Abstract
Let $\Omega\in\mathbb{R}^n$, $n=2,3$, be the region occupied by a hyperelastic body in its reference configuration. Let $E(\cdot)$ be the stored energy functional, and let $x_0$ be a flaw point in $\Omega$ (i.e., a point of possible discontinuity for admissible deformations of the body). For $V>0$ fixed, let $u_V$ be a minimizer of $E(\cdot)$ among the set of discontinuous deformations $u$ constrained to form a hole of prescribed volume $V$ at $x_0$ and satisfying the homogeneous boundary data $u(x)=Ax$ for $x\in\partial \Omega$. In this paper we describe a regularization scheme for the computation of both $u_V$ and $E(u_V)$ and study its convergence properties. In particular, we show that as the regularization parameter goes to zero, (a subsequence) of the regularized constrained minimizers converge weakly in $W^{1,p}(\Omega\setminus{{\mathcal{B}}_{\delta}(x_0)})$ to a minimizer $u_{V}$ for any $\delta>0$. We obtain various sensitivity results for the dependence of the energies and Lagrange multipliers of the regularized constrained minimizers on the boundary data $A$ and on the volume parameter $V$. We show that both the regularized constrained minimizers and $u_V$ satisfy suitable weak versions of the corresponding EulerLagrange equations. In addition we describe the main features of a numerical scheme for approximating $u_V$ and $E(u_V)$ and give numerical examples for the case of a stored energy function of an elastic fluid and in the case of the incompressible limit.
Original language  English 

Pages (fromto)  119141 
Journal  SIAM Journal on Applied Mathematics 
Volume  80 
Issue number  1 
Early online date  8 Jan 2020 
DOIs  
Publication status  Published  2020 
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Jeyabal Sivaloganathan
 Department of Mathematical Sciences  Professor
Person: Research & Teaching