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June  2015, 35(6): 2565-2590. doi: 10.3934/dcds.2015.35.2565

Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

Christian Heinemann 1, and  Christiane Kraus 1,

1. 

Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany, Germany

Received  December 2013 Revised  February 2014 Published  December 2014

In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [12]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.
Keywords: weak solutions, phase separation, hyperbolic-parabolic systems, Cahn-Hilliard system, rate-dependent systems., linear elasticity, doubly nonlinear differential inclusions, existence results, energetic solutions.
Mathematics Subject Classification: 35L20, 35L51, 35K86, 35K55, 49J40, 49S05, 74A45, 74G25, 34A12, 82B26, 82C26, 35K92, 35K3.
Citation: Christian Heinemann, Christiane Kraus. Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2565-2590. doi: 10.3934/dcds.2015.35.2565
References:
[1]

L. Bartkowiak and I. Pawlow, The Cahn-Hilliard-Gurtin system coupled with elasticity, Control Cybernet., 34 (2005), 1005-1043.  Google Scholar

[2]

J. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math., Ser. B, 32 (2011), 823-846. doi: 10.1007/s11401-011-0682-z.  Google Scholar

[3]

T. Böhme, W. Dreyer, F. Duderstadt and W. Müller, A Higher Gradient Theory of Mixtures for Multi-component Materials with Numerical Examples for Binary Alloys, WIAS-Preprint No. 1286, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2007. Google Scholar

[4]

T. Böhme, W. Dreyer and W. Müller, Determination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys, Contin. Mech. Thermodyn., 18 (2007), 411-441. doi: 10.1007/s00161-006-0037-2.  Google Scholar

[5]

E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65. doi: 10.1016/S0167-2789(02)00373-1.  Google Scholar

[6]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Contin. Mech. Thermodyn., 16 (2004), 319-335. doi: 10.1007/s00161-003-0152-2.  Google Scholar

[7]

E. Bonetti, G. Schimperna and A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials, J. Differential Equations, 218 (2005), 91-116. doi: 10.1016/j.jde.2005.04.015.  Google Scholar

[8]

M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic media, Adv. Math. Sci. Appl., 10 (2000), 539-569.  Google Scholar

[9]

M. Frémond, Non-smooth Thermomechanics, Berlin: Springer, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[10]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Int. J. Solids Structures, 33 (1996), 1083-1103. doi: 10.1016/0020-7683(95)00074-7.  Google Scholar

[11]

H. Garcke, On Mathematical Models for Phase Separation in Elastically Stressed Solids, Habilitation thesis, University Bonn, 2000. Google Scholar

[12]

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.  Google Scholar

[13]

C. Heinemann and C. Kraus, A degenerating Cahn-Hilliard system coupled with complete damage processes, WIAS preprint no. 1759. Google Scholar

[14]

C. Heinemann and C. Kraus, Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains, WIAS preprint no. 1890. Google Scholar

[15]

C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math., 24 (2013), 179-211. doi: 10.1017/S095679251200037X.  Google Scholar

[16]

D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616. doi: 10.1142/S021820251250056X.  Google Scholar

[17]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 177-209. doi: 10.1142/S021820250600111X.  Google Scholar

[18]

A. Mielke and M. Thomas, Damage of nonlinearly elastic materials at small strain - Existence and regularity results, ZAMM Z. Angew. Math. Mech, 90 (2010), 88-112. doi: 10.1002/zamm.200900243.  Google Scholar

[19]

I. Pawlow and W. M. Zajączkowski, Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids, Colloquium Mathematicum, 112 (2008), 313-334. doi: 10.4064/cm112-2-7.  Google Scholar

[20]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. arXiv:1205.3578v1. doi: 10.1142/S021820251450002X.  Google Scholar

[21]

G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion, J. Differential Equations, 254 (2013), 779-803. doi: 10.1016/j.jde.2012.09.018.  Google Scholar

[22]

G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63. doi: 10.1137/110835608.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96. doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

L. Bartkowiak and I. Pawlow, The Cahn-Hilliard-Gurtin system coupled with elasticity, Control Cybernet., 34 (2005), 1005-1043.  Google Scholar

[2]

J. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math., Ser. B, 32 (2011), 823-846. doi: 10.1007/s11401-011-0682-z.  Google Scholar

[3]

T. Böhme, W. Dreyer, F. Duderstadt and W. Müller, A Higher Gradient Theory of Mixtures for Multi-component Materials with Numerical Examples for Binary Alloys, WIAS-Preprint No. 1286, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2007. Google Scholar

[4]

T. Böhme, W. Dreyer and W. Müller, Determination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys, Contin. Mech. Thermodyn., 18 (2007), 411-441. doi: 10.1007/s00161-006-0037-2.  Google Scholar

[5]

E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65. doi: 10.1016/S0167-2789(02)00373-1.  Google Scholar

[6]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Contin. Mech. Thermodyn., 16 (2004), 319-335. doi: 10.1007/s00161-003-0152-2.  Google Scholar

[7]

E. Bonetti, G. Schimperna and A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials, J. Differential Equations, 218 (2005), 91-116. doi: 10.1016/j.jde.2005.04.015.  Google Scholar

[8]

M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic media, Adv. Math. Sci. Appl., 10 (2000), 539-569.  Google Scholar

[9]

M. Frémond, Non-smooth Thermomechanics, Berlin: Springer, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[10]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Int. J. Solids Structures, 33 (1996), 1083-1103. doi: 10.1016/0020-7683(95)00074-7.  Google Scholar

[11]

H. Garcke, On Mathematical Models for Phase Separation in Elastically Stressed Solids, Habilitation thesis, University Bonn, 2000. Google Scholar

[12]

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.  Google Scholar

[13]

C. Heinemann and C. Kraus, A degenerating Cahn-Hilliard system coupled with complete damage processes, WIAS preprint no. 1759. Google Scholar

[14]

C. Heinemann and C. Kraus, Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains, WIAS preprint no. 1890. Google Scholar

[15]

C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math., 24 (2013), 179-211. doi: 10.1017/S095679251200037X.  Google Scholar

[16]

D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616. doi: 10.1142/S021820251250056X.  Google Scholar

[17]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 177-209. doi: 10.1142/S021820250600111X.  Google Scholar

[18]

A. Mielke and M. Thomas, Damage of nonlinearly elastic materials at small strain - Existence and regularity results, ZAMM Z. Angew. Math. Mech, 90 (2010), 88-112. doi: 10.1002/zamm.200900243.  Google Scholar

[19]

I. Pawlow and W. M. Zajączkowski, Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids, Colloquium Mathematicum, 112 (2008), 313-334. doi: 10.4064/cm112-2-7.  Google Scholar

[20]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. arXiv:1205.3578v1. doi: 10.1142/S021820251450002X.  Google Scholar

[21]

G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion, J. Differential Equations, 254 (2013), 779-803. doi: 10.1016/j.jde.2012.09.018.  Google Scholar

[22]

G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63. doi: 10.1137/110835608.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96. doi: 10.1007/BF01762360.  Google Scholar

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