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

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

1. 

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

Received  May 2011 Revised  July 2011 Published  October 2012

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.
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.

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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