Singular limits for inhomogeneous equations of elasticity (original) (raw)
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Quarterly of Applied Mathematics, 1997
Discontinuous solutions with shocks for a family of almost incompressible hyperelastic materials are studied. An almost incompressible material is one whose deformations are not a priori constrained but whose stress response reacts strongly (of order ε − 1 {\varepsilon ^{ - 1}} ) to deformations that change volume. The material class considered is isotropic and admits motions that are self-similar, exhibit cavitation, and are energy minimizing. For the initial-value problem when considering the entire material, the solutions converge (as ε \varepsilon tends to zero) to an isochoric solution of the limit (incompressible) system with the corresponding arbitrary hydrostatic pressure being the singular limit of the pressures in the almost incompressible materials. The shocks, if they exist, disappear: their speed tends to infinity and their strength tends to zero.
On the Zero Relaxation Limit for a System Modeling the Motions of a Viscoelastic Solid
SIAM Journal on Mathematical Analysis, 1999
We consider a simple model of the motions of a viscoelastic solid. The model consists of a two by two system of conservation laws including a strong relaxation term. We establish the existence of a BV-solution of this system for any positive value of the relaxation parameter. We also show that this solution is stable with respect to the perturbations of the initial data in L 1 . By deriving the uniform bounds, with respect to the relaxation parameter, on the total variation of the solution, we obtain the convergence of the solutions of the relaxation system towards the solutions of a scalar conservation law as the relaxation parameter goes to zero. Due to the Lip + bound on the solutions of the relaxation system, an estimate on the rate of convergence towards equilibrium is derived. In particular, an O( p ) bound on the L 1 -error is established.
Transactions of the American Mathematical Society, 2004
In this work we establish the limiting absorption principle for the two-dimensional steady-state elasticity system in an inhomogeneous anisotropic medium. We then use the limiting absorption principle to prove the existence of a radiation solution to the exterior Dirichlet or Neumann boundary value problems for such a system. In order to define the radiation solution, we need to impose certain appropriate radiation conditions at infinity. It should be remarked that even though in this paper we assume that the medium is homogeneous outside of a large domain, it still preserves anisotropy. Thus the classical Kupradze's radiation conditions for the isotropic system are not suitable in our problem and new radiation conditions are required. The uniqueness of the radiation solution plays a key role in establishing the limiting absorption principle. To prove the uniqueness of the radiation solution, we make use of the unique continuation property, which was recently obtained by the authors. The study of this work is motivated by related inverse problems in the anisotropic elasticity system. The existence and uniqueness of the radiation solution are fundamental questions in the investigation of inverse problems.
A construction of infinitely many singular weak solutions to the equations of nonlinear elasticity
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2002
Radial deformations of a ball composed of a nonlinear elastic material and corresponding to cavitation have been much studied. In this paper we use rescalings to show that each such deformation can be used to construct infinitely many non-symmetric singular weak solutions of the equations of nonlinear elasticity for the same displacement boundary-value problem. Surprisingly, this property appears to have been unnoticed in the literature to date.
Journal of the Association of Arab Universities for Basic and Applied Sciences, 2012
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