Rolf Mahnken

A unified approach for parameter identification of inelastic material models in the frame of the finite element method

This work is concerned with identification of parameters for inelastic material models. In order to account for possible non-uniformness of stress and strain distributions, the identification is performed in the frame of the finite element method. In particular, linearization procedures are described in a systematic manner for the case of complex material models within a geometric linear theory. This unified approach allows one to apply the Newton method for solving the associated direct problem and to apply gradient based methods for solving the associated inverse problem, which is considered as an optimization problem. Two numerical examples demonstrate the versatility of our approach: firstly, we consider Cooks membrane problem based on simulated data for re-identification of material parameters for a viscoplastic power law. Furthermore, material data for J2-flow theory are determined, based on experimental data obtained by a grating method for a compact specimen, and we will investigate the results by using different starting values and stochastic perturbation of the experimental data.

Parameter identification for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigations

In this work a unified strategy for identification of material parameters of viscoplastic constitutive equations from uniaxial test data is presented. Gradient-based descent methods (e.g. Gauss-Newton method, Quasi-Newton method) are used for minimization of a least-squares functional, thus requiring the associative gradient. The corresponding sensitivity analysis is explained in detail, where as a main result a recursion formula is obtained. Furthermore, the stability of the numerical results for the material parameters is investigated by use of the eigenvalues for the Hessian of the least-squares functional. Numerical examples are presented in the context of monotonic and cyclic loading. In particular, comparative results with a genetic algorithm reflect the efficiency of our strategy with respect to execution time, and we study the effect of perturbations of the experimental data on the stability of the parameters. In one example we demonstrate how possible instabilities can be circumvented by a regularization of the basic least-squares functional.

Parameter identification for finite deformation elasto-plasticity in principal directions

This work is concerned with identification of material parameters based on experimental data, which represent nonuniform distributions of stresses and deformations within the volume of the specimen. Both elastic and inelastic material non-linearities in the frame of a finite deformation theory are taken into account. Gradient-based descent methods (e.g. Gauss-Newton method, Quasi-Newton method) are used for minimization of a least-squares function. To this end a sensitivity analysis is performed, and the resulting expressions are presented in a spatial and a material setting. In particular, the cases of an isotropic hyperelastic model and a multiplicative plasticity model with an exponential type integration scheme, both formulated in principal directions, are considered. Two numerical examples, based on simulated data and experimental data obtained by a grating method, demonstrate the versatility of our approach.

Theoretical, numerical and identification aspects of a new model class for ductile damage

Abstract In this contribution we compare the yield function of Gurson (Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth—I. Yield criteria and flow rules for porous ductile media. Engineering Materials Technology 99, 2–15), its extensions by Tvergaard (Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. 17, 389–407) and Tvergaard and Needleman (Tvergaard, V. Needleman, A., 1984. Analysis of the cup-cone fracture in a round tensile bar. Acta Metallurgica 32, 157–169), and the yield function by Rousselier (Rousselier, G., 1987. Ductile fracture models and their potential in local approach of fracture. Nuclear Engineering and Design 105, 97–111), formulated for modeling metallic materials with varying induced porosity, with the single-surface model of Ehlers (Ehlers, W., 1995. A single-surface yield function for geomaterials. Archive of Applied Mechanics 65, 246–259), formulated for modeling geomaterials with constant porosity. It is shown, that though obtained from rather different perspectives, all formulations can be casted into a similar mathematical structure. Based on this analogy a new model class of yield functions for simulating isotropic ductile damage is proposed. As a special case the model structure of Greens (Green, R.J., 1972. A plasticity theory for porous solids. Int. J. Mech. Sci. 14, 215–224) ellipsoid is considered, with coefficients dependent on the void volume fraction. The formulation of the rate equations is performed in the spatial configuration based on the multiplicative decomposition of the deformation gradient. Furthermore numerical aspects are addressed concerning the integration of the constitutive relations and the finite element equilibrium iteration, and additionally the sensitivity terms for parameter identification are derived. Two examples illustrate the performance of the proposed strategy.

Aspects on the finite-element implementation of the Gurson model including parameter identification

Abstract In this contribution, various aspects on the finite-element implementation of the Gurson model are considered. In particular, a linear representation for the plastic potential is used, which shows superior convergence property in the local iteration procedure compared to the original quadratic representation. The formulation of the model is performed in the spatial configuration based on the multiplicative decomposition of the deformation gradient, and for integration an exponential map scheme is used. A further important aspect is the sensitivity analysis consistent with the underlying integration scheme necessary for minimizing a least-squares functional for parameter identification by use of a gradient-based optimization algorithm. In a numerical example the local convergence behavior for the two versions of the Gurson model, linear and quadratic are compared. Furthermore material parameters are determined by least-squares minimization based on experimental data obtained for an axisymmetric tensile bar for a ferritic steel.

Parameter identification of gradient enhanced damage models with the finite element method

In this contribution an algorithm for parameter identification of gradient enhanced damage models is proposed, in which non-uniform distributions of the state variables such as stresses, strains and damage variables are taken into account. To this end a least-squares functional consisting of experimental data and simulated data is minimized, whereby the latter are obtained with the Finite-Element-Method. In order to improve the efficiency of the minimization process, a gradient-based optimization algorithm is applied, and therefore the corresponding sensitivity analysis for the coupled variational problem is described in a systematic manner. For illustrative purpose, the performance of the algorithm is demonstrated for a square panel under tension, in which an isotropic gradient damage model is used.

A comprehensive study of a multiplicative elastoplasticity model coupled to damage including parameter identification

Abstract In this contribution various aspects for a plasticity model coupled to damage are considered. The formulation of the model is performed in the intermediate configuration which occurs as a consequence of the multiplicative decomposition of the deformation gradient. We will resort to thermodynamic consistency, continuous tangent operator, algorithmic tangent operator and sensitivity analysis for parameter identification. Furthermore, for the discretized constitutive problem a robust iteration scheme with a two-level algorithm is proposed. In the numerical example material parameters are determined by least-squares minimization based on experimental data obtained with an optical method.

Parameter estimation for a viscoplastic damage model using a gradient‐based optimization algorithm

In this work a gradient‐based optimization method is applied in order to determine material parameters for a viscoplastic model with dynamic yield surface coupled to damage as presented in 1997. To this end a sensitivity analysis consistent with the integration scheme presented previously is performed in a systematic manner, both for strain and stress controlled experiments. The algorithm is tested in two numerical examples: first, simulated data are used, in order to re‐obtain parameters for the case of damage under monotonic loading. In the second example material parameters are obtained based on experimental data for lcf‐testing of an austenetic stainless steel, thus showing a very good agreement with respect to hardening, rate and damage effects.

Strength difference in compression and tension and pressure dependence of yielding in elasto-plasticity

Abstract A characteristic of most metallic materials is nonlinear work-hardening for the flow stress, equal in tension and compression at first loading. On the contrary, triaxial experimental tests for high-strength steels by Spitzig et al. [W.A. Spitzig, R.J. Sober, O. Richmond, Pressure dependence of yielding and associated volume expansion in tempered martensite, Acta Metall. 23 (1975) 885–893] and others also show a pronounced strength-differential effect between compression and tension (S-D effect), and furthermore a sensitivity of the flow stress to the hydrostatic pressure. These macroscopic phenomena are related to volume expansion of the material by the authors in [W.A. Spitzig, R.J. Sober, O. Richmond, Pressure dependence of yielding and associated volume expansion in tempered martensite, Acta Metall. 23 (1975) 885–893] as a consequence of plastic deformation. For simulation of the above observations in this work a yield function dependent on the three basic invariants of a reduced stress tensor and a process vector is proposed. Additionally the yield function is generalized towards the incorporation of a damaging void growth effect due to hydrostatic stresses. Concerning numerical aspects, the resulting local problem with 15 unknowns is reduced to an equivalent three-dimensional problem, and for the finite-element equilibrium iteration the algorithmic tangent operator is derived. Two examples illustrate the capability of the proposed model.

Anistropic creep modeling based on elastic projection operators with applications to CMSX-4 superalloy

Abstract This paper presents a unified framework for creep modeling of anisotropic materials, which is specified in more detail to the cases of isotropy, cubic symmetry and transverse isotropy. To this end an additive decomposition of the elastic and inelastic strain tensors into dilational and isochoric Kelvin modes is assumed. Each of these modes is obtained from fourth-order projection operators, resulting from solution of the eigenvalue problem for the anisotropic fourth-order elasticity tensor. The amount of strain rate for each mode is modeled with a Norton-type ansatz in terms of an equivalent stress. The formulation for the equivalent stress in terms of quadratic forms with aid of the projection operators is compared with polynomial expressions from the literature. The experimental phenomenon of primary creep is taken into account by a back stress tensor of Armstrong–Frederick type, which is also decomposed into Kelvin modes. As a consequence of the mode decomposition the classical radial-return method of isotropic elasto-plasticity is generalized to the different cases of anisotropy. Furthermore the implications on parameter identification are addressed. Two numerical examples are concerned with a superalloy CMSX-4.