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2009, Acta Mathematica Scientia
Based on the framework introduced in [4] or [5], the singular limits of stiff relaxation and dominant diffusion for the Cauchy problem of inhomogeneous equations of elasticity is studied. We are able to reach equilibrium even though the nonlinear stress term is not strictly increasing.
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.
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.
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.
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
In this paper we consider the linear Lame´equations in a non homogeneous three-dimensional domain Q composed of two homogeneous bodies Q + and Q À with the boundary condition: contact without friction-Dirichlet. We first establish the existence and uniqueness results for weak solutions. Then using cylindrical coordinates and assuming that the neighborhood of the edge A is sufficiently small, we give the transcendental equations governing the singular behavior in the spatial case. In the end, applying the results of , we obtain an explicit description of the singularities for the variational solution of the boundary value problem in the homogeneous case, i.e. the two bodies have the same elasticity coefficients.
Communications in Applied and Industrial Mathematics
The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represented by a wedge continuous function. Indeed, the linear integro-differential viscoelasticity equation, characterised by a suitable wedge continuous relaxation function, is shown to give the classical linear wave equation via a limit procedure.
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2015
In this work we consider the non-simple theory of elastic material with voids and we investigate how the coupling of these two aspects of the material affects the behavior of the solutions. We analyze only two kind of different behavior, slow or exponential decay. We introduce four different dissipation mechanisms in the system and we study, in each case, the effect of this mechanism in the behavior of the solutions.
Boundary Value Problems
The quasistatic problem of a viscoelastic body in a three-dimensional thin domain with Tresca’s friction law is considered. The viscoelasticity coefficients and data for this system are assumed to vary with respect to the thickness ε. The asymptotic behavior of weak solution, when ε tends to zero, is proved, and the limit solution is identified in a new data system. We show that when the thin layer disappears, its traces form a new contact law between the rigid plane and the viscoelastic body. In which case, a generalized weak form equation is formulated, the uniqueness result for the limit problem is also proved.
Continuum Mechanics and Thermodynamics, 2011
From non-linear elasticity to linear elasticity with initial stress via Γ -convergence. Abstract We consider a initially-stressed hyperelastic body in equilibrium in its undeformed configuration under a system of dead loads. We give sufficient conditions on the stored energy which guarantee that, when the loads undergo a small perturbation, the energy functional Γ -converges, after some re-scaling, to the energy functional of linear elasticity with initial stress. We also show, under stronger conditions, that quasi-minimizers of the non-linear problem converge to a minimizer of the incremental problem.
In the paper a new proposition of an energy-based hypothesis of material effort is introduced. It is based on the concept of influence functions introduced by Burzyński [3] and on the concept of decomposition of elastic energy density introduced by Rychlewski [18]. A new proposition enables description of a wide class of linearly elastic materials of arbitrary symmetry exhibiting strength differential effect.
We consider a initially stressed hyperelastic body in equilibrium in its undeformed configuration under a system of dead loads. We give sufficient conditions on the stored energy which guarantee that when the loads undergo a small perturbation, the energy functional converges, after some re-scaling, to the energy functional of linear elasticity with initial stress. We also show, under stronger conditions, that quasi-minimizers of the non-linear problem converge to a minimizer of the incremental problem.
Quarterly of Applied Mathematics
2012
We study the diffusion equation in the absence of instantaneous elasticity ut − Z t 0 g(t − τ)∆u(τ) dτ = 0, (x, t) ∈ Ω × (0, +∞), where Ω ⊂ R n , subjected to nonlinear boundary conditions. We prove that if the relaxation function g decays exponentially, then the solutions is exponential stable.
Mathematics of Computation, 2000
We study the effect of approximation matrices to semi-discrete relaxation schemes for the equations of one-dimensional elastodynamics. We consider a semi-discrete relaxation scheme and establish convergence using the L p theory of compensated compactness. Then we study the convergence of an associated relaxation-diffusion system, inspired by the scheme. Numerical comparisons of fully-discrete schemes are carried out.
Journal de Mathématiques Pures et Appliquées, 2005
We construct oscillating-decaying solutions for the general inhomogeneous anisotropic elasticity system. We also prove the Runge approximation property for the inhomogeneous transversely isotropic elasticity system. We apply the oscillating-decaying solutions and the Runge approximation property to the inverse problem of identifying an inclusion or a cavity embedded in a transversely isotropic elastic medium.
Archive for Rational Mechanics and Analysis, 2006
In the theory of solid-solid phase transitions the deformation u : Ω ⊂ R d → R d of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study
Calc Var Partial Differ Equat, 1994
In this paper we study the motion of slightly compressible inviscid fluids. We prove that the solution of the corresponding system of nonlinear partial differential equations converges (uniformly) in the strong norm (that of the data space) to the solution of the incompressible equations, as the Mach number goes to zero (see Theorem 1.2). Actually, our proof applies to a large class of singular limit problems as shown in the Theorem 2.2.
EMS Surveys in Mathematical Sciences, 2014
In order to understand nonlinear responses of materials to external stimuli of different sort, be they of mechanical, thermal, electrical, magnetic, or of optical nature, it is useful to have at one's disposal a broad spectrum of models that have the capacity to describe in mathematical terms a wide range of material behavior. It is advantageous if such a framework stems from a simple and elegant general idea. Implicit constitutive theory of materials provides such a framework: while being built upon simple ideas, it is able to capture experimental observations with the minimum number of physical quantities involved. It also provides theoretical justification in the full three-dimensional setting for various models that were previously proposed in an ad hoc manner. From the perspective of the theory of nonlinear partial differential equations, implicit constitutive theory leads to new classes of challenging mathematical problems. This study focuses on implicit constitutive models for elastic solids in general, and on its subclass consisting of elastic solids with limiting small strain. After introducing the basic concepts of implicit constitutive theory, we provide an overview of results concerning modeling within the framework of continuum mechanics. We then concentrate on the mathematical analysis of relevant boundary-value problems associated with models with limiting small strain, and we present the first analytical result concerning the existence of weak solutions in general three-dimensional domains.
1996
Abstract We study the Cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to zero. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws.
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