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Déplacements à déformations bornées et champs de contrainte mesures. (Displacements at bounded deformations and field of constraint measures). (French) Zbl 0611.73024

Let \(\Omega\) be a connected bounded open set in \({\mathbb{R}}^ N\) \((N=2,3\) mainly) with a Lipschitz boundary \(\partial \Omega\). Let X be a finite dimensional Euclidean space, and \(L^ q(\Omega,X)\) denotes the space of X valued, indefinitely differentiable functions with compact support in \(\Omega\) (respectively of functions with integrable q-th power on \(\Omega\) for the Lebesgue measure dx). The derivatives are understood in the distributions sense. Let E be the \(N(N+1)/2\) dimensional space of symmetric tensors of order 2 on \({\mathbb{R}}^ N\) and let M be the space of bounded measures functions on \(\Omega\) with values in X.
The mechanical problem of locking materials has been formulated by the author. The convenient spaces for displacements and for the stresses to get the variational solutions are studied. The properties of the following spaces \[ (1)\quad U^ q(\Omega)=\{u\in L^ q(\Omega,X),\quad \epsilon (u)\in L^ q(\Omega,X,)\} \]
\[ (2)\quad W^{1,q}(\Omega)=\{u\in L^ q(\Omega,X),\quad Du\in L^ q(\Omega,X)\} \]
\[ (3)\quad Z(\Omega,E)=\{\sigma \in M^ 2(\Omega,E),\quad div\quad \sigma \in L^ 2(\Omega,X)\} \] are analyzed in detail, when \(q=\infty\) and \(\epsilon_{ij}(u)=(u_{i,j}+u_{j,i})/2\).
Reviewer: I.Ecsedi

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
49J99 Existence theories in calculus of variations and optimal control

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References:

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