On the Representation of Energy and Momentum in Elasticity

Acta Mechanica, 2014

Elasticity is the prototype of constitutive models in Continuum Mechanics. In the nonlinear range, the elastic model claims for a geometrically consistent physico-mathematical formulation providing also the logical premise for linearized approximations. A theoretic framework is envisaged here with the aim of contributing a conceptually clear, physically consistent, and computationally convenient formulation. A reasoning about the physics of the model, from a geometric point of view, leads to conceive constitutive relations as instantaneous incremental responses to a finite set of tensorial state variables and to their time rates along the space-time motion. Integrability of the tangent elastic compliance, existence of an elastic stress potential, and conservativeness of the elastic response, under the conservation of mass, are given a brand new treatment. Finite elastic strains have no physical interpretation in the new rate theory, and referential local placements are appealed to, just as loci for operations of linear calculus. Frame invariance is assessed with a consistent geometric treatment, and the clear distinction between the new notion and the property of isotropy is pointed out, thus overcoming the improper statement of material frame indifference. Extension of the theory to elasto-visco-plastic constitutive models is briefly addressed. Basic computational steps are described to illustrate feasibility and convenience of calculations according to the new theory of elasticity.

This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors (Gil et al., 2014) [48], where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the new use of a tensor cross product [61] greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov-Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney-Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke-Hu-Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes. namely, a reduced order of convergence for strains and stresses , poor performance in nearly incompressible bending dominated scenarios [7-9] and numerical instabilities in the form of shear locking, volumetric locking and spurious hydrostatic pressure fluctuations . In addition, displacement-based methods used in conjunction with Newmark-type time integrators have a tendency to introduce high frequency noise and accuracy is degraded once numerical artificial damping is employed .

Acta Mechanica, 1990

Eshelby's elastic energy-momentum tensor is shown to satisfy a differential identity which, in the general ease of a uniform elastic body with inhomogeneities, is expressible in terms of the torsion of the material connection.

Acta Mechanica, 2008

The essential criterion for a good mathematical model for hyperelasticity is its ability to match the measured strain energy curves under different deformations over a large range. One group of models for hyperelasticity is to express the strain energy as a function of I 1 , I 2 , I 3 , the invariants of the right Cauchy-Green deformation tensor. Under the assumption of incompressibility, it can be proved that all valid (I 1 , I 2) pairs fall in a region bounded by the I 1 − I 2 locus from deformations under simple extension and equalbiaxial extension (or, equivalently, simple compression). I 1 − I 2 locus from planar extension lies inside the region. Since the strain energy curves from simple deformation modes can be measured from experiments, it is possible to approximately obtain the strain energy under other (I 1 , I 2) values by interpolating data from the three measured curves. The proposed model for hyperelasticity is an interpolation algorithm with all mathematical details. The new model is implemented into a user-defined material subroutine in commercial FEA software. It can not only accurately reproduce the measured data from these simple deformation modes but also predict the stress-strain curve under planar extension in a reasonable good accuracy even without using the measured data from planar extension.

The principal focus of this paper is the formulation of a general approach to hyperelastic strain energy functions that does not rely on the use of scalar invariants of tensors. We call this an invariant-free formulation of hyperelasticity. This essentially requires the conversion of the strain energy function from one of scalar products of scalar tensor invariants (all zeroth-order) into one of quadruple contractions between fourth-order tensors, thus preserving directional distinctions through to energy. We begin with an analysis of a range of hyperelastic properties in order to eliminate some non-physical models. In the section after, we are left with the Simo and Pister model and the Compressible neo-Hookean model, and decide on Simo and Pister's model [1] for further study. Presented is a general form of invariant-free hyperelasticity (the so-called generalized strain energy function), and the fitting of the Simo and Pister model into that framework. The novelty of this invariant-free formulation is threefold: first allowing the presentation of strain energy as a fourth-order tensor that explicitly provides the origin of energy contributions from a possible 81 combinations through the simple exchange of the quadruple contraction operator with the Hadamard product; second is a new ability to seamlessly integrate micropolar effects into existing hyperelastic functions (a cursory look); and third is the direction-preserving nature of the formulation, which satisfies the original charter of this work in providing a primer for the natural extension of advanced conventional hyperelastic functions from isotropic materials to anisotropic materials.

Journal of Elasticity, 1996

The strain energy density of a hyperelastic anisotropic body which is rotated before being subjected to a given but arbitrary deformation is viewed as a smooth function defined on the group of rotations, parametrized by the deformation gradient. It is shown that the critical points of this function correspond to rotations which, when composed with the prescribed deformation, yield a total strain tensor which is coaxial with the corresponding stress. For any type of material symmetry, there are at least two such rotations. Coaxiality of stress and strain for all deformations is shown to be a sufficient condition for the isotropicity of hyperelastic materials.

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...

Journal of Computational Physics, 2009

Eulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to non-linear elastic theory. In this paper one such class of solver is examined based upon characteristic tracing in conjunction with high-order monotonicity preserving weighted essentially non-oscillatory (MPWENO) reconstruction. Furthermore, a new iterative method for finding exact solutions of the Riemann problem in non-linear elasticity is presented. Access to exact solutions enables an assessment of the performance of the numerical techniques with focus on the resolution of the seven wave structure. The governing model represents a special case of a more general theory describing additional physics such as material plasticity. The numerical scheme therefore provides a firm basis for extension to simulate more complex physical phenomena. Comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations is presented.

Archive for Rational Mechanics and Analysis, 1990

2009

Modern theories of complex materials rely on the construction of specific potentials for inner actions; such a procedure may be linked with an approach by Poincaré. In this contribution we examine Poincaré’s approach to the ordinary theory of linear elasticity and find out how it is related to the standard molecular model by Navier, Cauchy and Poisson and to the refined molecular model by Voigt. In the end, we comment on the relationships among these approaches and the contemporary theories of complex material behaviour.