Linearization of elasticity models for incompressible materials

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2012

The energy functional of linear elasticity is obtained as Γ-limit of suitable rescalings of the energies of finite elasticity. The quadratic control from below of the energy density W (∇v) for large values of the deformation gradient ∇v is replaced here by the weaker condition W (∇v) |∇v| p , for some p > 1. Energies of this type are commonly used in the study of a large class of compressible rubber-like materials.

Calculus of Variations and Partial Differential Equations

We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.

1999

The first variation condition for the potential energy in nonlinear elasticity for incompressible materials provides a linear functional which vanishes on an appropriately constrained set of variations. We prove a representation theorem for such linear functionals which forms the basis for the existence of a constraint reaction (Lagrange multiplier) field.

2010

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress–strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incom-pressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incom-pressible elasticity l, A, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.

Computer Methods in Applied Mechanics and Engineering, 1979

2010

Rubberlike materials are characterized by high deformability and reversibility of deformation. From the continuum viewpoint, a strain energy density function is postulated for modeling the behavior of these materials. In this paper, a general form for the strain energy density of these materials is proposed from a phenomenological point of view. Based on the Valanis-Landel hypothesis, the strain energy density of incompressible materials is expressed as the sum of independent functions of the principal stretches meeting the essential requirements on the form of the strain energy density. It is cleared that the appropriate mathematical expressions for constitutive modeling of these materials are polynomial, logarithmic, and particularly exponential functions. In addition, the material parameters are calculated using a novel procedure that is based on the correlation between the values of the strain energy density (rather than the stresses) cast from the test data and the theory. In order to evaluate the performance of the proposed strain energy density functions, some test data of rubberlike materials with pure homogeneous deformations are used. It is shown that there is a good agreement between the test data and predictions of the models for incompressible isotropic materials.

1997

One of the unresolved issues on Saint-Venant's principle concerns the energy decay estimates established in the literature for the traction boundary-value problem of three-dimensional linear isotropic elastostatics for a cylinder. For the semi-infinite cylinder with traction-free lateral surface and self-equilibrated loads at the near end, it has been shown that the stresses decay exponentially from the end and results are available for the estimated decay rate, which is a lower bound for the exact decay rate. These results are, however, generally conservative in that they underestimate the exact decay rate. Another shortcoming, which motivated the present investigation, is that the estimated decay rates tend to zero as the Poisson's ratio tends to the value 1 2. Thus for the limiting case of an incompressible material, these methods fail to establish exponential decay. The purpose of the present paper is to remedy this defect. In particular, an exponential decay estimate is established with estimated decay rate independent of Poisson's ratio. Thus, in particular, the results here hold in the incompressible limit as ! 1 2. An alternative treatment directly for the incompressible case has been given recently. It should be noted that the stresses in the three-dimensional traction boundary-value problem do depend on Poisson's ratio and that stress decay estimates for the cylinder problem with estimated decay rates dependent on are, in fact, to be expected. However, in the absence of such results that do not deteriorate as ! 1 2 , we obtain here an estimated decay rate that is independent of .

2017

In this chapter we suggest new models for the study of deformations of elastic media through the minimization of distortion functionals. This provides a “holistic” approach to this problem and a potential mathematical underpinning of a number of phenomena which are actually observed. The functionals we study are conformally invariant and give measures of the local anisotropic deformation of the material which can be tuned by varying p-norms and weights. That these functions are conformally invariant offers the opportunity to address multiscale problems in an integrative manner there is no natural scale. When applied to the problem of deforming elastic bodies through stretching we see phenomena such as tearing naturally arising through bad delocalisation of energy. More formally, we find that deformations either exist within certain ranges or fail to exist outside these ranges – a material can only be stretched so far. We dub this the Nitsche phenomenon after a similar phenomenon was...

Archive for Rational Mechanics and Analysis, 1970

Numerische Mathematik, 2002

In linear elasticity problems, the pressure is usually introduced for computing the incompressible state. In this paper is presented a technique which is based on a power series expansion of the displacement with respect to the inverse of Lamé's coefficient λ. It does not require to introduce the pressure as an auxiliary unknown. Moreover, low degree finite elements can be used. The same technique can be applied to Stokes or Navier-Stokes equations, and can be extended to more general parameterized partial differential equations. Discretization error and convergence are analyzed and illustrated by some numerical results. Subject Classification (1991): 74S05, 74B05, 35Q72, 35Q30, 30E15