Discrete approximation of dynamic phase-field fracture in visco-elastic materials

Abstract

This contribution deals with the analysis of models for phasefield fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively 1homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

Dynamic Phase‐Field Fracture in Viscoelastic Materials using a First‐Order Formulation

“…h is attained by solving a Riemann problem locally between faces of neighbouring finite element cells to find a good approximation for the crossing flux (see details in [1], derivation for linear elasticity in [15,16]). Depending on the phase field z h ∈ V cf h the operators…”

“…This method has been developed and tested in [9]. In [1] we provide a mathematical rigorous convergence analysis for the algorithm of [9]. These analytical results we present here in the following.…”

“…Assuming initial values u 0 and ε 0 with ε 0 = sym(Du 0 ), the displacement u can be recovered such that u(t) = u 0 + t 0 v(s) ds. Then, the second-order approach in [8] is reformulated with a first-order * Corresponding author: e-mail sven.tornquist@fu-berlin.de, phone +49 30 838 65259, fax +49 30 838 465259…”

“…Note that in[1] a variable time step is discussed where the step size is adjusted depending on the energy release, see also[9, Elastic time step, Dissipative time step, p. 6].4 Analytical resultsThe following results are presented here without proofs. The detailed arguments can be found in[1]. At first, we introduce the notation△ε n h = ε n h −ε n−1 h and △z n h = z n h −z n−1 h and define interpolants (z h , v h , ε h , σ h ) ∈ L 2 (0, T ; V cf h ×V dg h ×W dg h ×W dg h ) by (z h , v h , ε h , σ h )(t) = (z n h , v n h , ε n h , σ n h ) in (t n−1 h , t n h ) and ( żh , εh ) ∈ L 2 (0, T ; V cf h ×W dg h ) by ( żh , εh )(t) = 1 △t n h (△z n h , △ε n h ) for t ∈ (t n−1 h , t n h ).To establish existence of discrete solutions, for (S2) a linear system of equations has to be solved while the damage evolution (S1) is more involved due to the nonlinearities coming from the Yosida term and the z-dependence of the elastic term.Lemma 4.1 (Existence of discrete solutions) Let the assumptions stated in Sect.…”

“…

In this contribution we present analytical results on a model for dynamic fracture in viscoelastic materials at small strains that have been obtained in full depth in [1]. In the model, the sharp crack interface is regularized with a phase-field approximation, and for the phase-field variable a viscous evolution with a quadratic dissipation potential is employed.

…”