This paper presents a new computational formulation for large strain polyconvex elasticity. The formulation is based on using the deformation gradient (the fibre map), its adjoint (the area map) and its determinant (the volume map) as independent kinematic variables of a convex strain energy function. Compatibility relationships between these variables and the deformed geometry are enforced by means of a multi-field variational principle with additional constraints. This process allows the use of different approximation spaces for each variable. The paper also introduces conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the Hellinger-Reissner type, where the new conjugate stresses are primary variables together with the deformed geometry. Both compressible and incompressible or nearly incompressible elastic models are considered. A key element to the developments presented in the paper is the definition of a new tensor cross product which facilitates the algebra associated with the adjoint of the deformation gradient. Two particular choices of interpolation spaces are considered to illustrate the formulation. The first is based on quadratic displacements, constant pressure discontinuous across element edges and linear discontinuous element stresses conjugate to the deformation gradient and its adjoint. The second choice involves linear displacement and continuous linear pressure stabilised using a Petrov-Galerkin technique.

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 [1], 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 use of a new tensor cross product 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-1 Corresponding author: j.bonet@swansea.ac.uk 2 Corresponding author: a.j.gil@swansea.ac.uk 3 Corresponding author: c.h.lee@swansea.ac.uk Preprint submitted to Computer Methods in Applied Mechanics and EngineeringSeptember 16, 2014Rivlin 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.

A first order hyperbolic framework for large strain computational solid dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity Antonio J. Gil, Chun Hean Lee, Javier Bonet, Rogelio Ortigosa PII: S0045-7825(15) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.A first order hyperbolic framework for large strain computational solid dynamics. AbstractIn Part I of this series, Bonet et al.[1] introduced a new computational framework for the analysis of large strain isothermal fast solid dynamics, where a mixed set of Total Lagrangian conservation laws was presented in terms of the linear momentum and an extended set of strain measures, namely the deformation gradient, its co-factor and its Jacobian. The main aim of this paper is to expand this formulation to the case of nearly incompressible and truly incompressible materials. The paper is further enhanced with three key novelties. First, the use of polyconvex nearly incompressible strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes. Two variants of the same formulation can then be obtained, namely, conservation-based and entropy-based, depending on the unknowns of the system. Crucially, the study of the eigenvalue structure of the system is carried out in order to demonstrate its hyperbolicity and, thus, obtain the correct time step bounds for explicit time integrators. Second, 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. Third, an adapted fractional step method, built upon the work presented in Gil et al. range of applications towards the incompressibility limit. Finally, a series of numerical examples are presented in order to assess the applicability and robustness of the proposed formulation. The overall scheme shows excellent behaviour in compressible, nearly incompressible and truly incompressible scenarios, yielding equal order of convergence for velocities and stresses.

Following the recent work of Bonet et al.[1], this paper postulates a new convex multi-variable variational framework for the analysis of Electro Active Polymers (EAP) in the context of reversible nonlinear electro-elasticity. This extends the concept of polyconvexity [2] to strain energies which depend on non-strain based variables introducing other physical measures such as the electric displacement. Six key novelties are incorporated in this work. First, a new definition of the electro-mechanical internal energy is introduced expressed as a convex multi-variable function of a new extended set of electromechanical arguments. Crucially, this new definition of the internal energy enables the most accepted constitutive inequality, namely ellipticity, to be extended to the entire range of deformations and electric fields and, in addition, to incorporate the electro-mechanical energy of the vacuum, and hence that for ideal dielectric elastomers, as a degenerate case. Second, a new extended set of variables, work conjugate to those characterising the new definition of multi-variable convexity, is introduced in this paper. Third, both new sets of variables enable the definition of novel extended Hu-Washizu type of mixed variational principles which are presented in this paper for the first time in the context of nonlinear electro-elasticity. Fourth, some simple strategies to create appropriate convex multi-variable energy functionals (in terms of convex multi-variable invariants) by incorporating minor modifications to a priori non-convex multi-variable functionals are also presented. Fifth, a tensor cross product operation [3] used in [1] to facilitate the algebra associ

This paper describes in detail the formulation of large strain solid mechanics based on the tensor cross product, originally presented by de Boer [1], page 76, and recently re-introduced by Bonet et al. in [2] and [3]. The paper shows how the tensor cross product facilitates the algebra associated with the area and volume maps between reference and final configurations. These maps, together with the fibre map, make up the fundamental kinematic variables in polyconvex elasticity. The algebra proposed leads to novel expressions for the tangent elastic operator which neatly separates material from geometrical dependencies. The paper derives new formulas for the spatial and material stress and their corresponding elasticity tensors. These are applied to the simple case of a Mooney-Rivlin material model. The extension to transversely isotropic material models is also considered.