Quasistatic damage evolution with spatial \mathrm{BV}-regularization

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February 2013, 6(1): 235-255. doi: 10.3934/dcdss.2013.6.235

Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization

Marita Thomas ^1,

1.	Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Received May 2011 Revised July 2011 Published October 2012

Abstract
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An existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [23] an existence result in the small strain setting was obtained under the assumption that the damage variable $z$ satisfies $z\in W^{1,r}(\Omega)$ with $r\in(1,\infty)$ for $\Omega⊂ \mathbb{R}^d.$ We now cover the case $r=1$. The lack of compactness in $W^{1,1}(\Omega)$ requires to do the analysis in $\mathrm{BV}(\Omega)$. This setting allows it to consider damage variables with values in {0,1}. We show that such a brittle damage model is obtained as the $\Gamma$-limit of functionals of Modica-Mortola type.

Keywords: Partial damage, damage evolution with spatial regularization, energetic formulation, $\Gamma$-convergence of rate-independent systems., functions of bounded variation.

Mathematics Subject Classification: Primary: 74C05, 74R05, 49J45; Secondary: 49S05, 74R2.

Citation: Marita Thomas. Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 235-255. doi: 10.3934/dcdss.2013.6.235

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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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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