Computational modeling of fracture in concrete: A review

Abstract

This paper presents a review of fracture modeling of concrete. The complex material, such as concrete, has been widely used in construction industries and become trending issue in the last decades. Based on comprehensive literature review, there are two main approaches considered to-date of concrete fracture modeling, such as macroscopic and micromechanical models. The purpose of this review is to provide insight comparison from different techniques in modeling of fracture in concrete which are available. In the first section, an overview of fracture modeling in general is highlighted. Two different approaches both of macroscopic and micromechanical models will be reviewed. As heterogeneity of concrete material is major concern in micromechanical-based concrete modeling, one section will discuss this approach. Finally, the summary from all of reviewed techniques will be pointed out before the future perspective is given.

Keywords

concrete fracture / macroscopic / micromechanical / heterogeneity

References

[1] Bažant Z P, Tabbara M R, Kazemi M T, Pijaudier-Cabot G. Random particle model for fracture of aggregate or fiber composites. Journal of Engineering Mechanics, 1990, 116(8): 1686–1705

[2] Bolander J E Jr, Saito S. Fracture analyses using spring networks with random geometry. Engineering Fracture Mechanics, 1998, 61(5–6): 569–591

[3] Ragab Mohamed A, Hansen W. Micromechanical modeling of crack-aggregate interaction in concrete materials. Cement and Concrete Composites, 1999, 21(5–6): 349–359

[4] Schlangen E, van Mier J. Micromechanical analysis of fracture of concrete. International Journal of Damage Mechanics, 1992, 1(4): 435–454

[5] van Mier J, Vervuurt A. Numerical analysis of interface fracture in concrete using a lattice-type fracture model. International Journal of Damage Mechanics, 1997, 6(4): 408–432

[6] Grassl P, Jirásek M. Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension. International Journal of Solids and Structures, 2010, 47(7–8): 957–968

[7] López C M, Carol I, Aguado A. Meso-structural study of concrete fracture using interface elements. I: Numerical model and tensile behavior. Materials and Structures, 2008, 41(3): 583–599

[8] López C M, Carol I, Aguado A. Meso-structural study of concrete fracture using interface elements. II: Compression, biaxial and brazilian test. Materials and Structures, 2008, 41(3): 601–620

[9] Labanda N A, Giusti S M, Luccioni B M. Meso-scale fracture simulation using an augmented Lagrangian approach. International Journal of Damage Mechanics, 2016, 27(1): 1056789516671092

[10] Rabczuk T, Eibl J. Modelling dynamic failure of concrete with mesh-free methods. International Journal of Impact Engineering, 2006, 32(11): 1878–1897

[11] Rabczuk T, Zi G. Numerical fracture analysis of prestressed concrete beams. International Journal of Concrete Structures and Materials, 2008, 2(2): 153–160

[12] Rabczuk T, Xiao S P, Sauer M. Coupling of mesh-free methods with finite elements: basic concepts and test results. International Journal for Numerical Methods in Biomedical Engineering, 2006, 22(10): 1031–1065

[13] Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760

[14] Rabczuk T, Belytschko T. A three-dimensional large deformation mesh-free method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799

[15] Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495

[16] Zi G, Rabczuk T, Wall W. Extended meshfree methods without branch enrichment for cohesive cracks. Computational Mechanics, 2007, 40(2): 367–382

[17] Rabczuk T, Areias P. A Meshfree Thin Shell for Arbitrary Evolving Cracks Based on An Extrinsic Basis. Christchurch: University of Canterbury, 2006

[18] Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315

[19] Areias P, Msekh M, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[20] Areias P, Reinoso J, Camanho P P, César de Sá J, Rabczuk T. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 2018, 189: 339–360

[21] Areias P, Rabczuk T. A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement. Computational Mechanics, 2016, 58(6): 1003–1018

[22] Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350

[23] Msekh M A, Cuong N, Zi G, Areias P, Zhuang X, Rabczuk T. Fracture properties prediction of clay/epoxy nanocomposites with interphase zones using a phase field model. Engineering Fracture Mechanics, 2018, 188: 287–299

[24] Areias P, Rabczuk T, Dias-da Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[25] Areias P, Rabczuk T, Camanho P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947

[26] Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122

[27] Rabczuk T, Belytschko T. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1–4): 19–49

[28] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455

[29] Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71

[30] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[31] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[32] Anitescu C, Hossain M N, Rabczuk T. Recovery-based error estimation and adaptivity using high-order splines over hierarchical t-meshes. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 638–662

[33] Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 1157–1178

[34] Nguyen B, Tran H, Anitescu C, Zhuang X, Rabczuk T. An isogeometric symmetric galerkin boundary element method for two-dimensional crack problems. Computer Methods in Applied Mechanics and Engineering, 2016, 306: 252–275

[35] Thai T Q, Rabczuk T, Bazilevs Y, Meschke G. A higher-order stress based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604

[36] Nguyen-Thanh N, Valizadeh N, Nguyen M, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

[37] Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146

[38] Silani M, Talebi H, Hamouda A M, Rabczuk T. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23

[39] Talebi H, Silani M, Rabczuk T. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92

[40] Talebi H, Silani M, Bordas S P, Kerfriden P, Rabczuk T. A computational library for multiscale modeling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071

[41] Budarapu P R, Gracie R, Bordas S P, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148

[42] Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143

[43] Amiri F, Anitescu C, Arroyo M, Bordas S P A, Rabczuk T. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57

[44] Chen L, Rabczuk T, Bordas S P A, Liu G, Zeng K, Kerfriden P. Extended finite element method with edge-based strain smoothing (ESM-XFEM) for linear elastic crack growth. Computer Methods in Applied Mechanics and Engineering, 2012, 209–212: 250–265

[45] Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23–24): 1391–1411

[46] Areias P, Rabczuk T, Camanho P. Finite strain fracture of 2d problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63

[47] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[48] Areias P M, Rabczuk T. Quasi-static crack propagation in plane and plate structures using set-valued traction-separation laws. International Journal for Numerical Methods in Engineering, 2008, 74(3): 475–505

[49] De Borst R. Some recent developments in computational modelling of concrete fracture. International Journal of Fracture, 1997, 86(1–2): 5–36

[50] Murthy A R C, Palani G, Iyer N R. State-of-the-art review on fracture analysis of concrete structural components. Sadhana, 2009, 34(2): 345–367

[51] Wu M, Johannesson B, Geiker M. A review: Self-healing in cementitious materials and engineered cementitious composite as a self-healing material. Construction & Building Materials, 2012, 28(1): 571–583

[52] Van Tittelboom K, De Belie N. Self-healing in cementitious materials: A review. Materials (Basel), 2013, 6(6): 2182–2217

[53] Talaiekhozan A, Keyvanfar A, Shafaghat A, A review of self-healing concrete research development. Journal of Environmental Treatment Techniques, 2014, 2(1): 1–11

[54] Lv Z, Chen D. Overview of recent work on self-healing in cementitious materials. Materiales de Construccion, 2014, 64(316): 034

[55] Ahn E, Kim H, Sim S H, Shin S W, Shin M. Principles and applications of ultrasonic-based nondestructive methods for self-healing in cementitious materials. Materials (Basel), 2017, 10(3): 278

[56] Mauludin L, Oucif C. Modeling of self-healing concrete: A review. Journal of Applied and Computational Mechanics, 2017, 5: 526–539

[57] Mauludin L M, Oucif C. The effects of interfacial strength on fractured microcapsule. Frontiers of Structural and Civil Engineering, 2019, 13(2): 353–363

[58] Mauludin L M, Oucif C. Interaction between matrix crack and circular capsule under uniaxial tension in encapsulation-based self-healing concrete. Underground Space, 2018, 3(3): 181–189

[59] Mauludin L M, Zhuang X, Rabczuk T. Computational modeling of fracture in encapsulation-based self-healing concrete using cohesive elements. Composite Structures, 2018, 196: 63–75

[60] Oucif C, Mauludin L. Continuum damage-healing and super healing mechanics in brittle materials: A state-of-the-art review. Applied Sciences (Basel, Switzerland), 2018, 8(12): 2350

[61] Oucif C, Ouzaa K, Mauludin L M. Cyclic and monotonic behavior of strengthened and unstrengthened square reinforced concrete columns. Journal of Applied and Computational Mechanics, 2019, 5: 517–525

[62] Rabczuk T. Computational methods for fracture in brittle and quasi brittle solids: State-of-the-art review and future perspectives, ISRN. Applied Mathematics, 2013: 332–369

[63] Djoković J M, Nikolić R R, Bujnak J. Fundamental problems of modeling the fracture processes in concrete I: Micromechanics and localization of damages. Procedia Engineering, 2013, 65: 186–195

[64] Djoković J M, Nikolić R R, Bujnak J. Fundamental problems of modeling the fracture processes in concrete II: Size effect and selection of the solution approach. Procedia Engineering, 2013, 65: 196–205

[65] Jendele L, Cervenka J, Saouma V, Pukl R. On the choice between discrete or smeared approach in practical structural fe analyses of concrete structures. In: The Fourth International Conference on Analysis of Discontinuous Deformation. Glasgow: University of Glasgow, 2001

[66] Hillerborg A, Modéer M, Petersson P E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 1976, 6(6): 773–781

[67] Jirásek M, Zimmermann T. Rotating crack model with transition to scalar damage. Journal of Engineering Mechanics, 1998, 124(3): 277–284

[68] Xu X P, Needleman A. Numerical simulations of dynamic crack growth along an interface. International Journal of Fracture, 1996, 74(4): 289–324

[69] Samaniego Alvarado E. Contributions to the Continuum Modelling of Strong Discontinuities in Two-dimensional Solids. Dissertation for the Doctoral Degree. Barcelona: Universitat Politécnica de Catalunya, 2003

[70] Belytschko T, Lu Y Y, Gu L. Element-free galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256

[71] Wells G N, Sluys L. A new method for modelling cohesive cracks using finite elements. International Journal for Numerical Methods in Engineering, 2001, 50(12): 2667–2682

[72] Rabczuk T, Akkermann J, Eibl J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5–6): 1327–1354

[73] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[74] Hrennikoff A. Solution of problems of elasticity by the framework method. Journal of Applied Mechanics, 1941, 8(4): 169–175

[75] Schlangen E, van Mier J. Simple lattice model for numerical simulation of fracture of concrete materials and structures. Materials and Structures, 1992, 25(9): 534–542

[76] Schlangen E, Garboczi E. Fracture simulations of concrete using lattice models: Computational aspects. Engineering Fracture Mechanics, 1997, 57(2–3): 319–332

[77] Lilliu G, van Mier J G M. 3D lattice type fracture model for concrete. Engineering Fracture Mechanics, 2003, 70(7–8): 927–941

[78] Cusatis G, Pelessone D, Mencarelli A. Lattice discrete particle model (LDPM) for failure behavior of concrete. I: Theory. Cement and Concrete Composites, 2011, 33(9): 881–890

[79] Cusatis G, Mencarelli A, Pelessone D, Baylot J. Lattice discrete particle model (LDPM) for failure behavior of concrete. II: Calibration and validation. Cement and Concrete Composites, 2011, 33(9): 891–905

[80] Cusatis G, Bažant Z P, Cedolin L. Confinement-shear lattice model for concrete damage in tension and compression: I. Theory. Journal of Engineering Mechanics, 2003, 129(12): 1439–1448

[81] Cusatis G, Bažant Z P, Cedolin L. Confinement-shear lattice model for concrete damage in tension and compression: II. Computation and validation. Journal of Engineering Mechanics, 2003, 129(12): 1449–1458

[82] Cusatis G, Bažant Z P, Cedolin L. Confinement-shear lattice CSL model for fracture propagation in concrete. Computer Methods in Applied Mechanics and Engineering, 2006, 195(52): 7154–7171

[83] Pelessone D. Discrete Particle Method, Technical Report. Engineering and Software System Solutions, Inc., 2005

[84] Menetrey P, Willam K. Triaxial failure criterion for concrete and its generalization. Structural Journal, 1995, 92(3): 311–318

[85] Grassl P, Lundgren K, Gylltoft K. Concrete in compression: A plasticity theory with a novel hardening law. International Journal of Solids and Structures, 2002, 39(20): 5205–5223

[86] Papanikolaou V K, Kappos A J. Confinement-sensitive plasticity constitutive model for concrete in triaxial compression. International Journal of Solids and Structures, 2007, 44(21): 7021–7048

[87] Červenka J, Papanikolaou V K. Three dimensional combined fracture-plastic material model for concrete. International Journal of Plasticity, 2008, 24(12): 2192–2220

[88] Folino P, Etse G. Performance dependent model for normal and high strength concretes. International Journal of Solids and Structures, 2012, 49(5): 701–719

[89] Ortiz M. A constitutive theory for the inelastic behavior of concrete. Mechanics of Materials, 1985, 4(1): 67–93

[90] Carol I, Rizzi E, Willam K. On the formulation of anisotropic elastic degradation. I. Theory based on a pseudo-logarithmic damage tensor rate. International Journal of Solids and Structures, 2001, 38(4): 491–518

[91] Tao X, Phillips D V. A simplified isotropic damage model for concrete under bi-axial stress states. Cement and Concrete Composites, 2005, 27(6): 716–726

[92] Voyiadjis G Z, Kattan P I. A comparative study of damage variables in continuum damage mechanics. International Journal of Damage Mechanics, 2009, 18(4): 315–340

[93] Jason L, Huerta A, Pijaudier-Cabot G, Ghavamian S. An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model. Computer Methods in Applied Mechanics and Engineering, 2006, 195(52): 7077–7092

[94] Grassl P, Jirásek M. Damage-plastic model for concrete failure. International Journal of Solids and Structures, 2006, 43(22–23): 7166–7196

[95] Nguyen G D, Korsunsky A M. Development of an approach to constitutive modelling of concrete: Isotropic damage coupled with plasticity. International Journal of Solids and Structures, 2008, 45(20): 5483–5501

[96] Nguyen G D, Houlsby G T. A coupled damage-plasticity model for concrete based on thermodynamic principles: Part I: Model formulation and parameter identification. International Journal for Numerical and Analytical Methods in Geomechanics, 2008, 32(4): 353–389

[97] Voyiadjis G Z, Taqieddin Z N, Kattan P I. Anisotropic damage-plasticity model for concrete. International Journal of Plasticity, 2008, 24(10): 1946–1965

[98] Grassl P. On a damage-plasticity approach to model concrete failure. Proceedings of the Institution of Civil Engineers, 2009, 162(em4): 221–231

[99] Sánchez P, Huespe A, Oliver J, Diaz G, Sonzogni V. A macroscopic damage-plastic constitutive law for modeling quasi-brittle fracture and ductile behavior of concrete. International Journal for Numerical and Analytical Methods in Geomechanics, 2012, 36(5): 546–573

[100] Hofstetter B V G. Review and enhancement of 3D concrete models for large-scale numerical simulations of concrete structures. International Journal for Numerical and Analytical Methods in Geomechanics, 2013, 37(3): 221–246

[101] Oliver J, Huespe A E, Samaniego E, Chaves E. Continuum approach to the numerical simulation of material failure in concrete. International Journal for Numerical and Analytical Methods in Geomechanics, 2004, 28(78): 609–632

[102] Tailhan J, Rossi P, Dal Pont S. Macroscopic probabilistic modeling of concrete cracking: First 3D results. In: The 7th International Conference on Fracture Mechanics of Concrete and Concrete Structures. Seoul: Korea Concrete Institute, 2010, 238–242

[103] Abu Al-Rub R K, Kim S M. Computational applications of a coupled plasticity-damage constitutive model for simulating plain concrete fracture. Engineering Fracture Mechanics, 2010, 77(10): 1577–1603

[104] Bažant Z P, Gambarova P G. Crack shear in concrete: Crack band microplane model. Journal of Structural Engineering, 1984, 110(9): 2015–2035

[105] Bažant Z P, Xiang Y, Prat P C. Microplane model for concrete. I: Stress-strain boundaries and finite strain. Journal of Engineering Mechanics, 1996, 122(3): 245–254

[106] Bažant Z P, Oh B H. Microplane Model for Fracture Analysis of Concrete Structures, Technical Report. Northwestern University, Technological Institute, 1983

[107] Ožbolt J, Li Y, Kožar I. Microplane model for concrete with relaxed kinematic constraint. International Journal of Solids and Structures, 2001, 38(16): 2683–2711

[108] Yang Z, Su X, Chen J F, Liu G. Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials. International Journal of Solids and Structures, 2009, 46(17): 3222–3234

[109] Su X, Yang Z, Liu G. Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials: A 3D study. International Journal of Solids and Structures, 2010, 47(17): 2336–2345

[110] Teng J, Zhu W, Tang C. Mesomechanical model for concrete. Part II: Applications. Magazine of Concrete Research, 2004, 56(6): 331–345

[111] Zhu H, Zhou S, Yan Z, Ju W, Chen Q. A 3D analytical model for the probabilistic characteristics of self-healing model for concrete using spherical microcapsule. Computers and Concrete, 2015, 15(1): 37–54

[112] Caballero A, L’opez C, Carol I. 3D meso-structural analysis of concrete specimens under uniaxial tension. Computer Methods in Applied Mechanics and Engineering, 2006, 195(52): 7182–7195

[113] Trias D, Costa J, Fiedler B, Hobbiebrunken T, Hurtado J E. A two scale method for matrix cracking probability in fibre-reinforced composites based on a statistical representative volume element. Composites Science and Technology, 2006, 66(11–12): 1766–1777

[114] Al-Ostaz A, Diwakar A, Alzebdeh K I. Statistical model for characterizing random microstructure of inclusion-matrix composites. Journal of Materials Science, 2007, 42(16): 7016–7030

[115] Ren W, Yang Z, Sharma R, Zhang C, Withers P J. Two-dimensional X-ray CT image based meso-scale fracture modelling of concrete. Engineering Fracture Mechanics, 2015, 133: 24–39

[116] Huang Y, Yang Z, Ren W, Liu G, Zhang C. 3D meso-scale fracture modelling and validation of concrete based on in-situ X-ray Computed Tomography images using damage plasticity model. International Journal of Solids and Structures, 2015, 67–68: 340–352

[117] Du X, Jin L, Ma G. Numerical modeling tensile failure behavior of concrete at mesoscale using extended finite element method. International Journal of Damage Mechanics, 2014, 23(7): 872–898

[118] Zemskov S V, Jonkers H M, Vermolen F J. A mathematical model for bacterial self-healing of cracks in concrete. Journal of Intelligent Material Systems and Structures, 2014, 25(1): 4–12

[119] Zhou X, Hao H. Mesoscale modelling of concrete tensile failure mechanism at high strain rates. Computers & Structures, 2008, 86(21–22): 2013–2026

[120] Wang X, Yang Z, Yates J, Jivkov A, Zhang C. Monte Carlo simulations of mesoscale fracture modelling of concrete with random aggregates and pores. Construction & Building Materials, 2015, 75: 35–45

[121] Wang X, Yang Z, Jivkov A P. Monte Carlo simulations of mesoscale fracture of concrete with random aggregates and pores: A size effect study. Construction & Building Materials, 2015, 80: 262–272

[122] Wang X, Zhang M, Jivkov A P. Computational technology for analysis of 3D meso-structure effects on damage and failure of concrete. International Journal of Solids and Structures, 2016, 80: 310–333

[123] Wang X, Jivkov A P. Combined numerical-statistical analyses of damage and failure of 2D and 3D mesoscale heterogeneous concrete. Mathematical Problems in Engineering, 2015, 2015: 1–12

[124] Koutsourelakis P S, Deodatis G. Simulation of multidimensional binary random fields with application to modeling of two-phase random media. Journal of Engineering Mechanics, 2006, 132(6): 619–631

[125] Xu X F, Graham-Brady L. A stochastic computational method for evaluation of global and local behavior of random elastic media. Computer Methods in Applied Mechanics and Engineering, 2005, 194(42–44): 4362–4385

[126] Graham-Brady L, Xu X F. Stochastic morphological modeling of random multiphase materials. Journal of Applied Mechanics, 2008, 75(6): 061001

[127] Most T. Stochastic crack growth simulation in reinforced concrete structures by means of coupled finite element and meshless methods. Dissertation for the Doctoral Degree. Weimar: Bauhaus-Universität Weimar, 2005

[128] Bruggi M, Casciati S, Faravelli L. Cohesive crack propagation in a random elastic medium. Probabilistic Engineering Mechanics, 2008, 23(1): 23–35

[129] Yang Z, Xu X F. A heterogeneous cohesive model for quasi-brittle materials considering spatially varying random fracture properties. Computer Methods in Applied Mechanics and Engineering, 2008, 197(45–48): 4027–4039

[130] Grassl P, Grégoire D, Rojas Solano L, Pijaudier-Cabot G. Meso-scale modelling of the size effect on the fracture process zone of concrete. International Journal of Solids and Structures, 2012, 49(13): 1818–1827

[131] Hamdia K M, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[132] Hamdia K M, Zhuang X, He P, Rabczuk T. Fracture toughness of polymeric particle nanocomposites: Evaluation of models performance using Bayesian method. Composites Science and Technology, 2016, 126: 122–129

[133] Hamdia K M, Lahmer T, Nguyen-Thoi T, Rabczuk T. Predicting the fracture toughness of PNCS: A stochastic approach based on ann and anfis. Computational Materials Science, 2015, 102: 304–313

[134] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[135] Hamdia K M, Msekh M A, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T. Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Composite Structures, 2015, 133: 1177–1190

RIGHTS & PERMISSIONS

Higher Education Press