Article
MSC:
34A12,
35A01,
35D30,
35J50,
35K55,
35K65,
35K85,
35K92,
35M30,
49S05,
74A45,
74G25,
82B26 | MR 3238842 | Zbl 06362261 | DOI: 10.21136/MB.2014.143857
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Cahn-Hilliard system; phase separation; complete damage; elliptic-parabolic degenerating system; linear elasticity; energetic solution; weak solution; doubly nonlinear differential inclusion; existence result; rate-dependent system
Cahn-Hilliard system; phase separation; complete damage; elliptic-parabolic degenerating system; linear elasticity; energetic solution; weak solution; doubly nonlinear differential inclusion; existence result; rate-dependent system
Summary:
This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field. The analysis is performed on a time-dependent domain which characterizes the nondegenerated elastic material regions. We choose a notion of weak solutions which consists of weak formulations of the Cahn-Hilliard system and the momentum balance equation, a variational inequality for the damage evolution and an energy inequality. For the introduced degenerating system, we prove global-in-time existence of weak solutions. The main results are sketched from our recent paper [WIAS preprint no. 1759 (2012)].
References:
[1] Bartkowiak, L., Pawłow, I.: The Cahn-Hilliard-Gurtin system coupled with elasticity. Control Cybern. 34 1005-1043 (2005). MR 2256463 | Zbl 1119.35099
[2] Bonetti, E., Colli, P., Dreyer, W., Gilardi, G., Schimperna, G., Sprekels, J.: On a model for phase separation in binary alloys driven by mechanical effects. Physica D 165 48-65 (2002). DOI 10.1016/S0167-2789(02)00373-1 | MR 1910617 | Zbl 1008.74066
[3] Bonetti, E., Schimperna, G., Segatti, A.: On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Differ. Equations 218 91-116 (2005). DOI 10.1016/j.jde.2005.04.015 | MR 2174968 | Zbl 1078.74048
[4] Bouchitté, G., Mielke, A., Roubíček, T.: A complete-damage problem at small strains. Z. Angew. Math. Phys. 60 205-236 (2009). DOI 10.1007/s00033-007-7064-0 | MR 2486153 | Zbl 1238.74005
[5] Carrive, M., Miranville, A., Piétrus, A.: The Cahn-Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. 10 539-569 (2000). MR 1807441 | Zbl 0987.35156
[6] Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33 1083-1103 (1996). DOI 10.1016/0020-7683(95)00074-7 | MR 1370124
[7] Garcke, H.: On Mathematical Models for Phase Separation in Elastically Stressed Solids. Habilitation thesis. University Bonn (2000).
[8] Gurtin, M. E.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92 178-192 (1996). DOI 10.1016/0167-2789(95)00173-5 | MR 1387065 | Zbl 0885.35121
[9] Heinemann, C., Kraus, C.: Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 321-359 (2011). MR 2953122 | Zbl 1253.35179
[10] Heinemann, C., Kraus, C.: Existence results for diffuse interface models describing phase separation and damage. Eur. J. Appl. Math. 24 179-211 (2013). DOI 10.1017/S095679251200037X | MR 3031777 | Zbl 1290.35265
[11] Heinemann, C., Kraus, C.: A degenerating Cahn-Hilliard system coupled with complete damage processes. WIAS preprint no. 1759, 23 pages (2012). MR 3280841
[12] Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23 565-616 (2013). DOI 10.1142/S021820251250056X | MR 3021776 | Zbl 1262.74030
[13] Mielke, A.: Complete-damage evolution based on energies and stresses. Discrete Contin. Dyn. Syst., Ser. S 4 423-439 (2011). DOI 10.3934/dcdss.2011.4.423 | MR 2746382 | Zbl 1209.74010
[14] Mielke, A., Roubíček, T., Zeman, J.: Complete damage in elastic and viscoelastic media and its energetics. Comput. Methods Appl. Mech. Eng. 199 1242-1253 (2010). DOI 10.1016/j.cma.2009.09.020 | MR 2601392 | Zbl 1227.74058
[15] Nor, F. M., Keat, L. W., Kamsah, N., Tamin, M. N.: Damage mechanics model for interface fracture process in solder interconnects. 10th Electronics Packaging Technology Conference 821-827 (2008).
[16] Rocca, E., Rossi, R.: A degenerating PDE system for phase transitions and damage. Math. Models Methods Appl. Sci., arXiv:1205.3578v1 (2012), 53 pages. MR 3192591
[17] Thomas, M., Mielke, A.: Damage of nonlinearly elastic materials at small strain-existence and regularity results. ZAMM, Z. Angew. Math. Mech. 90 88-112 (2010). DOI 10.1002/zamm.200900243 | MR 2640367 | Zbl 1191.35159
Search
Partner of
