Irreversibility and alternate minimization in phase field fracture: a viscosity approach
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Irreversibility and alternate minimization in phase field fracture: a viscosity approach Stefano Almi Abstract. This work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by a nonlinear splitting of tensile and compressive strains, featuring non-interpenetration of the fracture lips. The alternating scheme is coupled with an L2 -penalization in the phase field variable, driven by a viscous parameter δ > 0, and with an irreversibility constraint, forcing the monotonicity of the phase field only w.r.t. time, but not along the whole iterative minimization. We show first the convergence of such a scheme to a viscous evolution for δ > 0 and then consider the vanishing viscosity limit δ → 0. Mathematics Subject Classification. 35Q74, 49J45, 74R05, 74R10. Keywords. Phase field, Fracture mechanics, Alternate minimization, Vanishing viscosity.
1. Introduction In the seminal work [16], the quasi-static propagation of brittle fractures in linearly elastic bodies is approximated in terms of equilibrium states of the Ambrosio–Tortorelli functional 1 Gε (u, z) := 12 (z 2 + ηε )σ(u) : (u) dx + Gc ε|∇z|2 + 4ε (z − 1)2 dx , (1.1) Ω
Ω
where Ω is an open bounded subset of Rn with Lipschitz boundary ∂Ω, u ∈ H 1 (Ω; Rn ) is the displacement, (u) denotes the symmetric part of the gradient of u, σ(u) := C(u) is the stress, C being the usual elasticity tensor, ε and ηε are two small positive parameters, and Gc is the toughness, a positive constant related to the physical properties of the material under consideration (from now on we impose Gc = 1). The so-called phase field function z ∈ H 1 (Ω) is supposed to take values in [0, 1], where z(x) = 1 if the material is completely sound at x, while z(x) = 0 means that the elastic body Ω has developed a crack at x. Hence, the zero level set of z represents the fracture and z can be interpreted as a regularization of the crack set. In the static framework, the connection between (1.1) and fracture mechanics has been drawn in [9,19,21,29], where the authors showed the Γ-convergence of Gε as ε → 0 to the functional 1 G(u) := 2 σ(u) : (u) dx + Hn−1 (Ju ) for u ∈ GSBD2 (Ω; Rn ) , (1.2) Ω
where Hn−1 denotes the (n − 1)-dimensional Hausdorff measure and Ju is the discontinuity set of u. From the computational point of view, the study of the functional (1.1) is very convenient in combination with the so-called alternate minimization algorithm [12,15–18]: equilibrium configurations of the energy are computed iteratively by minimizing Gε first w.r.t. u and then w.r.t. z. A fracture irreversibility is imposed by forcing z to be non-increasing in time. Exploiting the separate convexity of Gε , the above scheme guarantees the convergence to a critical point of Gε , whose direct computation would be rather time-consuming because of the non-convexity of the functional. 0123456789().: V,-vol
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