A surface energy approach to the mass reduction problem for elastic bodies (original) (raw)

On the existence of elastic minimizers for initially stressed materials

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors sa...

On the Uniqueness of Energy Minimizers in Finite Elasticity

Journal of Elasticity, 2018

The uniqueness of absolute minimizers of the energy of a compressible, hyperelastic body subject to a variety of dead-load boundary conditions in two and three dimensions is herein considered. Hypotheses under which a given solution of the corresponding equilibrium equations is the unique absolute minimizer of the energy are obtained. The hypotheses involve uniform polyconvexity and pointwise bounds on derivatives of the stored-energy density when evaluated on the given equilibrium solution. In particular, an elementary proof of the uniqueness result of Fritz John [Comm. Pure Appl. Math. 25 (1972), 617-634] is obtained for uniformly polyconvex stored-energy densities.

An energy gap functional: Cavity identification in linear elasticity

Journal of Inverse and Ill-posed Problems, 2017

The aim of this work is an analysis of some geometrical inverse problems related to the identification of cavities in linear elasticity framework. We rephrase the inverse problem into a shape optimization one using an energetic least-squares functional. The shape derivative of this cost functional is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by several numerical results.

On the Existence of Minimizers with Prescribed Singular Points in Nonlinear Elasticity

Advances in Continuum Mechanics and Thermodynamics of Material Behavior, 2000

Experiments on elastomers have shown that sufficiently-large triaxial tensions induce the material to exhibit holes that were not previously evident. In this paper conditions are presented that allow one to use the direct method of the calculus of variations to deduce the existence of hole creating deformations that are global minimizers of a nonlinear, purely-elastic energy. The crucial physical assumption used is that there are a finite (possibly large) number of material points in the undeformed body that constitute the only points at which cavities can form. Each such point can be viewed as a preexisting flaw or an infinitesimal microvoid in the material.

Energy change due to the appearance of cavities in elastic solids

International Journal of Solids and Structures, 2003

The paper presents an overview of the problem of assessing an increment of strain energy due to the appearance of small cavities in elastic solids. The following approaches are discussed: the compound asymptotic method by Mazja et al., the Eshelby-like method used in the classical works on the mechanics of composites, the homogenization method, and the topological derivative method proposed by Sokołowski and _ Z Zochowski. The increment of energy is expressed by a quadratic form with respect to strains referring to the virgin solid. All the methods lead to the same formula for the increment of energy. It is expressed by a quadratic form with respect to strains referring to the virgin solid. This quadratic form turns out to be unconditionally positive definite. Explicit formulae are derived for an elliptical hole and for a spherical cavity. The results derived determine the characteristic function of the bubble method of the optimal shape design of elastic 2D and 3D structures.

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

Archive for Rational Mechanics and Analysis, 2009

Consider an incompressible, nonlinear, hyperelastic material which occupies the region A ⊂ R n , n ≥ 2, in its reference configuration, where A denotes the annular region A = {x ∈ R n : a < |x| < b } , 0 < a < b. Deformations of A are therefore isochoric maps u : A → R n and so satisfy the incompressibility constraint det ∇u = 1. The boundary of the annulus ∂A is separated into two disjoint pieces ∂A = ∂A o ∪ ∂A I , where ∂A I = {x ∈ R n : |x| = a} and ∂A o = {x ∈ R n : |x| = b} denote the inner and outer boundary components respectively. We study displacement and mixed displacement/zero-traction boundaryvalue problems in which we impose a displacement boundary condition of the form u(x) = σx on one of the boundary components (where σ > 0 is a given constant) and the displacement on the remaining boundary component is either prescribed (in the case of the pure displacement boundaryvalue problem) or left unspecified (in the case of the mixed boundary-value problem). In this paper we use isoperimetric arguments to prove that the radially symmetric solutions to these problems are global energy minimisers in various classes of (possibly non-symmetric) isochoric deformations of the annulus.