On a tensor cross product based formulation of large strain solid mechanics

Bonet, Javier; Gil, Antonio J.; Ortigosa, Rogelio

doi:10.1016/j.ijsolstr.2015.12.030

Cited by 73 publications

(148 citation statements)

References 39 publications

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“…In references [12,32], the authors employ an alternative definition of the cofactor H, namely H = 1 2 F × F, which simplifies considerably the algebra [33]. The new definition oftheareamapH is based on a tensor cross product × introduced in [34] and applied within the context of nonlinear elasticity for the first time in [12].…”

Section: Continuum Kinematicsmentioning

confidence: 99%

“…It is then possible [12,32,33] to express the first PiolaKirchhoff stres tensor P in terms of the extended strain measures {F, H, J } and conjugate stresses { F , H , J } as…”

Section: Polyconvex Elasticitymentioning

confidence: 99%

“…A tangent elasticity operator D 2 [δu; u] is usually needed [16] in order to ensure quadratic convergence of a NewtonRaphson type of solution process, derived in [12,32,33]a s follows …”

Section: Polyconvex Elasticitymentioning

confidence: 99%

“…However, the classical approach consists of ensuring that the strain energy satisfies the polyconvexity condition first but then proceeds towards a computational solution by re-expressing the energy in terms of the deformation gradient alone. Pioneered in [29] and [12,32,33], an alternative framework is proposed based on maintaining as independent variables the extended kinematic set on which the strain energy is expressed as a convex function, namely, the deformation gradient, its cofactor and its determinant.…”

Section: Introductionmentioning

confidence: 99%

“…The paper also presents the work conjugates of the extended kinematic set in the context of beam theory as well as novel expressions for the tangent operators. The present formulation utilises a novel algebra based on a tensor cross product operation pioneered in [34] and reintroduced and exploited for the first time in [12,32,33] in the context of solid mechanics. The new tensor cross product operation is particularly helpful when dealing with polyconvex constitutive laws, where invariants of the cofactor and the determinant of the deformation feature heavily in the representation of the strain energy functional.…”

Section: Introductionmentioning

confidence: 99%

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A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

et al. 2015

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In this paper, a new computational framework is presented for the analysis of nonlinear beam finite elements subjected to large strains. Specifically, the methodology recently introduced in Bonet et al. (Comput Methods Appl Mech Eng 283:1061-1094 in the context of three dimensional polyconvex elasticity is extended to the geometrically exact beam model of Simo (Comput Methods Appl Mech Eng 49:55-70, 1985), the starting point of so many other finite element beam type formulations. This new variational framework can be viewed as a continuum degenerate formulation which, moreover, is enhanced by three key novelties. First, in order to facilitate the implementation of the sophisticated polyconvex constitutive laws particularly associated with beams undergoing large strains, a novel tensor cross product algebra by Bonet et al. (Comput Methods Appl Mech Eng 283:1061-1094) is adopted, leading to an elegant and physically meaningful representation of an otherwise complex computational framework. Second, the paper shows how the novel algebra facilitates the reexpression of any invariant of the deformation gradient, its cofactor and its determinant in terms of the classical beam strain measures. The latter being very useful whenever a classical beam implementation is preferred. This is particularised for the case of a Mooney-Rivlin model although the technique can be straightforwardly generalised to other more complex isotropic and anisotropic polyconvex models. Third, the connection between the two most accepted restrictions for the definition of constitutive models in three dimensional elasticity and beams is shown, bridging the gap between the continuum and its degenerate beam description. This is carried out via a novel insightful representation of the tangent operator.

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Section: Continuum Kinematicsmentioning

confidence: 99%

“…It is then possible [12,32,33] to express the first PiolaKirchhoff stres tensor P in terms of the extended strain measures {F, H, J } and conjugate stresses { F , H , J } as…”

Section: Polyconvex Elasticitymentioning

confidence: 99%

“…A tangent elasticity operator D 2 [δu; u] is usually needed [16] in order to ensure quadratic convergence of a NewtonRaphson type of solution process, derived in [12,32,33]a s follows …”

Section: Polyconvex Elasticitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

et al. 2015

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A first‐order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme

Haider

Lee

Gil

et al. 2016

Numerical Meth Engineering

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Citation for this version held on GALA:Haider, Jibran, Lee, Chun Hean, Gil, Antonio J. and Bonet, Javier (2016) A first order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme. London: Greenwich Academic Literature Archive. , a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Després [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications.

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An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics

Franke

Janz

Schiebl

et al. 2018

Numerical Meth Engineering

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The present contribution provides a new approach to the design of energy momentum consistent integration schemes in the field of nonlinear thermo-elastodynamics. The method is inspired by the structure of polyconvex energy density functions and benefits from a tensor cross product that greatly simplifies the algebra. Furthermore, a temperature-based weak form is used, which facilitates the design of a structure-preserving time-stepping scheme for coupled thermoelastic problems. This approach is motivated by the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework for open systems. In contrast to complex projection-based discrete derivatives, a new form of an algorithmic stress formula is proposed. The spatial discretization relies on finite element interpolations for the displacements and the temperature. The superior performance of the proposed formulation is shown within representative quasi-static and fully transient numerical examples. KEYWORDSfinite element method, nonlinear thermo-elastodynamics, polyconvexity-based framework, structure-preserving discretization, tensor cross product INTRODUCTIONThe present contribution aims for the consistent discretization of nonlinear thermo-elastodynamics. The emphasis of this paper is on both theoretical and numerical aspects. In recent decades, thermomechanical constitutive models have been addressed in numerous works (see, eg, the works of Miehe, 1 Holzapfel and Simo, 2 and Reese and Govindjee 3 as well as textbooks by Holzapfel 4 and Gonzalez and Stuart 5 ). Classically, the hyperelastic Helmholtz free-energy density function depends only on the deformation gradient and the temperature. Furthermore, the weak form is deduced from its strong form. Dependent on the chosen material model, eg, for a Mooney-Rivlin model, the consistent linearization may lead to cumbersome expressions. In contrast to the classic approach, the present work is inspired by the concept of polyconvexity (see, eg, the works of Ball 6 and Ciarlet 7 ), where the Helmholtz free energy is a convex function of the deformation gradient, its cofactor, and its determinant and is concave with respect to absolute temperature. In addition to the polyconvexity-based framework, the present work relies on the cross product between second-order tensors, as introduced in the work of de Boer 8 (see also Appendix B 4.9.3 in another work of de Boer 9 ). This tensor cross product has been used in the context of large-strain Int J Numer Methods Eng. 2018;115:549-577. wileyonlinelibrary.com/journal/nme 549 550 FRANKE ET AL.hyperelasticity in the works of Bonet et al 10,11 and remarkably simplifies the algebraic manipulation of the large-strain continuum formulation and is extended herein to the thermomechanical formulation. Furthermore, the polyconvexity-based framework makes possible a wide range of advanced finite element technologies in the context of continuum mechanics. For example, the introduction of mixed finite elements, 10,12 phase-field fracture models, 13 anisotropic...

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On a tensor cross product based formulation of large strain solid mechanics

Cited by 73 publications

References 39 publications

A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

A first‐order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme

An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics

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