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August  2017, 10(4): 867-893. doi: 10.3934/dcdss.2017044

## An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat

 Institute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, CZ-182 00 Praha 8, Czech Republic

Received  April 2016 Revised  November 2016 Published  April 2017

Fund Project: This research has been partially supported from the grants 16-03823S "Homogenization and multi-scale computational modelling of flow and nonlinear interactions in porous smart structures" and 14-15264S "Experimentally justified multiscale modelling of shape memory alloys" of Czech Science Foundation, and from the institutional support RVO:61388998 (ČR)

The model of brittle cracks in elastic solids at small strains is approximated by the Ambrosio-Tortorelli functional and then extended into evolution situation to an evolutionary system, involving viscoelasticity, inertia, heat transfer, and coupling with Cahn-Hilliard-type diffusion of a fluid due to Fick's or Darcy's laws. Damage resulting from the approximated crack model is considered rate independent. The fractional-step Crank-Nicolson-type time discretisation is devised to decouple the system in a way so that the energy is conserved even in the discrete scheme. The numerical stability of such a scheme is shown, and also convergence towards suitably defined weak solutions. Various generalizations involving plasticity, healing in damage, or phase transformation are mentioned, too.

Citation: Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
##### References:
 [1] L. Ambrosio and V. M. Tortorelli, Approximation of functional depending on jumps via by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math., 43 (1990), 999-1036. [2] L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Bollettino Unione Mat. Italiana, 7 (1992), 105-123. [3] M. Artina, M. Fornasier, S. Micheletti and S. Perotto, Anisotropic mesh adaptation for crack detection in brittle materials, SIAM J. Sci. Comput., 37 (2015), B633-B659. doi: 10.1137/140970495. [4] F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. Plasticity J., 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012. [5] M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164. doi: 10.1063/1.1712886. [6] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. [7] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655. [8] E. Bonetti, C. Heinemann, C. Kraus and A. Segatti, Modeling and analysis of a phase field system for damage and phase separation processes in solids, J. Diff. Equations, 258 (2015), 3928-3959. [9] B. Bourdin, G. A. Francfort and J. -J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, 48 (2000), 797-826. doi: 10.1016/S0022-5096(99)00028-9. [10] B. Bourdin, G. A. Francfort and J. -J. Marigo, The variational approach to fracture, Elasticity J., 91 (2008), 5-148. doi: 10.1007/978-1-4020-6395-4. [11] B. Bourdin, C. J. Larsen and C. L. Richardson, A time-discrete model for dynamic fracture based on crack regularization, Int. J. of Fracture, 168 (2011), 133-143. doi: 10.1007/s10704-010-9562-x. [12] S. Burke, C. Ortner and E. Süli, An adaptive finite element approximation of a generalised Ambrosio-Tortorelli functional, Math. Meth. Models Appl. Sci., 23 (2013), 1663-1697. [13] J. W. Cahn and J. E. Hilliard, Free energy of a uniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [14] J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Camb. Phil. Soc., 43 (1947), 50-67. [15] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. doi: 10.1137/S0036141094267662. [16] I. Faragó, Splitting methods and their application to the abstract Cauchy problems, In Numerical Analysis and Its Application, Lect. Notes Comp. Sci. 3401, pages 35-45. Springer, Berlin, 2005. [17] E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Diff. Equations Nonlin. Mech. , pages 14pp. (electronic), Art. ID 90616,2006. [18] M. Frost, B. Benešová and P. Sedlák, A microscopically motivated constitutive model for shape memory alloys: formulation, analysis and computations, Math. Mech. of Solids, 2014. doi: 10.1177/1081286514522474. [19] A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Diff. Eqs., 22 (2005), 129-172. doi: 10.1007/s00526-004-0269-6. [20] R. Glowinski, J. -L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. (French original Dunod, Paris, 1976). [21] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [22] Y. Hamiel, V. Lyakhovsky and A. Agnon, Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks, Geophys. J. Int., 156 (2004), 701-713. doi: 10.1111/j.1365-246X.2004.02172.x. [23] C. Heinemann and C. Kraus, Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., 21 (2011), 321-359. [24] C. Heinemann and C. Kraus, Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes, Mathematica Bohemica, 139 (2014), 315-331. [25] C. Heinemann and C. Kraus, Phase Separation Coupled with Damage Processes -Analysis of Phase Field Models in Elastic Media, Springer Fachmedien, Wiesbaden, 2014. [26] C. Heinemann and C. Kraus, Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems -Series A, 35 (2015), 2565-2590. doi: 10.3934/dcds.2015.35.2565. [27] C. Heinemann, C. Kraus, E. Rocca and R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Archive for Rational Mechanics and Analysis, arXiv: 1510.03755v1, 2017. doi: 10.1007/s00205-017-1102-7. [28] H. M. Hilber, T. J. R. Hughes and R. L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Eng. Struct. Dyn., 5 (1977), 283-292. doi: 10.1002/eqe.4290050306. [29] L. Jakabčin, A visco-elasto-plastic evolution model with regularized fracture, ESAIM Control, Optim., Calc. Var., 22 (2016), 148-168. [30] J. Kruis, T. Koudelka and T. Krejčí, Multi-physics analyses of selected civil engineering concrete structure, Commun. Comput. Phys., 12 (2012), 885-918. doi: 10.4208/cicp.031110.080711s. [31] C. J. Larsen, Models for dynamic fracture based on griffith's criterion, In K. Hackl, editor, IUTAM Symp. on Variational Concepts with Appl. to the Mech. of Mater. , pages 131-140. Springer, 2010. doi: 10.1007/978-90-481-9195-6_10. [32] C. J. Larsen, C. Ortner and E. Süli, Existence of solution to a regularized model of dynamic fracture, Math. Models Meth. Appl. Sci., 20 (2010), 1021-1048. doi: 10.1142/S0218202510004520. [33] V. Lyakhovsky and Y. Hamiel, Damage evolution and fluid flow in poroelastic rock, Izvestiya, Physics of the Solid Earth, 43 (2007), 13-23. doi: 10.1134/S106935130701003X. [34] V. Lyakhovsky, Y. Hamiel and Y. Ben-Zion, A non-local visco-elastic damage model and dynamic fracturing, J. Mech. Phys. Solids, 59 (2011), 1752-1776. doi: 10.1016/j.jmps.2011.05.016. [35] G. Marchuk, plitting and alternating direction methods, Handbook of Numerical Analysis, 1 (1990), 197-462. doi: 10.1016/S1570-8659(05)80035-3. [36] A. Mielke, Evolution in rate-independent systems (Ch. 6), In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Evolutionary Equations, , vol. 2, pages 461-559. Elsevier B. V. , Amsterdam, 2005. [37] A. Mielke and T. Roubíček, Rate-Independent Systems -Theory and Application, Springer, New York, 2015. [38] A. Mielke, T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Part. Diff. Eqns., 31 (2008), 387-416. [39] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. [40] A. Miranville, A model of Cahn-Hilliard equation based on a microforce balance, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1247-1252. doi: 10.1016/S0764-4442(99)80448-0. [41] L. Modica and S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici, Boll. Unione Mat. Italiana A, 14 (1977), 526-529. [42] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [43] A. Novick-Cohen, The Cahn-Hilliard equation, In C. M. Dafermos and E. Feireisl, editors, Handbook of Differential Equations -Evolutionary Equations, chapter 4, pages 201-228. Elsevier, 2008. [44] R. Rossi, On two classes of generalized viscous Cahn-Hilliard equations, Comm. Pure Appl. Anal., 4 (2005), 405-430. doi: 10.3934/cpaa.2005.4.405. [45] T. Roubíček, Rate independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862. [46] T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297. [47] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013. [48] T. Roubíček, Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM J. Math. Anal., 45 (2013), 101-126. [49] T. Roubíček and C. G. Panagiotopoulos, Energy-conserving time-discretisation of abstract dynamical problems with applications in continuum mechanics of solids, Numer. Funct. Anal. Optim. , arXiv: 1605.09762, 2016. [50] T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discr. Cont. Dyn. Syst. B, 14 (2014), 2313-2333. [51] T. Roubíček and G. Tomassetti, Thermomechanics of damageable materials under diffusion: Modeling and analysis, Zeit. angew. Math. Phys., 66 (2015), 3535-3572. [52] T. Roubíček and J. Valdman, Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation, SIAM J. Appl. Math., 75 (2016), 314-340. [53] A. Sadjadpour and K. Bhattacharya, A micromechanics inspired constitutive model for shapememory alloys, Smart Mater. Structures, 16 (2007), 1751-1765. doi: 10.1088/0964-1726/16/5/030. [54] C. H. Scholz, The Mechanics of Earthquakes and Faulting, Cambridge Univ. Press, Cambridge, 2nd edition, 2002. doi: 10.1017/CBO9780511818516. [55] P. Sedlák, M. Frost, B. Benešová, T. B. Zineb and P. Šittner, Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings, Intl. Plasticity J., 39 (2012), 132-151. [56] J. C. Simo and J. R. Hughes, Computational Inelasticity, Springer, Berlin, 1998. [57] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre ô coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. [58] L. Wang, Foundations of Stress Waves, Elsevier, Amsterdam, 2007. [59] N. N. Yanenko, The Method of Fractional Steps, Springer, Berlin, 1971. doi: 10.1007/978-3-642-65108-3.

show all references

##### References:
 [1] L. Ambrosio and V. M. Tortorelli, Approximation of functional depending on jumps via by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math., 43 (1990), 999-1036. [2] L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Bollettino Unione Mat. Italiana, 7 (1992), 105-123. [3] M. Artina, M. Fornasier, S. Micheletti and S. Perotto, Anisotropic mesh adaptation for crack detection in brittle materials, SIAM J. Sci. Comput., 37 (2015), B633-B659. doi: 10.1137/140970495. [4] F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. Plasticity J., 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012. [5] M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164. doi: 10.1063/1.1712886. [6] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. [7] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655. [8] E. Bonetti, C. Heinemann, C. Kraus and A. Segatti, Modeling and analysis of a phase field system for damage and phase separation processes in solids, J. Diff. Equations, 258 (2015), 3928-3959. [9] B. Bourdin, G. A. Francfort and J. -J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, 48 (2000), 797-826. doi: 10.1016/S0022-5096(99)00028-9. [10] B. Bourdin, G. A. Francfort and J. -J. Marigo, The variational approach to fracture, Elasticity J., 91 (2008), 5-148. doi: 10.1007/978-1-4020-6395-4. [11] B. Bourdin, C. J. Larsen and C. L. Richardson, A time-discrete model for dynamic fracture based on crack regularization, Int. J. of Fracture, 168 (2011), 133-143. doi: 10.1007/s10704-010-9562-x. [12] S. Burke, C. Ortner and E. Süli, An adaptive finite element approximation of a generalised Ambrosio-Tortorelli functional, Math. Meth. Models Appl. Sci., 23 (2013), 1663-1697. [13] J. W. Cahn and J. E. Hilliard, Free energy of a uniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [14] J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Camb. Phil. Soc., 43 (1947), 50-67. [15] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. doi: 10.1137/S0036141094267662. [16] I. Faragó, Splitting methods and their application to the abstract Cauchy problems, In Numerical Analysis and Its Application, Lect. Notes Comp. Sci. 3401, pages 35-45. Springer, Berlin, 2005. [17] E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Diff. Equations Nonlin. Mech. , pages 14pp. (electronic), Art. ID 90616,2006. [18] M. Frost, B. Benešová and P. Sedlák, A microscopically motivated constitutive model for shape memory alloys: formulation, analysis and computations, Math. Mech. of Solids, 2014. doi: 10.1177/1081286514522474. [19] A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Diff. Eqs., 22 (2005), 129-172. doi: 10.1007/s00526-004-0269-6. [20] R. Glowinski, J. -L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. (French original Dunod, Paris, 1976). [21] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [22] Y. Hamiel, V. Lyakhovsky and A. Agnon, Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks, Geophys. J. Int., 156 (2004), 701-713. doi: 10.1111/j.1365-246X.2004.02172.x. [23] C. Heinemann and C. Kraus, Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., 21 (2011), 321-359. [24] C. Heinemann and C. Kraus, Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes, Mathematica Bohemica, 139 (2014), 315-331. [25] C. Heinemann and C. Kraus, Phase Separation Coupled with Damage Processes -Analysis of Phase Field Models in Elastic Media, Springer Fachmedien, Wiesbaden, 2014. [26] C. Heinemann and C. Kraus, Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems -Series A, 35 (2015), 2565-2590. doi: 10.3934/dcds.2015.35.2565. [27] C. Heinemann, C. Kraus, E. Rocca and R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Archive for Rational Mechanics and Analysis, arXiv: 1510.03755v1, 2017. doi: 10.1007/s00205-017-1102-7. [28] H. M. Hilber, T. J. R. Hughes and R. L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Eng. Struct. Dyn., 5 (1977), 283-292. doi: 10.1002/eqe.4290050306. [29] L. Jakabčin, A visco-elasto-plastic evolution model with regularized fracture, ESAIM Control, Optim., Calc. Var., 22 (2016), 148-168. [30] J. Kruis, T. Koudelka and T. Krejčí, Multi-physics analyses of selected civil engineering concrete structure, Commun. Comput. Phys., 12 (2012), 885-918. doi: 10.4208/cicp.031110.080711s. [31] C. J. Larsen, Models for dynamic fracture based on griffith's criterion, In K. Hackl, editor, IUTAM Symp. on Variational Concepts with Appl. to the Mech. of Mater. , pages 131-140. Springer, 2010. doi: 10.1007/978-90-481-9195-6_10. [32] C. J. Larsen, C. Ortner and E. Süli, Existence of solution to a regularized model of dynamic fracture, Math. Models Meth. Appl. Sci., 20 (2010), 1021-1048. doi: 10.1142/S0218202510004520. [33] V. Lyakhovsky and Y. Hamiel, Damage evolution and fluid flow in poroelastic rock, Izvestiya, Physics of the Solid Earth, 43 (2007), 13-23. doi: 10.1134/S106935130701003X. [34] V. Lyakhovsky, Y. Hamiel and Y. Ben-Zion, A non-local visco-elastic damage model and dynamic fracturing, J. Mech. Phys. Solids, 59 (2011), 1752-1776. doi: 10.1016/j.jmps.2011.05.016. [35] G. Marchuk, plitting and alternating direction methods, Handbook of Numerical Analysis, 1 (1990), 197-462. doi: 10.1016/S1570-8659(05)80035-3. [36] A. Mielke, Evolution in rate-independent systems (Ch. 6), In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Evolutionary Equations, , vol. 2, pages 461-559. Elsevier B. V. , Amsterdam, 2005. [37] A. Mielke and T. Roubíček, Rate-Independent Systems -Theory and Application, Springer, New York, 2015. [38] A. Mielke, T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Part. Diff. Eqns., 31 (2008), 387-416. [39] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. [40] A. Miranville, A model of Cahn-Hilliard equation based on a microforce balance, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1247-1252. doi: 10.1016/S0764-4442(99)80448-0. [41] L. Modica and S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici, Boll. Unione Mat. Italiana A, 14 (1977), 526-529. [42] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [43] A. Novick-Cohen, The Cahn-Hilliard equation, In C. M. Dafermos and E. Feireisl, editors, Handbook of Differential Equations -Evolutionary Equations, chapter 4, pages 201-228. Elsevier, 2008. [44] R. Rossi, On two classes of generalized viscous Cahn-Hilliard equations, Comm. Pure Appl. Anal., 4 (2005), 405-430. doi: 10.3934/cpaa.2005.4.405. [45] T. Roubíček, Rate independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862. [46] T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297. [47] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013. [48] T. Roubíček, Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM J. Math. Anal., 45 (2013), 101-126. [49] T. Roubíček and C. G. Panagiotopoulos, Energy-conserving time-discretisation of abstract dynamical problems with applications in continuum mechanics of solids, Numer. Funct. Anal. Optim. , arXiv: 1605.09762, 2016. [50] T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discr. Cont. Dyn. Syst. B, 14 (2014), 2313-2333. [51] T. Roubíček and G. Tomassetti, Thermomechanics of damageable materials under diffusion: Modeling and analysis, Zeit. angew. Math. Phys., 66 (2015), 3535-3572. [52] T. Roubíček and J. Valdman, Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation, SIAM J. Appl. Math., 75 (2016), 314-340. [53] A. Sadjadpour and K. Bhattacharya, A micromechanics inspired constitutive model for shapememory alloys, Smart Mater. Structures, 16 (2007), 1751-1765. doi: 10.1088/0964-1726/16/5/030. [54] C. H. Scholz, The Mechanics of Earthquakes and Faulting, Cambridge Univ. Press, Cambridge, 2nd edition, 2002. doi: 10.1017/CBO9780511818516. [55] P. Sedlák, M. Frost, B. Benešová, T. B. Zineb and P. Šittner, Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings, Intl. Plasticity J., 39 (2012), 132-151. [56] J. C. Simo and J. R. Hughes, Computational Inelasticity, Springer, Berlin, 1998. [57] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre ô coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. [58] L. Wang, Foundations of Stress Waves, Elsevier, Amsterdam, 2007. [59] N. N. Yanenko, The Method of Fractional Steps, Springer, Berlin, 1971. doi: 10.1007/978-3-642-65108-3.
Summary of the basic notation used through this paper
 $u$ displacements $e(u)=\frac12(\nabla u)^\top+\frac12\nabla u$ small strain tensor $v$ velocity $M$ Biot modulus $z$ damage scalar variable $\beta$ Biot coefficient $c$ concentration $\kappa$ coefficient for the ratio Fick/Darcy flow $\theta$ temperature $\varkappa$ capillarity coefficient $\vartheta$ heat content $a$ energy released per unit volume by damage $\sigma$ stress $\psi=\varphi+\phi$ free energy $\mu$ chemical potential $\varphi, \phi$ chemo-mechanical and thermal energies $\mathbb{C}$ elastic-moduli tensor $\mathfrak{u}$ internal energy $\mathbb{D}$ viscous-moduli tensor $c_{_{\rm E}}$ equilibrium concentration $\mathbb{M}$ the mobility matrix $g$ bulk force (gravity) $\mathbb{K}$ the heat-conductivity matrix $f$ traction force $c_{\rm v}$ heat capacity $h_{_{\rm{B}}}$ prescribed boundary heat flux $\varrho$ mass density $j_{_{\rm{B}}}$ prescribed boundary diffusant flux $r$ heat-production rate $\varepsilon >0$ a fixed regularization parameter $\mathfrak{s}$ entropy $\tau>0$ a time step for discretisation
 $u$ displacements $e(u)=\frac12(\nabla u)^\top+\frac12\nabla u$ small strain tensor $v$ velocity $M$ Biot modulus $z$ damage scalar variable $\beta$ Biot coefficient $c$ concentration $\kappa$ coefficient for the ratio Fick/Darcy flow $\theta$ temperature $\varkappa$ capillarity coefficient $\vartheta$ heat content $a$ energy released per unit volume by damage $\sigma$ stress $\psi=\varphi+\phi$ free energy $\mu$ chemical potential $\varphi, \phi$ chemo-mechanical and thermal energies $\mathbb{C}$ elastic-moduli tensor $\mathfrak{u}$ internal energy $\mathbb{D}$ viscous-moduli tensor $c_{_{\rm E}}$ equilibrium concentration $\mathbb{M}$ the mobility matrix $g$ bulk force (gravity) $\mathbb{K}$ the heat-conductivity matrix $f$ traction force $c_{\rm v}$ heat capacity $h_{_{\rm{B}}}$ prescribed boundary heat flux $\varrho$ mass density $j_{_{\rm{B}}}$ prescribed boundary diffusant flux $r$ heat-production rate $\varepsilon >0$ a fixed regularization parameter $\mathfrak{s}$ entropy $\tau>0$ a time step for discretisation
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