Quantification of uncertainty modelling in stochastic analysis of FRP composites

Composites Part A: Applied Science and Manufacturing

Volumn 40, Issue 11, 2009, Pages 1673-1684

  Sriramula, Srinivas ^a   Chryssanthopoulos, Marios K ^a  

^a UNIVERSITY OF SURREY   (United Kingdom)

Author keywords

A. Polymer matrix composites (PMCs); B. Mechanical properties; C. Micro mechanics; C. Statistical properties methods

Indexed keywords

A. POLYMER-MATRIX COMPOSITES (PMCS); COMPONENT LEVELS; DESIGN OBJECTIVES; FRP COMPOSITE; GLOBAL RESPONSE; INHERENT VARIABILITY; LENGTH SCALE; LOCAL DAMAGE; MULTI-SCALE MODELLING; RANDOM COMPOSITES; STATISTICAL PROPERTIES/METHODS; STOCHASTIC ANALYSIS; STOCHASTIC MODELLING; TEST DATA; UNCERTAINTY MODELLING; UNCERTAINTY PROPAGATION;

BUILDING COMPONENTS; COMPOSITE MICROMECHANICS; COMPUTATIONAL MECHANICS; DEGREES OF FREEDOM (MECHANICS); FIBER REINFORCED PLASTICS; FRACTURE MECHANICS; MECHANICAL PROPERTIES; RANDOM ERRORS; RANDOM PROCESSES; RANDOM VARIABLES; REINFORCED PLASTICS; STOCHASTIC SYSTEMS; STRUCTURE (COMPOSITION); UNCERTAINTY ANALYSIS;

POLYMER MATRIX COMPOSITES;

EID: 71749084601     PISSN: 1359835X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compositesa.2009.08.020     Document Type: Review

References (147)

1. Clarke J.L. Structural design of polymer composites. Eurocomp design code and handbook (1996), E & F N Spon, London
2. Lekou D.J., and Philippidis T.P. Mechanical property variability in FRP laminates and its effect on failure prediction. Composites: Part B (2008) 10.1016/j.compositesb.2008.01.00
3. Polymer matrix composites: Guidelines for characterization of structural materials. Composite materials handbook, vol. 1. Washington (DC): Department of Defence; 2002 [Report no: DOD-MIL-HDBK-17-1F].
4. Long MW, Narciso JD. Probabilistic design methodology for composite aircraft structures. DOT/FAA/AR-99/2: Federal Aviation Administration, Office of Aviation Research Washington, DC; 1999.
5. Shiao MC, Singhal SN, Chamis CC. A method for the probabilistic design assessment of composite structures. NASA TM - 106384; 1994.
6. Ellingwood B.R. Toward load and resistance factor design for fibre-reinforced polymer composite structures. J Struct Eng 129 4 (2003) 449-458
7. Atadero R.A., and Karbhari V.M. Calibration of resistance factors for reliability based design of externally-bonded FRP composites. Composites: Part B 39 (2008) 665-679
8. Ostoja-Starzewski M. Material spatial randomness: from statistical to representative volume element. Probabilist Eng Mech 21 2 (2006) 112-132
9. Kaminski M.M. Computational mechanics of composite materials (2005), Springer-Verlag, London
10. Jeong H.K., and Shenoi R.A. Probabilistic strength analysis of rectangular FRP plates using Monte Carlo simulation. Comput Struct 76 3 (2000) 219-235
11. Onkar A.K., Upadhyay C.S., and Yadav D. Probabilistic failure of laminated composite plates using the stochastic finite element method. Compos Struct 77 1 (2007) 79-91
12. Potter K.D., Campbell M., Langer C., and Wisnom M.R. The generation of geometrical deformations due to tool/part interaction in the manufacture of composite components. Compos Part A: Appl Sci Manuf 36 2 (2005) 301-308
13. Potter K., Langer C., Hodgkiss B., and Lamb S. Sources of variability in uncured aerospace grade unidirectional carbon fibre epoxy preimpregnate. Compos Part A: Appl Sci Manuf 38 3 (2007) 905-916
14. Potter K., Khan B., Wisnom M., Bell T., and Stevens J. Variability, fibre waviness and misalignment in the determination of the properties of composite materials and structures. Composites: Part A 39 9 (2008) 1343-1354
15. Shiao M.C., and Chamis C.C. Probabilistic evaluation of fuselage-type composite structures. Probabilist Eng Mech 14 1-2 (1999) 179-187
16. Vinckenroy G.V., and de Wilde W.P. The use of Monte Carlo techniques in statistical finite element methods for the determination of the structural behaviour of composite materials structural components. Compos Struct 32 1-4 (1995) 247-253
17. Fernlund G., Rahman N., Courdji R., Bresslauer M., Poursartip A., Willden K., et al. Experimental and numerical study of the effect of cure cycle, tool surface, geometry, and lay-up on the dimensional fidelity of autoclave-processed composite parts. Compos Part A: Appl Sci Manuf 33 3 (2002) 341-351
18. Yushanov S.P., and Bogdanovich A.E. Stochastic theory of composite materials with random waviness of the reinforcements. Int J Solids Struct 35 22 (1998) 2901-2930
19. Wisnom M.R. Size effects in the testing of fibre-composite materials. Compos Sci Technol 59 13 (1999) 1937-1957
20. Oberkampf WL, Helton JC, Sentz K. Mathematical representation of uncertainty. In: 42nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference and exhibit. Seattle, WA; 2001.
21. Agarwal H., Renaud J.E., Preston E.L., and Padmanabhan D. Uncertainty quantification using evidence theory in multidisciplinary design optimization. Reliab Eng Syst Safe 85 1-3 (2004) 281-294
22. Chamis C.C. Probabilistic simulation of multi-scale composite behavior. Theor Appl Fract Mech 41 (2004) 51-61
23. Elseifi MA, Khalessi MR, Lin H-Z, Rogers G, Torng TY. Probabilistic analysis of thick composite plates with manufacturing and material uncertainties. In: 42nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference and exhibit. Seattle, WA; 2001.
24. Ghosh S., Nowak Z., and Lee K. Quantitative characterization and modeling of composite microstructures by Voronoi cells. Acta Mater 45 6 (1997) 2215-2234
25. Car E., Zalamea F., Oller S., Miquel J., and Oñate E. Numerical simulation of fibre reinforced composite materials - two procedures. Int J Solids Struct 39 7 (2002) 1967-1986
26. Graham-Brady L.L., Siragy E.F., and Baxter S.C. Analysis of heterogeneous composites based on moving-window techniques. J Eng Mech 129 4 (2003) 1054-1064
27. Carbillet S., Richard F., and Boubakar L. Reliability indicator for layered composites with strongly non-linear behaviour. Compos Sci Technol (2007) 10.1016/j.compscitech.2007.10.05
28. Nader J.W., Dagher H.J., Lopez-Anido R., Chiti F.E., Fayad G.N., and Thomson L. Probabilistic finite element analysis of modified ASTM D3039 tension test for marine grade polymer matrix composites. J Reinf Plast Comp 27 6 (2008) 583-597
29. Rollet Y., Bonnet M., Carrere N., Leroya F.-H., and Maire J.-F. Improving the reliability of material databases using multiscale approaches. Compos Sci Technol (2007) 10.1016/j.compscitech.2007.10.04
30. Soize C. Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probabilist Eng Mech (2008) 23 307-323
31. Guilleminot J., Soize C., Kondo D., and Binetruy C. Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites. Int J Solids Struct (2008) 10.1016/j.ijsolstr.2008.06.00
32. Graham-Brady L.L., Arwade S.R., Corr D.J., Gutíerrez M.A., Breysse D., Grigoriu M., et al. Probability and materials: from Nano- to Macro-Scale: A summary. Probabilist Eng Mech 21 3 (2006) 193-199
33. Trias D., Costa J., Fiedler B., Hobbiebrunken T., and Hurtado J.E. A two-scale method for matrix cracking probability in fibre-reinforced composites based on a statistical representative volume element. Compos Sci Technol 66 (2006) 1766-1777
34. Baxter S., and Graham L. Characterization of random composites using a moving window technique. J Eng Mech 126 4 (2000) 389-404
35. Chamis C.C., and Abumeri G.H. Probabilistic dynamic buckling of composite shell structures. Compos Part A: Appl Sci Manuf 36 10 (2005) 1368-1380
36. Noor A.K., Starnes J.H., and Peters J.M. Uncertainty analysis of composite structures. Comput Methods Appl Mech Eng 185 2-4 (2000) 413-432
37. Noor A.K., Starnes J.H., and Peters J.M. Uncertainty analysis of stiffened composite panels. Compos Struct 51 2 (2001) 139-158
38. Tsai SW. Structural behaviour of composite materials. NASA CR-71 NASA, Washington DC; 1964.
39. Huang Z. Micromechanical prediction of ultimate strength of transversely isotropic fibrous composites. Int J Solids Struct 38 (2001) 4147-4172
40. Luccioni B.M. Constitutive model for fiber-reinforced composite laminates. J Appl Mech 73 1 (2006) 901-910
41. Soden P.D., Kaddour A.S., and Hinton M.J. Recommendations for designers and researchers resulting from the world-wide failure exercise. Compos Sci Technol 64 3-4 (2004) 589-604
42. Frantziskonis G.N. Stochastic modeling of heterogeneous materials - a process for the analysis and evaluation of alternative formulations. Mech Mater 27 3 (1998) 165-175
43. Graham L.L., and Baxter S.C. Simulation of local material properties based on moving-window GMC. Probabilist Eng Mech 16 4 (2001) 295-305
44. Spanos P.D., and Kontsos A. A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites. Probabilist Eng Mech 23 4 (2008) 456-470
45. Hori M., and Nemat-Nasser S. On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mech Mater 31 10 (1999) 667-682
46. Xia Z., Zhou C., Yong Q., and Wang X. On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites. Int J Solids Struct 43 2 (2006) 266-278
47. Drago A., and Pindera M.-J. Micro-macromechanical analysis of heterogeneous materials: macroscopically homogeneous vs periodic microstructures. Compos Sci Technol 67 6 (2007) 1243-1263
48. Trias D., Costa J., Mayugo J.A., and Hurtado J.E. Random models versus periodic models for fibre reinforced composites. Comput Mater Sci 38 (2006) 316-324
49. Hill R. Elastic properties of reinforced solids: Some theoretical principles. J Mech Phys Solids 11 5 (1963) 357-372
50. Pensée V., and He Q.-C. Generalized self-consistent estimation of the apparent isotropic elastic moduli and minimum representative volume element size of heterogeneous media. Int J Solids Struct 44 7-8 (2007) 2225-2243
51. Williams T.O., and Baxter S.C. A framework for stochastic mechanics. Probabilist Eng Mech 21 3 (2006) 247-255
52. Terada K., Miura T., and Kikuchi N. Digital image-based modeling applied to the homogenization analysis of composite materials. Comput Mech 20 4 (1997) 331-346
53. Shan Z., and Gokhale A.M. Digital image analysis and microstructure modeling tools for microstructure sensitive design of materials. Int J Plasticity 20 7 (2004) 1347-1370
54. Gusev A.A., Hine P.J., and Ward I.M. Fibre packing and elastic properties of a transversely random unidirectional glass/epoxy composite. Compos Sci Technol 60 4 (2000) 535-541
55. Sheehan N., and Torquato S. Generating microstructures with specified correlation functions. J Appl Phys 89 1 (2001) 53-60
56. Ostoja-Starzewski M. Random field models of heterogeneous materials. Int J Solids Struct 35 19 (1998) 2429-2455
57. Trias D., Costa J., Turon A., and Hurtado J.E. Determination of the critical size of a statistical representative volume element (SRVE) for carbon reinforced polymers. Acta Mater 54 13 (2006) 3471-3484
58. Ostoja-Starzewski M., and Wang X. Stochastic finite element as a bridge between random material microstructure and global response. Comput Methods Appl Mech Eng 168 (1999) 35-49
59. Gusev A.A. Representative volume element size for elastic composites: a numerical study. J Mech Phys Solids 45 9 (1997) 1449-1459
60. Terada K., Hori M., Kyoya T., and Kikuchi N. Simulation of the multi-scale convergence in computational homogenization approaches. Int J Solids Struct 37 16 (2000) 2285-2311
61. Shan Z., and Gokhale A.M. Representative volume element for non-uniform micro-structure. Comput Mater Sci 24 3 (2002) 361-379
62. Willis JR. Elasticity theory of composites, mechanics of solids. The R. Hill 60th Anniversary Volume. Oxford: Pergamon Press; 1982. p. 653-86.
63. Zeman J., and Sejnoha M. Numerical evaluation of effective elastic properties of graphite fibre tow impregnated by polymer matrix. J Mech Phys Solids 49 1 (2001) 69-90
64. Pyrz R. Correlation of microstructure variability and local stress field in two-phase materials. Mater Sci Eng A 117 1-2 (1994) 253-259
65. Pyrz R. Quantitative description of the microstructure of composites. Part I: Morphology of unidirectional composite systems. Compos Sci Technol 50 2 (1994) 197-208
66. Ghosh S., Nowak Z., and Lee K. Tessellation-based computational methods for the characterization and analysis of heterogeneous microstructures. Compos Sci Technol 57 9-10 (1997) 1187-1210
67. Ripley B.D. The second order analysis of stationary point processes. J Appl Probab 13 (1976) 255-266
68. Trias D., Garcia R., Costa J., Blanco N., and Hurtado J.E. Quality control of CFRP by means of digital image processing and statistical point pattern analysis. Compos Sci Technol 67 (2007) 2438-2446
69. Grufman C., and Ellyin F. Determining a representative volume element capturing the morphology of fibre reinforced polymer composites. Compos Sci Technol 67 3-4 (2007) 766-775
70. Drugan W., and Willis J. A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J Mech Phys Solids 44 4 (1996) 497-524
71. Hollister S.J., and Kikuchi N. A comparison of homogenization and standard mechanics analysis for periodic porous composites. Comput Mech 10 (1992) 73-95
72. Markovic D., and Ibrahimbegovic A. On micro-macro interface conditions for micro scale based FEM for inelastic behavior of heterogeneous materials. Comput Methods Appl Mech Eng 193 48-51 (2004) 5503-5523
73. Nemat-Nasser S., and Hori M. Micromechanics: overall properties of heterogeneous materials (1993), Elsevier Science Publishers, Amsterdam
74. Torquato S. Random heterogeneous materials: microstructure and macroscopic properties (2002), Springer-Verlag, New York
75. Aboudi J. Mechanics of composite materials. A unified micromechanical approach (1991), Elsevier Science, Amsterdam
76. Sanchez-Palencia E. Non-homogeneous media and vibration theory. Lecture notes in physics vol. 127 (1980), Springer Verlag, Berlin
77. Graham L.L., Gurley K., and Masters F. Non-Gaussian simulation of local material properties based on a moving-window technique. Probabilist Eng Mech 18 3 (2003) 223-234
78. Zureick A.-H., Bennett R.M., and Ellingwood B.R. Statistical characterization of fibre-reinforced polymer composite material properties for structural design. J Struct Eng 132 8 (2006) 1320-1327
79. Tenn L. Statistical analysis of fibrous composite strength data ASTM STP 734 (1981), American Society for Testing and Materials
80. Sriramula S., and Chryssanthopoulos M.K. Probabilistic models for spatially varying mechanical properties of in-service GFRP cladding panels. J Compos Construct 13 2 (2009) 159-167
81. Philippidis T.P., Lekou D.J., and Aggelis D.G. Mechanical property distribution of CFRP filament wound composites. Compos Struct 45 1 (1999) 41-50
82. Nader J.W., Dagher H.J., Chiti F.E., and Lopez-Anido R. Probabilistic finite element analysis of ASTM D6641 compression test for marine grade polymer matrix composites. J Reinf Plastics Compos 28 8 (2009) 897-911
83. Zureick A., and Scott D. Short-term behavior and design of fibre-reinforced polymeric slender members under axial compression. J Compos Construct 1 4 (1997) 140-149
84. Lin S.C. Reliability predictions of laminated composite plates with random system parameters. Probabilist Eng Mech 15 4 (2000) 327-338
85. Lin S.C., and Kam T.Y. Probabilistic failure analysis of transversely loaded laminated composite plates using first-order second moment method. J Eng Mech 126 8 (2000) 812-820
86. Lin S.C. Buckling failure analysis of random composite laminates subjected to random loads. Int J Solids Struct 37 51 (2000) 7563-7576
87. Hwang T.-K., Hong C.-S., and Kim C.-G. Probabilistic deformation and strength prediction for a filament wound pressure vessel. Compos Part B: Eng 34 5 (2003) 481-497
88. Philippidis T.P., Lekou D.J., and Kalogiannakis G.A. On the stochastic nature of thermomechanical properties in glass reinforced polyester laminates. Compos Struct 49 3 (2000) 293-301
89. Alqam M., Bennett R.M., and Zureick A.-H. Three-parameter vs. two-parameter Weibull distribution for pultruded composite material properties. Compos Struct 58 4 (2002) 497-503
90. Rao C.G.K., and Bhar A. Reliability of composite laminates in ship hulls. J Reinf Plastics Compos 24 9 (2005) 977-991
91. Moy S., Shenoi R., and Allen H. Strength and stiffness of fibre reinforced plastic plates. Proc Inst of Civil Eng. Struct Build (1996) 204-220
92. Jeong H.K., and Shenoi R.A. Reliability analysis of mid-plane symmetric laminated plates using direct simulation method. Compos Struct 43 1 (1998) 1-13
93. Nakayasu H., and Maekawa Z. A comparative study of failure criteria in probabilistic fields and stochastic failure envelopes of composite materials. Reliab Eng Syst Safe 56 3 (1997) 209-220
94. Li H., Lu Z., and Zhang Y. Probabilistic strength analysis of bolted joints in laminated composites using point estimate method. Compos Struct (2008) 10.1016/j.compstruct.2008.03.04
95. Chen N.-Z., and Soares C.G. Reliability assessment for ultimate longitudinal strength of ship hulls in composite materials. Probabilist Eng Mech 22 (2007) 330-342
96. Wu W.-F., Cheng H.-C., and Kang C.-K. Random field formulation of composite laminates. Compos Struct 49 1 (2000) 87-93
97. Ngah M.F., and Young A. Application of the spectral stochastic finite element method for performance prediction of composite structures. Compos Struct 78 3 (2007) 447-456
98. Venini P., and Mariani C. Free vibrations of uncertain composite plates via stochastic Rayleigh-Ritz approach. Comput Struct 64 1-4 (1997) 407-423
99. Ganesan R., and Kowda V.K. Buckling of composite beam-columns with stochastic properties. J Reinf Plastics Compos 24 5 (2005) 513-543
100. Ganesan R., and Kowda V.K. Free-vibration of composite beam-columns with stochastic material and geometric properties subjected to random axial loads. J Reinf Plastics Compos 24 1 (2005) 69-91
101. Sutherland L.S., and Guedes Soares C. Review of probabilistic models of the strength of composite materials. Reliab Eng Syst Safe 56 3 (1997) 183-196
102. Hinton M.J., and Soden P.D. Predicting failure in composite laminates: the background to the exercise. Compos Sci Technol 58 7 (1998) 1001-1010
103. Orifici A.C., Herszberg I., and Thomson R.S. Review of methodologies for composite material modelling incorporating failure. Compos Struct 86 (2008) 194-210
104. Weibull W. A statistical theory of the strength of materials. Ingeniorsvetenscapsakademien Handlingar 151 (1939) 1-29
105. Barbero E., Fernández-Sáez J., and Navarro C. Statistical analysis of the mechanical properties of composite materials. Composites: Part B 31 (2000) 375-381
106. Gurvich M.R. Strength/size effect for anisotropic brittle materials under a random stress state. Compos Sci Technol 59 11 (1999) 1701-1711
107. Daniels H.E. The statistical theory of the strength of bundles of threads. Proc Roy Soc Lond 183A (1945) 405-435
108. Sentler L. Reliability of high performance fibre composites. Reliab Eng Syst Safe 56 3 (1997) 249-256
109. Phoenix S.L. Modeling the statistical lifetime of glass fiber/polymer matrix composites in tension. Compos Struct 48 1-3 (2000) 19-29
110. Tabiei A., and Sun J. Statistical aspects of strength size effect of laminated composite materials. Compos Struct 46 3 (1999) 209-216
111. Harlow D.G. Statistical properties of hybrid composites: asymptotic distributions for strain. Reliab Eng Syst Safe 56 3 (1997) 197-208
112. Mignolet M.P., and Mallick K. Random inelastic behavior of composite materials with local load sharing. Probabilist Eng Mech 10 2 (1995) 83-93
113. Ibnabdeljalil M., and Curtin W.A. Strength and reliability of notched fiber-reinforced composites. Acta Mater 45 9 (1997) 3641-3652
114. Wada A., and Fukuda H. Approximate upper and lower bounds for the strength of unidirectional composites. Compos Sci Technol 59 1 (1999) 89-96
115. Gurvich M.R., and Pipes R.B. Probabilistic analysis of multi-step failure process of a laminated composite in bending. Compos Sci Technol 55 4 (1995) 413-421
116. Goda K. A strength reliability model by Markov process of unidirectional composites with fibers placed in hexagonal arrays. Int J Solids Struct 40 24 (2003) 6813-6837
117. Paramonov Y., and Andersons J. A family of weakest link models for fiber strength distribution. Composites: Part A 38 (2007) 1227-1233
118. Soares C.G. Reliability of components in composite materials. Reliab Eng Syst Safe 55 2 (1997) 171-177
119. Frangopol D.M., and Recek S. Reliability of fiber-reinforced composite laminate plates. Probablist Eng Mech 18 2 (2003) 119-137
120. Liu X., and Mahadevan S. Ultimate strength failure probability estimation of composite structures. J Reinf Plastics Compos 19 5 (2000) 403-426
121. Miki M., Murotsu Y., Tanaka T., and Shao S. Reliability-based optimization of fibrous laminated composites. Reliab Eng Syst Safe 56 3 (1997) 285-290
122. Boyer C., Béakou A., and Lemaire M. Design of a composite structure to achieve a specified reliability level. Reliab Eng Syst Safe 56 3 (1997) 273-283
123. Elhewy A.H., Mesbahi E., and Pu Y. Reliability analysis of structures using neural network method. Probabilist Eng Mech 21 1 (2006) 44-53
124. Carrere N., Rollet Y., Retel V., Boubakar L., and Maire J.-F. Composites structural modelling with uncertain data. Compos Sci Technol (2007) 10.1016/j.compscitech.2007.10.05
125. Sciuva M.D., and Lomario D. A comparison between Monte Carlo and FORMs in calculating the reliability of a composite structure. Compos Struct 59 1 (2003) 155-162
126. Engelstad S.P., and Reddy J.N. Probabilistic nonlinear finite element analysis of composite structures. AIAA J 31 2 (1993) 362-369
127. Gajdošík J., Zeman J., and Šejnoha M. Qualitative analysis of fiber composite microstructure: Influence of boundary conditions. Probabilist Eng Mech 21 4 (2006) 317-329
128. Kanit T., Forest S., Galliet I., Mounory V., and Jeulin D. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40 13-14 (2003) 3647-3679
129. Jeulin D. Mathematical morphology and multiphase materials. In: Proceedings of 3rd European symposium of stereology. Streol Iugosl; 1981. p. 265-86.
130. Jeulin D. Caractérisation Morphologique et Modèles de Structures Aléatoires. Homogénéisation en mécanique des matériaux (tome1) (2001), Hermès, France
131. Torquato S., and Lado F. Effective properties of two phase disordered composite media: ii. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres. Phys Rev B 33 (1986) 6428-6434
132. Torquato S., and Stell G. Microstructure of two-phase random media iii. The n-point matrix probability functions for fully penetrable spheres. J Chem Phys 79 3 (1983) 1505-1510
133. Torquato S. Random heterogeneous media: microstructure and improved bounds on effective properties. Appl Mech Rev 44 (1991) 37-76
134. Suquet P. Local and global aspects in the mathematical theory of plasticity. In: Sawczuk A., and Bianchi G. (Eds). Plasticity today - modeling methods and applications (1997), Elsevier, London
135. Chaboche J.L., Kruch S., Maire J.F., and Potter T. Towards a micromechanics based inelastic and damage modeling of composites. Int J Plasticity 17 (2001) 411-439
136. Jeulin D., and Ostoja-Starzewski M. Mechanics of random and multiscale microstructures (2001), Springer-Verlag, New York
137. Ghosh S., Lee K., and Moorthy S. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method. Int J Solids Struct 32 1 (1995) 27-62
138. Ghosh S., and Moorthey S. Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite element method. Comput Methods Appl Mech Eng 121 (1995) 373-409
139. Swaminathan S., Ghosh S., and Pagano N.J. Statistically equivalent representative volume elements for unidirectional composite microstructures: Part I - without damage. J Compos Mater 40 7 (2006) 583-604
140. Kaminski M., and Kleiber M. Stochastic structural interface defects in fiber composites. Int J Solids Struct 33 20-22 (1996) 3035-3056
141. Kaminski M., and Kleiber M. Numerical homogenization of n-phase composites including stochastic interface defects. Int J Numer Methods Eng 47 (2000) 1001-1025
142. Kaminski M., and Kleiber M. Perturbation based stochastic finite element method for homogenization of two-phase elastic composites. Comput Struct 78 6 (2000) 811-826
143. Kaminski M. Wavelet-based homogenization of unidirectional multiscale composites. Comput Mater Sci 27 (2003) 446-460
144. Dvorak G.J. Transformation field analysis of inelastic composite materials. Proc: Math Phys Sci, Roy Soc 437 1900 (1992) 311-327
145. Dvorak G., and Benveniste Y. On transformation strains and uniform fields multiscale elastic media. Proc Roy Soc Lond A437 (1992) 291-310
146. Tay T.E., Liu G., Tan V.B.C., Sun X.S., and Pham D.C. Progressive failure analysis of composites. J Compos Mater 42 18 (2008) 1921-1966
147. Kaminski M. Generalized perturbation-based stochastic finite element method in elastostatics. Comput Struct 85 10 (2007) 586-594