scholarly journals Dynamic cavitation with shocks in nonlinear elasticity

1997 ◽  
Vol 127 (4) ◽  
pp. 837-857 ◽  
Author(s):  
K. A. Pericak-Spector ◽  
Scott J. Spector
Keyword(s):  
Conservation Laws ◽  
Equilibrium Solution ◽  
Slow Growth ◽  
Discontinuous Solution ◽  
Stored Energy ◽  
Energy Functions ◽  
Nonlinearly Elastic Body ◽  
Entropy Criterion ◽  
Rate Criterion ◽  
Nonlinearly Elastic

The hyperbolic system of conservation laws that govern the motion of a homogeneous isotropic, nonlinearly elastic body is shown to have a discontinuous solution for a class of stored-energy functions of slow growth. This solution is admissible by the usual entropy criterion and is in fact preferred by the entropy-rate criterion over the smooth equilibrium solution to the same problem. The existence of such a dissipative solution shows that the equilibrium solution is dynamically unstable. This instability cannot be ascertained by linearisation.

Isotropic polyconvex electromagnetoelastic bodies

2018 ◽  
Vol 24 (3) ◽  
pp. 738-747
Author(s):  
M Šilhavý
Keyword(s):  
Magnetic Field ◽  
Mechanical Properties ◽  
Electromagnetic Field ◽  
Smart Materials ◽  
Equilibrium Solution ◽  
Deformation Gradient ◽  
Electric Displacement ◽  
The Body ◽  
Sufficient Condition ◽  
Energy Functions

The recent renewal of interest in nonlinear electromagnetoelastic interactions comes from the technological importance of electro- or magnetosensitive elastomers, smart materials whose mechanical properties change instantly on the application of an electric or magnetic field. We consider materials with free energy functions of the form [Formula: see text], where F is the deformation gradient, d is the electric displacement, and b is the magnetic induction. It was recently shown by the author that such an energy function is polyconvex if and only if it is of the form [Formula: see text] where [Formula: see text] is a convex function (of 31 scalar variables). Moreover, an existence theorem was proved for the equilibrium solution for a system consisting of a polyconvex electromagnetoelastic solid plus the vacuum electromagnetic field outside the body. The condition (8), is not just the combination of Ball’s polyconvexity of elastomers [Formula: see text] with the convexity in the electromagnetic variables. The differential constraints div [Formula: see text], div [Formula: see text] allow for the cross mechanical–electric and mechanical–magnetic terms Fd and Fb which substantially enlarge the class of energies covered by the theory. The result (*), applies to a material of any symmetry; this paper analyzes the condition in the case of isotropic materials. A broad sufficient condition for the polyconvexity is given in that case. Further, it is shown that the commonly used isotropic electroelastic or magnetoelastic invariants are polyconvex except for the biquadratic ones; the paper explicitly determines their quasiconvex envelopes and shows that they are polyconvex.

scholarly journals MATHEMATICAL ANALYSIS OF NONLINEAR BONDED JOINT MODELS

2004 ◽  
Vol 14 (04) ◽  
pp. 535-556 ◽  
Author(s):  
FRANCOISE KRASUCKI ◽  
ARNAUD MÜNCH ◽  
YVES OUSSET
Keyword(s):  
Mathematical Analysis ◽  
Energy Function ◽  
Three Dimensional ◽  
Expansion Method ◽  
Limit Problem ◽  
Stored Energy ◽  
Joint Models ◽  
Numerical Application ◽  
Energy Functions ◽  
The One

Within the framework of nonlinear elasticity, we consider the problem of two adherents joined along their common surface by a thin soft adhesive. Two stored energy functions are considered: the stored energy function of Saint Venant–Kirchhoff and the stored energy function of Ciarlet–Geymonat. Using the asymptotic expansion method, the limit energy associated to each of these stored energy functions is obtained. The aim of this paper is to give a rigorous mathematical analysis of the formally derived limit problem. We show that the limit problem associated to the Saint Venant–Kirchhoff case admits at least one solution and the limit problem associated to the Ciarlet–Geymonat case admits exactly one solution. An analytical comparison in the one-dimensional case and a three-dimensional numerical application are also presented.

On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

2006 ◽  
Vol 86 (3) ◽  
pp. 235-243 ◽  
Author(s):  
Albrecht Bertram ◽  
Thomas Böhlke ◽  
Miroslav Šilhavý
Keyword(s):  
Stored Energy ◽  
Stress Strain ◽  
Energy Functions ◽  
Rank 1 Convexity

Stresses and Birefringence in Rubber Subjected to General Homogeneous Strain

1949 ◽  
Vol 22 (1) ◽  
pp. 53-63 ◽  
Author(s):  
L. R. G. Treloar
Keyword(s):  
Physical Mechanism ◽  
Molecular Model ◽  
Statistical Treatment ◽  
Additional Term ◽  
General Term ◽  
Theoretical Interpretation ◽  
Stored Energy ◽  
Energy Functions ◽  
3 Dimensional ◽  
Homogeneous Strain

Abstract The conclusion to be drawn from the preceding analysis of the stress-strain relations may be summarized in the following way. As a first approximation the equations derived from the statistical treatment of a molecular model provide a basis for the interpretation of the elastic properties of rubber. A closer approximation is obtained by including an additional term in the stored-energy function, i.e., by the use of Mooney's Equation (3). This appears to give an accurate representation of the properties of a swollen rubber, but is still inadequate when applied to a dry rubber, for which a third approximation, including at least one more general term, is likely to be required. It is necessary to emphasize that the use of the second or higher approximations does not in itself throw any light on the physical mechanism responsible for the behavior observed. It is, in fact, the 3-dimensional analog of simple curve-fitting. It must also be borne in mind that any physical model that might be postulated must necessarily lead to results which may be represented by some combination of the stored-energy functions discussed in the section on the theoretical interpretation of the data.

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