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.
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Acknowledgments
This material is based on work supported by the Italian Ministry of Education, University, and Research under the Project “Calculus of Variations” (PRIN 2010-11) and by the European Research Council under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors thank the referee for some useful remarks about the role of the viscosity tensor \(A_1\).
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Appendix
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
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
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
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
By (3.14) we have
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
by the Dominated Convergence Theorem. The conclusion follows from (6.1) and (6.2).
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Maso, G.D., Scala, R. Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes. J Dyn Diff Equat 26, 915–954 (2014). https://doi.org/10.1007/s10884-014-9409-7
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DOI: https://doi.org/10.1007/s10884-014-9409-7