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A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class
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 |
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.
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. |
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