A comparison between inverse form finding and shape optimization methods for anisotropic hyperelasticity in logarithmic strain space

Germain, Sandrine; Steinmann, Paul

doi:10.1002/pamm.201110175

Cited by 14 publications

(24 citation statements)

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“…The inverse mechanical formulation was first introduce in [1] and extended in [2] to anisotropic hyperelasticity in the logarithmic strain space. This method is based on the mechanical equilibrium of the deformed body in the spatial configuration B t under the following assumptions: body forces and inertia are omitted, the material is homogeneous and the surface tractions are independent of the inverse deformation map Φ.…”

Section: Inverse Mechanical Formulationmentioning

confidence: 99%

“…An algorithmic solution is presented in [4] and consists in dividing the total applied force in small increments and updating the reference configuration of the functional component between two incremental steps. The shape optimization formulation is not suitable for hyperelastic behavior because of higher computational costs compared to the inverse mechanical formulation [5] but is necessary for elastoplastic materials. Another method for avoiding the mesh distortions could be a remeshing of the functional part between two optimization iterations.…”

Section: Section 6: Materials Modelling In Solid Mechanicsmentioning

confidence: 99%

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On two different inverse form finding methods for hyperelastic and elastoplastic materials

Germain

Steinmann

2012

Proc Appl Math and Mech

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This contribution focuses on two different developments of mechanical-computational methods for the optimal determination of the initial shape of formed functional components knowing the deformed configuration, the applied loads and the boundary conditions. The first method uses an inverse mechanical formulation and can be applied to materials with hyperelastic behaviors. For materials with elastoplastic properties this method is not advocated, without knowing the final plastic strains, due to the non uniqueness of the solution. The second method uses a shape optimization formulation in the sense of an inverse problem via successive iterations of the direct problem. For hyperelastic materials the inverse mechanical formulation is preferred for its velocity and the non exhibition of possible mesh distortions. In the shape optimization formulation mesh distortions can be avoided by an update of the reference configuration of the functional part. Both methods are using a formulation in the logarithmic strain space. A numerical example for materials with isotropic elastoplastic behaviors illustrates the shape optimization formulation.

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Section: Inverse Mechanical Formulationmentioning

confidence: 99%

Section: Section 6: Materials Modelling In Solid Mechanicsmentioning

confidence: 99%

On two different inverse form finding methods for hyperelastic and elastoplastic materials

Germain

Steinmann

2012

Proc Appl Math and Mech

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“…In order to overcome the large computational costs ( [20,21]) in shape optimisation and the fact that the set of internal variables is unknown at the deformed state, we propose, in this contribution, a new method for solving inverse form finding problems in isotropic elastoplasticity based on the inverse mechanical formulation originally proposed in [1].…”

Section: Introductionmentioning

confidence: 99%

On a recursive formulation for solving inverse form finding problems in isotropic elastoplasticity

Germain

Landkammer

Steinmann

2014

Advanced Modeling and Simulation in Engineering Sciences

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Background: Inverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. The deformed configuration of the functional component, the applied forces and boundary conditions are given and the undeformed configuration of this component is sought. Methods:In this paper we present a new recursive formulation for solving inverse form finding problems for isotropic elastoplastic materials, based on an inverse mechanical formulation written in the logarithmic strain space. First, the inverse mechanical formulation is applied to the target deformed configuration of the workpiece with the set of internal variables set to zero. Subsequently a direct mechanical formulation is performed on the resulting undeformed configuration, which will capture the path-dependency in elastoplasticity. The so obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the inverse mechanical formulation is started again with the target deformed configuration and the current state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached. Results: In our three numerical examples in isotropic elastoplasticity, the convergence was reached after five, six and nine iterations, respectively, when the set of internal variables is initialised to zero at the beginning of the computation. It was also found that when the initial set of internal variables is initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. Different starting values for the set of internal variables have no influence on the obtained undeformed configuration, if convergence can be achieved. Conclusions:With the presented recursive formulation we are able to find an appropriate undeformed configuration for isotropic elastoplastic materials, when only the deformed configuration, the applied forces and boundary conditions are given. An initial homogeneous set of internal variables equal to zero should be considered for such problems.

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“…This problem is inverse to the standard (direct) static analysis, in which the undeformed shape is known and the deformed unknown. In [1] the method originally proposed in [2] is extended to anisotropic hyperelasticity that is based on logarithmic (Hencky) strains. In this contribution, we present an extension of our work [1] to anisotropic elastoplasticity.…”

Section: Introductionmentioning

confidence: 99%

“…In [1] the method originally proposed in [2] is extended to anisotropic hyperelasticity that is based on logarithmic (Hencky) strains. In this contribution, we present an extension of our work [1] to anisotropic elastoplasticity. The logarithmic strains are decomposed into an elastic and a plastic part.…”

Section: Introductionmentioning

confidence: 99%

On Inverse Form Finding for Anisotropic Elastoplastic Materials

Germain¹,

Steinmann²

2011

AIP Conference Proceedings

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A rheological model of the dynamic behavior of magnetorheological elastomers J. Appl. Phys. 110, 013513 (2011) Enhanced friction of elastomer microfiber adhesives with spatulate tips Appl. Phys. Lett. 91, 221913 (2007) Compression stress relaxation apparatus for the long-time monitoring of the incremental modulus Rev. Sci. Instrum. 74, 4737 (2003) High-frequency ultrasound array element using thermoelastic expansion in an elastomeric film Appl. Phys. Lett. 79, 548 (2001) Model of magnetorheological elastomers J. Appl. Phys. 85, 3348 (1999) Additional information on AIP Conf. Proc. Abstract. The goal of this contribution is to present an extension of the work of Germain et al.[1] on inverse form finding for anisotropic hyperelastic materials to the case of anisotropic elastoplastic materials formulated in the logarithmic strain space. A short review of the theoretical aspects is presented. Based on some numerical examples we show that our method for inverse form finding in elastoplasticity can be used if the plastic strains are previously given as it is the case when a desired hardening state is prescribed.

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A comparison between inverse form finding and shape optimization methods for anisotropic hyperelasticity in logarithmic strain space

Cited by 14 publications

References 1 publication

On two different inverse form finding methods for hyperelastic and elastoplastic materials

On two different inverse form finding methods for hyperelastic and elastoplastic materials

On a recursive formulation for solving inverse form finding problems in isotropic elastoplasticity

On Inverse Form Finding for Anisotropic Elastoplastic Materials

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