A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity

Zdunek, Adam; Rachowicz, Waldemar; Eriksson, Thomas

doi:10.1016/j.camwa.2016.04.022

Cited by 15 publications

(10 citation statements)

References 21 publications

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“…Since the stiffness of fiber reinforced polymers along fiber directions is considerably higher compared to that in transverse direction, a similar mathematical problem arises in the inextensibility limit for the fiber reinforced polymers and biological tissues. 3,4 One solution to the problem is to use h-or p-refinement strategies. Locking response is known to vanish for high-order triangles p > 4.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, Hu-Washizu-type mixed variational principles for the treatment of the inextensibility limit in fiber-reinforced materials and biological tissues have been studied in related works. 3,4,55,56 The element formulation of Zdunek et al 3,4 is based on the kinematic split of the deformation gradient into purely unimodular extensional part, a purely spherical part, and an extension free unimodular tensor. The Lagrangian element formulation is similar to the mean dilatational approach 32 (MDA) as it uses scalar conjugate pairs ( p, ) and ( , ) for pressure-dilatation and fiber stress-stretch, respectively.…”

Section: Discussionmentioning

confidence: 99%

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A quasi‐incompressible and quasi‐inextensible element formulation for transversely isotropic materials

Dal

2018

Numerical Meth Engineering

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The contribution presents a new finite element formulation for quasiinextensible and quasi-incompressible finite hyperelastic behavior of transeversely isotropic materials and addresses its computational aspects. The material formulation is presented in purely Eulerian setting and based on the additive decomposition of the free energy function into isotropic and anisotropic parts, where the former is further decomposed into isochoric and volumetric parts. For the quasi-incompressible response, the Q1P0 element formulation is outlined briefly, where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu-type mixed variational potential is introduced, where the volume averaged fiber stretch and fiber stress are additional field variables. Within this context, the resulting Euler-Lagrange equations and the element formulation resulting from the extended variational principle are derived. The numerical implementation exploits the underlying variational structure, leading to a canonical symmetric structure. The efficiency of the proposed approached is demonstrated through representative boundary value problems. The superiority of the proposed element formulation over the standard Q1 and Q1P0 element formulation is studied through convergence analyses. The proposed finite element formulation is modular and exhibits very robust performance for fiber reinforced elastomers in the inextensibility limit. KEYWORDSanisotropy, hyperelasticity, mixed finite element design, mixed variational principles, quasi-incompressiblity, quasi-inextensibility

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

A quasi‐incompressible and quasi‐inextensible element formulation for transversely isotropic materials

Dal

2018

Numerical Meth Engineering

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“…It gives rise to the so-called generalised metric method. A unified theory for a number of verified finite element applications [14,[27][28][29][30][31] is presented. Possible extensions are indicated, for example, formulation of a preservation of the angle between two families of material line elements alone or in combination with inextensibility or incompressibility.…”

Section: Summary Conclusion and Outlookmentioning

confidence: 99%

“…The 3-field, displacement, dilatation and pressure formulation by Simo, Taylor and Pister [14, (1985)] for near incompressibility is the first to be mentioned. The following formulations [27][28][29][30] for near incompressibility and/or strongly transversely isotropic finite hyperelasticity are further realizations.…”

Section: Introductionmentioning

confidence: 99%

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On Purely Mechanical Simple Kinematic Internal Constraints

Zdunek

2019

J Elast

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The classic purely mechanical approach to materials with simple kinematic internal constraints is supplemented. A right Cauchy-Green tensor which locally represents the kinematically admissible restricted domain of a finite hyperelastic stress response function is constructed explicitly. It satisfies all imposed constraints identically. It is obtained by a procedure which annihilates the banned modes of deformation in the actual placement. A unique direct sum based stress decomposition is obtained. Further, a procedure is provided, to seamlessly relax constraints, or reversibly, that allows constraints to smoothly develop under loading starting from an unconstrained description. The involved relaxation of internal constraints is briefly illustrated herein. References to published full feathered applications are given where the method is used and verified in finite element form. A. Zdunek internal constraints is laid down. Notably, it leaves the mechanical interpretation of the introduced Lagrange multipliers undetermined. Later, the so-called constraint manifold theory by Podio-Guidugli [3], Gurtin and Podio-Guidugli [5], Podio-Guidugli and Vianello [4] and recently Vianello [6], and by Carlson et al. [7], to mention a few, updated the classic theory using geometric-algebraic analogies from linear-algebra. The additive stress decomposition emerges from spanning the workless stress on the normals to the set of constraints. The work-performing stress is isolated as the orthogonal complement, spanned by the tangent set.The normalisation condition removing the indeterminate part from the extra stress, which makes the stress decomposition unique, is only mentioned as an option in [1, Sect. 30]. It is invoked firmly in the constraint manifold theory (see, e.g., [7]), where orthogonality is assumed between the workless and the work-performing stresses. The constraint manifold theory concludes making the direct sum based stress decomposition unique by constructing the orthogonal projection to the set of constraint normals. It is concluded that the normalisation condition as presented in [1, Sect. 30] and in [7] is an auxiliary assumption to make the stress decomposition unique.According to Steigmann [11] the subject of materials with internal constraints is not especially well treated in text and monographs. The necessity and usefulness of a normalisation condition, introduced in [1, Sect. 30], has been questioned, see for example Bertram and Glüge [9]. Negahban [10] discusses different assumed stress decompositions and normalisations. Normalised work performing stresses are more seldom used in analytic works. See, for example, the presentation of the classic families of controllable non-homogeneous solutions of arbitrary isotropic incompressible materials in Truesdell and Noll [1, Sects. 56 and 57]. In computational mechanics, on the other hand, the advantage of using the normalised deviatoric description of the work-performing stresses in nearly incompressible hyperelasticity was early recognised by Simo, Taylor and ...

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Finite element approximations for near‐incompressible and near‐inextensible transversely isotropic bodies

Rasolofoson

Grieshaber

Reddy

2018

Numerical Meth Engineering

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This work comprises a detailed theoretical and computational study of the boundary value problem for transversely isotropic linear elastic bodies. General conditions for well-posedness are derived in terms of the material parameters. The discrete form of the displacement problem is formulated for conforming finite element approximations. The error estimate reveals that anisotropy can play a role in minimising or even eliminating locking behaviour for moderate values of the ratio of Young's moduli in the fibre and transverse directions.In addition to the standard conforming approximation, an alternative formulation, involving under-integration of the volumetric and extensional terms in the weak formulation, is considered. The latter is equivalent to either a mixed or a perturbed Lagrangian formulation, analogously to the well-known situation for the volumetric term. A set of numerical examples confirms the locking-free behaviour in the near-incompressible limit of the standard formulation with moderate anisotropy, with locking behaviour being clearly evident in the case of near-inextensibility. On the other hand, under-integration of the extensional term leads to extensional locking-free behaviour, with convergence at superlinear rates. KEYWORDS elasticity, finite element methods, solids Int J Numer Methods Eng. 2019;117:693-712.wileyonlinelibrary.com/journal/nme

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A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity

Cited by 15 publications

References 21 publications

A quasi‐incompressible and quasi‐inextensible element formulation for transversely isotropic materials

A quasi‐incompressible and quasi‐inextensible element formulation for transversely isotropic materials

On Purely Mechanical Simple Kinematic Internal Constraints

Finite element approximations for near‐incompressible and near‐inextensible transversely isotropic bodies

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