Reiner Kreißig

A material model for finite elasto-plastic deformations considering a substructure

Abstract Developing further the substructure models proposed by Mandel and Dafalias a thermodynamically consistent system of differential and algebraic equations is derived to describe anisotropic elasto-plastic material behavior at finite deformations. Based on the multiplicative split of the deformation gradient an appropriate material law is formulated applying the principle of the maximum of plastic dissipation. Generalized basic relations of this material model containing a relation of hyperelasticity, evolutional equations for the internal variables describing different kinds of hardening, and the yield condition are presented. The capacity of the proposed material model is demonstrated on the example of a sheet with a hole. Presenting the evolution of yield surfaces the capability of the model to describe anisotropic hardening behavior is shown.

Anisotropic hardening – numerical application of a cubic yield theory and consideration of variable r-values for sheet metal

Abstract The first part of this paper contains a polynomial yield condition of third order connected with evolution equations for material tensors of higher orders. They are formulated by formal generalisation of an approach by Danilov. The second part presents a possibility of taking into account the rotation of the yield surface as a result of a variable planar anisotropy ( r -value) in sheet metal. This is done by an extension of the evolution equations, based on a quadratic yield function. The corresponding deformation law and the set of evolution equations are numerically integrated for selected loading paths in the subspaces σ 1 , σ 2 and σ , τ . Some of the experimentally observed effects, such as the increasing curvature of the yield locus curve in the loading direction or the specific rotation of the yield surface, are correctly reproduced.

Development of a generalized material interface for the simulation of finite elasto-plastic deformations

Suitable material models have to be chosen to achieve realistic numerical simulations of the mechanical behavior of components and structures as well as a reliable identification of material parameters for finite deformations analyzing inhomogeneous displacement fields. The fundamental relations of the kinematics of an elasto-plastic continuum are formulated using the multiplicative split of the deformation gradient, the concept of dual variables and the covariance principle. A thermodynamically consistent system of differential and algebraic equations (DAE) is derived in the reference configuration to describe the rate-independent inelastic material behavior. It contains the associated flow rule, evolutional equations for the internal variables describing different kinds of hardening and the yield condition. An algorithm for the solution of the DAE based on a suitable time discretization of the differential equations is presented as well as a numerical example using the experimental FE-code PMHP on parallel computers developed at the Chemnitz University of Technology. An important advantage of the presented algorithm is its high efficiency in case of the determination of the consistent material matrix as well as within the scope of the semianalytical sensitivity analysis as a deciding part of the parameter optimization process.

Höhere Technische Mechanik

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Hierarchical Adaptive FEM at Finite Elastoplastic Deformations

The simulation of non-linear problems of continuum mechanics was a crucial point within the framework of the subproject “Efficient parallel algorithms for the simulation of the deformation behaviour of components of inelastic materials”. Nonlinearity appears with the occurence of finite deformations as well as with special material behaviour as e.g. elastoplasticity. To solve such kind of problems by means of a Finite Element simulation for reliable predictions of the response of components and structures to mechanical loading, the insertion of suitable material models into FE codes is a basic requirement. Additionally, their efficient numerical implementation plays also an important role. Likewise, a realistic numerical simulation essentially depends on appropriate FE meshes. According to the motto “as fine as necessary, as coarse as possible”, several techniques of mesh adaptation have been developed in the last decades. These days the feature of adaptivity is a high-performance attribute for FE codes. We want to present the results of our research within the framework of the subproject D1 concerning the efficient treatment of non-linear problems of continuum mechanics:

About an Efficient and Consistent Numerical Strategy for the Solution of the Initial-Boundary Value Problem for Anisotropic Finite Elastoplasticity

Geometrically and physically nonlinear field problems of solid mechanics are numerically processed using suitable spatial discretization methods solving the boundary value problem and time discretization procedures treating the embedded initial value problem (IVP). In case of anisotropic elastoplasticity the IVP is usually defined by a system of differential and algebraic equations (DAE): a stress-strain relation in rate formulation, evolutional equations for the internal variables, and a yield condition.

Analysis of Inhomogeneous Displacement Fields for the Identification of Parameters for Elasto-Plastic Deformation Laws

The design and development stages of the production process have increasingly changed using numerical methods for strength analysis and the optimization of components and structures. Costly experimental test series have been superseded by more economical numerical calculations the results of which can be experimentally verified on optimized prototypes. Especially the Finite Element Method (FEM) has been established in the industrial practice as an effective numerical aid for engineers. The results of the numerical simulation of the mechanical behaviour essentially depend on the implemented material laws which describe the real material behaviour sufficiently exactly.

Towards the Direct and Inverse Adaptive Mixed Finite Element Formulations for Nearly Incompressible Elasticity at Large Strains

This contribution presents advanced numerical models for the solution of the direct and inverse problems of nearly incompressible hyperelastic processes at large strains. The discussed mixed finite element approach contributes to the numerical simulation of coupled multiphysics problems, including the calibration of appropriate material models (parameter identification). The presented constitutive approach is based on the multiplicative decomposition of the deformation gradient resulting in a two-field formulation with displacement components and hydrostatic pressure as primary variables. The ill-posed inverse problem of parameter identification analyzing inhomogeneous displacement fields is solved using deterministic trust-region optimization techniques.Within this context, a semi-analytical approach for sensitivity analysis represents an efficient and accurate method to determine the gradient of the objective function. Themixed boundary value problem is based on the spatial discretization of the weak formulations of the linear momentum balance and the incompressibility condition. Its linearization serves as basis for the solution of the direct problem, while the implicit differentiation of the weak formulations with respect to material parameters provides the necessary relations for the semi-analytical sensitivity analysis. Adaptive mesh refinement and mesh coarsening are realized controlled by a residual a posteriori error estimator. Efficiency and accuracy of the presented direct and inverse numerical techniques are demonstrated on a typical example.

Grundgleichungen der linearen Elastizitätstheorie

Die Beschreibung der stoffunabhangigen und der stoffabhangigen Gleichungen der linearen Elastizitatstheorie erfolgt unter der Voraussetzung, dass diese im Wesentlichen bereits aus der Festigkeitslehre bekannt sind. Obwohl zumeist ein anderer Zugang Verwendung findet, stellen die sich anschliesenden Ausfuhrungen quasi eine Wiederholung unter Nutzung des im Kapitel 1 behandelten Tensorkalkuls dar.

Einführung in die Tensorrechnung

Der Tensorkalkul stellt ein wichtiges mathematisches Hilfsmittel in der Physik sowie zunehmend in den theoretischen Ingenieurwissenschaften dar. Seine Anwendung ermoglicht bei nur wenigen Rechenregeln eine kompakte Darstellung komplizierter Zusammenhange. Da z. B. in den Handbuchern fur kommerzielle Software (Methode der finiten Elemente, der Randelemente u. a.) die kontinuumsmechanischen Grundlagen unter Verwendung dieses Kalkuls beschrieben werden, sind entsprechende Kenntnisse auch im Anwendungsbereich erforderlich.