Symmetric-Hyperbolic Equations of Motion for a Hyperelastic Material

Computer Methods in Applied Mechanics and Engineering

Bonet

et al. 2014

198

A mixed second order stabilised Petrov-Galerkin finite element framework was recently introduced by the authors (C.H.Lee, A.J.Gil and J.Bonet. "Development of a stabilised Petrov-Galerkin formulation for conservation laws in Lagrangian fast solid dynamics", CMAME, 268:40-64, 2014). The new mixed formulation, written as a system of conservation laws for the linear momentum and the deformation gradient, performs extremely well in bending dominated scenarios (even when linear tetrahedral elements are used) yielding equal order of convergence for displacements and stresses. In this paper, this formulation is further enhanced for nearly and truly incompressible deformations with three key novelties. First, a new conservation law for the Jacobian of the deformation is added into the system providing extra flexibility to the scheme. Second, a variationally consistent PetrovGalerkin stabilisation methodology is derived. Third, an adapted fractional step method is presented for both incompressible and nearly incompressible materials in the context of nonlinear elastodynamics. For completeness and ease of understanding, these three improvements are presented both in small and large strain regimes, studying the eigenstructure of the resulting systems. A series of numerical examples are presented in order to demonstrate the robustness of the enhanced methodology with respect to the work previously published by the authors.

Section: Discussionmentioning

confidence: 99%

A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

Computer Methods in Applied Mechanics and Engineering

Bonet

et al. 2014

198

“…Essentially, the strain energy Ψ per unit undeformed volume must be a function of the deformation gradient F via a convex multi-valued function W [62] as:…”

Section: Constitutive Laws: Polyconvex Elasticitymentioning

confidence: 99%

“…The three strain measures F , H and J have work conjugate stresses Σ F , Σ H and Σ J defined by [45,46,62,63]:…”

Section: Constitutive Laws: Polyconvex Elasticitymentioning

confidence: 99%

“…The enhanced mixed methodology is formulated in the form of a system of first order conservation laws [62,63], where the linear momentum p conservation equation is now supplemented with three geometric conservation laws for the minors of the deformation tensor (i.e. one for the deformation gradient F , one for the area map H and one for the Jacobian of the deformation J).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity

Bonet

Computer Methods in Applied Mechanics and Engineering

et al. 2015

257

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.

“…Additionally, following Gil and Ortigosa [1,2], we propose an extended Hu-Washizu [44,52,54,54,[69][70][71][71][72][73][74][75][76][77][78][79][80] mixed variational principle for nearly and truly incompressible scenarios. Considering now interpolation spaces which satisfy the LBB condition, a comparison of this enhanced methodology is carried out in this paper against the three-field formulation, where both LBB compliant and non-compliant interpolation spaces are considered.…”

Section: Introductionmentioning

confidence: 99%

A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies

Ortigosa

Computer Methods in Applied Mechanics and Engineering

2016

Please cite this article as: R. Ortigosa, A.J. Gil, C.H. Lee, A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies, Comput. Methods Appl. Mech. Engrg. (2016), http://dx.doi.org/10. 1016/j.cma.2016.06.025 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 computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies. AbstractThe series of papers published by Gil and Ortigosa [1-3] introduced a new convex multi-variable variational and computational framework for the numerical simulation of Electro Active Polymers (EAPs) in scenarios characterised by extreme deformations and/or extreme electric fields. Building upon this body of work, five key novelties are incorporated in this paper. First, a generalisation of the concept of multi-variable convexity to energy functionals additively decomposed into isochoric and volumetric components. This decomposition is typical of nearly and truly incompressible materials, group which represents the majority of the most relevant EAPs. Second, convexification or regularisation strategies are applied to a priori non-convex multi-variable isochoric functionals to yield physically meaningful convex multi-variable functionals. Third, based on the mixed variational principles introduced in Reference [1] in the context of compressible electro-elasticity, a novel extended Hu-Washizu mixed variational principle for nearly and truly incompressible scenarios is presented. From the computational standpoint, a static condensation procedure is applied in order to condense out the element-wise extra fields, the resulting formulation having a comparable cost to the more standard three-field displacement-potential-pressure mixed formulation. Fourth, the computational framework for the three-field mixed variational principle in nearly and truly incompressible scenarios is also presented. In this case, the novelty resides in the consideration of convex multi-