A novel computational formulation for nearly incompressible and nearly inextensible finite hyperelasticity

Zdunek, Adam; Rachowicz, Waldemar; Eriksson, Thomas

doi:10.1016/j.cma.2014.08.008

Cited by 20 publications

(14 citation statements)

References 24 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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“…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%

“…Small examples are used to illustrate concepts and show applications throughout the note. Finite element applications are found in [27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

“…Its tangent set is known as the set of admissible variations/test functions. Consequently, it is restricted by the first variations of the constraints, i.e., with the tangent set to the constraint manifold.Putting it all together, this contribution presents the unified theoretical development published as verified finite element implementations [27,29,31] which extends the classic theory of kinematic internal constraints, [1,7]. The claimed extensions concern, governing theory and procedures for,…”

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confidence: 98%

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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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ERRATUM: Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p‐version finite elements

2015

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We found out that there is a mistake in the derivative of the strain-energy function ψ ani (C, M 0 ) with respect to the right Cauchy-Green tensor,T = 2∂ψ(C, M 0 )/C. The correct derivative leads, however, to an elasticity tensor, where there is no connection between the shear strains and the shear stresses in the undeformed state. This leads to a singular tangential stiffness matrix in the computations. Thus, a new strain-energy w(I C r ), instead of Eq. (40), has to be found, which yields on the one hand to a stress-free initial configuration, and on the other hand to a reasonable elasticity tensor. We made a few attempts but solved not the problem. In the meanwhile, a similar approach has been published by [1] to which we refer to.This circumstance has to be apologized and we express regret for not working carefully.Acknowledgement We would like to thank Chris Leistner, who found out the inconsistence.

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A novel computational formulation for nearly incompressible and nearly inextensible finite hyperelasticity

Cited by 20 publications

References 24 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

ERRATUM: Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p‐version finite elements

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