Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes

Abstract

We introduce a model of dynamic visco-elasto-plastic evolution in the linearly elastic regime and prove an existence and uniqueness result. Then we study the limit of (a rescaled version of) the solutions when the data vary slowly. We prove that they converge, up to a subsequence, to a quasistatic evolution in perfect plasticity.

Appendix

This section contains the proof of two technical results concerning the convergence of suitable Riemann sums for functions with values in Banach spaces.

Lemma 6

Let $X$ be a Banach space, let $\phi \in W^{1,1}([0,T];X)$, let $S\subset (0,T]$ be a set of full measure containing $T$ and let $\psi :S\rightarrow X'$ be a bounded weakly* continuous function. For every $k>0$ let $\{t^k_i\}_{0\le i\le k}$ be a subset of $S\cup \{0\}$ such that $0=t^k_0<t^k_1<\dots <t^k_k=T$ and $\max _{i=1}^k|t^k_i-t^k_{i-1}|\rightarrow 0$ as $k\rightarrow +\infty$. Then

$$\begin{aligned} \lim _{k\rightarrow \infty }\sum _{i=1}^k\langle \psi (t^k_i),\phi (t^k_{i})- \phi (t^k_{i-1})\rangle =\int _0^T\langle \psi (t),\dot{\phi }(t)\rangle dt, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ denotes the duality product between $X'$ and $X$.

Proof

Let $\psi _k:[0,T]\rightarrow X'$ be the piecewise constant function defined by $\psi _k(t)=\psi (t^k_i)$ for $t^k_{i-1}< t\le t^k_i$. Then

$$\begin{aligned} \sum _{i=1}^k\langle \psi (t^k_i),\phi (t^k_{i})-\phi (t^k_{i-1})\rangle =\int _0^T \langle \psi _k(t),\dot{\phi }(t)\rangle dt. \end{aligned}$$

Since $\psi _k(t)\rightharpoonup \! \psi (t)$ weakly* for every $t\in S$ we have $\langle \psi _k(t),\dot{\phi }(t)\rangle \rightarrow \langle \psi (t),\dot{\phi }(t)\rangle$ for a.e. $t\in [0,T]$. The conclusion follows from the Dominated Convergence Theorem.

The next lemma extends the previous result to the case of the duality product introduced in (3.13).

Lemma 7

Let $\varrho$ be the function introduced in the safe-load condition (3.23)–(3.24) and let $p:[0,T]\rightarrow \mathcal M_b(\varOmega \cup \varGamma _0;{\mathbb {M}}^{n \times n}_D)$ be a bounded function. Assume that there exists a set $S\subset (0,T]$ of full measure containing $T$ such that for every $t\in S$ the function $p$ is continuous at $t$ with respect to the strong topology of $\mathcal M_b(\varOmega \cup \varGamma _0;{\mathbb {M}}^{n \times n}_D)$ and $p(t)\in \varPi _{\varGamma _0}(\varOmega )$. For every $k>0$ let $\{t^k_i\}_{0\le i\le k}$ be a subset of $S\cup \{0\}$ such that $0=t^k_0<t^k_1<\dots <t^k_k=T$ and $\max _{i=1}^k|t^k_i-t^k_{i-1}|\rightarrow 0$ as $k\rightarrow +\infty$. Then

$$\begin{aligned} \lim _{k\rightarrow \infty }\sum _{i=1}^k\langle \varrho _D(t^k_{i})-\varrho _D(t^k_{i-1}),p(t^k_i)\rangle =\int _0^T\langle \dot{\varrho }_D(t),p(t)\rangle dt, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ denotes the duality product introduced in (3.13).

Proof

Let $p_k:[0,T]\rightarrow \varPi _{\varGamma _0}(\varOmega )$ be the piecewise constant function defined by $p_k(t)=p(t^k_i)$ for $t^k_{i-1}< t\le t^k_i$. Using (3.27) and (3.28) we obtain that

$$\begin{aligned}&\sum _{i=1}^k\langle \varrho _D(t^k_{i})-\varrho _D(t^k_{i-1}),p(t^k_i)\rangle =\int _0^T\langle \dot{\varrho }_D(t),p_k(t)\rangle dt=\nonumber \\&=\int _0^T\langle \dot{\varrho }_D(t),p_k(t)-p(t)\rangle dt + \int _0^T\langle \dot{\varrho }_D(t),p(t)\rangle dt. \end{aligned}$$

(6.1)

By (3.14) we have

$$\begin{aligned} \int _0^T|\langle \dot{\varrho }_D(t),p_k(t)-p(t)\rangle | dt\le \int _0^T\Vert \dot{\varrho }_D(t)\Vert _{L^\infty } \Vert p_k(t)-p(t)\Vert _{\mathcal M_b} dt \end{aligned}$$

Since $\Vert p_k(t)-p(t)\Vert _{\mathcal M_b}\rightarrow 0$ for a.e. $t\in S$ by our continuity assumption and $t\mapsto \Vert \dot{\varrho }(t)\Vert _{L^\infty }$ belongs to $L^1([0,T])$ (see [6, Theorem 7.1]), we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _0^T|\langle \dot{\varrho }_D(t),p_k(t)-p(t)\rangle | dt=0 \end{aligned}$$

(6.2)

by the Dominated Convergence Theorem. The conclusion follows from (6.1) and (6.2).