Algorithms for computation of stresses and elasticity moduli in terms of Seth–Hill's family of generalized strain tensors

Miehé, Christian; Lambrecht, Matthias

doi:10.1002/cnm.404

Cited by 115 publications

(60 citation statements)

References 17 publications

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“…These tensors are defined as derivatives of the logarithmic elastic strain measure (85) with respect to the current metric g. The recent works Miehe and Lambrecht [73] and Miehe et al [39] give details of the computation. …”

Section: Incremental Variational Formulation Of Additive Plasticitymentioning

confidence: 99%

Anisotropic elastic–plastic analysis of shells at large strains. A comparison of multiplicative and additive approaches to enhanced finite element design and constitutive modelling

Miehé

Apel

2004

Numerical Meth Engineering

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SUMMARYThe article presents new aspects of large-strain anisotropic elastoplastic analyses of shells. On the side of computational shell analysis we focus on recently developed brick-type mixed finite shell elements based on trilinear displacement interpolations equipped with enhanced strain modes and shell-typical assumed strain modifications. Here, we compare two classes of this shell element design based on a multiplicative and an additive definition of an enhanced strain measure, respectively. Furthermore, for both types of continuum-based shell elements we outline unified interfaces to straindriven constitutive stress update algorithms of anisotropic finite plasticity. On the side of computational plasticity we investigate two classes of constitutive models suitable for the description of anisotropic elastoplastic response at finite strains. The first framework uses a multiplicative definition of an elastic strain measure based on a plastic map, the second an additively defined elastic strain measure based on a plastic metric. Both constitutive frameworks use Hencky-type logarithmic strain measures and are specified to a model problem of orthotropic elastic-plastic response of a standard material. The algorithmic treatment of the two constitutive models is based on an incremental variational formulation that defines the stresses and consistent tangent moduli in terms of a function evaluation. We compare the performances of the two finite element formulations and the two constitutive models by means of numerical simulations of sheet drawing processes. For the range of metal plasticity at moderate strains it is shown that results obtained with the different approaches are close to each other, while the additive approaches provide simpler and more efficient settings.

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Section: Incremental Variational Formulation Of Additive Plasticitymentioning

confidence: 99%

Anisotropic elastic–plastic analysis of shells at large strains. A comparison of multiplicative and additive approaches to enhanced finite element design and constitutive modelling

Miehé

Apel

2004

Numerical Meth Engineering

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“…Generalized measures have been used e.g. by Doyle and Ericksen [4], Seth [20], Hill [8], Ogden [15] and Miehe and Lambrecht [13] in the case of nonlinear elasticity.…”

Section: Introductionmentioning

confidence: 99%

A simple orthotropic finite elasto-plasticity model based on generalized stress-strain measures

Schröder

Gruttmann

Löblein

2002

Computational Mechanics

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In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based on generalized stress-strain measures, which reduces for one special case to the so-called Green-Naghdi theory. The main goal is the representation of the governing constitutive equations within the invariant theory. Introducing additional argument tensors, the so-called structural tensors, the anisotropic constitutive equations, especially the free energy function, the yield criterion, the stress-response and the flow rule, are represented by scalar-valued and tensor-valued isotropic tensor functions. The proposed model is formulated in terms of generalized stressstrain measures in order to maintain the simple additive structure of the infinitesimal elasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detail and some representative numerical examples are discussed. IntroductionThe complex mechanical behaviour of elasto-plastic materials at large strains with an oriented internal structure can be described with tensor-valued functions in terms of several tensor variables, usually deformation tensors and additional structural tensors. General invariant forms of the constitutive equations lead to rational strategies for the modelling of the complex anisotropic response functions. Based on representation theorems for tensor functions the general forms can be derived and the type and minimal number of the scalar variables entering the constitutive equations can be given. For an introduction to the invariant formulation of anisotropic constitutive equations based on the concept of structural tensors and their representations as isotropic tensor functions see Spencer [26], Boehler [3], Betten [2] and for some specific model problems see also Schröder [19]. These invariant forms of the constitutive equations satisfy automatically the symmetry relations of the considered body. Thus, they are automatically invariant under coordinate transformations with elements of the material symmetry group. For the representation of the scalar-valued and tensor-valued functions the set of scalar invariants, the integrity bases, and the generating set of tensors are required. For detailed representations of scalar-and tensor-valued functions we refer to Wang [31, 32], Smith [23, 24]. The integrity bases for polynomial isotropic scalar-valued functions are given by Smith [23] and the generating sets for the tensor functions are derived by Spencer [26]. In this work we formulate a model for anisotropic elasto-plasticity at large strains following the line of Papadopoulos and Lu [16].Here we use a representation of the free energy function and the flow rule which fulfill the material symmetry conditions with respect to the reference configuration a priori.Papadopoulos and Lu [16] proposed a rate-independent finite elasto-plasticity model within the framework of a Green-Naghdi type theory, see e.g. Green and Naghdi [5], using a family of generalized stress-strain measures. An extension of this work to anisotr...

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“…squares of the principal stretches and M i=1,2,3 the associated eigenvalue bases (see for example [38]). The total strains are then decomposed into an elastic and a plastic part using an additive Lagrangian formulation…”

Section: Kinematics Of Geometrically Nonlinear Continuum Mechanicsmentioning

confidence: 99%

“…The first and second derivatives of the logarithmic strain with respect to the right Cauchy-Green strain [38] are defined by…”

Section: Kinematics Of Geometrically Nonlinear Continuum Mechanicsmentioning

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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Algorithms for computation of stresses and elasticity moduli in terms of Seth–Hill's family of generalized strain tensors

Cited by 115 publications

References 17 publications

Anisotropic elastic–plastic analysis of shells at large strains. A comparison of multiplicative and additive approaches to enhanced finite element design and constitutive modelling

Anisotropic elastic–plastic analysis of shells at large strains. A comparison of multiplicative and additive approaches to enhanced finite element design and constitutive modelling

A simple orthotropic finite elasto-plasticity model based on generalized stress-strain measures

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

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