From Damage to Delamination in Nonlinearly Elastic Materials at Small Strains

Abstract

Brittle Griffith-type delamination of compounds is deduced by means of Γ-convergence from partial, isotropic damage of three-specimen-sandwich-structures by flattening the middle component to the thickness 0. The models used here allow for nonlinearly elastic materials at small strains and consider the processes to be unidirectional and rate-independent. The limit passage is performed via a double limit: first, we gain a delamination model involving the gradient of the delamination variable, which is essential to overcome the lack of a uniform coercivity arising from the passage from partial damage to delamination. Second, the delamination gradient is suppressed. Noninterpenetration- and transmission-conditions along the interface are obtained.

Appendix: Abstract Γ-Convergence Result

In [25] the theory of Γ-convergence was adapted to the framework of the energetic formulation of rate-independent processes. In the following we introduce sufficient conditions guaranteeing that a subsequence of energetic solutions of the approximating systems $(\mathcal{Q},{\mathcal{E}}_{j},{\mathcal{R}}_{j})$ converges to an energetic solution of the limit system $(\mathcal{Q},{\mathcal{E}}_{\infty},{\mathcal{R}}_{\infty})$. Let the topology for the convergence of the energetic solutions be denoted by $\mathcal{T}$. Then we want to obtain that $q_{j}(t)\overset{\mathcal{T}}{\to}q(t)$ for all t∈[0,T].

For all $j\in\mathbb{N}_{\infty}=\mathbb{N}\cup\{\infty\}$ we introduce the stable sets ${\mathcal{S}}_{j}(t):=\{q\in\mathcal{Q}\,|\,{\mathcal {E}}_{j}(t,q)<\infty ,\,\forall\tilde{q}=(\tilde{u},\tilde{z}):{\mathcal{E}}_{j}(t,q_{j})\leq{\mathcal{E}}_{j}(t,\tilde{q})+\mathcal {R}_{j}(\tilde{z}-z_{j})\}$.

In order to ensure the Γ-convergence of the systems $(\mathcal{Q},{\mathcal{E}}_{j},{\mathcal{R}}_{j})_{j\in\mathbb{N}}$ the following conditions have to be satisfied by the energy functionals ${\mathcal{E}}_{j}:[0,T]\times\mathcal{Q}\to \mathbb {R}_{\infty}$ for all $j\in\mathbb{N}_{\infty}$.

(A.1-E1)

(A.1-E2)

(A.1-E3)

Furthermore the dissipation distances $\mathcal{D}_{j}:\mathcal {Z}\times\mathcal{Z}\to [0,\infty]$ with $\mathcal{D}_{j}(z,\tilde{z})={\mathcal{R}}_{j}(\tilde{z}-z)$ for all $z,\tilde{z}\in\mathcal{Z}$ must fulfill for all $j\in\mathbb{N}_{\infty}$:

(A.2-D1)

(A.2-D2)

(A.2-D3)

Additionally the following compatibility conditions have to be satisfied:

For all t _j →t in [0,T], $q_{j}=(u_{j},z_{j})\overset{\mathcal{T}}{\to}q=(u,z)$ with $q_{j}\in{\mathcal{S}}_{j}(t_{j})$ for all $j\in\mathbb{N}$ it holds

(A.3-C1)

(A.3-C2)

(A.3-C3)

(A.3-C4)

The theorem below states the convergence result. A proof is given in [25, Th. 3.1].

Theorem 7

(Γ-convergence of $(\mathcal{Q},{\mathcal {E}}_{j},{\mathcal{R}}_{j})_{j\in\mathbb{N}}$)

Let conditions (A.1-E1)–(A.1-E3), (A.2-D1)–(A.2-D3) and (A.3-C1)–(A.3-C4) hold and for all $j\in\mathbb{N}$ let $q_{j}:[0,T]\to\mathcal{Q}$ be an energetic solution of $(\mathcal {Q},{\mathcal{E}}_{j},{\mathcal{R}}_{j})$ in the sense of Def. 1. If $q_{j}(t)\overset{\mathcal{T}}{\to}q(t)$ for all t∈[0,T] and if ${\mathcal{E}}_{j}(0,q_{j}(0))\to{\mathcal{E}}_{\infty}(0,q(0))$ then $q:[0,T]\to\mathcal{Q}$ is an energetic solution of $(\mathcal{Q},{\mathcal{E}}_{\infty},{\mathcal{R}}_{\infty})$.

Moreover, for all t∈[0,T] it is ${\mathcal{E}}_{j}(t,q_{j}(t)) \to{\mathcal{E}}(t,q(t))$, $\mathrm{Diss}_{{\mathcal{R}}_{j}}(q_{j},[0,t]) \to \mathrm {Diss}_{{\mathcal{R}}}(q,[0,t])$ and $\partial_{t}{\mathcal{E}}_{j}(t,q_{j}(t)) \to\partial_{t}\mathcal {E}(t,q(t))$ for a.a. t∈[0,T]. Furthermore, for $\mathcal{Q}$ being a separable, reflexive Banach space, the energetic solution q is measurable with respect to time.