underlying physical phenomena span time and length scales over several orders of magnitude, which a robust and flexible modeling framework must efficiently account for. With this purpose as the eventual goal, we propose a waveletbased multiscale numerical framework and demonstrate it on the coupling of two prototype methods for the problem of species generated on a chemically reactive boundary and diffusing through the bulk. The two methods consider different time and length scales. The first method in this coupling, termed "fine," models the chemical reactions on the reactive boundary stochastically by the kinetic Monte Carlo method and the diffusion in the medium deterministically using relatively small time increments and small spatial discretization mesh size. The second method, termed "coarse," models both the reaction and the diffusion deterministically and uses drastically larger time increments and spatial discretization size than the fine model. The two methods are coupled by forming a spatiotemporal compound wavelet matrix that combines information about the time and spatial scales contained in them. © 2006 by Begell House, Inc.
 Frantziskonis, G., Mishra, S. K., Pannala, S., Simunovic, S., Daw, C. S., Nukala, P., Fox, R. O., & Deymier, P. A. (2006). Waveletbased spatiotemporal multiscaling in diffusion problems with chemically reactive boundary. International Journal for Multiscale Computational Engineering, 4(56).
 FRANTZISKONIS, G., & VARDOULAKIS, . (2003). ON THE MICROSTRUCTURE OF SURFACE EFFECTS AND RELATED INSTABILITIES. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 11(1), 2134.More infoBased on Mindlin's theory for material microstructure interesting surface effects under conditions of equilibrium are studied in this paper. The governing field equations for uniaxial plane deformations are established; surface instability analysis shows non uniform deformations for a layer of specified distance from the surface. Experimental as well as fracture mechanics based considerations show that this surfacelayer is extremely thin for metals while for brittle materials its magnitude is of the order of 1 cm. Material microstructure introduces a singular perturbation to the original Hill and Hutchinson problem; here we introduce a single perturbation parameter and we obtain a "dispersion" law for the surface buckling load. It is found that surface degradation and skin effects can be attributed to localized surface buckling instabilities. Experimental information on skin effects can provide an estimation of the internal material length.
 Frantziskonis, G., & Breysse, D. (2003). Influence of soil variability on differential settlements of structures. Computers and Geotechnics, 30(3), 217230.
 Frantziskonis, G., & Breysse, D. (2003). Influence of soil variability on differential settlements of structures. Computers and Geotechnics, 30(3), 217230.More infoAbstract: The paper addresses the interaction of a structure with spatially varying soil properties. In particular, the problem of a twospan continuous beam founded on a heterogeneous soil is solved analytically. The geometrical and stiffness characteristics of the structure interact strongly with the spatial properties of the heterogeneous soil. For a certain value of the correlation distance, a feature of the heterogeneous soil formation, the uncertainty and the risk of high values, not predicted with deterministic models, in estimating differential settlements and forces (bending moments, shear forces, etc.) on the structure becomes maximum. The analytical solution uses a series expansion of the soil properties relative to those of the structure. The error in the solution, due to the truncation of the series expansion, is estimated by relevant numerical results. The paper shows clearly that the forces on the structure founded on a heterogeneous soil can differ widely from those usually predicted by a deterministic model. Furthermore, a usual deterministic approach can underestimate the safety level of the structure significantly. © 2003 Elsevier Science Ltd. All rights reserved.
 Frantziskonis, G., & Denis, A. (2003). Complementary Entropy and Wavelet Analysis of DrillingAbility Data. Mathematical Geology, 35(1), 89103.More infoAbstract: The paper combines wavelet and entropy analysis of nonstationary drilling ability data in order to obtain optimum information on the mechanical behavior of different geological formations. Both methods are multiscale in nature, and while entropy analysis provides information on stationary subdomains, wavelet analysis identifies dominant scales or range of scales at which the entropy analysis is most useful. The combination of the techniques yields relatively simple results that can aid the drilling process by providing information on delays to be expected.
 Frantziskonis, G., & Denis, A. (2003). Complementary entropy and wavelet analysis of drillingability data. Mathematical geology, 35(1), 89103.
 Frantziskonis, G., & Deymier, P. (2003). Structure, structural phase transitions, mechanical properties, defects, etc.Waveletbased spatial and temporal multiscaling: Bridging the atomistic and continuum space and time scales. Physical ReviewSection BCondensed Matter, 68(2), 2410524105.
 Frantziskonis, G., & Deymier, P. (2003). The effects of stress concentrators on strength of materials at nanoscale: A molecular dynamics study. MECHANICS RESEARCH COMMUNICATIONS, 33(3), 352358.More infoThere is evidence that when at least one spatial dimension of a material component is in the nanometer range, the effects of nanosize stress concentrators (NSCs) such as impurities, inclusions, pores, and cracks are either eliminated or significantly reduced. The aim of the paper is to examine such evidence using atomistic simulation techniques for a crystalline metal and identify the critical dimensions below which the effects of NSCs are minimal or even nonexistent. The preliminary results reported herein show that for Cu single crystals subjected to constant external strain rate, such critical dimensions are larger than about 30 rim. Since atomistic details are crucial in understanding material behavior at such scales, the paper points to the need for multiscale simulations techniques, presently being developed, for identifying critical dimensions and for examining slow strain rates. Based on the results, the paper presents simulationbased explanations why NSCs may be insignificant at nanoscales. (c) 2005 Elsevier Ltd. All rights reserved.
 Frantziskonis, G., & Deymier, P. (2003). Waveletbased spatial and temporal multiscaling: Bridging the atomistic and continuum space and time scales. Physical Review B  Condensed Matter and Materials Physics, 68(2), 241051241058.More infoAbstract: A waveletbased multiscale methodology is presented that naturally addresses time scaling in addition to spatial scaling. The method combines recently developed atomisticcontinuum models and wavelet analysis. An atomistic onedimensional harmonic crystal is coupled to a onedimensional continuum. The methodology is illustrated through analysis of the dispersion relation, which is highly dispersive at small spatial scales and, as usual, nondispersive at large (continuum) scales. It is feasible to obtain the complete dispersion relation through the combination of the atomistic and the continuum analyses. Wavelet analysis in this work is not only used for bridging the atomistic and continuum scales but also for efficiently extracting the dispersion relation from the solution of wave propagation problems.
 Frantziskonis, G., & Deymier, P. (2003). Waveletbased spatial and temporal multiscaling: Bridging the atomistic and continuum space and time scales. Physical Review B, 68(2), 024105.
 FRANTZISKONIS, G. (2002). 8 ON THE RELATION BETWEEN FRACTURE SURFACE CHARACTERISTICS AND MATERIAL PROPERTIES. SizeScale Effects in the Failure Mechanisms of Materials and Structures, 109.
 FRANTZISKONIS, G. (2002). ON SCALING PHENOMENA IN FRACTURE OF HETEROGENEOUS SOLIDS. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 13(1), 7392.More infoSeveral studies address the scaling geometrical properties of fracture surfaces/networks in solid material. This paper examines the implications of the underlying heterogeneous fields before fracture on the properties of the fracture network. Three loading conditions are examined analytically as twodimensional problems  uniaxial load, pure shear, and dilatation. The relevant kinematic quantities before (macro) fracture are described statistically as random fields or as scaling fields. It is shown that fracture networks with scaling properties can develop. Notably, some similarity is identified between the case of fast crack propagation using scaling fields, and the case of ''slow'' crack propagation. Under low rate force or displacement controlled external load application, the slow crack propagation case is the only one consistent with relevant physical reasoning. The present analytical solutions can be used for identification of material properties through post mortem information, i.e. by studying the geometrical properties of fracture networks/surfaces. They also provide a better understanding of the mechanisms contributing to fracture, and show the important contribution of material microstructure on fracture toughness.
 Frantziskonis, G. (2002). Multiscale characterization of materials with distributed pores and inclusions and application to crack formation in an aluminum alloy. Probabilistic Engineering Mechanics, 17(4), 359367.More infoAbstract: The paper reports a study on the mechanical behavior of materials containing pores and inclusions distributed over a wide range of length scales. Utilizing a waveletbased multiscale process such microstructures are characterized and their effects on material properties is studied. In order to present the process in a semianalytical fashion, the variance of the strain field for an approximated onedimensional deformation problem is examined in detail. It is shown that with respect to crack initiation, there is a strong interplay between the distribution of pores and inclusions. Furthermore, their interaction with boundaries proves to be paramount. The process is applied to a particular cast aluminum alloy where pores are, in general, about two orders of magnitude larger than the silicon particles (inclusions). Results agree well with recent experimental reports on crack initiation where the interplay of pores, inclusions, and boundaries is observed, yet not explained on a fundamental basis. The present work extends recent efforts on porous materials [Frantziskonis G. Waveletbased analysis of multiscale phenomena  application to material porosity and identification of dominant scales. Prob Eng Mech (this issue). PII: S02668920(2)000322] to include the interaction of pores at certain scales with inclusions at other scales as well as the interaction of both with boundaries. © 2002 Elsevier Science Ltd. All rights reserved.
 Frantziskonis, G. (2002). Multiscale characterization of materials with distributed pores and inclusions and application to crack formation in an aluminum alloy. Probabilistic engineering mechanics, 17(4), 359367.
 Frantziskonis, G. (2002). Waveletbased analysis of multiscale phenomena: Application to material porosity and identification of dominant scales. Probabilistic Engineering Mechanics, 17(4), 349357.More infoAbstract: The paper presents a general process that utilizes wavelet analysis in order to link information on material properties at several scales. In the particular application addressed analytically and numerically, multiscale porosity is the source of material structure or heterogeneity, and the waveletbased analysis of multiscale information shows clearly its role on properties such as resistance to mechanical failure. Furthermore, through the statistical properties of the heterogeneity at a hierarchy of scales, the process clearly identifies a dominant scale or range of scales. Special attention is paid to porosity appearing at two distinct scales far apart from each other since this demonstrates the process in a lucid fashion. Finally, the paper suggests ways to extend the process to general multiscale phenomena, including time scaling. © 2002 Elsevier Science Ltd. All rights reserved.
 Frantziskonis, G., & Aifantis, E. C. (2002). On the stochastic interpretation of gradientdependent constitutive equations. European Journal of Mechanics, A/Solids, 21(4), 589596.More infoAbstract: The paper elaborates on the statistical interpretation of a class of gradient models by resorting to both microscopic and macroscopic considerations. The microscopic stochastic representation of stress and strain fields reflects the heterogeneity inherently present in engineering materials at small scales. A physical argument is advanced to conjecture that stress shows small fluctuations and strong spatial correlations when compared to those of strain; then, a series expansion in the respective constitutive equations renders unimportant stress gradient terms, in contrast to strain gradient terms, which should be retained. Each higherorder strain gradient term is given a physically clear interpretation. The formulation also allows for the underlying microstrain field to be statistically nonstationary, e.g., of fractal character. The paper concludes with a comparison between surface effects predicted by gradient and stochastic formulations. © 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
 Frantziskonis, G., & Aifantis, E. C. (2002). On the stochastic interpretation of gradientdependent constitutive equations. European Journal of MechanicsA/Solids, 21(4), 589596.
 Frantziskonis, G., & Deymier, P. (2002). Waveletbased spatial and temporal multiscaling: Bridging the atomistic and continuum space and time scales. PHYSICAL REVIEW B, 68(2).More infoA waveletbased multiscale methodology is presented that naturally addresses time scaling in addition to spatial scaling. The method combines recently developed atomisticcontinuum models and wavelet analysis. An atomistic onedimensional harmonic crystal is coupled to a onedimensional continuum. The methodology is illustrated through analysis of the dispersion relation, which is highly dispersive at small spatial scales and, as usual, nondispersive at large (continuum) scales. It is feasible to obtain the complete dispersion relation through the combination of the atomistic and the continuum analyses. Wavelet analysis in this work is not only used for bridging the atomistic and continuum scales but also for efficiently extracting the dispersion relation from the solution of wave propagation problems.
 Frantziskonis, G., & Frantziskonis, G. (2002). Waveletbased analysis of multiscale phenomenaapplication to material porosity and identification of dominant scales. Prob Eng Mech,(this issue). PII: S 02668920 (02) 000322.
 Proczka, J. J., Muralidharan, K., Villela, D., Simmons, J. H., & Frantziskonis, G. (2002). Guidelines for the pressure and efficient sizing of pressure vessels for compressed air energy storage. ENERGY CONVERSION AND MANAGEMENT, 65, 597605.More infoThe paper reports guidelines for the efficient design and sizing of SmallScale Compressed Air Energy Storage (SSCAES) pressure vessels, including guidelines for pressures that should be used in the SSCAES system to minimize the cost of the pressure vessel. Under a specified energy storage capacity and specified maximum and minimum operating pressures in CAES, the volume of the vessel(s) can be evaluated. The present study provides guidelines for choosing appropriate shape and size for the vessels that minimize material and manufacturing cost for cylindrical vessels. The two main contributions of the paper are that it provides a methodology to determine: (a) an optimum pressure; (b) the shape, size, and number of vessel to be used in a particular application. Results suggest that pressure vessels with a length to diameter ratio of roughly three are the most economical, and that a system should be designed for a pressure of roughly three times the minimum pressure of the expansion device. (C) 2012 Elsevier Ltd. All rights reserved.
 Avlonitis, M., Ioannidou, T., Frantziskonis, G., & Aifantis, E. C. (2001). Statistical Aspects of Gradient Theory. Journal of the Mechanical Behavior of Materials, 12(2), 7784.
 Frantziskonis, G. (2001). Characterization of materials with pores and inclusions at diverse scales. 6th NATIONAL CONGRESS OF MECHANICS, 359.
 Frantziskonis, G. N. (2001). Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids (vol 83, 066703, 2011). PHYSICAL REVIEW E.
 Frantziskonis, G. N., Konstantinidis, A. A., & Aifantis, E. C. (2001). Scaledependent constitutive relations and the role of scale on nominal properties. European Journal of Mechanics, A/Solids, 20(6), 925936.More infoAbstract: Size effects in strength and fracture energy of heterogeneous materials is considered within a context of scaledependent constitutive relations. Using tools of wavelet analysis, and considering the failure state of a onedimensional solid, constitutive relations which include scale as a parameter are derived from a 'background' gradient formulation. In the resulting theory, scale is not a fixed quantity independent of deformation, but rather directly dependent on the global deformation field. It is shown that strength or peak nominal stress (maximum point at the engineering stressstrain diagram) decreases with specimen size while toughness or total work to fracture per nominal area (area under the curve in the engineering stressstrain diagram integrated along the length of the considered onedimensional specimen) increases. This behavior is in agreement with relevant experimental findings on heterogeneous materials where the overall mechanical response is determined by variations in local material properties. The scaledependent constitutive relations are calibrated from experimental data on concrete specimens.
 Frantziskonis, G. N., Konstantinidis, A. A., & Aifantis, E. C. (2001). Scaledependent constitutive relations and the role of scale on nominal properties. European Journal of MechanicsA/Solids, 20(6), 925936.
 Konstantinidis, A., Frantziskonis, G., Carpinteri, A., & Aifantis, E. (2001). Size effects on tensile strength and fracture energy in concrete: wavelet vs. fractal approach. Journal of the Mechanical Behavior of Materials, 12(2), 6376.
 FRANTZISKONIS, G., & DESAI, C. (2000). DEGRADATION INSTABILITIES IN BRITTLE MATERIAL STRUCTURES. MECHANICS RESEARCH COMMUNICATIONS, 17(3), 135141.
 FRANTZISKONIS, G., TANG, F., & DESAI, C. (2000). BOREHOLE SCALE EFFECTS AND RELATED INSTABILITIES. ENGINEERING FRACTURE MECHANICS, 39(2), 377389.More infoA new mechanics based approach is proposed for scale effects and instabilities on borehole problems. In borehole types of structural systems, two types of instabilities can take place. The first is due to surface degradation growth and results into spalling of layers at the hole wall. The second is due to damage progression, and results into globally unstable response of the structure. The hole size has been found experimentally to be an important parameter in breakout instability initiation. Laboratory size holes may overestimate instability initiation properties by a large factor. At the same time, material properties such as peak stress depend largely on the size and shape of a specimen subjected to uniaxial or triaxial compression. This work attempts to incorporate size and scale effects into the instability initiation conditions. The important task of transferring information from laboratory experiments to actual large scale engineering problems is analysed and discussed. The potential of the theory is demonstrated. The need for further experimental and theoretical work is identified.
 Frantziskonis, G. N., Simon, L. B., Woo, J., & Matikas, T. E. (2000). Multiscale characterization of pitting corrosion and application to an aluminum alloy. European Journal of Mechanics, A/Solids, 19(2), 309318.More infoAbstract: This paper documents a novel method for characterizing pitting corrosion damage in structural materials such as Al 2024T3. Specimens of such alloys are corroded in a controlled environment and the pits' geometry is captured digitally using white light interference microscopy. The digital data are then processed with waveletbased analysis, thus making possible a multiresolution description of the geometrical features. The analysis reveals several interesting features of the pits that are similar for all the experimental data analyzed herein, and independent of the process followed for creating them (time the material is exposed to corroding environment, concentration of the corroding agent, surface area exposed to the agent, etc.). The first property identified as common to all pits is their geometrical scaling with a (Hurst) exponent of 0.63±0.12. Furthermore, the ratio ω of the surface area of the pit as represented at coarse scales through the wavelet representation, over the area of its intersection with the plane at zero depth is found to be 1.17±0.07 consistently. The ratio of the total surface area over its intersection is found to be 1.6±0.2. Either one of these ratios together with the Hurst exponent provide sufficient information for obtaining a pit's geometry from images capturing its twodimensional shape only, a capability important for efficient characterization. Additionally, such a characterization is paramount for rigorously addressing fatigue crack initiation and propagation.
 Frantziskonis, G. N., Simon, L. B., Woo, J., & Matikas, T. E. (2000). Multiscale characterization of pitting corrosion and application to an aluminum alloy. European Journal of MechanicsA/Solids, 19(2), 309318.
 Frantziskonis, G., & Deymier, P. (2000). Compound Wavelet Matrix for Bridging Simulations at Complementary ScalesApplication to Microstructure Evolution.
 Frantziskonis, G., & Deymier, P. (2000). Surface effects at the nanoscale significantly reduce the effects of stress concentrators. PROBABILISTIC ENGINEERING MECHANICS, 21(3), 277286.More infoAt nanoscales, the large surface over volume ratio is shown to be instrumental in eliminating or significantly reducing the adverse effects of nanoscale stress concentrators (NSCs) such as impurities, inclusions, pores, and cracks. Using molecular dynamics (MD) simulations, Cu crystals with and without NSCs are strained in tension and in shear, at two strain rates, one being an order of magnitude larger than the other. Cubeshape crystals with periodic boundary conditions show sensitivity to NSCs similar to macrosize samples where fracture mechanics works well. For such crystals, atomistic defects cluster near the loaded surfaces, the clustering being stimulated significantly by the NSCs. Crystals with nonperiodic boundary conditions, however, show insensitivity to NSCs, for the sample sizes examined herein, i.e., cubes up to about 30 nm side length. Atomistic defects do not cluster near the loading surfaces but rather distribute over the entire sample. Even though the spatial distribution of atomistic defects depends on the presence of NSCs, the total number of such defects is found to be independent of the presence of NSCs for the cubic crystals. The reason for this is the presence of a "vast" amount of surfaces, for the nonperiodic boundary conditions case, where numerous atomistic defects initiate, making the number of defects initiating from NSCs insignificant. Provided that the average energy in creating these defects is constant, a robust explanation of the insignificance of NSCs emerges. (C) 2005 Elsevier Ltd. All rights reserved.
 Frantziskonis, G., & Deymier, P. (2000). Wavelet methods for analysing and bridging simulations at complementary scalesthe compound wavelet matrix and application to microstructure evolution. Modelling and Simulation in Materials Science and Engineering, 8(5), 649.
 Frantziskonis, G., & Deymier, P. A. (2000). Wavelet methods for analysing and bridging simulations at complementary scales  the compound wavelet matrix and application to microstructure evolution. Modelling and Simulation in Materials Science and Engineering, 8(5), 649664.More infoAbstract: We introduce a novel waveletbased compound matrix to bridge models (LennardJones model simulated with molecular dynamics and a lattice Qstates Potts model with a Monte Carlo simulation) that describe grain growth over different ranges of spatial and time scales. The compound wavelet matrix provides full statistical information on the microstructure at the range of scales that is the union of those handled by the two models.
 Frantziskonis, G., & Hansen, A. (2000). Waveletbased multiscaling in selfaffine random media. Fractals, 8(04), 403411.
 Frantziskonis, G., & Hansen, A. (2000). Waveletbased multiscaling in selfaffine random media. Fractals, 8(4), 403411.More infoAbstract: Wavelet analysis is used to determine particular properties of selfaffine random media, i.e. ones with homogeneous increments. When such media are filtered through introducing upper and lower cutoffs  corresponding to the domain and sample support scales interesting properties result from wavelet analysis. In particular, we show that the filtered media can be used for transferring information along scales. Furthermore, we show that a waveletbased approach for determining the underlying Hurst exponent is highly efficient. Examples relating to size and surface effects in materials demonstrate applications of the relevant multiscaling. Finally, we show that the waveletbased approach and a method based on mode superposition complement each other nicely.
 Budhu, M., Ramakrishnan, S., & Frantziskonis, G. (1999). A lattice type model for particulate media. International Journal for Numerical and Analytical Methods in Geomechanics, 23(7), 647671.More infoAbstract: In this paper, a latticetype model to simulate the micromechanical behaviour of particulate/granular media is presented. In this numerical model, a particulate assembly is simulated as a lattice/truss. Nodes located at contacts between a particle and its neighbours are linked by bars to each other. Each particle is represented by a lattice within its microstructure and particle interact through load transfer at the nodes. Constraints are prescribed at the nodes to describe active, deactivated and reactivated contacts. When a particulate assembly develops into a mechanism (deformation with zero incremental load), further deformation is simulated through a framework that describes the kinematics of the particles (sliding, rolling and rotation of particles). This framework is formed by introducing nodes at the particle centroids and linking them with bars. Barslinking particles with a nonsliding contact are assigned large stiffnesses relative to bars linking particles with a sliding contact. Numerical tests are conducted on twodimensional assemblies of disks, arranged as very loose and very dense packing under simple shear loading conditions. The results concord with the results of numerical tests conducted using the discrete element method and with photoelastic experiments. Additionally, the model is applied to study the effects of initial imperfections caused by particles with low elastic modulus. Copyright © 1999 John Wiley & Sons, Ltd.
 Budhu, M., Ramakrishnan, S., & Frantziskonis, G. (1999). A lattice type model for particulate media. International journal for numerical and analytical methods in geomechanics, 23(7), 647671.
 Frantziskonis, G. N., Frantziskonis, G. N., Matikas, T. E., & Matikas, T. E. (1999). Multiscale WaveletBased Analysis and Characterization of Fretting Fatigue Damage in Titanium Alloys. MATERIALS TRANSACTIONS, 50(7), 17581767.More infoWavelet analysis is used to rationalize information at various scales in several branches of science, including particle physics, biology, electrical engineering, fluid mechanics, and medicine. However, this powerful technique has not been applied extensively to characterize structures of materials, fretting damage for the present case, even though many critical questions could be addressed. In particular, the following unsolved problems are considered in this paper: (a) The first problem deals with the quantitative characterization of fretted surfaces in a Ti6Al4V alloy. This is investigated by analyzing profilometric digital images of fretted surfaces obtained in a range of magnifications. Wavelet analysis of the data is able to identify, by examining the wavelet coefficients, dominant length scales as those regions in scalespace where the energy of the wavelet transform and/or peaks of local concentration dominate. For the range of magnifications examined, i.e. from 1.25 x to 100x, the similar to 20x magnification is identified as the one with the most useful information. (b) An alternative procedure is employed for the second use of wavelets which deals with the nonuniformity of the contact regions. Wavelet analysis is employed to identify partially slipping regions, which result in the "pattern" of the fretted surface morphology. [doi: 10.2320/matertrans.MRA2008378]
 Frantziskonis, G. N., Shell, E., Woo, J., Matikas, T. E., & Nicolaou, P. D. (1999). Wavelet analysis of fretting experimental data. Proceedings of SPIE  The International Society for Optical Engineering, 3585, 1127.More infoAbstract: Wavelet analysis is applied to the characterization of the fretting damage of materials. Two cases are considered. In the first case, fretted surfaces in a Ti6Al4V alloy are quantitatively characterized by analyzing profilometric digital images of fretted surfaces. Through wavelet analysis, dominant length scales are determined as those regions in the scalespace where the energy of the wavelet transform and/or peaks of local concentration dominate. The second use of wavelets deals with the nonuniformity of the contact regions. In particular, wavelet analysis is employed to identify those regions.
 Frantziskonis, G. N., Simon, L. B., Woo, J., & Matikas, T. E. (1999). Characterization of pitting corrosion damage through multiscale analysis. Proceedings of SPIE  The International Society for Optical Engineering, 3585, 4858.More infoAbstract: The paper documents a novel method for characterizing pitting corrosion damage in structural materials such as Al 2024T3. Specimens of such alloys are corroded in a controlled environment and the pits' geometry is captured digitally using white light interference microscopy. The digital data are then processed with waveletbased analysis thus making possible a multiresolution description of the geometrical features. The analysis reveals several interesting features of the pits that are similar for all the experimental data analyzed herein, and independent of the process followed for creating them (time material is exposed to corroding environment, concentration of the corroding agent, surface are exposed to the agent, etc.). The first property identified as common to all pits is their geometrical scaling with a (Hurst) exponent of 0.63±0.12. Furthermore, the ratio ω of the surface area of the pit as represented at coarse scales through the wavelet representation, over the area of its intersection with the plane at zero depth is found to be 1.17±0.07 consistently. The ratio of the total surface area over its intersection is found to be 1.6±0.2. Either one of these ratios together with the Hurst exponent provide sufficient information for obtaining a pit's geometry from images capturing its twodimensional shape only, a capability important for efficient characterization. Additionally, such a characterization is paramount for rigorously addressing fatigue crack initiation and propagation.
 Frantziskonis, G., & Aifantis, E. (1999). On the stochastic interpretation of gradientdependent constitutive equations. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 21(4), 589596.More infoThe paper elaborates on the statistical interpretation of a class of gradient models by resorting to both microscopic and macroscopic considerations. The microscopic stochastic representation of stress and strain fields reflects the heterogeneity inherently present in engineering materials at small scales. A physical argument is advanced to conjecture that stress shows small fluctuations and strong spatial correlations when compared to those of strain; then, a series expansion in the respective constitutive equations renders unimportant stress gradient terms, in contrast to strain gradient terms, which should be retained. Each higherorder strain gradient term is given a physically clear interpretation. The formulation also allows for the underlying microstrain field to be statistically nonstationary, e.g., of fractal character. The paper concludes with a comparison between surface effects predicted by gradient and stochastic formulations. (C) 2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.
 FRANTZISKONIS, G. (1998). HETEROGENEITY AND IMPLICATED SURFACE EFFECTS  STATISTICAL, FRACTAL FORMULATION AND RELEVANT ANALYTICAL SOLUTION. ACTA MECHANICA, 108(14), 157178.More infoThe aim of this paper is to examine the implications of material heterogeneity on brittle material response and on relevant surface effects. Statistical and fractal concepts are used for this purpose. In the statistical formulation the displacement gradients of the micromedium are considered to be random fields characterized by stationary exponential or Gaussian autocovariance and by the relevant correlation length or scale of fluctuation. Through Taylor series expansion around the mean of the random field, an important analogy is found between the statistical formulation and the microstructural theory, originally introduced by Mindlin, where higher order gradients of deformation appear in the constitutive equations. The analogy is valid only when fluctuations are small, so that some simplifications are allowed. It is found that the socalled internal length appearing in the microstructural formulation is analogous to the correlation length in the statistical one. In the statistical approach there are no extra boundary conditions in the formulation, as is the case when higher order gradients are introduced. However, what is known as ''conditioning'' of the held at the boundaries effects its behavior near/on them. The statistical approach can provide further information in the form of higher order moments not captured by the gradient theory. Material/structure response is strongly dependent on the aforementioned scale. Its effect is most pronounced near the boundaries of a structure where its role on surface related phenomena is paramount. In order to study heterogeneity at a hierarchy of scales, i.e. absence of characteristic length, complex disorderly system, fractal concepts and relevant power decay laws are considered. The formulation introduces the fractal dimension of the heterogeneous displacement gradient of the micromedium, a length describing the overall size of the structure, and the lower cutoff of the scaling law. The physical interpretation of the lower cutoff is the lower limit of applicability of the power scaling law. Mathematically it is important since in this case the fractal can be ''followed'' in the spatial domain. Similarly to the statistical case, an analogy between the fractal formulation and gradient theories is identified. No extra boundary conditions appear in the fractal formulation. However, there are still open questions with respect to the behavior of a fractal after conditioning, as is the case on boundaries. The analytical solution of a relevant surface instability problem for the gradient, statistical, and fractal formulation is presented. The solution was obtained through symbolic computations by computer because the analytical work is tedious and error prone. The analytical solution provides significant insight into the problem of heterogeneity and skin effects in brittle materials, internal length estimation, and the role of fractal scaling properties. Finally, the concepts introduced herein are discussed with respect to experimental information and numerical implementation.
 FRANTZISKONIS, G. (1998). RESEARCH NOTE ON THE EQUIVALENCE OF ISOTROPIC DAMAGE MECHANICS AND PLASTICITY. INTERNATIONAL JOURNAL OF PLASTICITY, 10(3), 303308.More infoIt is shown that certain isotropic damage mechanics models are mathematically equivalent to plasticitybased ones. From the equivalence, a macroscopic physical interpretation of damage evolution and restrictions on the partial stresses are derived.
 Frantziskonis, G. (1998). Probabilities and materials. Proc. of the NATO Advanced Research Workshop PROBAMAT 21st CENTURY. Kluwer, Dordrecht.
 Frantziskonis, G. N. (1998). Stochastic modeling of heterogeneous materials  A process for the analysis and evaluation of alternative formulations. Mechanics of Materials, 27(3), 165175.More infoAbstract: The paper addresses the problem of understanding and modeling the constitutive behavior of complex, heterogeneous solid materials. Several issues in the area of stochastic material modeling remain open, thus a process for the analysis of different formulations is proposed and examined in detail. Using the proposed process, it is shown that a number of basic models are stochastically similar to each other. Furthermore, often, small apparently unimportant changes in a formulation may result in significant changes in the predicted response. For some formulations, inequalities are established that when satisfied it is guaranteed that the entropy of a structure/system cannot decrease. Attention is given to stochastic formulations within the context of nonlocal theories because such formulations provide new information to the subject of localization of deformation and fracture toughness. © 1998 Elsevier Science Ltd.
 Frantziskonis, G. N. (1998). Stochastic modeling of heterogeneous materialsa process for the analysis and evaluation of alternative formulations. Mechanics of Materials, 27(3), 165175.
 Frantziskonis, G., Konstantinidis, A., & Aifantis, E. (1998). Scaledependent constitutive relations and the role of scale on nominal properties. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 20(6), 925936.More infoSize effects in strength and fracture energy of heterogeneous materials is considered within a context of scaledependent constitutive relations. Using tools of wavelet analysis, and considering the failure state of a onedimensional solid, constitutive relations which include scale as a parameter are derived from a 'background' gradient formulation. In the resulting theory, scale is not a fixed quantity independent of deformation, but rather directly dependent on the global deformation field. It is shown that strength or peak nominal stress (maximum point at the engineering stressstrain diagram) decreases with specimen size while toughness or total work to fracture per nominal area (area under the curve in the engineering stressstrain diagram integrated along the length of the considered onedimensional specimen) increases. This behavior is in agreement with relevant experimental findings on heterogeneous materials where the overall mechanical response is determined by variations in local material properties. The scaledependent constitutive relations are calibrated from experimental data on concrete specimens. (C) 2001 Editions scientifiques et medicales Elsevier SAS.
 Gustafson, S. C., Mehta, J. R., & Frantziskonis, G. N. (1998). Practical optical characterization of fretted surfaces. Materials Research Society Symposium  Proceedings, 503, 2530.More infoAbstract: Nominally smooth surfaces subjected to repeated contact may develop characteristic fretted textures. An important example concerns fretted surfaces that may develop on disk dovetail joints that secure fan blades in turbine engines. These surfaces may form sites for crack nucleation and subsequent catastrophic disk and engine failure. Thus a practical and reliable method for the nondestructive evaluation of such fretted surfaces is critically needed. In this paper straightforward optical techniques are considered for identifying and quantifying fretted (and other surface damaged) regions. For example, digital images of candidate surfaces may be processed to separate fretted from unfretted regions and to evaluate the relative degree of fretting. Ideally, optical characterization should be rapid, convenient, inexpensive, reliable, and insensitive to details of lighting and image capture.
 Meisner, M., & Frantziskonis, G. (1998). Multifractal fracturetoughness properties of brittle heterogeneous materials. JOURNAL OF PHYSICS AMATHEMATICAL AND GENERAL, 29(11), 26572670.More infoThis paper documents a study of novel fracturetoughness properties of brittle heterogeneous materials. Through simulations of the rupture process based on a lattice discretization of the material, the spatial variation of dissipated energy due to fracture is evaluated. Under certain conditions, its distribution is characterized by a multifractal spectrum f(alpha). Importantly, f(alpha) depends not only on the initial heterogeneity present in the material but also on the nature of the externally applied load. This provides a renewed loadpathdependent definition of fracture roughness material properties. It avoids the difficulties associated with 'traditional' continuum/fracture mechanics definitions where the macroscopic fracture mode must be known a priori.
 Vasseur, J. O., Deymier, P. A., Frantziskonis, G., Hong, G., DjafariRouhani, B., & Dobrzynski, L. (1998). Experimental evidence for the existence of absolute acoustic band gaps in twodimensional periodic composite media. Journal of Physics Condensed Matter, 10(27), 60516064.More infoAbstract: Transmission of acoustic waves in twodimensional binary solid/solid composite media composed of arrays of Duralumin cylindrical inclusions embedded in an epoxy resin matrix is studied. The experimental transmission spectrum and theoretical band structure of two periodic arrays of cylinders organized on a square lattice and on a centred rectangular network are reported. Absolute gaps extending throughout the first twodimensional Brillouin zone are predicted. The measured transmission is observed to drop to noise level throughout frequency intervals in reasonable agreement with the calculated forbidden frequency bands.
 Vasseur, J., Deymier, P., Frantziskonis, G., Hong, G., DjafariRouhani, B., & Dobrzynski, L. (1998). Experimental evidence for the existence of absolute acoustic band gaps in twodimensional periodic composite media. Journal of Physics: Condensed Matter, 10(27), 6051.
 Budhu, M., Ramakrishnan, S., & Frantziskonis, G. (1997). Modeling of granular materials  a numerical model using lattices. ASMEASCESES Joint Summer Meeting, 7790.More infoAbstract: In this paper, a numerical model is presented in which the particles in a granular assembly are modeled as lattices. In a two dimensional simulation, nodes are introduced at particle contacts and are linked by bar elements. A granular assembly is then transformed into a statically indeterminate truss and is analyzed using standard techniques in structural mechanics. The areas of bars are derived by computing the displacements of an elastic disk under contact forces and then the bar areas that will give equivalent displacements in the direction of the bars are determined. The lattice type model incorporates bonding, debonding, sliding and rotation of particles. The lattice type model has been applied to loose and dense packing of disks. The two packings were subjected to simple shear deformation after an initial vertical load was applied. The results from the lattice type model agree with the results from numerical tests using the discrete element method and with results of photoelastic experiments.
 FRANTZISKONIS, G., & JOSHI, S. (1997). DAMAGE EVOLUTION AND CONSTITUTIVE BEHAVIOR OF ADVANCED COMPOSITES. COMPOSITE STRUCTURES, 16(4), 341357.
 Frantziskonis, G. N. (1997). Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids. PHYSICAL REVIEW E, 83(6).More infoThis paper reports the lattice Boltzmann method (LBM) based formulation for viscoelastic fluids with both volumetric and shear viscoelasticity. The relaxation limit of the viscoelastic fluid formulation yields the LBM for elastic solids with both volumetric or pressure (p) and shear (s) wave propagation modes. The reflection of a twodimensional p wave from an obstacle (wedge) inclined to the propagation direction of the p wave is studied together with the convergence and stability behavior of the LBM as the lattice size and lattice time step decrease. The model is capable of accurately predicting the mode change (p to s) due to the reflection. The model provides a unique unified approach capable of simulating fluids, viscoelastic fluids, and solids within a single LBM framework, thus avoiding interface problems between different simulation methods. The paper concentrates on the wave propagation part of the model, in the quasielastic regime.
 Frantziskonis, G. N., Breysse, D., Casciati, F., & Mishnaevsky Jr, L. (1997). NATO✍✍ ARW960993.
 Frantziskonis, G., Renaudin, P., & Breysse, D. (1997). Heterogeneous solids  Part I: Analytical and numerical 1D results on boundary effects. European Journal of Mechanics, A/Solids, 16(3), 409423.More infoAbstract: This twopart study explores the consequences of material heterogeneity on the response of an externally loaded structure/specimen near its boundaries. The single deviation from the framework of classical (homogeneous) linear elasticity is that the elastic properties of the material form a stationary random field. It is shown that the consequences near boundaries can be quite adverse. Specifically, in the first part, investigation of a bar subjected to uniaxial load demonstrates that the strain field near the two ends is significantly and unfavorably different than in the bulk. Here, `unfavorably' concurs to fracture initiation. Further, it is shown that for brittle materials, both size effects and data scatter of peak strength are closely related to boundary phenomena. The (approximate) analytical solution of the problem provides significant insight, and allows examination of the efficiency of numerical solution schemes for problems of the type studied herein.
 Frantziskonis, G., Renaudin, P., & Breysse, D. (1997). Heterogeneous solidsPart I: analytical and numerical 1D results on boundary effects. European journal of mechanics. A. Solids, 16(3), 403423.
 Gustafson, S., Mehta, J., & Frantziskonis, G. (1997). Optical Nondestructive Evaluation of Fretting in Titanium Turbine Engine Fan Disks. Fifth TriService Titanium Contracts Review, WrightPatterson AFB, OH, 79.
 Gustafson, S., Mehta, J., & Frantziskonis, G. (1997). Practical Optical Characterization of Fretted Surfaces. MRS Proceedings, 503, 25.
 Meisner, M. J., & Frantziskonis, G. N. (1997). Heterogeneous materials  Scaling phenomena relevant to fracture and to fracture toughness. Chaos, Solitons and Fractals, 8(2 SPEC. ISS.), 151170.More infoAbstract: The paper documents a study on novel fracture and fracture toughness properties of brittle heterogeneous materials. Before any external load is applied on a structure/specimen, certain material relevant variables are considered to form a random field. The implications of these underlying heterogeneous fields before fracture on the properties of the fracture network developed from external load application are examined first. Then, emphasis is given on the distribution of the spatial variation of the dissipated energy due to fracture which shows, under certain yet general conditions, multifractal scaling properties. Importantly such scaling depends not only on the initial heterogeneity present, but also on the externally applied load and on the nature and extend (depth) of relevant surface effects. From the engineering point of view, such properties of dissipated energy provide a renewed loadpath as well as structure dependent definition of fracture toughness. © 1997 Elsevier Science Ltd All rights reserved.
 Meisner, M. J., & Frantziskonis, G. N. (1997). Heterogeneous materials—Scaling phenomena relevant to fracture and to fracture toughness. Chaos, Solitons \& Fractals, 8(2), 151170.
 Mishra, S. K., Muralidharan, K., Pannala, S., Simunovic, S., Daw, C. S., Nukala, P., Fox, R., Deymier, P. A., & Frantziskonis, G. N. (1997). Spatiotemporal compound wavelet matrix framework for multiscale/multiphysics reactor simulation: Case study of a heterogeneous reaction/diffusion system. INTERNATIONAL JOURNAL OF CHEMICAL REACTOR ENGINEERING, 6.More infoWe present a mathematical method for efficiently compounding information from different models of species diffusion from a chemically reactive boundary. The proposed method is intended to serve as a key component of a multiscale/multiphysics framework for heterogeneous chemically reacting processes. An essential feature of the method is the merging of wavelet representations of the different models and their corresponding time and length scales. Upanddownscaling of the information between the scales is accomplished by application of a compounding wavelet operator, which is assembled by establishing limited overlap in scales between the models. We show that the computational efficiency gain and potential error associated with the method depend on the extent of scale overlap and wavelet filtering used. We demonstrate the method for an example problem involving a twodimensional chemically reactive boundary and first order reactions involving two species.
 Renaudin, P., Breysse, D., & Frantziskonis, G. (1997). Heterogeneous solids  Part II: Numerical 2D results on boundary and other relevant phenomena. European Journal of Mechanics, A/Solids, 16(3), 425443.More infoAbstract: This twopart study explores the consequences of material heterogeneity on the response of an externally loaded structure/specimen near its boundaries. The single deviation from the framework of classical (homogeneous) linear elasticity is that the elastic properties of the material form a stationary random field. In this second part, it is first shown how the internal variables like stresses, strains and energy are correlated with the Young's modulus, and conclusions about the effect of a local failure criterion upon global failure are drawn. Then, boundary effects for which an analytical solution is not straightforward are enlightened. The effects yet found in one dimension are now coupled with those of the free boundary effect. The consequences of the free boundary on size related phenomena and the implications of spatial correlations on failure are analyzed and discussed in relation to the socalled characteristic material length and nonlocal models.
 Renaudin, P., Breysse, D., & Frantziskonis, G. (1997). Heterogeneous solids .2. Numerical 2D results on boundary and other relevant phenomena. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 16(3), 425443.
 Renaudin, P., Breysse, D., & Frantziskonis, G. (1997). Heterogeneous solidsPart II: numerical 2D results on boundary and other relevant phenomena. European journal of mechanics. A. Solids, 16(3), 425443.
 Frantziskonis, G. (1996). Stochastic approaches for damage evolution in standard and nonstandard continua. International Journal of Solids and Structures, 33(15), 22612265.
 Frantziskonis, G. N. (1996). ON THE STRENGTH RELIABILITY OF STATISTICALLY HETEROGENEOUS MATERIALS WITH MICROSTRUCTURE AT DIVERSE SCALES. INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING, 12(3), 249255.More infoThe paper examines the effects of microstructure realized at diverse scales on the overall strength reliability of the material. Reliability, or more precisely indirect reliability evaluated herein through the spatial fluctuations of strength, is shown to depend strongly on the scalewise distribution of heterogeneity as well as on the size of a specimen or structure. In particular, for a polycrystalline material with pores, the overall (indirect) strength reliability increases as (a) the average grain size decreases relevant to the that of the pores, (b) the variance of the grain size decreases relevant to that of the pores, and (c) the size of a specimen or structure or material building block decreases. tinder certain conditions, it is possible to reach nearzero or even zero variance, implying perfect reliability. The major conclusion is that scales interact with each other and that affects the overall fluctuations in material properties, thus providing renewed opportunities for tailoring the reliability of materials. The major conclusion is amenable to experimental investigation and verification for different heterogeneity scenarios.
 Frantziskonis, G., & Hansen, A. (1996). Waveletbased multiscaling in selfaffine random media. FRACTALSCOMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY, 8(4), 403411.More infoWavelet analysis is used to determine particular properties of selfaffine random media, i.e. ones with homogeneous increments. When such media are filtered through introducing upper and lower cutoffs  corresponding to the domain and sample support scales  interesting properties result from wavelet analysis. In particular, we show that the filtered media can be used for transferring information along scales. Furthermore, we show that a waveletbased approach for determining the underlying Hurst exponent is highly efficient. Examples relating to size and surface effects in materials demonstrate applications of the relevant multiscaling. Finally, we show that the waveletbased approach and a method based on mode superposition complement each other nicely.
 HONG, G., YALIZIS, A., & FRANTZISKONIS, G. (1996). HYGROTHERMAL DEGRADATION IN GLASS EPOXY  EVALUATION VIA STRESS WAVE FACTORS. COMPOSITE STRUCTURES, 30(4), 407417.More infoThe paper documents an experimental study on nondestructive evaluation of hygrothermally degraded glass fiber/epoxy matrix composite laminates. Specimens were subjected to accelerated life tests, i.e. high temperature and high pressure steam. The efficiency of the stress (ultrasonic) wave propagation through the specimens was evaluated quantitatively, at various stages of degradation, via the socalled stress wave factors (SWFs). Scanning electron micrographs were taken at some stages of degradation. In addition, the specimens were tested in bending mode. All measurements are correlated with the SWFs.
 Meisner, M. J., & Frantziskonis, G. N. (1996). Dissipated Energy as a Fundtion of Material Microstructure: Towards Optimum Fracture Toughness. Journal of the Mechanical Behavior of Materials, 6(4), 285300.
 Meisner, M. J., & Frantziskonis, G. N. (1996). Dissipated energy as a function of material microstructure. Proceedings of Engineering Mechanics, 2, 10301033.More infoAbstract: The paper addresses the issue of how material microstructure relates to fracturetoughness, the focus being on brittle disordered materials. The problem is investigated numerically through lattice discretization simulations of the rupture process. The spatial variation of dissipated energy due to fracture is evaluated, and under certain conditions, it is found that its distribution is characterized by a multifractal spectrum. The multifractal spectrum, f(α), depends not only on the initial disorder present in the material, but also on the nature of the externally applied load. It is found that when the disorder of the material increases, the relative total dissipated energy decreases, and vise versa.
 Meisner, M. J., & Frantziskonis, G. N. (1996). Multifractal fracturetoughness properties of brittle heterogeneous materials. Journal of Physics A: Mathematical and General, 29(11), 26572670.More infoAbstract: This paper documents a study of novel fracturetoughness properties of brittle heterogeneous materials. Through simulations of the rupture process based on a lattice discretization of the material, the spatial variation of dissipated energy due to fracture is evaluated. Under certain conditions, its distribution is characterized by a multifractal spectrum f (α). Importantly, f (α) depends not only on the initial heterogeneity present in the material but also on the nature of the externally applied load. This provides a renewed loadpathdependent definition of fracture toughness material properties. It avoids the difficulties associated with 'traditional' continuum/fracture mechanics definitions where the macroscopic fracture mode must be known a priori. © 1996 IOP Publishing Ltd.
 Meisner, M. J., & Frantziskonis, G. N. (1996). Multifractal fracturetoughness properties of brittle heterogeneous materials. Journal of Physics A: Mathematical and General, 29(11), 2657.
 Ramakrishnan, S., Budhu, M., & Frantziskonis, G. (1996). Constitutive behavior of granular media using a lattice type model. Proceedings of Engineering Mechanics, 2, 713716.More infoAbstract: In this paper, a lattice type model is used to gain an insight into the mechanical behavior of granular media. In a two dimensional simulation, nodes are introduced at grain contacts and the nodes are linked by bars. An assembly of grains, is transformed into truss elements and the truss is then analyzed using finite element analysis. The areas of bars are derived by computing the displacements of an elastic disk under contact forces and then determining the bar areas that will give equivalent displacements in the direction of the bars. Bonding, debonding, sliding and rotation of particles are incorporated in this model. The lattice type model is applied on a loose and dense packing of disks. The two packings are subjected to an initial vertical load followed by a simple shear deformation. The results of numerical tests are in general agreement with the behavior of sands in laboratory simple shear tests.
 FRANTZISKONIS, G. (1995). DAMAGE AND EDGE DELAMINATION IN COMPOSITES. SOLID MECHANICS ARCHIVES, 13(3), 129146.
 Frantziskonis, G. (1995). Heterogeneity and implicated surface effects: statistical, fractal formulation and relevant analytical solution. Acta Mechanica, 108(14), 157178.More infoAbstract: The aim of this paper is to examine the implications of material heterogeneity on brittle material response and on relevant surface effects. Statistical and fractal concepts are used for this purpose. In the statistical formulation the displacement gradients of the micromedium are considered to be random fields characterized by stationary exponential or Gaussian autocovariance and by the relevant correlation length or scale of fluctuation. Through Taylor series expansion around the mean of the random field, an important analogy is found between the statistical formulation and the microstructural theory, originally introduced by Mindlin, where higher order gradients of deformation appear in the constitutive equations. The analogy is valid only when fluctuations are small, so that some simplifications are allowed. It is found that the socalled internal length appearing in the microstructural formulation is analogous to the correlation length in the statistical one. In the statistical approach there are no extra boundary conditions in the formulation, as is the case when higher order gradients are introduced. However, what is know as "conditioning" of the field at the boundaries effects its behavior near/on them. The statistical approach can provide further information in the form of higher order moments not captured by the gradient theory. Material/structure response is strongly dependent on the aforementioned scale. Its effect is most pronounced near the boundaries of a structure where its role on surface related phenomena is paramount. In order to study heterogeneity at a hierarchy of scales, i.e. absence of characteristic length, complex disorderly system, fractal concepts and relevant power decay laws are considered. The formulation introduces the fractal dimension of the heterogeneous displacement gradient of the micromedium, a length describing the overall size of the structure, and the lower cutoff of the scaling law. The physical interpretation of the lower cutoff is the lower limit of applicability of the power scaling law. Mathematically it is important since in this case the fractal can be "followed" in the spatial domain. Similarly to the statistical case, an analogy between the fractal formulation and gradient theories is identified. No extra boundary conditions appear in the fractal formulation. However, there are still open questions with respect to the behavior of a fractal after conditioning, as is the case on boundaries. The analytical solution of a relevant surface instability problem for the gradient, statistical, and fractal formulation is presented. The solution was obtained through symbolic computations by computer because the analytical work is tedious and error prone. The analytical solution provides significant insight into the problem of heterogeneity and skin effects in brittle materials, internal length estimation, and the role of fractal scaling properties. Finally, the concepts introduced herein are discussed with respect to experimental information and numerical implementation. © 1995 SpringerVerlag.
 Frantziskonis, G. (1995). Heterogeneity and implicated surface effects: statistical, fractal formulation and relevant analytical solution. Acta mechanica, 108(14), 157178.
 Frantziskonis, G., & Loret, B. (1995). Scale dependent constitutive relationsinformation from wavelet analysis and application to localization problems. European journal of mechanics. A. Solids, 14(6), 873892.
 Frantziskonis, G., Simon, L., Woo, J., & Matikas, T. (1995). Multiscale characterization of pitting corrosion and application to an aluminum alloy. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 19(2), 309318.More infoThis paper documents a novel method for characterizing pitting corrosion damage in structural materials such as Al 2024T3. Specimens of such alloys are corroded in a controlled environment and the pits' geometry is captured digitally using white light interference microscopy. The digital data are then processed with waveletbased analysis, thus making possible a multiresolution description of the geometrical features. The analysis reveals several interesting features of the pits that are similar for all the experimental data analyzed herein, and independent of the process followed for creating them (time the material is exposed to corroding environment, concentration of the corroding agent, surface area exposed to the agent, etc.). The first property identified as common to all pits is their geometrical scaling with a (Hurst) exponent of 0.63 +/ 0.12. Furthermore, the ratio omega of the surface area of the pit as represented at coarse scales through the wavelet representation, over the area of its intersection with the plane at zero depth is found to be 1.17 +/ 0.07 consistently. The ratio of the total surface area over its intersection is found to be 1.6 +/ 0.2. Either one of these ratios together with the Hurst exponent provide sufficient information for obtaining a pit's geometry from images capturing its twodimensional shape only, a capability important for efficient characterization. Additionally, such a characterization is paramount for rigorously addressing fatigue crack initiation and propagation. (C) 2000 Editions scientifiques et medicales Elsevier SAS.
 Hong, G., Yalizis, A., & Frantziskonis, G. N. (1995). Hygrothermal degradation in glass/epoxy  evaluation via stress wave factors. Composite Structures, 30(4), 407417.More infoAbstract: The paper documents an experimental study on nondestructive evaluation of hygrothermally degraded glass fiber/epoxy matrix composite laminates. Specimens were subjected to accelerated life tests, i.e. high temperature and high pressure steam. The efficiency of the stress (ultrasonic) wave propagation through the specimens was evaluated quantitatively, at various stages of degradation, via the socalled stress wave factors (SWFs). Scanning electron micrographs were taken at some stages of degradation. In addition, the specimens were tested in bending mode. All measurements are correlated with the SWFs. © 1995.
 Hong, G., Yalizis, A., & Frantziskonis, G. N. (1995). Hygrothermal degradation in glass/epoxy—evaluation via stress wave factors. Composite structures, 30(4), 407417.
 Bruno, M., Machcha, A., NematNasser, S., Sofronis, P., McMeeking, R., Ragab, A., Alawi, H., Sorein, K., Lamouroux, C., Debat, P., & others, . (1994). Modelling of semibrittle MgO ceramic behaviour under compression T. Sadowski 1 On the prediction of interfacial behavior in fiberreinforced composites using a new contact element Mallikarjuna and M. Fafard 17. Mechanics of Materials, 18, 371372.
 Budhu, M., Ramakrishnan, S., & Frantziskonis, G. (1994). Mechanics of particulate mediaA lattice type approach. Proceedings: Workshop on Mechanics and Statistical Physics of Particulate Materials, Institute for Mechanics and Materials, University of California, San Diego, CA.
 Dai, H., & Frantziskonis, G. (1994). Heterogeneity, spatial correlations, size effects and dissipated energy in brittle materials. Mechanics of Materials, 18(2), 103118.More infoAbstract: The present study was inspired by recent experiments performed by the second author and coworkers where heterogeneity in a brittle material was investigated by ultrasonic scanning. The experiments showed an irregular pattern of material heterogeneity even before any external load was applied on the specimens. Under increasing uniaxial compressive load, the initial heterogeneity pattern evolved, and finally a macro crack network formed. In a previous study the finite element method was used in conjunction with a theory for distributed damage to study the effects of material heterogeneity numerically. Both experiments and finite element analysis showed that initially "strong" regions dissipated energy at a much higher rate than "weak" ones. However, the FEM is more suitable and efficient when material response can be homogenized and deformation gradients are not prominent. If this is not the case, or if one is interested in understanding and/or modeling the effects of heterogeneity and crack network formation, a lattice approach may be more suitable. In this study, the predictions of the latticebased numerical approach are compared to experimental data on crack formation. Recently, a branch of statistical physics has focused on statistical modeling of materials. Here, it is attempted to "connect" this approach to continuum solid mechanics theories. Important connections are believed to be spatial correlations and localization phenomena in materials, as discussed herein. Using random fields to represent initial heterogeneity, the effects of spatial correlations on size effects, dissipated energy and on crack formation are studied. Results from two random field generation algorithms are reported, and, surprisingly, dissipated energy and crack network characteristics were dependent on the algorithm. Both provide a good description of the wellknown size effect observed in brittle materials. However, notable differences between the two algorithms with respect to localization of deformation were identified. © 1994.
 Dai, H., & Frantziskonis, G. (1994). Heterogeneity, spatial correlations, size effects and dissipated energy in brittle materials. Mechanics of materials, 18(2), 103118.
 FRANTZISKONIS, G., & DESAI, C. (1994). SURFACE DEGRADATION MECHANISMS IN BRITTLE MATERIAL STRUCTURAL SYSTEMS. INTERNATIONAL JOURNAL OF FRACTURE, 48(4), 231244.More infoA mechanics based theory for surface degradation in brittle material systems is introduced. Surface degradation is directly related to damage progression. For this reason the mechanics of damage evolution is presented first. Subsequently, relations governing surface degradation mechanisms are derived and discussed in detail. It is shown that surface degradation can capture important properties of brittle materials such as scale (size) and shape effects, surface damage growth and subsequent bursting instabilities. Finally, the problem of transferring information from laboratory experiments to large scale problems is discussed; the need for further experimental and theoretical research is pointed out.
 Frantziskonis, G. (1994). On scaling phenomena in fracture of heterogeneous solids. European journal of mechanics. A. Solids, 13(1), 7392.
 Frantziskonis, G. (1994). On the possibly multifractal properties of dissipated energy in brittle materials. Applied Mechanics Reviews, 47(1S), S132S140.
 Frantziskonis, G. (1994). Research note on the equivalence of isotropic damage mechanics and plasticity. International Journal of Plasticity, 10(3), 303308.More infoAbstract: It is shown that certain isotropic damage mechanics models are mathematically equivalent to plasticitybased ones. From the equivalence, a macroscopic physical interpretation of damage evolution and restrictions on the partial stresses are derived. © 1994.
 Frantziskonis, G. (1994). Research note on the equivalence of isotropic damage mechanics and plasticity. International journal of plasticity, 10(3), 303308.
 Frantziskonis, G. (1994). Stochastic modeling of heterogeneous materials  A process for the analysis and evaluation of alternative formulations. MECHANICS OF MATERIALS, 27(3), 165175.More infoThe paper addresses the problem of understanding and modeling the constitutive behavior of complex, heterogeneous solid materials. Several issues in the area of stochastic material modeling remain open, thus a process for the analysis of different formulations is proposed and examined in detail. Using the proposed process, it is shown that a number of basic models are stochastically similar to each other. Furthermore, often, small apparently unimportant changes in a formulation may result in significant changes in the predicted response. For some formulations, inequalities are established that when satisfied it is guaranteed that the entropy of a structure/system cannot decrease. Attention is given to stochastic formulations within the context of nonlocal theories because such formulations provide new information to the subject of localization of deformation and fracture toughness. (C) 1998 Elsevier Science Ltd.
 Frantziskonis, G. N., Karpur, P., Matikas, T. E., Krishnamurthy, S., & Jero, P. D. (1994). Fibermatrix interface  information from experiments via simulation. Composite Structures, 29(3), 231247.More infoAbstract: This study explores a novel procedure for obtaining quantitative information on the mechanical properties of the fibermatrix interface in composite materials. The method, based on lattice discretization of a medium, simulates actual experiments in detail, including fiber breakage, matrix yield and/or cracking, and interface failure. The paper concentrates on two experiments performed commonly, the socalled fragmentation test for metal matrix, and the pushout/pullout test for metal as well as ceramic matrix composites. Based on the documented capability of the method to simulate actual experimental data, reliable values of (homogenized) interface properties can be obtained. In addition, the simulations provide further understanding of the mechanisms involved during the relevant testing. Although this study presents results from basic problems, the method is general enough to include effects of residual stress, of high temperature environment, and of dynamic crack propagation, as well as threedimensional details of the interface failure process. The potential exists for simulating nondestructive wavebased techniques aimed at evaluating interface properties. © 1994.
 Frantziskonis, G. N., Karpur, P., Matikas, T. E., Krishnamurthy, S., & Jero, P. D. (1994). Fibermatrix interface—information from experiments via simulation. Composite structures, 29(3), 231247.
 JOSHI, S., & FRANTZISKONIS, G. (1994). DAMAGE EVOLUTION IN LAMINATED ADVANCED COMPOSITES. COMPOSITE STRUCTURES, 17(2), 127139.
 Mishra, S. K., Deymier, P. A., Muralidharan, K., Frantziskonis, G., Pannala, S., & Simunovic, S. (1994). Modeling the coupling of reaction kinetics and hydrodynamics in a collapsing cavity. ULTRASONICS SONOCHEMISTRY, 17(1), 258265.More infoWe introduce a model of cavitation based on the multiphase Lattice Boltzmann method (LBM) that allows for coupling between the hydrodynamics of a collapsing cavity and supported solute chemical species. We demonstrate that this model can also be coupled to deterministic or stochastic chemical reactions. In a twospecies model of chemical reactions (with a major and a minor species). the major difference observed between the deterministic and stochastic reactions takes the form of random fluctuations in concentration of the minor species. We demonstrate that advection associated with the hydrodynamics of a collapsing cavity leads to highly inhomogeneous concentration of solutes. In turn these inhomogeneities in concentration may lead to significant increase in concentrationdependent reaction rates and can result in a local enhancement in the production of minor species. Published by Elsevier B.V.
 FRANTZISKONIS, G., DESAI, C., & SOMASUNDARAM, S. (1993). CONSTITUTIVE MODEL FOR NONASSOCIATIVE BEHAVIOR. JOURNAL OF ENGINEERING MECHANICSASCE, 112(9), 932946.
 Frantziskonis, G. (1993). Heterogeneity and Its ImplicationsMicromechanical, Statistical, Fractal Approach and Their Similarity. Elsevier Science Publishers B. V., Damage in Composite Materials(Netherlands), 1993,, 137160.
 Buban, D. M., & Frantziskonis, G. N. (1992). Shape memory alloy fracture as a deployment actuator. SMART MATERIALS AND STRUCTURES, 22(11).More infoThe paper reports an experimental investigation into using shape memory alloy (SMA) as a deployment actuator. SMA specimens were heated and pulled to failure or pulled and heated to failure, thus developing an environmental and structural operating envelope for application as deployment mechanisms. The experimental results strongly suggest that SMAs can be implemented as deployment actuators. Recorded data shows that SMA fracture is possible over a wide range of temperatures and strains, filling a material performance gap not found in the literature. The obtained information allows design engineers to appropriately size SMAs given the design requirements for achieving the desired deployment effects. The major conclusion of the reported work is that SMAs as actuators are strong competitors to typical existing deployment efforts that use explosive or nonexplosive actuators having implementation drawbacks such as the expense associated with special handling and the volume encountered in mounting the devices.
 Daniewicz, D., & Frantziskonis, G. (1992). Edge delamination in laminated composites. Composite Structures, 21(3), 141153.More infoAbstract: A damage evolution theory for the effects of the rather complicated edge delamination phenomena on composite laminate response is developed. The theory is a mechanicsbased formulation which quantifies the damage development of a laminate under general loading, and incorporates it directly into the laminate constitutive equations. The theoretical development attempts to describe the basic features of the edge delamination phenomena by introducing a single laminatespecific constant φ. The methodology of φ selection is presented for a [± 45/0/90]s T300 graphite/epoxy laminate. Edge effects are demonstrated by comparing the state of the above laminate with and without edge delamination damage. The capability of the theory in predicting the volume scaling effect (the phenomenon whereby the onset of edge delamination is observed to be a function of ply thickness, when the stacking sequence is invariant) is shown through consideration of the failure strengths and strains of the [± 25 90n]s laminate series. © 1992.
 Daniewicz, D., & Frantziskonis, G. (1992). Edge delamination in laminated composites. Composite structures, 21(3), 141153.
 FRANTZISKONIS, G., & DESAI, C. (1992). ELASTOPLASTIC MODEL WITH DAMAGE FOR STRAIN SOFTENING GEOMATERIALS. ACTA MECHANICA, 68(34), 151170.
 Frantziskonis, G. (1992). Heterogeneity, microstructural surface effects and internal length estimation. American Society of Mechanical Engineers, Applied Mechanics Division, AMD, 135, 5166.More infoAbstract: Recently, the concept of state of degradation has been employed by the author in order to study the effect of initial material (structure) inhomogeneity and the implications of surface (skin) effects on brittle material response. The analysis has revealed the importance of the internal material length, e.g. for assigning the initial random variables according to a material dependent fluctuation scale. Different possibilities for its estimation and/or evolution have been suggested. Here, symbolic computations by computer that resulted in the analytical solution of an instability problem are presented. Such analytical solution without computer had not been obtained in the past because the analytical work is tedious and error prone making it very difficult to pursue. The analytical solution, made possible through symbolic computations, provides significant insight into the problem of skin effects in brittle materials and internal length estimation. This paper summarizes the previous work of the author and coworkers, presents the results of symbolic computations of an instability problem, and interprets the results with respect to material length estimation.
 Frantziskonis, G. (1992). Heterogeneity, microstructural surface effects and internal length estimation. Proc. ASMEAMD, 135, 5166.
 Frantziskonis, G. (1992). Multiscale characterization of materials with distributed pores and inclusions and application to crack formation in an aluminum alloy. PROBABILISTIC ENGINEERING MECHANICS, 17(4), 359367.More infoThe paper reports a study on the mechanical behavior of materials containing pores and inclusions distributed over a wide range of length scales. Utilizing a waveletbased multiscale process such microstructures are characterized and their effects on material properties is studied. In order to present the process in a semianalytical fashion, the variance of the strain field for an approximated onedimensional deformation problem is examined in detail. It is shown that with respect to crack initiation, there is a strong interplay between the distribution of pores and inclusions. Furthermore, their interaction with boundaries proves to be paramount. The process is applied to a particular cast aluminum alloy where pores are, in general, about two orders of magnitude larger than the silicon particles (inclusions). Results agree well with recent experimental reports on crack initiation where the interplay of pores, inclusions, and boundaries is observed, yet not explained on a fundamental basis. The present work extends recent efforts on porous materials [Frantziskonis G. Waveletbased analysis of multiscale phenomenaapplication to material porosity and identification of dominant scales. Prob Eng Mech (this issue). PII: S02668920(2)000322] to include the interaction of pores at certain scales with inclusions at other scales as well as the interaction of both with boundaries. (C) 2002 Elsevier Science Ltd. All rights reserved.
 Frantziskonis, G. (1992). Surface effects in brittle materials and internal length estimation. Applied Mechanics Reviews, 45(3 pt 2), S62S70.More infoAbstract: The concept of state of degradation is employed as a tool to study the effect of initial material (structure) inhomogeneity and the implications of surface (skin) effects on brittle material response. The initial heterogeneity pattern and its growth under external load is studied experimentally via ultrasonic scanning measurements on prismatic simulated rock samples subjected to mechanical load. Numerical solutions are compared with the experimental results. The skin effects are also studied experimentally through the spatial variation of dissipated energy within the samples tested. In addition, microstructure considerations highlight the skin effect and its implications. The material length for the simulated rock is estimated (for load levels up to peak load) through the experimental observations which yielded 1≅0.24 inches (0.61 cm). The analytical solution of a simple problem including microstructure considerations for the same material yielded 1≅0.21 inches (0.53 cm). This paper integrates and extends the recent work of the author and coworkers.
 Frantziskonis, G. (1992). Surface effects in brittle materials and internal length estimation. Applied Mechanics Reviews, 45(3S), S62S70.
 Frantziskonis, G., & Vardoulakis, I. (1992). On the microstructure of surface effects and related instabilities. European Journal of Mechanics, A/Solids, 11(1), 2134.More infoAbstract: Based on Mindlin's theory for material microstructure interesting surface effects under conditions of equilibrium are studied in this paper. The governing field equations for uniaxial plane deformations are established; surface instability analysis shows non uniform deformations for a layer of specified distance from the surface. Experimental as well as fracture mechanics based considerations show that this surfacelayer is extremely thin for metals while for brittle materials its magnitude is of the order of 1 cm. Material microstructure introduces a singular perturbation to the original Hill and Hutchinsonproblem; here we introduce a single perturbation parameter and we obtain a 'dispersion' law for the surface buckling load. It is found that surface degradation and skin effects can be attributed to localized surface buckling instabilities. Experimental information on skin effects can provide an estimation of the internal material length.
 Frantziskonis, G., & Vardoulakis, I. (1992). On the microstructure of surface effects and related instabilities. European journal of mechanics. A. Solids, 11(1), 2134.
 Frantziskonis, G., Abdulraheem, A., & others, . (1992). Mechanics of Reservoir Rocks, Theory and Verification. Society of Petroleum Engineers, paper, 50.
 Frantziskonis, G., Desai, C. S., Tang, F. F., & Daniewicz, D. (1992). Degradation mechanisms in brittle materials investigated by ultrasonic scanning. Engineering Fracture Mechanics, 42(2), 347369.More infoAbstract: An experimental study of degradation mechanisms and patterning in brittle materials is reported. Specimens of a castin brittle material were subjected to external mechanical load. At various prepeak load levels, through transmission ultrasonic measurements were taken at several locations of the sample. The damage evolution and its patterning were studied through analysis of the attenuation of the transmitted wave. For monotonically increasing load significant attenuation was observed at low load levels, while for unloading the attenuation changes were relatively small. The test results indicate random and nonsymmetric distribution of attenuation within the sample. However, in general, higher attenuation was observed close to the loadfree surfaces than in the interior of the sample. This observation signifies the socalled skin effect. Occasionally, at a few locations of various samples the energy of the received wave increased with increasing external load. This may indicate unloading of partially formed cracks. Measured surface strains are compared to the overall ones and the degradation patterning within the samples. The ultrasonic measurements are examined with respect to estimating the internal material length. The relation between ultrasonic wave attenuation and mechanical dissipated energy is examined. © 1992.
 Frantziskonis, G., Desai, C., Tang, F., & Daniewicz, D. (1992). Degradation mechanisms in brittle materials investigated by ultrasonic scanning. Engineering fracture mechanics, 42(2), 347369.
 Frantziskonis, G., Mishra, S. K., Pannala, S., Simunovic, S., Daw, C. S., Nukala, P., Fox, R. O., & Deymier, P. A. (1992). Waveletbased spatiotemporal multiscaling in diffusion problems with chemically reactive boundary. INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING, 4(56), 755770.More infoChemically reacting flows over catalytic and noncatalytic surfaces are one of the elementary operations in chemical processing plants. The underlying physical phenomena span time and length scales over several orders of magnitude, which a robust and flexible modeling framework must efficiently account for. With this purpose as the eventual goal, we propose a waveletbased multiscale numerical framework and demonstrate it on the coupling of two prototype methods for the problem of species generated on a chemically reactive boundary and diffusing through the bulk. The two methods consider different time and length scales. The first method in this coupling, termed "fine," models the chemical reactions on the reactive boundary stochastically by the kinetic Monte Carlo method and the diffusion in the medium deterministically using relatively small time increments and small spatial discretization mesh size. The second method, termed "coarse," models both the reaction and the diffusion deterministically and uses drastically larger time increments and spatial discretization size than the fine model. The two methods are coupled by forming a spatiotemporal compound wavelet matrix that combines information about the time and spatial scales contained in them.
 Meisner, M., & Frantziskonis, G. (1992). Heterogeneous materialsscaling phenomena relevant to fracture and to fracture toughness. CHAOS SOLITONS & FRACTALS, 8(2), 151170.More infoThe paper documents a study on novel fracture and fracture toughness properties of brittle heterogeneous materials. Before any external load is applied on a structure/specimen, certain material relevant variables are considered to form a random field. The implications of these underlying heterogeneous fields before fracture on the properties of the fracture network developed from external load application are examined first. Then, emphasis is given on the distribution of the spatial variation of the dissipated energy due to fracture which shows, under certain yet general conditions, multifractal scaling properties. Importantly such scaling depends not only on the initial heterogeneity present, but also on the externally applied load and on the nature and extend (depth) of relevant surface effects. From the engineering point of view, such properties of dissipated energy provide a renewed loadpath as well as structure dependent definition of fracture toughness. (C) 1997 Elsevier Science Ltd All rights reserved.
 TANG, F., FRANTZISKONIS, G., & DESAI, C. (1992). Instabilities of damage and surface degradation mechanisms in brittle material structural systems(Final Report, 1 Jul. 1989 31 Dec. 1991).
 Tang, F. F., Desai, C. S., & Frantziskonis, G. (1992). Heterogeneity and degradation in brittle materials. Engineering Fracture Mechanics, 43(5), 779796.More infoAbstract: The concept of state of degradation is employed in order to consider the initial material (structure) inhomogeneity. Then, degradation and its patterning in simulated rock specimens subjected to external load are modeled numerically by implementing the theory for damage and surface degradation with incorporation of the initial state. The theoretical results are compared with experimental observations obtained through ultrasonic scanning tests. © 1992.
 Tang, F., Desai, C., & Frantziskonis, G. (1992). Heterogeneity and degradation in brittle materials. Engineering fracture mechanics, 43(5), 779796.
 Vardoulakis, I., & Frantziskonis, G. (1992). Microstructure in kinematichardening plasticity. European Journal of Mechanics, A/Solids, 11(4), 467486.More infoAbstract: A gradient regularization of the classical kinematichardening plasticity is presented. The underlying continuum model is formally related to Mindlin's elasticity theory with microstructure. The evolution law for the back stress is identical to Mindlin's higher order equilibrium equation. For consistency reasons the flow rule of classical plasticity is modified by incorporating the Laplacian of the plastic multiplier. The variational formulation of the problem with appropriate boundary conditions is given and an expression for the dissipated energy is established. Shearband analysis shows that the theory provides the band thickness, and regularizes the governing equations. Microstructure introduces a singular perturbation to the classical surface instability analysis, and the internal length l is the perturbation parameter. In addition, microstructure effects tend to reduce the wavelength at onset of surface instability.
 Vardoulakis, I., & Frantziskonis, G. (1992). Microstructure in kinematichardening plasticity.
 Yuan, X., Tang, F. F., & Frantziskonis, G. (1992). Random initial heterogeneity and degradation in brittle materials. Proceedings of Engineering Mechanics, 616619.More infoAbstract: The question of implementation of the initial state of the material is addressed. The problem is tackled by assuming random initial values for a relevant variable, within a prespecified range. The numerical implementation shows that the lower the initial assigned degradation in the material the higher the rate of dissipated energy. This numerical result agrees with relevant experiments. The analysis has revealed the importance of the internal material length, e.g. for assigning the initial random variables according to a material dependent fluctuation scale. Different possibilities for its estimation and/or evolution have been suggested. Symbolic computations by computer that resulted in the analytical solution of an instability problem are presented. Such analytical solution without computer had not been obtained in the past because the analytical work is tedious and error prone making it very difficult to pursue. The analytical solution, made possible through symbolic computations, provides significant insight into the problem of skin effects in brittle materials and internal length estimation.
 DAI, H., & FRANTZISKONIS, G. (1991). HETEROGENEITY, SPATIAL CORRELATIONS, SIZE EFFECTS AND DISSIPATED ENERGY IN BRITTLE MATERIALS. MECHANICS OF MATERIALS, 18(2), 103118.More infoThe present study was inspired by recent experiments performed by the second author and coworkers where heterogeneity in a brittle material was investigated by ultrasonic scanning. The experiments showed an irregular pattern of material heterogeneity even before any external load was applied on the specimens. Under increasing uniaxial compressive load, the initial heterogeneity pattern evolved, and finally a macro crack network formed. In a previous study the finite element method was used in conjunction with a theory for distributed damage to study the effects of material heterogeneity numerically. Both experiments and finite element analysis showed that initially ''strong'' regions dissipated energy at a much higher rate than ''weak'' ones. However, the FEM is more suitable and efficient when material response can be homogenized and deformation gradients are not prominent. If this is not the case, or if one is interested in understanding and/or modeling the effects of heterogeneity and crack network formation, a lattice approach may be more suitable. In this study, the predictions of the latticebased numerical approach are compared to experimental data on crack formation. Recently, a branch of statistical physics has focused on statistical modeling of materials. Here, it is attempted to ''connect'' this approach to continuum solid mechanics theories. Important connections are believed to be spatial correlations and localization phenomena in materials, as discussed herein. Using random fields to represent initial heterogeneity, the effects of spatial correlations on size effects, dissipated energy and on crack formation are studied. Results from two random field generation algorithms are reported, and, surprisingly, dissipated energy and crack network characteristics were dependent on the algorithm. Both provide a good description of the wellknown size effect observed in brittle materials. However, notable differences between the two algorithms with respect to localization of deformation were identified.
 Frantziskonis, G., & Breysse, D. (1991). Influence of soil variability on differential settlements of structures. COMPUTERS AND GEOTECHNICS, 30(3), 217230.More infoThe paper addresses the interaction of a structure with spatially varying soil properties. In particular, the problem of a twospan continuous beam founded on a heterogeneous soil is solved analytically. The geometrical and stiffness characteristics of the structure interact strongly with the spatial properties of the heterogeneous soil. For a certain value of the correlation distance, a feature of the heterogeneous soil formation, the uncertainty and the risk of high values, not predicted with deterministic models, in estimating differential settlements and forces (bending moments, shear forces, etc.) on the structure becomes maximum. The analytical solution uses a series expansion of the soil properties relative to those of the structure. The error in the solution, due to the truncation of the series expansion, is estimated by relevant numerical results. The paper shows clearly that the forces on the structure founded on a heterogeneous soil can differ widely from those usually predicted by a deterministic model. Furthermore, a usual deterministic approach can underestimate the safety level of the structure significantly. (C) 2003 Elsevier Science Ltd. All rights reserved.
 Frantziskonis, G., & Desai, C. (1991). Surface degradation mechanisms in brittle material structural systems. International Journal of Fracture, 48(4), 231244.
 Frantziskonis, G., & Desai, C. S. (1991). Surface degradation mechanisms in brittle material structural systems. International Journal of Fracture, 48(4), 231244.More infoAbstract: A mechanics based theory for surface degradation in brittle material systems is introduced. Surface degradation is directly related to damage progression. For this reason the mechanics of damage evolution is presented first. Subsequently, relations governing surface degradation mechanisms are derived and discussed in detail. It is shown that surface degradation can capture important properties of brittle materials such as scale (size) and shape effects, surface damage growth and subsequent bursting instabilities. Finally, the problem of transferring information from laboratory experiments to large scale problems is discussed; the need for further experimental and theoretical research is pointed out. © 1991 Kluwer Academic Publishers.
 Frantziskonis, G., & Deymier, P. (1991). Wavelet methods for analysing and bridging simulations at complementary scales  the compound wavelet matrix and application to microstructure evolution. MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING, 8(5), 649664.More infoWe introduce a novel waveletbased compound matrix to bridge models (LennardJones model simulated with molecular dynamics and a lattice Qstates Potts model with a Monte Carlo simulation) that describe grain growth over different ranges of spatial and time scales. The compound wavelet matrix provides full. statistical information on the microstructure at the range of scales that is the union of those handled by the two models.
 Frantziskonis, G., Tang, F. F., & Desai, C. S. (1991). Borehole scale effects and related instabilities. Engineering Fracture Mechanics, 39(2), 377389.More infoAbstract: A new mechanics based approach is proposed for scale effects and instabilities on borehole problems. In borehole types of structural systems, two types of instabilities can take place. The first is due to surface degradation growth and results into spalling of layers at the hole wall. The second is due to damage progression, and results into globally unstable response of the structure. The hole size has been found experimentally to be an important parameter in breakout instability initiation. Laboratory size holes may overestimate instability initiation properties by a large factor. At the same time, material properties such as peak stress depend largely on the size and shape of a specimen subjected to uniaxial or triaxial compression. This work attempts to incorporate size and scale effects into the instability initiation conditions. The important task of transferring information from laboratory experiments to actual large scale engineering problems is analysed and discussed. The potential of the theory is demonstrated. The need for further experimental and theoretical work is identified. © 1991.
 Frantziskonis, G., Tang, F., & Desai, C. (1991). Borehole scale effects and related instabilities. Engineering fracture mechanics, 39(2), 377389.
 Herrmann, H., Meisner, M. J., Frantziskonis, G. N., Alava, M., Ritala, R., Krajcinovic, D., Vujosevic, M., Politi, A., Zei, M., Alava, M. J., & others, . (1991). Article cit\'e par. Physica Scripta, 38, 13.
 Joshi, S. P., & Frantziskonis, G. (1991). Damage evolution in laminated advanced composites. Composite Structures, 17(2), 127139.More infoAbstract: This paper extends the theory for damage evolution in a single ply presented earlier by the authors. Simultaneous damage evolution in individual plies, at different rates, is accounted for; reduction of the rate of damage growth in a ply due to deformational restrictions imposed by adjacent plies is implicit in the formulation. The theory is valid for any sequence of external loading. However, unloading and cyclic loading cases are not considered in this paper. This is achieved by completely updating the material parameters in a ply that has reached a saturated damage state. This parameter updating is based on the identified damage mode (at failure) of the ply. Predicted results are compared with available experimental data for T300 GraphiteEpoxy. Further results that demonstrate the capabilities of the theory are also presented. © 1990.
 Joshi, S., & Frantziskonis, G. (1991). Damage evolution in laminated advanced composites. Composite Structures, 17(2), 127139.
 Yuan, X. (1991). Random initial inhomogeneity in brittle materials.
 FRANTZISKONIS, G. (1990). DISTRIBUTED DAMAGE IN COMPOSITES, THEORY AND VERIFICATION. COMPOSITE STRUCTURES, 10(2), 165184.
 Frantziskonis, G. (1990). Degradation instabilities in brittle material structural systems. American Society of Mechanical Engineers, Applied Mechanics Division, AMD, 109, 203209.More infoAbstract: The paper focuses on energy based instability analysis of brittle material structural systems. In such structures, two types of degradation instabilities are possible. The first is due to surface degradation growth and results into spalling of layers at the stressfree surfaces. The second is due to damage progression, and results into globally unstable response of the structure. Simple problems are treated analytically, while more complicated ones are treated computationally. It is concluded that surface degradation instability and/or damage progression instability occurs before the peak strength is reached. Thus a homogeneous post peak material response may be impossible, since basic energy criteria are violated. Numerical results are presented and discussed in detail.
 Frantziskonis, G., & Desai, C. (1990). Degradation instabilities in brittle material structures. Mechanics research communications, 17(3), 135141.
 Frantziskonis, G., & Desai, C. S. (1990). Degradation instabilities in brittle material structures. Mechanics Research Communications, 17(3), 135141.
 Frantziskonis, G., & Joshi, S. (1990). Damage evolution and constitutive behavior of advanced composites. Composite Structures, 16(4), 341357.
 Frantziskonis, G., & Joshi, S. P. (1990). Damage evolution and constitutive behavior of advanced composites. Composite Structures, 16(4), 341357.More infoAbstract: The aim of this paper is to present a mechanics based theory that accounts for damage evolution in advanced composites. Singleply behavior, under various loading conditions, is studied in detail herein. It is shown that the proposed theory can capture the essential deformational characteristics of single composite material plies, subjected to external load. Emphasis is placed on the properties of the theory and its capability to predict the observed behavior; further, the present approach is compared to classical analysis procedures. The paper extends previous work by the first author; it considers loading path effects, relation of damage growth to apparent Poisson ratio, strain based distinction between tension and compression, failure criteria under different loading conditions, and thermodynamic restrictions. © 1990.
 Desai, C. S., Saadatmanesh, H., & Frantziskonis, G. (1989). Development of construction materials like concrete from lunar soils without water.
 Frantziskonis, G. (1989). Damage and free edge effects in laminated composites. Energy and stability propositions. Acta Mechanica, 77(34), 213230.More infoAbstract: The paper considers certain important aspects of a theory that accounts for distributed damage and edge delamination effects in laminated composites. The proposed model considers the mechanics of damage, resulting from the application of external loads. Also edge delamination, an important phenomenon of laminated anisotropic composites, is included in the theory. The irreversible part, due to damage and edge effects, of rate of global energy shows an important contribution to the overall energy equations. Instabilities due to damage growth, edge delamination growth or both the above are considered and relations for their development are established. © 1989 SpringerVerlag.
 Frantziskonis, G. (1989). Damage and free edge effects in laminated composites. Energy and stability propositions. Acta mechanica, 77(34), 213230.
 Frantziskonis, G. N. (1989). Mechanics of tectonic faulting, G. Mandl, Developments in Structural Geology Vol. 1 (Series Editor HJ Zwart), Elsevier, Amsterdam, 1988. No. of pages: 407. Price US $79.00/Dfl. 150.00. ISBN 0444429468. International Journal for Numerical and Analytical Methods in Geomechanics, 13(3), 339339.
 VARDOULAKIS, ., & FRANTZISKONIS, G. (1989). MICROSTRUCTURE IN KINEMATICHARDENING PLASTICITY. EUROPEAN JOURNAL OF MECHANICS ASOLIDS, 11(4), 467486.More infoA gradient regularization of the classical kinematichardening plasticity is presented. The underlying continuum model is formally related to Mindlin's elasticity theory with microstructure. The evolution law for the back stress is identical to Mindlin's higher order equilibrium equation. For consistency reasons the flow rule of classical plasticity is modified by incorporating the Laplacian of the plastic multiplier. The variational formulation of the problem with appropriate boundary conditions is given and an expression for the dissipated energy is established. Shearband analysis shows that the theory provides the band thickness, and regularizes the governing equations. Microstructure introduces a singular perturbation to the classical surface instability analysis, and the internal length l is the perturbation parameter. In addition, microstructure effects tend to reduce the wavelength at onset of surface instability.
 FRANTZISKONIS, G. (1988). DAMAGE AND FREE EDGE EFFECTS IN LAMINATED COMPOSITES  ENERGY AND STABILITY PROPOSITIONS. ACTA MECHANICA, 77(34), 213230.
 Frantziskonis, G. (1988). Damage and edge delamination in composites. SM archives, 13(3), 129146.
 Frantziskonis, G. (1988). Damage and edge delamination in composites. SM archives, 13(3), 129146.More infoAbstract: The aspect of damage development in laminated composites is considered. The recent work of the author is generalized and extended. The theory is developed in the rate form of the governing equations and the properties of the theory are analyzed and discussed. Chapter 3 focuses on the subject of edge delamination which is of importance in the analysis of laminated composites. The concept of effective free edge damage distribution area is proposed and verified by numerical examples.
 Frantziskonis, G. (1988). Distributed damage in composites, theory and verification. Composite Structures, 10(2), 165184.More infoAbstract: A model that accounts for the effect of structural changes, resulting from the application of loads, in composite materials is developed. Such changes are incorporated in the theory through a tensor form of a damage variable. The model extends the theory of elasticity and it accounts for certain important properties observed in composite materials. The parameters involved in the theory are identified and determined from available experimental data. Back prediction of test results, for two different composite materials, verifies the theory. © 1988.
 Frantziskonis, G. (1988). Distributed damage in composites, theory and verification. Composite structures, 10(2), 165184.
 Frantziskonis, G., & Desai, C. (1988). Analysis of a strain softening constitutive model. Mathematical and Computer Modelling, 10(10), 795.
 Frantziskonis, G., & Desai, C. (1988). Constitutive model with strain softening. Mathematical and Computer Modelling, 10(10), 794.
 Frantziskonis, G., & Desai, C. S. (1988). Analysis of a strain softening constitutive model. Mathematical and Computer Modelling, 10(10), 795.More infoAbstract: The paper first examines the question of uniqueness of a basic yet simplified constitutive model with strain softening. It is shown that the constitutive equations lead to a unique solution for the case of ratedependent as well as rateindependent formulation and its implementation in finite element analysis shows mesh size insensitivity in the hardening and softening regimes. Conditions for formation of narrow shear bands are developed and discussed. It is shown that as the damage accumulates the material approaches localization of deformation. © 1988.
 Frantziskonis, G., & Desai, C. S. (1988). Constitutive model with strain softening. Mathematical and Computer Modelling, 10(10), 794.More infoAbstract: The aim of this paper is to propose a simple yet realistic model for the mechanical behavior of geologic materials such as concrete and rock. The effect of structural changes in such materials is addressed and incorporated in the theory through a tensor form of a damage variable. It is shown that formation of damage is responsible for the softening in strength observed in experiments, for the degradation of the elastic shear modulus, and for induced anisotropy. A generalized plasticity model is incorporated for the socalled topical or continuum part of the behavior, whereas the damage part is represented by the socalled stressrelieved behavior. The parameters required to define the model are identified and determined from multiaxial testing of a concrete. The predictions are compared with observed behavior for a number of stress paths. The model shows very good agreement with the observed response. © 1988.
 TANG, F., DESAI, C., & FRANTZISKONIS, G. (1988). HETEROGENEITY AND DEGRADATION IN BRITTLE MATERIALS. ENGINEERING FRACTURE MECHANICS, 43(5), 779796.More infoThe concept of state of degradation is employed in order to consider the initial material (structure) inhomogeneity. Then, degradation and its patterning in simulated rock specimens subjected to external load are modeled numerically by implementing the theory for damage and surface degradation with incorporation of the initial state. The theoretical results are compared with experimental observations obtained through ultrasonic scanning tests.
 DESAI, C., SOMASUNDARAM, S., & FRANTZISKONIS, G. (1987). A HIERARCHICAL APPROACH FOR CONSTITUTIVE MODELING OF GEOLOGIC MATERIALS. INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, 10(3), 225257.
 FRANTZISKONIS, G., & DESAI, C. (1987). Paper: ANALYSIS OF A STRAIN SOFTENING CONSTITUTIVE MODEL.
 FRANTZISKONIS, G., & DESAI, C. (1987). Paper: CONSTITUTIVE MODEL WITH STRAIN SOFTENING.
 FRANTZISKONIS, G., DESAI, C., & SOMASUNDARAM, S. (1986). Paper: CONSTITUTIVE MODEL FOR NONASSOCIATIVE BEHAVIOUR.
 Frantziskonis, G. (1987). Waveletbased analysis of multiscale phenomena: application to material porosity and identification of dominant scales. PROBABILISTIC ENGINEERING MECHANICS, 17(4), 349357.More infoThe paper presents a general process that utilizes wavelet analysis in order to link information on material properties at several scales. In the particular application addressed analytically and numerically, multiscale porosity is the source of material structure or heterogeneity, and the waveletbased analysis of multiscale information shows clearly its role on properties such as resistance to mechanical failure. Furthermore, through the statistical properties of the heterogeneity at a hierarchy of scales, the process clearly identifies a dominant scale or range of scales. Special attention is paid to porosity appearing at two distinct scales far apart from each other since this demonstrates the process in a lucid fashion. Finally, the paper suggests ways to extend the process to general multiscale phenomena, including time scaling. (C) 2002 Elsevier Science Ltd. All rights reserved.
 Frantziskonis, G., & Desai, C. (1987). Analysis of a strain softening constitutive model. International journal of solids and structures, 23(6), 751767.
 Frantziskonis, G., & Desai, C. (1987). Constitutive model with strain softening. International Journal of Solids and Structures, 23(6), 733750.
 Frantziskonis, G., & Desai, C. (1987). Elastoplastic model with damage for strain softening geomaterials. Acta Mechanica, 68(34), 151170.
 Frantziskonis, G., & Desai, C. S. (1987). Analysis of a strain softening constitutive model. International Journal of Solids and Structures, 23(6), 751767.More infoAbstract: The paper first examines the question of uniqueness of a basic yet simplified constitutive model with strain softening. It is shown that the constitutive equations lead to a unique solution for the case of ratedependent as well as rateindependent formulation and its implementation in finite element analysis shows mesh size insensitivity in the hardening and softening regimes. Conditions for formation of narrow shear bands are developed and discussed. It is shown that as the damage accumulates the material approaches localization of deformation. © 1987.
 Frantziskonis, G., & Desai, C. S. (1987). Constitutive model with strain softening. International Journal of Solids and Structures, 23(6), 733750.More infoAbstract: The aim or this paper is to propose a simple yet realistic model for the mechanical behavior of geologic materials such as concrete and rock. The effect of structural changes in such materials is addressed and incorporated in the theory through a tensor form of a damage variable. It is shown that formation of damage is responsible for the softening in strength observed in experiments, for the degradation of the elastic shear modulus, and for induced anisotropy. A generalized plasticity model is incorporated for the socalled topical or continuum part of the behavior, whereas the damage part is represented by the socalled stressrelieved behavior. The parameters required to define the model are identified and determined from multiaxial testing of a concrete. The predictions are compared with observed behavior for a number of stress paths. The model shows very good agreement with the observed response. © 1987.
 Frantziskonis, G., & Desai, C. S. (1987). Elastoplastic model with damage for strain softening geomaterials. Acta Mechanica, 68(34), 151170.More infoAbstract: The paper examines certain important aspects of a rate independent model that accounts for distributed damage due to microcrack growth. Material behavior is considered as a mixture of two elasticplastic interacting components, one termed topical (undamaged), and the other termed damaged. Energy considerations show the equivalence of the twocomponent body to an elasticplastic body containing cracks; the equivalence is considered in the Griffith sense. The mechanisms of failure are considered and discussed with respect to multiaxial stress paths. An explanation of failure, at the microlevel, is given. A series of laboratory tests on a concrete are used to illustrate the development of failure. © 1987 SpringerVerlag.
 Budhu, M., Ramakrishnan, S., & Frantziskonis, G. (1986). A lattice type model for particulate media. INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, 23(7), 647671.More infoIn this paper, a latticetype model to simulate the micromechanical behaviour of particulate/granular media is presented. In this numerical model, a particulate assembly is simulated as a lattice/truss. Nodes located at contacts between a particle and its neighbours are linked by bars to each other. Each particle is represented by a lattice within its microstructure and particle interact through load transfer at the nodes. Constraints are prescribed at the nodes to describe active, deactivated and reactivated contacts. When a particulate assembly develops into a mechanism (deformation with zero incremental load), further deformation is simulated through a framework that describes the kinematics of the particles (sliding, rolling and rotation of particles). This framework is formed by introducing nodes at the particle centroids and linking them with bars. Barslinking particles with a nonsliding contact are assigned large stiffnesses relative to bars linking particles with a sliding contact. Numerical tests are conducted on twodimensional assemblies of disks, arranged as very loose and very dense packing under simple shear loading conditions. The results concord with the results of numerical tests conducted using the discrete element method and with photoelastic experiments. Additionally, the model is applied to study the effects of initial imperfections caused by particles with low elastic modulus. Copyright (C) 1999 John Wiley & Sons, Ltd.
 Desai, C. S., Somasundaram, S., & Frantziskonis, G. (1986). HIERARCHICAL APPROACH FOR CONSTITUTIVE MODELLING OF GEOLOGIC MATERIALS.. International Journal for Numerical and Analytical Methods in Geomechanics, 10(3), 225257.More infoAbstract: A hierarchical concept is proposed for the development of constitutive models to account for various factors that influence behavior of (geologic) materials. It permits evolution of models of progressively higher grades from the basic model representing isotropic hardening with associative behavior. Factors such as nonassociativeness and induced anisotropy due to friction and cyclic loading, and softening are introduced as corrections or perturbations to the basic model. The influence of these factors is captured through nonassociativeness manifested by deviation from normality of the plastic strain increments to the yield surface, F. Details of four models: isotropic hardening with associative behavior, isotropic hardening with nonassociative behavior, general anisotropic hardening and strainsoftening with a damage variable are presented.
 Desai, C., Somasundaram, S., & Frantziskonis, G. (1986). A hierarchical approach for constitutive modelling of geologic materials. International Journal for Numerical and Analytical Methods in Geomechanics, 10(3), 225257.
 Desal, C., Somasundaram, S., & Frantziskonis, G. (1986). Hierach al approach for constitutive mode\~ ng of geologic material. International Journal of Numerical Methods in Geomeehanics, 10(3), 225257.
 FRANTZISKONIS, G. N. (1986). PROGRESSIVE DAMAGE AND CONSTITUTIVE BEHAVIOR OF GEOMATERIALS INCLUDING ANALYSIS AND IMPLEMENTATION..
 Frantziskonis, G., Desai, C. S., & Somasundaram, S. (1986). CONSTITUTIVE MODEL FOR NONASSOCIATIVE BEHAVIOR.. Journal of Engineering Mechanics, 112(9), 932946.More infoAbstract: A simplified approach for the nonassociative behavior of granular (geological) materials is proposed. The basic concept is that the plastic potential can be obtained by controlling or correcting the yield function used for associative plasticity. This is achieved by correcting the hardening, evolution or growth function, which involves only one additional parameter above those for the associative model. Determination of constants and verification with respect to two granular soils tested under a variety of multiaxial stress paths are presented. It is shown that incorporation of the nonassociative behavior through the correction of the growth function yields remarkably satisfactory predictions for volumetric and stressstrain responses of the material response.
 Frantziskonis, G., Desai, C., & Somasundaram, S. (1986). Constitutive model for nonassociative behavior. Journal of engineering mechanics, 112(9), 932946.
 Swanson, S., Burns, J., Frantziskonis, G., Joshi, S., Yeh, H., Yeh, H., Joo, S., Hong, C., Kim, C., Gargiulo, C., & others, . (1986). ANALYSIS OF PIN LOADING OF CARBON/EPOXY PLATES..
 Desai, C. S., Frantziskonis, G., & Somasundaram, S. (1985). CONSTITUTIVE MODELLING FOR GEOLOGICAL MATERIALS.. Array, 1, 1934.More infoAbstract: A number of isotropic and anisotropic hardening constitutive models for geological materials are identified on the basis of a generalized procedure. Attention is given to determination of material constants from laboratory tests, verification and implementation in finite element procedures.
 Desai, C. S., Frantziskonis, G., & Somasundaram, S. (1985). CONSTITUTIVE MODELLING FOR GEOLOGICAL MATERIALS.. Array, 1934.More infoAbstract: A number of isotropic and anisotropic hardening constitutive models for geological materials are identified on the basis of a generalized procedure. Attention is given to determination of material constants from laboratory tests, verification and implementation in finite element procedures. (Author abstract. ) Refs.
Proceedings Publications
 Gur, S., Sadat, M. R., Frantziskonis, G. N., Muralidharan, K., Bringuier, S., & Zhang, L. (2019, October). Brittle Fracture of Polycrystaline 3CSic: A Case Study Using Molecular DynamicsPeridynamics Multiscale Models. In 19th International Conference on New Trends in Fatigue and Fracture.
 Gur, S., Muralidharan, K., & Frantziskonis, G. N. (2018, March 1115). Effect of porosity on the stressstrain response and hysteretic energy dissipation capacity of NiTi SMA. In 2018 TMS Annual Meeting & Exhibition.
 Gur, S., Muralidharan, K., & Frantziskonis, G. N. (2018, March 1115). Scaleparity Preserving Multiscale Models for Investigating the Mechanical Properties of Geopolymers. In 2018 TMS Annual Meeting & Exhibition.
 Meisner, M., Frantziskonis, G., Lin, Y., & Su, T. (1996, May/1996). Dissipated energy as a function of material microstructure. In ENGINEERING MECHANICS: PROCEEDINGS OF THE 11TH CONFERENCE, VOLS 1 AND 2, 10301033.
 S. K. Mishra, ., & Pannala, S. (2008). Wavelet Based Spatial Scaling of Coupled Reaction Diffusion Fields. In 8th International Conference in Computational Science ICCS 2008, 301310.
 Villela, D., Kasinathan, V. V., De Valle, S., Alvarez, M., Frantziskonis, G., Deymier, P., Muralidharan, K., & , . (2008). COMPRESSEDAIR ENERGY STORAGE SYSTEMS FOR STANDALONE OFFGRID PHOTOVOLTAIC MODULES. In 35TH IEEE PHOTOVOLTAIC SPECIALISTS CONFERENCE, 962967.More infoIn this work, a lowcost, lowvolume, lowmaintenance, smallscale compressedair energy storage system (SSCAES) is proposed, which can be used in conjunction with offgrid standalone photovoltaic panels, for powering appliances and residential units in order to minimize the dependency on centralized power system grids. As a first step towards achieving this objective, we have designed and examined the compression efficiency of a singlestage, isothermal compression system that utilizes a fluid piston. Preliminary results clearly establish that the prototype holds enormous promise as energy storage systems that are compatible with renewable energy sources such as solar.
 FRANTZISKONIS, G., BEER, G., BOOKER, ., & CARTER, J. (2006). DEGRADATION INSTABILITIES IN BRITTLE MATERIAL STRUCTURES. In COMPUTER METHODS AND ADVANCES IN GEOMECHANICS, VOL 1, 617623.
 FRANTZISKONIS, G., Anagnostopoulos, A., Schlosser, F., Kalteziotis, N., & Frank, R. (2003). HETEROGENEITY IN BRITTLE MATERIALS  STATISTICAL APPROACH AND RELATION TO FRACTURE PROPERTIES. In GEOTECHNICAL ENGINEERING OF HARD SOILS  SOFT ROCKS, VOL 2, 16011608.
 Frantziskonis, G. N., Shell, E. B., Woo, J., Matikas, T. E., & Nicolaou, P. D. (1999). Wavelet analysis of fretting experimental data. In Nondestructive Evaluation Techniques for Aging Infrastructures \& Manufacturing, 1127.
 Frantziskonis, G. N., Simon, L. B., Woo, J., & Matikas, T. E. (1999). Characterization of pitting corrosion damage through multiscale analysis. In Nondestructive Evaluation Techniques for Aging Infrastructures \& Manufacturing, 4858.
 Mishra, S. K., Muralidharan, K., Deymier, P., Frantziskonis, G., Simunovic, S., Pannala, S., Bubak, M., VanAlbada, G., Dongarra, J., & Sloot, P. (1999). Wavelet based spatial scaling of coupled reaction diffusion fields. In COMPUTATIONAL SCIENCE  ICCS 2008, PT 2, 5102, 301310.More infoMultiscale schemes for transferring information from fine to coarse scales are typically based on some sort of averaging. Such schemes smooth the fine scale features of the underlying fields, thus altering the fine scale correlations. As a superior alternative to averaging, a wavelet based scheme for the exchange of information between a reactive and diffusive field in the context of multiscale reactiondiffusion problems is proposed and analyzed. The scheme is shown to be efficient in passing information along scales, from fine to coarse, i.e. upscaling as well as from coarse to fine, i.e. downscaling. In addition, it retains fine scale statistics, mainly due to the capability of wavelets to represent fields hierarchically. Critical to the success of the scheme is the identification of dominant scales containing the majority of useful information. The scheme is applied in detail to the analysis of a diffusive system with chemically reacting boundary. Reactions are simulated using kinetic Monte Carlo (KMC) and diffusion is solved by finite differences. Spatial scale differences are present at the interface of the KMC sites and the diffusion grid. The computational efficiency of the scheme is compared to results obtained by local averaging, and to results from a benchmark model. The spatial scaling scheme ties to wavelet based schemes for temporal scaling, presented elsewhere by the authors.
 Budhu, M., Ramakrishnan, S., & Frantziskonis, G. (1997). Modeling of granular materials: A numerical model using lattices. In Mechanics of deformation and flow of particulate materials, 7790.
 Frantziskonis, G., Blodgett, M., & Frantziskonis, G. (1996). Multiscale material characterization and applications. In PROBAMAT21ST CENTURY: PROBABILITIES AND MATERIALS, 46, 367378.More infoGiven the hierarchical structure of engineering materials, it is natural to seek multiscale characterization tools. This study explores wavelet analysis, a recently developed mathematical tool, for this purpose, In particular, multiscale microstructure characterization of certain alloys is studied on an exploratory basis. Ultrasonic Bscan data are used for multiscale characterization of various Ti 64 microstructures. Two such microstructures are studied in detail for the purpose of quantitative characterization as well as for extraction of microstructural features from the Bscan data.
 Meisner, M. J., & Frantziskonis, G. N. (1996). Dissipated Energy as a Function of Material Microstructure. In Engineering Mechanics (1996), 10301033.
 Ramakrishnan, S., Budhu, M., & Frantziskonis, G. (1996). Constitutive Behavior of Granular Media Using a Lattice Type Model. In Engineering Mechanics (1996), 713716.
 Budhu, M., Ramakrishnan, S., Frantziskonis, G., Chang, C., Misra, A., Liang, R., & Babic, M. (1995). Modeling of granular materials  A numerical model using lattices. In MECHANICS OF DEFORMATION AND FLOW OF PARTICULATE MATERIALS, 7790.More infoIn this paper, a numerical model is presented in which the particles in a granular assembly are modeled as lattices. In a two dimensional simulation, nodes are introduced at particle contacts and are linked by bar elements. A granular assembly is then transformed into a statically indeterminate truss and is analyzed using standard techniques in structural mechanics. The areas of bars are derived by computing the displacements of an elastic disk under contact forces and then the bar areas that will give equivalent displacements in the direction of the bars are determined. The lattice type model incorporates bonding, debonding sliding and rotation of particles. The lattice type model has been applied to loose and dense packing of disks. The two packings were subjected to simple shear deformation after an initial vertical load was applied. The results from the lattice type model agree with the results from numerical tests using the discrete element method and with results of photoelastic experiments.
 Frantziskonis, G., Shell, E., Woo, J., Matikas, T., Nicolaou, P., Baaklini, G., Lebowitz, C., & Boltz, E. (1994). Wavelet analysis of fretting experimental data. In NONDESTRUCTIVE EVALUATION OF AGING MATERIALS AND COMPOSITES III, 3585, 1127.More infoWavelet analysis is being used to rationalize information at various scales in several branches of science, including particle physics, biology, electrical engineering, fluid mechanics, and medicine. However, this powerful technique has not been applied to characterizing structures of materials, fretting damage for the present case, even though many critical questions could be addressed. In particular, the following unsolved problems are considered in this paper: (a) The first problem deals with the quantitative characterization of fretted surfaces in a Ti6Al4V alloy. This is investigated by analyzing profilometric digital images of fretted surfaces obtained at a range of magnifications. Wavelet analysis of the data is able to identify, by examining the wavelet coefficients, dominant length scales as those regions in the scalespace where the energy of the wavelet transform and/or peaks of local concentration dominate. For the range of magnifications examined, i.e., from 1.25x to 100x, the similar to 20x magnification is identified as the one with the most useful information.
 Gustafson, S., Mehta, ., Frantziskonis, G., Crane, R., Achenbach, J., Shah, S., Matikas, T., KhuriYakub, P., & Gilmore, R. (1993). Practical optical characterization of fretted surfaces. In NONDESTRUCTIVE CHARACTERIZATION OF MATERIALS IN AGING SYSTEMS, 503, 2530.More infoNominally smooth surfaces subjected to repeated contact may develop characteristic fretted textures. An important example concerns fretted surfaces that may develop on disk dovetail joints that secure fan blades in turbine engines. These surfaces may form sites for crack nucleation and subsequent catastrophic disk and engine failure. Thus a practical and reliable method for the nondestructive evaluation of such fretted surfaces is critically needed. In this paper straightforward optical techniques are considered for identifying and quantifying fretted (and other surface damaged) regions. For example, digital images of candidate surfaces may be processed to separate fretted from unfretted regions and to evaluate the relative degree of fretting. Ideally, optical characterization should be rapid, convenient, inexpensive, reliable, and insensitive to details of lighting and image capture.
 Yuan, X., Tang, F., & Frantziskonis, G. (1992). Random Initial Heterogeneity and Degradation in Brittle Materials. In Engineering Mechanics (1992), 616619.
 Ramakrishnan, S., Budhu, M., Frantziskonis, G., Lin, Y., & Su, T. (1991). Constitutive behavior of granular media using a lattice type model. In ENGINEERING MECHANICS: PROCEEDINGS OF THE 11TH CONFERENCE, VOLS 1 AND 2, 713716.
 Frantziskonis, G., Simon, L., Woo, J., Matikas, T., Baaklini, G., Lebowitz, C., & Boltz, E. (1990). Characterization of pitting corrosion damage through multiscale analysis. In NONDESTRUCTIVE EVALUATION OF AGING MATERIALS AND COMPOSITES III, 3585, 4858.More infoThe paper documents a novel method for characterizing pitting corrosion damage in structural materials such as Al 2024T3. Specimens of such alloys are corroded in a controlled environment and the pits' geometry is captured digitally using white light interference microscopy. The digital data are then processed with waveletbased analysis thus making possible a multiresolution description of the geometrical features. The analysis reveals several interesting features of the pits that are similar for all the experimental data analyzed herein, and independent of the process followed for creating them (time material is exposed to corroding environment, concentration of the corroding agent, surface area exposed to the agent, etc.). The first property identified as common to all pits is their geometrical scaling with a (Hurst) exponent of 0.63+/0.12. Furthermore, the ratio omega of the surface area of the pit as represented at coarse scales through the wavelet representation, over the area of its intersection with the plane at zero depth is found to be 1.17+/0.07 consistently. The ratio of the total surface area over its intersection is found to be 1.6+/0.2. Either one of these ratios together with the Hurst exponent provide sufficient information for obtaining a pit's geometry from images capturing its twodimensional shape only, a capability important for efficient characterization. Additionally, such a characterization is paramount for rigorously addressing fatigue crack initiation and propagation.
 FRANTZISKONIS, G., & Breysse, D. (1987). CRACK PATTERN RELATED UNIVERSAL CONSTANTS. In PROBABILITIES AND MATERIALS, 269, 361375.
 FRANTZISKONIS, G., & Voyiadjis, G. (1986). HETEROGENEITY AND ITS IMPLICATIONS  MICROMECHANICAL, STATISTICAL, FRACTAL APPROACH AND THEIR SIMILARITY. In DAMAGE IN COMPOSITE MATERIALS, 34, 137160.
 Desai, C., Frantziskonis, G., & Somasundaram, S. (1985). Constitutive modelling for geological materials. In International conference on numerical methods in geomechanics, 1934.
Presentations
 Frantziskonis, G. N. (2016, June). Thermally Modulated Superelastic SMA Dampers for Vibration Control. 6th Annual International Conference on Civil Engineering & 1st Annual International Conference on Structural Engineering and Mechanics. Athens, Greece: ATINER.
 Gur, S., Manga, V. R., Bringuier, S., Muralidharan, K., & Frantziskonis, G. N. (2016, April). The role of internal strain in the austenite phase stabilization during the martensitic phase transformation in NiTi shape memory alloys. MRS Spring meeting and exhibit. Phoenix, AZ: MRS.
 Gur, S., Manga, V. R., Bringuier, S., Muralidharan, K., & Frantziskonis, G. N. (2015, March). Anisotropy in the Transformation Dynamics of Austenite (B2) to Martensite (B19’) Associated with Superelasticity in NiTi. 2015 TMS Annual Meeting & Exhibition. Orlando, FL: TMS.
Reviews
 Vasseur, J., Deymier, P., Frantziskonis, G., Hong, G., & DjafariRouhani, B. (2006. Experimental evidence for the existence of absolute acoustic band gaps in twodimensional periodic composite media(pp 60516064).More infoTransmission of acoustic waves in twodimensional binary solid/solid composite media composed of arrays of Duralumin cylindrical inclusions embedded in an epoxy resin matrix is studied. The experimental transmission spectrum and theoretical band structure of two periodic arrays of cylinders organized on a square lattice and on a centred rectangular network are reported. Absolute gaps extending throughout the first twodimensional Brillouin zone are predicted. The measured transmission is observed to drop to noise level throughout frequency intervals in reasonable agreement with the calculated forbidden frequency bands.
Others
 Frantziskonis, G. (1996, JUN). Stochastic approaches for damage evolution in standard and nonstandard continua  Discussion. INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES.
 Yializis, A., & Frantziskonis, G. (2002). Method and apparatus for increasing hitting efficacy in a sporting implement.
 Budhu, M., Frantziskonis, G., & Ramakrishnan, S. (1996). MicroStructure in Particulate MediaA Lattice Type Approach and Its Validation..
 Frantziskonis, G. (1995). Materials propertiesinformation from nondestructive and destructive experiments via simulation. Final report, 27 July 199427 December 1995.
 Frantziskonis, G. N. (1995). Materials PropertiesInformation from NonDestructive and Destructive Experiments Via Simulation..
 Tang, F., Frantziskonis, G., & Desai, C. (1992). Instabilities of Damage and Surface Degradation Mechanisms in Brittle Material Structural Systems.