A variational approach for materially stable anisotropic hyperelasticity

Schröder, Jörg; Neff, Patrizio; Balzani, Daniel

doi:10.1016/j.ijsolstr.2004.11.021

Cited by 168 publications

(133 citation statements)

References 28 publications

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“…2. One advantage of the proposed formulation is that due to convexity also material stability is guaranteed, see [12]. In addition to that, mesh-independent solutions are observed in further numerical examples, see [3].…”

Section: Numerical Examplesmentioning

confidence: 77%

“…The relaxed model is conceptually in line with the generalized variational approach from [9] and obtained by constructing a rank-one convex hull of the incremental formulation numerically. For the effective energy a polyconvex function is chosen in order to guarantee material stability in the undamaged regime, see [12]. A series of polyconvex functions for fiber-reinforced materials is given in [2].…”

Section: Relaxed Variational Formulationmentioning

confidence: 99%

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Relaxed Incremental Variational Formulation for Damage in Fiber‐Reinforced Materials

Balzani

Ortiz

2012

Proc Appl Math and Mech

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An incremental variational formulation for damage at finite strains is proposed based on the classical continuum damage mechanics. Since loss of convexity is obtained at some critical deformations a relaxed incremental stress potential is constructed which convexifies the original non-convex problem. The resulting model can be interpreted as the homogenization of a micro-heterogeneous material bifurcated into a strongly and weakly damaged phase at the microscale. A one-dimensional relaxed formulation is derived and based thereon, a model for fiber-reinforced materials is given. Finally, some numerical examples illustrate the performance of the model.

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Section: Numerical Examplesmentioning

confidence: 77%

Section: Relaxed Variational Formulationmentioning

confidence: 99%

Relaxed Incremental Variational Formulation for Damage in Fiber‐Reinforced Materials

Balzani

Ortiz

2012

Proc Appl Math and Mech

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“…Here, we briefly describe the extension of these theories to the present application. In order to satisfy material stability restrictions, the strain energy functions used are polyconvex; see, for example, Itskov (2004) and Schroder et al (2005).…”

Section: Appendix A: Incremental Growth Analysismentioning

confidence: 99%

A nonlinear finite element model of cartilage growth

Davol

Bingham

Sah

et al. 2007

Biomech Model Mechanobiol

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The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MAT-LAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.

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“…A growing number of studies on the material stability of elas tic materials have required that the strain energy function be polyconvex (e.g., see, Schroder et al 2005). However, the strain energy function for the second-order model stud ied here does not need to be known, as the material con stants are defined as derivatives of the strain energy function (with respect to the invariants) and evaluated in the reference configuration.…”

Section: Appendix A: Second-order Elastic Materialsmentioning

confidence: 99%

A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage

Klisch

2006

Biomech Model Mechanobiol

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Cartilaginous tissues, such as articular cartilage and the annulus fibrosus, exhibit orthotropic behavior with highly asymmetric tensile-compressive responses. Due to this complex behavior, it is difficult to develop accurate stress constitutive equations that are valid for finite deformations. Therefore, we have developed a bimodular theory for finite deformations of elastic materials that allows the mechanical properties of the tissue to differ in tension and compres sion. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain ten sor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equa tion, with six to nine parameters, and a bimodular exponential equation, with seven parameters, were fit to an experimental dataset that captures the highly asymmetric and orthotropic mechanical response of cartilage. The results suggest that the bimodular second-order models may be appealing for some applications with cartilaginous tissues.

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A variational approach for materially stable anisotropic hyperelasticity

Cited by 168 publications

References 28 publications

Relaxed Incremental Variational Formulation for Damage in Fiber‐Reinforced Materials

Relaxed Incremental Variational Formulation for Damage in Fiber‐Reinforced Materials

A nonlinear finite element model of cartilage growth

A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage

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