Small strain plasticity: classical versus multifield formulation

Schröder, Bettina; Kuhl, Detlef

doi:10.1007/s00419-015-0984-9

Cited by 12 publications

(5 citation statements)

References 28 publications

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“…In this way, we obtain a principle of virtual power in mixed form, or a mixed JOURDAIN principle, respectively [3]. After integrating by parts, we arrive at the EULER-LAGRANGE equations.…”

Section: Continuous Variational Settingmentioning

confidence: 99%

Galerkin‐based energy‐momentum time‐stepping schemes for anisotropic hyperelastic materials

Groß

Ramesh

Dietzsch

2016

Proc Appl Math and Mech

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This paper is motivated by dynamic simulations of fiber-reinforced materials in light-weight structures, and has two goals. First of all, the introduction of energy-momentum schemes for nonlinear anisotropic materials, based on GALERKIN approximations in space and time, assumed strain approximations in time and superimposed algorithmic stress fields (compare [1]). Second, to show a variationally consistent design of energy-momentum schemes using a differential variational principle. We develop a discrete variational principle leading to energy-momentum schemes as discrete EULER-LAGRANGE equations.

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“…In this way, we obtain a principle of virtual power in mixed form, or a mixed JOURDAIN principle, respectively [3]. After integrating by parts, we arrive at the EULER-LAGRANGE equations.…”

Section: Continuous Variational Settingmentioning

confidence: 99%

Galerkin‐based energy‐momentum time‐stepping schemes for anisotropic hyperelastic materials

Groß

Ramesh

Dietzsch

2016

Proc Appl Math and Mech

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“…The time discretization is obtained by applying stiffly accurate RUNGE-KUTTA schemes, cf. [1,4,5]. Thus, in the end a linear system of equations…”

Section: Solution Proceduresmentioning

confidence: 99%

“…For an axisymmetric case of ideal isotropic elastoplasticity following the approach of, cf. [5], the matrices B e , K e σε p can be inverted on element level.…”

Section: Solution Proceduresmentioning

confidence: 99%

Static condensation within the context of multifield elastoplasticity

Schröder

Kuhl

2015

Proc Appl Math and Mech

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In the present paper a multifield approach towards ideal elastoplasticity is presented and its numerical implementation is touched. Furthermore, the strategy of static condensation is applied to this ansatz. With the help of an axisymmetric model problem its realization is demonstrated and compared to the pure multifield approach. MotivationDifferent industrial manufacturing processes are characterized by plastic deformations where dynamic effects cannot be longer neglected, cf. [6]. Thus, the analysis of time discretization schemes linked to plasticity comes to the fore. In this context alternative numerical implementations have to be taken into account and optimized with regard to their computational effort. In order to carry out the time discretization of an elastoplastic problem in one single step a multifield approach will be introduced and examined. Multifield formulation of elastoplasticityIn order to describe elastoplastic deformation processes mathematically, adequate equations have to be derived. Within the framework of a multifield approach this can be done by formulating the balance of power augmented by a dissipation potential, cf. [2]. The latter characterizes the occurring plastic effects. Furthermore, it is assumed that the physical state, where the virtual power reaches a stationary point, is the preferable one. Consequently, an enhanced version of the principle of JOURDAIN is exploited leading to δP = δĖ + δK + δẆ + δD = 0.(1)Therein δĖ represents the internal virtual power, δK the virtual power of inertia effects, δẆ the power due to external forces and δD the virtual dissipation potential. Following the approach in [5] and references therein the nonlinear system of equations can be derived, assuming that δγ ≥ 0 and f (σ) ≤ 0 hold.

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“…This does not obviously result in an equivalent structure compared with common FE implementations (for example, just to mention a few); the equation system may become larger and stronger coupled, and the reuse of existing material routines is probably impossible. For example, for a small strain elasto‐plastic problem. The differences will be highlighted in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…The aim is to benefit from goal‐oriented adaptivity while still maintaining structures of current FE implementations. The main idea is to first introduce a multifield formulation similar to and then discretize the primal problem as in current FE implementations with the interpretation of . Subsequently, we introduce the dual problem via a Lagrangian and perform a rearrangement of the dual equations to structurally match the discretized primal problem as in standard FE implementations, resulting in equations similar to for sensitivity analysis.…”

Section: Introductionmentioning

confidence: 99%

Dual‐based adaptive FEM for inelastic problems with standard FE implementations

Widany

Mahnken

2015

Numerical Meth Engineering

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SUMMARYGoal-oriented error estimation allows to refine meshes in space and time with respect to arbitrary quantities. The required dual problems that need to be solved usually require weak formulations and the Galerkin method in space and time to be established. Unfortunately, this does not obviously leads to structures of standard finite element implementations for solid mechanics. These are characterized by a combination of variables at nodes (e.g. displacements) and at integration points (e.g. internal variables) and are solved with a two-level Newton method because of local uncoupled and global coupled equations. Therefore, we propose an approach to approximate the dual problem while maintaining these structures. The primal and the dual problems are derived from a multifield formulation. Discretization in time and space with appropriate shape functions and rearrangement yields the desired result. Details on practical implementation as well as applications to elasto-plasticity are given. Numerical examples demonstrate the effectiveness of the procedure.

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Small strain plasticity: classical versus multifield formulation

Cited by 12 publications

References 28 publications

Galerkin‐based energy‐momentum time‐stepping schemes for anisotropic hyperelastic materials

Galerkin‐based energy‐momentum time‐stepping schemes for anisotropic hyperelastic materials

Static condensation within the context of multifield elastoplasticity

Dual‐based adaptive FEM for inelastic problems with standard FE implementations

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