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Thermalization of rate-independent processes by entropic regularization
Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization
1. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany |
References:
[1] |
L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives, Proceedings of the American Mathematical Society, 108 (1990), 691-702.
doi: 10.1090/S0002-9939-1990-0969514-3. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, 2005. |
[3] |
G. Alberti, Variational models for phase transitions, an approach via gamma-convergence, 1998, in "Differential Equations and Calculus of Variations" (eds. G. Buttazzo et al.), Springer-Verlag, 2000. |
[4] |
B. Bourdin, G. Francfort and J. J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148.
doi: 10.1007/s10659-007-9107-3. |
[5] |
G. Francfort and A. Garroni, A variational view of partial brittle damage evolution, Arch. Rational Mech. Anal., 182 (2006), 125-152.
doi: 10.1007/s00205-006-0426-5. |
[6] |
A. Fiaschi, D. Knees and U. Stefanelli, Young-measure quasi-static damage evolution, Arch. Ration. Mech. Anal., 203 (2012), 415-453. |
[7] |
G. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium, Eur. J. Mech., A/Solids, 12 (1993), 149-189. |
[8] |
G. Francfort and A. Mielke, Existence results for a class of rate-independent material models, with nonconvex elastic energies, ().
doi: 10.1515/CRELLE.2006.044. |
[9] |
M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Internat. J. Solids Structures, 33 (1996), 1083-1103.
doi: 10.1016/0020-7683(95)00074-7. |
[10] |
A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fracture, Calc. Var. Partial Differ. Equ., 22 (2005), 129-172. |
[11] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhäuser, Boston, 1984. |
[12] |
A. Garroni and C. Larsen, Threshold-based quasi-static brittle damage evolution, Arch. Ration. Mech. Anal., 194 (2009), 585-609.
doi: 10.1007/s00205-008-0174-9. |
[13] |
K. Hackl and H. Stumpf, Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-static brittle fracture, Int. J. Solids Structures, 30 (2003), 1567-1584. |
[14] |
A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2 (2005), 461-559. |
[15] |
A. Mielke, Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDE's and applications, 87-170, Lecture Notes in Math., 2028, Springer, Heidelberg, 2011. |
[16] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. U. Mat. Ital. B, 14 (1977), 285-299. |
[17] |
A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. PDEs, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[18] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[19] |
A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, M$^3$AS Math. Models Methods Appl. Sci., 16 (2006), 177-209.
doi: 10.1142/S021820250600111X. |
[20] |
A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differ. Equ., 31 (2008), 387-416. |
[21] |
A. Mielke, T. Roubíček and M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, J. Elasticity, 109 (2012), 235-273. |
[22] |
M. Thomas, "Rate-independent Damage Processes in Nonlinearly Elastic Materials," PhD thesis, Humboldt-Universität zu Berlin, 2010. |
[23] |
M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain: existence and regularity results, Zeit. angew. Math. Mech., 90 (2010), 88-112. |
show all references
References:
[1] |
L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives, Proceedings of the American Mathematical Society, 108 (1990), 691-702.
doi: 10.1090/S0002-9939-1990-0969514-3. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, 2005. |
[3] |
G. Alberti, Variational models for phase transitions, an approach via gamma-convergence, 1998, in "Differential Equations and Calculus of Variations" (eds. G. Buttazzo et al.), Springer-Verlag, 2000. |
[4] |
B. Bourdin, G. Francfort and J. J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148.
doi: 10.1007/s10659-007-9107-3. |
[5] |
G. Francfort and A. Garroni, A variational view of partial brittle damage evolution, Arch. Rational Mech. Anal., 182 (2006), 125-152.
doi: 10.1007/s00205-006-0426-5. |
[6] |
A. Fiaschi, D. Knees and U. Stefanelli, Young-measure quasi-static damage evolution, Arch. Ration. Mech. Anal., 203 (2012), 415-453. |
[7] |
G. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium, Eur. J. Mech., A/Solids, 12 (1993), 149-189. |
[8] |
G. Francfort and A. Mielke, Existence results for a class of rate-independent material models, with nonconvex elastic energies, ().
doi: 10.1515/CRELLE.2006.044. |
[9] |
M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Internat. J. Solids Structures, 33 (1996), 1083-1103.
doi: 10.1016/0020-7683(95)00074-7. |
[10] |
A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fracture, Calc. Var. Partial Differ. Equ., 22 (2005), 129-172. |
[11] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhäuser, Boston, 1984. |
[12] |
A. Garroni and C. Larsen, Threshold-based quasi-static brittle damage evolution, Arch. Ration. Mech. Anal., 194 (2009), 585-609.
doi: 10.1007/s00205-008-0174-9. |
[13] |
K. Hackl and H. Stumpf, Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-static brittle fracture, Int. J. Solids Structures, 30 (2003), 1567-1584. |
[14] |
A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2 (2005), 461-559. |
[15] |
A. Mielke, Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDE's and applications, 87-170, Lecture Notes in Math., 2028, Springer, Heidelberg, 2011. |
[16] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. U. Mat. Ital. B, 14 (1977), 285-299. |
[17] |
A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. PDEs, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[18] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[19] |
A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, M$^3$AS Math. Models Methods Appl. Sci., 16 (2006), 177-209.
doi: 10.1142/S021820250600111X. |
[20] |
A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differ. Equ., 31 (2008), 387-416. |
[21] |
A. Mielke, T. Roubíček and M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, J. Elasticity, 109 (2012), 235-273. |
[22] |
M. Thomas, "Rate-independent Damage Processes in Nonlinearly Elastic Materials," PhD thesis, Humboldt-Universität zu Berlin, 2010. |
[23] |
M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain: existence and regularity results, Zeit. angew. Math. Mech., 90 (2010), 88-112. |
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