This paper presents the state of the art of the theory of rock damage and healing mechanics, with a particular emphasis on the strategies available to relate the micro-scale of crystals, cracks and pores to the scale of a Representative Elementary Volume (REV). We focus on mechanical degradation and recovery of stiffness and strength. Damage and healing models formulated in the author’s group are used as examples to illustrate and compare the reviewed micro-macro approaches, which include fabric enrichment, micromechanical formulations and homogenization schemes. This manuscript was written for doctoral students or researchers relatively new to the field of damage mechanics of geomaterials. Equations are provided to explain how to formulate a thermodynamically consistent model from scratch. Reviewing damage and healing modeling strategies led to the following conclusions: (i) The framework of hyperplasticity, which does not require any postulate on the existence or uniqueness of yield functions and which automatically ensures thermodynamic consistency, was never applied to Continuum Damage Mechanics (CDM). There may be an avenue to improve state-of-the-art damage and healing models in a similar framework of “hyper-damage mechanics”. (ii) In damage softening models, the mesh dependence of the width of the damage localization zone is currently alleviated by non-local regularization. Perhaps the next step is to couple micro-macro damage and healing models at the REV scale to discrete fracture mechanics at a larger scale to understand how damage and healing localization occurs. (iii) There may be an opportunity to use fabric-enriched models to capture the effect of microstructure organization on both mechanical properties and permeability. (iv) Coupling chemo-mechanical damage and healing processes across the scales would be useful to model the competition between damagre and healing whenever both can occur at the same temperature and pressure conditions. (v) Many challenges still exist to implement healing models in the Finite Element Method, especially in regards to the mapping of net damage.

Revised: 2019-09-30

Accepted: 2019-11-20

Published online: 2020-02-04

Keywords: mechanics, healing processes, rock mechanics, fabric, micro-mechanics, homogenization, fracture mechanics, thermodynamics

@article{OGEO_2020__2__A1_0, author = {Arson, Chlo\'e}, title = {Micro-macro mechanics of damage and healing in rocks}, journal = {Open Geomechanics}, publisher = {Alert Geomaterials}, volume = {2}, year = {2020}, doi = {10.5802/ogeo.4}, language = {en}, url = {opengeomechanics.centre-mersenne.org/item/OGEO_2020__2__A1_0/} }

Arson, Chloé. Micro-macro mechanics of damage and healing in rocks. Open Geomechanics, Volume 2 (2020) , article no. 1, 41 p. doi : 10.5802/ogeo.4. https://opengeomechanics.centre-mersenne.org/item/OGEO_2020__2__A1_0/

[1] Mechanical characterization in unconventional reservoirs: a facies-based methodology, Petrophysics, Volume 54 (2013) no. 05, pp. 457-464

[2] The failure of brittle solids containing small cracks under compressive stress states, Acta Metallurgica, Volume 34 (1986) no. 3, pp. 497-510 | Article

[3] Homogenization of a cracked saturated porous medium: Theoretical aspects and numerical implementation, International Journal of Solids and Structures, Volume 94 (2016), pp. 222-237 | Article

[4] Dispersion analysis and element-free Galerkin solutions of second-and fourth-order gradient-enhanced damage models, International Journal for Numerical Methods in Engineering, Volume 49 (2000) no. 6, pp. 811-832 | Article | Zbl 1009.74082

[5] Generalized stress variables in continuum damage mechanics, Mechanics Research Communications, Volume 60 (2014), pp. 81-84 | Article | Zbl 1359.74366

[6] On the coupling of anisotropic damage and plasticity models for ductile materials, International Journal of Solids and Structures, Volume 40 (2003) no. 11, pp. 2611-2643 | Zbl 1051.74528

[7] Explicit and implicit gradient series in damage mechanics, European Journal of Mechanics-A/Solids, Volume 21 (2002) no. 3, pp. 379-390 | Article | Zbl 1023.74004

[8] On the definition of damage in time-dependent healing models for salt rock, Géotechnique Letters, Volume 2 (2012), pp. 67-71 | Article

[9] Hyper-elastoplastic/damage modeling of rock with application to porous limestone, International Journal of Solids and Structures, Volume 143 (2018), pp. 218-231 | Article

[10] A note on brittle crack growth in compression, Journal of Geophysical Research, Volume 68 (1963) no. 12, pp. 3709-3713 | Article

[11] Microplane model M5 with kinematic and static constraints for concrete fracture and anelasticity. I: Theory, Journal of engineering mechanics, Volume 131 (2005) no. 1, pp. 31-40 | Article

[12] Microplane model M4 for concrete. I: Formulation with work-conjugate deviatoric stress, Journal of Engineering Mechanics, Volume 126 (2000) no. 9, pp. 944-953 | Article

[13] Dislocation mechanics of creep, Materials Science and Engineering: A, Volume 510 (2009), pp. 7-13 | Article

[14] Transmission Electron Microscope Study of Biotite Weathering, Clays and Clay Minerals, Volume 36 (1988), pp. 47-60 | Article

[15] A new approach to the application of Mori-Tanaka’s theory in composite materials, Mechanics of materials, Volume 6 (1987) no. 2, pp. 147-157 | Article

[16] Continuum damage-healing mechanics with application to self-healing composites, International Journal of Damage Mechanics, Volume 14 (2005) no. 1, pp. 51-81 | Article

[17] Modelling of anisotropic damage by microcracks: towards a discrete approach, Archives of Mechanics, Volume 58 (2006) no. 2, pp. 93-123 | MR 2250979 | Zbl 1106.74007

[18] Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of Engineering Mechanics, Volume 128 (2002) no. 11, pp. 1119-1149 | Article

[19] Nonlocal smeared cracking model for concrete fracture, Journal of Structural Engineering, Volume 114 (1988) no. 11, pp. 2493-2510 | Article

[20] Elastic moduli of a cracked solid, International journal of Solids and structures, Volume 12 (1976) no. 2, pp. 81-97 | Article | Zbl 0318.73065

[21] Microplane model for progressive fracture of concrete and rock, Journal of Engineering Mechanics, Volume 111 (1985) no. 4, pp. 559-582 | Article

[22] Nonlocal microplane model for fracture, damage, and size effect in structures, Journal of Engineering Mechanics, Volume 116 (1990) no. 11, pp. 2485-2505 | Article | Zbl 0868.73066

[23] Microplane model for brittle-plastic material: I. Theory, Journal of Engineering Mechanics, Volume 114 (1988) no. 10, pp. 1672-1688 | Article

[24] Nonlocal continuum damage, localization instability and convergence, Journal of applied mechanics, Volume 55 (1988) no. 2, pp. 287-293 | Zbl 0663.73075

[25] Measurement of characteristic length of nonlocal continuum, Journal of Engineering Mechanics, Volume 115 (1989) no. 4, pp. 755-767 | Article | Zbl 0663.73075

[26] On yielding of oriented solids, Acta Mechanica, Volume 27 (1977) no. 1-4, pp. 185-204 | Article

[27] Weathering of the Rio Blanco quartz diorite, Luquillo Mountains, Puerto Rico: Coupling oxidation, dissolution, and fracturing, Geochimica et Cosmochimica Acta, Volume 72 (2008) no. 18, pp. 4488 -4507 http://www.sciencedirect.com/science/article/pii/S0016703708003955 | Article

[28] Compaction and failure in high porosity carbonates: Mechanical data and microstructural observations, Pure and Applied Geophysics, Volume 166 (2009) no. 5-7, pp. 869-898 | Article

[29] Dislocations in minerals, Dislocations in Solids, Volume 16 (2010), pp. 171-232 | Article

[30] Microplane model for concrete. I: Stress-strain boundaries and finite strain, Journal of Engineering Mechanics, Volume 122 (1996) no. 3, pp. 245-254 | Article

[31] An extension of the self-consistent scheme to plastically-flowing polycrystals, Journal of the Mechanics and Physics of Solids, Volume 26 (1978) no. 5-6, pp. 325-344 | Article | Zbl 0395.73033

[32] Microplane model M7 for plain concrete. I: Formulation, Journal of Engineering Mechanics, Volume 139 (2012) no. 12, pp. 1714-1723 | Article

[33] Damage and plasticity in microplane theory, International Journal of Solids and Structures, Volume 34 (1997) no. 29, pp. 3807-3835 | Article | Zbl 0942.74631

[34] Recovery and healing of damage in WIPP salt, International Journal of Damage Mechanics, Volume 7 (1998) no. 2, pp. 143-166 | Article

[35] Théorie des corps déformables, A. Hermann et fils,, 1909 | Zbl 40.0862.02

[36] A paraboloid failure surface for transversely isotropic materials, Mechanics of materials, Volume 31 (1999) no. 6, pp. 381-393 | Article

[37] Plastic continuum with microstructure, local second gradient theories for geomaterials: localization studies, International Journal of Solids and Structures, Volume 38 (2001) no. 46-47, pp. 8503-8527 | Article | Zbl 1047.74522

[38] A new anisotropic failure criterion for transversely isotropic solids, Mechanics of Cohesive-frictional Materials: An International Journal on Experiments, Modelling and Computation of Materials and Structures, Volume 3 (1998) no. 1, pp. 89-103 | Article

[39] A strain space gradient plasticity theory for finite strain, Computer Methods in Applied Mechanics and Engineering, Volume 193 (2004) no. 27-29, pp. 2797-2826 | Article | MR 2060915 | Zbl 1329.74049

[40] Poroplastic damage model for claystones, Applied Clay Science, Volume 26 (2004) no. 1-4, pp. 473-487 | Article

[41] Temperature and rate dependent deformation of halite, American Journal of Science, Volume 269 (1970) no. 3, pp. 193-249 | Article

[42] Application of thermomechanical principles to the modelling of geotechnical materials, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Volume 453 (1997) no. 1964, pp. 1975-2001 | Article | Zbl 0933.74045

[43] Damage induced anisotropy: on the difficulties associated with the active/passive unilateral condition, International Journal of Damage Mechanics, Volume 1 (1992) no. 2, pp. 148-171 | Article

[44] Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage, International Journal of Damage Mechanics, Volume 2 (1993) no. 4, pp. 311-329 | Article

[45] A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses, International Journal of Solids and Structures, Volume 38 (2001) no. 17, pp. 2921-2931 | Article | Zbl 0999.74012

[46] Deformation and strength anisotropy of Asan gneiss, Boryeong shale, and Yeoncheon schist, International Journal of Rock Mechanics and Mining Sciences, Volume 50 (2012), pp. 158-169

[47] Self-consistent techniques for heterogeneous media, Journal of the Engineering Mechanics Division, Volume 106 (1980) no. 5, pp. 861-887

[48] Mechanisms of compaction of quartz sand at diagenetic conditions, Earth and Planetary Science Letters, Volume 220 (2004) no. 3-4, pp. 435-451 | Article

[49] An extended FE strategy for transition from continuum damage to mode I cohesive crack propagation, International Journal for Numerical and Analytical Methods in Geomechanics, Volume 31 (2007) no. 2, pp. 213-238 | Article | Zbl 1108.74052

[50] The relationship between the elasticity tensor and the fabric tensor, Mechanics of Materials, Volume 4 (1985) no. 2, pp. 137-147 | Article

[51] Fracture energy based bi-dissipative damage model for concrete, International journal of solids and structures, Volume 38 (2001) no. 36-37, pp. 6427-6454 | Article | Zbl 0996.74069

[52] Damage induced elastic anisotropy, Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes, Springer, 1982, pp. 761-774 | Article | Zbl 0516.73013

[53] Modeling of elastoplastic damage behavior of a claystone, International Journal of plasticity, Volume 19 (2003) no. 1, pp. 23-45 | Article | Zbl 1032.74649

[54] Coupled elastoplastic damage modeling of anisotropic rocks, Computers and Geotechnics, Volume 37 (2010) no. 1-2, pp. 187-194 | Article

[55] Induced anisotropic damage and plasticity in initially anisotropic sedimentary rocks, International Journal of Rock Mechanics and Mining Sciences, Volume 51 (2012), pp. 13-23 | Article

[56] The effect of spatial distribution on the effective behavior of composite materials and cracked media, Journal of the Mechanics and Physics of Solids, Volume 43 (1995) no. 12, pp. 1919-1951 | Article | MR 1360832 | Zbl 0919.73061

[57] A continuum damage mechanics framework for modeling micro-damage healing, International Journal of Solids and Structures, Volume 49 (2012) no. 3-4, pp. 492-513 | Article

[58] Localized deformation induced by heterogeneities in porous carbonate analysed by multi-scale digital image correlation, Tectonophysics, Volume 503 (2011) no. 1-2, pp. 100-116 | Article

[59] On coupled gradient-dependent plasticity and damage theories with a view to localization analysis, European Journal of Mechanics-A/Solids, Volume 18 (1999) no. 6, pp. 939-962 | Article | Zbl 0968.74007

[60] On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials, Computational Mechanics, Volume 17 (1995) no. 1-2, pp. 130-141 | Article | Zbl 0840.73047

[61] Pressure solution: theory and experiments, Tectonophysics, Volume 39 (1977) no. 1-3, pp. 287-301 | Article

[62] Positivity of intrinsic dissipation of a class of nonstandard anisotropic damage models, Comptes Rendus Mécanique, Volume 334 (2006) no. 10, pp. 587-592 | Article | Zbl 1177.74056

[63] A micromechanical analysis of damage propagation in fluid-saturated cracked media, Comptes Rendus Mecanique, Volume 334 (2006) no. 7, pp. 440-446 | Article | Zbl 1177.74356

[64] Microporomechanics, John Wiley & Sons, 2006 | Article | Zbl 1112.76002

[65] Simple plasticity sand model accounting for fabric change effects, Journal of Engineering mechanics, Volume 130 (2004) no. 6, pp. 622-634 | Article

[66] Sand plasticity model accounting for inherent fabric anisotropy, Journal of Engineering Mechanics, Volume 130 (2004) no. 11, pp. 1319-1333 | Article

[67] Model of dilatancy of brittle materials with cracks under compression, Mechanics of solids, Volume 22 (1987) no. 6, pp. 165-173

[68] Comparison of nonlocal approaches in continuum damage mechanics, Computers & Structures, Volume 55 (1995) no. 4, pp. 581-588 | Article

[69] A thermodynamically consistent framework for saturated viscoplastic rock-materials subject to damage, Mechanics Research Communications, Volume 45 (2012), pp. 15-21 | Article

[70] The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Volume 241 (1957) no. 1226, pp. 376-396 | MR 87326

[71] Micromechanisms of creep in clay-rich gouge from the Central Deforming Zone of the San Andreas Fault, Journal of Geophysical Research: Solid Earth, Volume 120 (2015) no. 2, pp. 827-849

[72] Fabric evolution within shear bands of granular materials and its relation to critical state theory, International Journal for numerical and analytical methods in geomechanics, Volume 35 (2011) no. 18, pp. 1918-1948

[73] Damage, gradient of damage and principle of virtual power, International Journal of Solids and Structures, Volume 33 (1996) no. 8, pp. 1083-1103 | Article | MR 1370124 | Zbl 0910.73051

[74] Effects of cyclic loading on mechanical properties of Maha Sarakham salt, Engineering Geology, Volume 112 (2010) no. 1-4, pp. 43-52 | Article

[75] Laboratory assessment of healing of fractures in rock salt, Bulletin of Engineering Geology and the Environment, Volume 70 (2011) no. 4, p. 665

[76] Damage micromechanics for constitutive relations and failure of microcracked quasi-brittle materials, International Journal of Damage Mechanics, Volume 19 (2010) no. 8, pp. 911-948 | Article

[77] Strain-based transient-gradient damage model for failure analyses, Computer methods in applied mechanics and engineering, Volume 160 (1998) no. 1-2, pp. 133-153 | Article | Zbl 0938.74006

[78] Validation and internal length scale determination for a gradient damage model: application to short glass-fibre-reinforced polypropylene, International Journal of Solids and Structures, Volume 36 (1999) no. 17, pp. 2557-2583 | Article | Zbl 0963.74502

[79] The method of virtual power in continuum mechanics. Part 2: Microstructure, SIAM Journal on Applied Mathematics, Volume 25 (1973) no. 3, pp. 556-575 | Article | Zbl 0273.73061

[80] The chemical, mechanical, and hydrological evolution of weathering granitoid, Journal of Geophysical Research: Earth Surface, Volume 121 (2016) no. 8, pp. 1410-1435

[81] A microcrack damage model for brittle materials, International Journal of Solids and Structures, Volume 30 (1993) no. 2, pp. 177-198 | Article | Zbl 0775.73196

[82] Mechanisms of brittle fracture of rock with pre-existing cracks in compression, Pure and Applied Geophysics, Volume 143 (1994) no. 1-3, pp. 117-149 | Article

[83] A critical state sand plasticity model accounting for fabric evolution, International journal for numerical and analytical methods in geomechanics, Volume 38 (2014) no. 4, pp. 370-390

[84] Failure criteria for unidirectional fiber composites, Journal of applied mechanics, Volume 47 (1980) no. 2, pp. 329-334 | Article | Zbl 0468.73123

[85] An anisotropic model of damage and frictional sliding for brittle materials, European Journal of Mechanics-A/Solids, Volume 17 (1998) no. 3, pp. 439-460 | Article | MR 1627155 | Zbl 0933.74055

[86] A modular damage model for quasi-brittle solids–interaction between initial and induced anisotropy, Archive of Applied Mechanics, Volume 72 (2002) no. 6-7, pp. 498-510 | Article | Zbl 1084.74543

[87] Experimental and theoretical study of the anisotropic properties of shale, International Journal of Rock Mechanics and Mining Sciences, Volume 74 (2015), pp. 58-68 | Article

[88] A theory of the yielding and plastic flow of anisotropic metals, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Volume 193 (1948) no. 1033, pp. 281-297 | MR 25919 | Zbl 0032.08805

[89] A self-consistent mechanics of composite materials, Journal of the Mechanics and Physics of Solids, Volume 13 (1965) no. 4, pp. 213-222 | Article | MR 1677111

[90] Continuum micro-mechanics of elastoplastic polycrystals, Journal of the Mechanics and Physics of Solids, Volume 13 (1965) no. 2, pp. 89-101 | Article | Zbl 0127.15302

[91] Thermodynamical modeling of elastic-plastic damage and experimental validation of damage potential, International Journal of damage mechanics, Volume 6 (1997) no. 4, pp. 333-363 | Article

[92] Overall moduli of solids with microcracks: load-induced anisotropy, Journal of the Mechanics and Physics of Solids, Volume 31 (1983) no. 2, pp. 155-171 | Article | Zbl 0506.73097

[93] Brittle failure in compression: splitting faulting and brittle-ductile transition, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 319 (1986) no. 1549, pp. 337-374 | Article | Zbl 0634.73109

[94] The brittle strength of orthotropic materials, Journal of Composite Materials, Volume 1 (1967) no. 2, pp. 200-206 | Article

[95] Mechanical and hydraulic behavior of rock salt in the excavation disturbed zone around underground facilities, International Journal of Rock Mechanics and Mining Sciences, Volume 40 (2003) no. 5, pp. 725-738

[96] Principles of hyperplasticity: an approach to plasticity theory based on thermodynamic principles, Springer Science & Business Media, 2007

[97] A thermodynamically consistent framework for theories of elastoplasticity coupled with damage, International Journal of Solids and Structures, Volume 31 (1994) no. 3, pp. 359-389 | Article

[98] Discrete equivalent wing crack based damage model for brittle solids, International Journal of Solids and Structures, Volume 110 (2017), pp. 279-293

[99] Anisotropic nonlocal damage model for materials with intrinsic transverse isotropy, International Journal of Solids and Structures, Volume 139 (2018), pp. 29-42

[100] Micromechanics based discrete damage model with multiple non-smooth yield surfaces: Theoretical formulation, numerical implementation and engineering applications, International Journal of Damage Mechanics, Volume 27 (2018) no. 5, pp. 611-639

[101] Nonlocal enrichment of a micromechanical damage model with tensile softening: Advantages and limitations, Computers and Geotechnics, Volume 94 (2018), pp. 196-206

[102] Computational Modeling of the Transition from Damage to Fracture in Intrinsically Anisotropic Porous Media (2018) (Ph. D. Thesis)

[103] Nonlocal models for damage and fracture: comparison of approaches, International Journal of Solids and Structures, Volume 35 (1998) no. 31-32, pp. 4133-4145 | Article | MR 1638122 | Zbl 0930.74054

[104] Micromechanical damage models for brittle solids. Part I: tensile loadings, Journal of Engineering Mechanics, Volume 117 (1991) no. 7, pp. 1495-1514 | Article

[105] On two-dimensinal self-consistent micromechanical damage models for brittle solids, International Journal of Solids and Structures, Volume 27 (1991) no. 2, pp. 227-258 | Article

[106] On micromechanical evolutionary damage models for polycrystalline ceramics, International Journal of Damage Mechanics, Volume 5 (1996) no. 2, pp. 113-137 | Article

[107] Computational model coupling mode II discrete fracture propagation with continuum damage zone evolution, International Journal for Numerical and Analytical Methods in Geomechanics, Volume 41 (2017) no. 2, pp. 223-250

[108] New strain-energy-based coupled elastoplastic two-parameter damage and healing models for earth-moving processes, International Journal of Damage Mechanics, Volume 21 (2012) no. 7, pp. 989-1019 | Article

[109] Continuum model of medium with cracks, Journal of the engineering mechanics division, Volume 106 (1980) no. 5, pp. 1039-1051

[110] A microcrack model of rock inelasticity part I: Frictional sliding on microcracks, Mechanics of Materials, Volume 1 (1982) no. 1, pp. 19-27 | Article

[111] A microcrack model of rock inelasticity part II: propagation of microcracks, Mechanics of Materials, Volume 1 (1982) no. 1, pp. 29-41 | Article

[112] Effective elastic properties of cracked solids: critical review of some basic concepts, Applied Mechanics Reviews, Volume 45 (1992) no. 8, pp. 304-335 | Article

[113] Elastic solids with many cracks and related problems, Advances in applied mechanics, Volume 30, Elsevier, 1993, pp. 259-445 | Article | Zbl 0803.73057

[114] Micromechanically inspired phenomenological damage model, Journal of Applied Mechanics, Volume 58 (1991) no. 2, pp. 305-310 | Article

[115] Zur plastischen verformung des vielkristalls, Acta metallurgica, Volume 9 (1961) no. 2, pp. 155-161 | Article

[116] A Mesomechanical Model for Brittle Deformation Processes: Part 2, Journal of Applied Mechanics, Volume 56 (1989), p. 57

[117] Localization limiters in transient problems, International Journal of Solids and Structures, Volume 24 (1988) no. 6, pp. 581-597 | Article | Zbl 0636.73021

[118] On micromechanical damage modeling in geomechanics: Influence of numerical integration scheme, Journal of Computational and Applied Mathematics, Volume 246 (2013), pp. 215-224 | Article | MR 3018110 | Zbl 06156340

[119] Hydro-mechanical modeling of damage due to underground excavation in hardened clay, Computational Geomechanics (2009), pp. 503-512

[120] Engineering damage mechanics: ductile, creep, fatigue and brittle failures, Springer Science & Business Media, 2005

[121] Anisotropic critical state theory: role of fabric, Journal of Engineering Mechanics, Volume 138 (2011) no. 3, pp. 263-275

[122] A model for intergranular pressure solution in open systems, Tectonophysics, Volume 245 (1995) no. 3-4, pp. 153-170 | Article

[123] Micromechanical damage models for brittle solids. Part II: compressive loadings, Journal of Engineering Mechanics, Volume 117 (1991) no. 7, pp. 1515-1536 | Article

[124] Tensorial representation of the effective elastic properties of the damaged material, International Journal of Damage Mechanics, Volume 3 (1994) no. 1, pp. 38-56 | Article

[125] On modelling of “winged” cracks forming under compression, International Journal of Fracture, Volume 77 (1996) no. 4, p. R69-R75 | Article | Zbl 1066.74052

[126] Damage model for brittle elastic solids with unequal tensile and compressive strengths, Engineering Fracture Mechanics, Volume 49 (1994) no. 5, pp. 681-697 | Article

[127] Micro-macro quantification of the internal structure of granular materials, Journal of engineering mechanics, Volume 135 (2009) no. 7, pp. 641-656

[128] Gradient concepts for evolution of damage, Mechanics of materials, Volume 31 (1999) no. 12, pp. 831-860 | Article

[129] Coupled plastic-damaged model, Computer methods in applied mechanics and engineering, Volume 129 (1996) no. 1-2, pp. 81-89 | Article | Zbl 0860.73047

[130] A fully coupled damage-plasticity model for unsaturated geomaterials accounting for the ductile–brittle transition in drying clayey soils, International Journal of solids and structures, Volume 91 (2016), pp. 102-114 | Article

[131] Porous media equivalents for networks of discontinuous fractures, Water Resources Research, Volume 18 (1982) no. 3, pp. 645-658 | Article

[132] Lineal-path function for random heterogeneous materials, Physical Review A, Volume 45 (1992) no. 2, p. 922

[133] Nearest-surface distribution functions for polydispersed particle systems, Physical Review A, Volume 45 (1992) no. 8, p. 5530

[134] On the self-consistent modeling of elastic-plastic behavior of polycrystals, Mechanics of Materials, Volume 26 (1997) no. 1, pp. 43-62 | Article

[135] An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals, Journal of the Mechanics and Physics of Solids, Volume 48 (2000) no. 6-7, pp. 1203-1227 | Article | MR 1766401 | Zbl 0984.74068

[136] Second gradient models as a particular case of microstructured models: a large strain finite elements analysis, Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics-Physics-Astronomy, Volume 328 (2000) no. 2, pp. 179-186 | Article | Zbl 1002.74009

[137] On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, Volume 4 (1968) no. 1, pp. 109-124 | Article

[138] Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, Volume 16 (1964) no. 1, pp. 51-78 | Article | MR 160356 | Zbl 0119.40302

[139] Homogenization of elastic–viscoplastic heterogeneous materials: Self-consistent and Mori-Tanaka schemes, International Journal of Plasticity, Volume 25 (2009) no. 6, pp. 1024-1048 | Article | Zbl 1162.74037

[140] Numerical simulation of damage–Permeability relationship in brittle geomaterials, Computers and Geotechnics, Volume 37 (2010) no. 5, pp. 619-628 | Article

[141] Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta metallurgica, Volume 21 (1973) no. 5, pp. 571-574 | Article

[142] Constitutive models for healing of materials with application to compaction of crushed rock salt, Journal of Engineering Mechanics, Volume 121 (1995) no. 10, pp. 1122-1129 | Article

[143] Self-consistent estimates for the rate-dependentelastoplastic behaviour of polycrystalline materials, Journal of the Mechanics and Physics of Solids, Volume 47 (1999) no. 7, pp. 1543-1568 | Article | Zbl 0976.74010

[144] On a concept of directional damage gradient in transversely isotropic materials, International Journal of Solids and Structures, Volume 88 (2016), pp. 56-67 | Article

[145] Plasticity: a treatise on finite deformation of heterogeneous inelastic materials, Cambridge University Press, 2004 | Zbl 1073.74001

[146] Micromechanics: overall properties of heterogeneous materials Volume 37, Elsevier, 2013 | Zbl 0704.45007

[147] A microcrack model of dilatancy in brittle materials, Journal of applied mechanics, Volume 55 (1988) no. 1, pp. 24-35 | Article

[148] Anisotropic strength and deformational behavior of Himalayan schists, International Journal of Rock Mechanics and Mining Sciences, Volume 40 (2003) no. 1, pp. 3-23 | Article

[149] Laboratory investigation of the mechanical behaviour of Tournemire shale, International Journal of Rock Mechanics and Mining Sciences, Volume 34 (1997) no. 1, pp. 3-16 | Article

[150] A homogenization approach for effective viscoelastic properties of porous media, Mechanics of Materials, Volume 100 (2016), pp. 175-185 | Article

[151] A constitutive theory for the inelastic behavior of concrete, Mechanics of materials, Volume 4 (1985) no. 1, pp. 67-93 | Article

[152] Correlation between mechanical behaviour and petrological properties of rock salt, The 32nd US Symposium on Rock Mechanics (USRMS) (1991)

[153] Some observations on localization in non-local and gradient damage models, European Journal of Mechanics, A/Solids, Volume 15 (1996) no. 6, pp. 937-953 | Zbl 0891.73055

[154] Nonlocal damage theory, Journal of engineering mechanics, Volume 113 (1987) no. 10, pp. 1512-1533 | Article | Zbl 0663.73075

[155] Gradient enhanced damage for quasi-brittle materials, International Journal for numerical methods in engineering, Volume 39 (1996) no. 19, pp. 3391-3403 | Article | Zbl 0882.73057

[156] Gradient-enhanced damage modelling of concrete fracture, Mechanics of Cohesive-frictional Materials: An International Journal on Experiments, Modelling and Computation of Materials and Structures, Volume 3 (1998) no. 4, pp. 323-342 | Article

[157] Estimation of influence tensors for eigenstressed multiphase elastic media with nonaligned inclusion phases of arbitrary ellipsoidal shape, Journal of engineering mechanics, Volume 136 (2010) no. 8, pp. 1043-1053 | Article

[158] A method for analyzing void distribution in air-entrained concrete, Cement, concrete and aggregates, Volume 5 (1983) no. 2, pp. 128-130

[159] Micromechanics of anisotropic brittle damage: comparative analysis between a stress based and a strain based formulation, Mechanics of materials, Volume 35 (2003) no. 8, pp. 747-761 | Article | Zbl 1058.74015

[160] Micromechanical analysis of anisotropic damage in brittle materials, Journal of Engineering Mechanics, Volume 128 (2002) no. 8, pp. 889-897 | Article | Zbl 1161.74371

[161] Plastic-damage modeling of saturated quasi-brittle shales, International Journal of Rock Mechanics and Mining Sciences, Volume 93 (2017), pp. 295-306 | Article

[162] Modelling of inherent anisotropy in sedimentary rocks, International Journal of Solids and Structures, Volume 39 (2002) no. 3, pp. 637-648 | Article | Zbl 0991.74515

[163] Formulation of anisotropic failure criteria incorporating a microstructure tensor, Computers and Geotechnics, Volume 26 (2000) no. 2, pp. 105-112 | Article

[164] On failure criteria for anisotropic cohesive-frictional materials, International Journal for Numerical and Analytical Methods in Geomechanics, Volume 25 (2001) no. 5, pp. 509-524 | Article | Zbl 0976.74060

[165] An interacting micro-crack damage model for failure of brittle materials under compression, Journal of the Mechanics and Physics of Solids, Volume 56 (2008) no. 3, pp. 896-923 | Article | Zbl 1152.74037

[166] Compaction creep of simulated anhydrite fault gouge by pressure solution: theory v. experiments and implications for fault sealing, Geological Society, London, Special Publications, Volume 409 (2015) no. 1, pp. 107-124 | Article

[167] Constitutive analysis of shale: a coupled damage plasticity approach, International Journal of Solids and Structures, Volume 75 (2015), pp. 88-98 | Article

[168] Linéarisation et homogénéisation en viscoélasticité, Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics-Physics-Astronomy, Volume 327 (1999) no. 4, pp. 365-370 | Article | Zbl 0931.74016

[169] Micro–macro approach of salt viscous fatigue under cyclic loading, Mechanics of Materials, Volume 93 (2016), pp. 13-31 | Article

[170] Self-consistent micromechanical approach for damage accommodation in rock-like polycrystalline materials, International Journal of Damage Mechanics, Volume 28 (2019) no. 1, pp. 134-161 | Article

[171] A numerical damage model for initially anisotropic materials, International Journal of Solids and Structures, Volume 100 (2016), pp. 245-256

[172] Damage and plastic friction in initially anisotropic quasi brittle materials, International Journal of Plasticity, Volume 82 (2016), pp. 260-282 | Article

[173] Creep in polycrystalline aggregates by matter transport through a liquid phase, Journal of Geophysical Research: Solid Earth, Volume 87 (1982) no. B6, pp. 4731-4739

[174] Distribution of sizes of spheres in a solid from a study of slices of the solid, Journal of Mathematics and Physics, Volume 34 (1955) no. 1-4, pp. 95-102 | Article | MR 70085 | Zbl 0065.11701

[175] Self consistent modelling of elastic-viscoplastic polycrystals, Comptes Rendus de l’Academie des Sciences Serie II, Volume 318 (1994), pp. 145-151 | Zbl 0787.73031

[176] Triaxial behaviour of transversely isotropic materials: Application to sedimentary rocks, International Journal for Numerical and Analytical Methods in Geomechanics, Volume 31 (2007) no. 13, pp. 1517-1535 | Article | Zbl 1196.74138

[177] The kinetics of rock deformation by pressure solution, Philos. Trans. R. Soc. London, Volume 238 (1976), pp. 43-54

[178] Pressure solution in nature, theory and experiment, Journal of the Geological Society, Volume 140 (1983) no. 5, pp. 725-740 | Article

[179] An isotropic self-consistent homogenization scheme for chemo-mechanical healing driven by pressure solution in halite, International Journal of Solids and Structures, Volume 161 (2019), pp. 96-110 | Article

[180] Simulation of salt cavity healing based on a micro-macro model of pressure-solution, Petroleum Geoscience (2019), pp. petgeo2018-094 | Article

[181] Experimental Characterization of Microstructure Development for Calculating Fabric and Stiffness Tensors in Salt Rock, 51st US Rock Mechanics/Geomechanics Symposium (2017)

[182] Mineral weathering and bedrock weakening: Modeling microscale bedrock damage under biotite weathering, Journal of Geophysical Research: Earth Surface, Volume (accepted) (2019)

[183] Densification behaviour of wet granular salt: Theory versus experiment, Seventh Symposium on salt, Volume 1 (1993), pp. 83-92

[184] Quantifying Damage, Saturation and Anisotropy in Cracked Rocks by Inverting Elastic Wave Velocities, Pure and Applied Geophysics, Volume 163 (2006) no. 5, pp. 947-973 | Article

[185] Dynamic and static recrystallization-related microstructures in halite samples from the Kłodawa salt wall (central Poland) as revealed by gamma-irradiation, Neues Jahrbuch für Mineralogie-Abhandlungen: Journal of Mineralogy and Geochemistry, Volume 184 (2007) no. 1, pp. 17-28

[186] Modelling progressive failure in fractured rock masses using a 3D discrete element method, International Journal of Rock Mechanics and Mining Sciences, Volume 52 (2012), pp. 18-30 | Article

[187] Crack growth under biaxial compression, Engineering Fracture Mechanics, Volume 69 (2002) no. 18, pp. 2187-2198 | Article

[188] Mechanisms controlling rupture shape during subcritical growth of joints in layered rocks, Geological Society of America Bulletin, Volume 117 (2005) no. 3-4, pp. 436-449 | Article

[189] Diffusional crack healing in quartz, Journal of Geophysical Research: Solid Earth, Volume 89 (1984) no. B6, pp. 4125-4135 | Article

[190] Mechanical behaviour of rock salt: phenomenology and micromechanisms, International journal of rock mechanics and mining sciences & geomechanics abstracts, Volume 29 (1992) no. 4, pp. 363-378

[191] A self-consistent model for microcrack-weakened solids, Mechanics of Materials, Volume 6 (1987) no. 1, pp. 39-52 | Article

[192] Constitutive theory of plasticity coupled with orthotropic damage for geomaterials, Applied Mathematics and Mechanics, Volume 22 (2001) no. 9, pp. 1028-1034 | Article | Zbl 1081.74548

[193] A microcrack-based continuous damage model for brittle geomaterials, Mechanics of Materials, Volume 32 (2000) no. 10, pp. 607-619 | Article

[194] Intergranular pressure solution in NaCl: Grain-to-grain contact experiments under the optical microscope, Oil & Gas Science and Technology, Volume 54 (1999) no. 6, pp. 729-750

[195] Experimental determination of constitutive parameters governing creep of rocksalt by pressure solution, Geological Society, London, Special Publications, Volume 54 (1990) no. 1, pp. 215-227 | Article | Zbl 1067.74567

[196] A coupled elastoplastic damage model for geomaterials, Computer methods in applied mechanics and engineering, Volume 193 (2004) no. 27-29, pp. 2625-2643 | Article | Zbl 1067.74567

[197] Bifurcation analysis in geomechanics, CRC Press, 2014

[198] An energy-based damage model of geomaterials – I. Deduction of damage evolution laws, International journal of solids and structures, Volume 36 (1999) no. 12, pp. 1735-1755 | Article | Zbl 0973.74009

[199] An energy-based damage model of geomaterials – I. Formulation and numerical results, International journal of solids and structures, Volume 36 (1999) no. 12, pp. 1719-1734 | Article | Zbl 0973.74008

[200] Mechanical properties of shale-gas reservoir rocks – Part 1: Static and dynamic elastic properties and anisotropy, Geophysics, Volume 78 (2013) no. 5, p. D381-D392 | Article

[201] Coupling between anisotropic damage and permeability variation in brittle rocks, International Journal for Numerical and Analytical Methods in Geomechanics, Volume 29 (2005) no. 12, pp. 1231-1247 | Article | Zbl 1140.74509

[202] Dependence of elastic constants of an anisotropic porous material upon porosity and fabric, Journal of materials science, Volume 22 (1987) no. 9, pp. 3178-3184 | Article

[203] Upper limits of power law creep of rocks, Tectonophysics, Volume 136 (1987) no. 1-2, pp. 1-26 | Article

[204] A second gradient elastoplastic cohesive-frictional model for geomaterials, Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics, Volume 329 (2001) no. 10, pp. 735-739 | Article | Zbl 1329.74049

[205] Chord-length distribution function for two-phase random media, Physical Review E, Volume 47 (1993) no. 4, p. 2950

[206] Nearest-neighbor distribution functions in many-body systems, Physical Review A, Volume 41 (1990) no. 4, p. 2059

[207] Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, Volume 17 (1964) no. 2, pp. 85-112 | Article | MR 169425 | Zbl 0131.22001

[208] Can damage mechanics explain temporal scaling laws in brittle fracture and seismicity?, Rock Damage and Fluid Transport, Part I, Springer, 2006, pp. 1031-1045 | Article

[209] A general theory of strength for anisotropic materials, Journal of composite materials, Volume 5 (1971) no. 1, pp. 58-80 | Article

[210] Deformation mechanisms operating in naturally deformed halite rocks as deduced from microstructural investigations, Geologie en Mijnbouw, Volume 66 (1987) no. 2, pp. 165-176

[211] Micromechanical investigations of the hydro-mechanical behaviour of argillite rocks by means of optical full field strain measurement and acoustic emission techniques, 11th ISRM Congress (2007)

[212] A practical guide to rock microstructure, Cambridge university press, 2018 | Article

[213] Mechanics of damage, healing, damageability, and integrity of materials: A conceptual framework, International Journal of Damage Mechanics, Volume 26 (2017) no. 1, pp. 50-103 | Article | Zbl 1159.74314

[214] Multi-scale micromorphic theory for hierarchical materials, Journal of the Mechanics and Physics of Solids, Volume 55 (2007) no. 12, pp. 2603-2651 | Article | MR 2373746 | Zbl 1159.74314

[215] A thermodynamic consistent damage and healing model for self healing materials, International Journal of Plasticity, Volume 27 (2011) no. 7, pp. 1025-1044 | Article | MR 2865165 | Zbl 06032961

[216] A theory of anisotropic healing and damage mechanics of materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 468 (2011) no. 2137, pp. 163-183 | Article | MR 2865165 | Zbl 1243.74125

[217] The effect of time on the weathering of silicate minerals: why do weathering rates differ in the laboratory and field?, Chemical Geology, Volume 202 (2003) no. 3, pp. 479 -506 http://www.sciencedirect.com/science/article/pii/S0009254103002560 (Controls on Chemical Weathering) | Article

[218] Microfracture analysis of fault growth and wear processes, Punchbowl Fault, San Andreas system, California, Journal of Structural Geology, Volume 25 (2003) no. 11, pp. 1855-1873

[219] Bounds and self-consistent estimates for the overall properties of anisotropic composites, Journal of the Mechanics and Physics of Solids, Volume 25 (1977) no. 3, pp. 185-202 | Article | Zbl 0363.73014

[220] Deformation of NaCl single crystals at $0.27\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}m<T<0.44\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}m$, Series on rock and soil mechanics, Volume 20 (1996), pp. 117-128

[221] A discrete viscoplastic damage model for time-dependent behaviour of quasi-brittle rocks, International Journal of Damage Mechanics, Volume 24 (2015) no. 1, pp. 21-40 | Article

[222] Crack healing in glass, Journal of the American Ceramic Society, Volume 53 (1970) no. 9, pp. 486-489 | Article | MR 3189698

[223] Anisotropic damage models for geomaterials: theoretical and numerical challenges, International Journal of Computational Methods, Volume 11 (2014) no. 02, p. 1342007 | MR 3189698 | Zbl 1359.74366

[224] Mechanistic analysis of rock damage anisotropy and rotation around circular cavities, Rock Mechanics and Rock Engineering, Volume 48 (2015) no. 6, pp. 2283-2299 | Article

[225] Fracture properties characterization of shale rocks, Unconventional Resources Technology Conference, San Antonio, Texas, 1-3 August 2016 (2016), pp. 1083-1095

[226] A review of the state of the art in modelling progressive mechanical breakdown and associated fluid flow in intact heterogeneous rocks, International Journal of Rock Mechanics and Mining Sciences, Volume 43 (2006) no. 7, pp. 1001-1022 | Article

[227] Micromechanical identification of anisotropic damage evolution laws, International journal of fracture, Volume 98 (1999) no. 1, pp. 55-76 | Article

[228] A thermo-mechanical damage model for rock stiffness during anisotropic crack opening and closure, Acta Geotechnica, Volume 9 (2014) no. 5, pp. 847-867

[229] A model of damage and healing coupling halite thermo-mechanical behavior to microstructure evolution, Geotechnical and Geological Engineering, Volume 33 (2015) no. 2, pp. 389-410

[230] An alternative model for anisotropic elasticity based on fabric tensors, Mechanics of Materials, Volume 21 (1995) no. 4, pp. 243-250 | Article

[231] A unified elastic–plastic and viscoplastic damage model for quasi-brittle rocks, International Journal of Rock Mechanics and Mining Sciences, Volume 45 (2008) no. 8, pp. 1237-1251 | Article

[232] Micromechanical analysis of coupling between anisotropic damage and friction in quasi brittle materials: role of the homogenization scheme, International Journal of Solids and Structures, Volume 45 (2008) no. 5, pp. 1385-1405

[233] Homogenization-based analysis of anisotropic damage in brittle materials with unilateral effect and interactions between microcracks, International Journal for Numerical and Analytical Methods in Geomechanics, Volume 33 (2009) no. 6, pp. 749-772 | Zbl 1273.74419

[234] Prediction of viscous cracking and cyclic fatigue of salt polycrystals using a joint-enriched finite element model, Mechanics of Materials, Volume 103 (2016), pp. 28-43

[235] A refined micromechanical damage–friction model with strength prediction for rock-like materials under compression, International Journal of Solids and Structures, Volume 60 (2015), pp. 75-83

[236] Modeling of viscoplastic deformation in geomaterials with a polycrystalline approach, International Journal of Rock Mechanics and Mining Sciences, Volume 72 (2014), pp. 182-190 | Article

[237] A micromechanical model for the elastic–plastic behavior of porous rocks, Computers and Geotechnics, Volume 70 (2015), pp. 130-137 | Article

[238] Shear banding analysis of geomaterials by strain gradient enhanced damage model, International journal of solids and structures, Volume 42 (2005) no. 20, pp. 5335-5355 | Article | Zbl 1119.74585

[239] A coupled elastoplastic damage model for brittle rocks and its application in modelling underground excavation, International Journal of Rock Mechanics and Mining Sciences, Volume 84 (2016), pp. 130-141 | Article

[240] Simulation of localization failure with strain-gradient-enhanced damage mechanics, International journal for numerical and analytical methods in geomechanics, Volume 26 (2002) no. 8, pp. 793-813 | Article | Zbl 1076.74050