Section 6: Material modelling in solid mechanics - GAMM 2012

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30 Section 6: Material modelling in solid mechanics idealized phenomena (e.g. nonlinear elastic behaviour and viscous or plastic yielding). Several of those elements are then assembled to one complex material model. This can be achieved by enforcing the equilibrium of stresses on the connecting points between rheological elements. Within the scope of nonlinear continuum mechanics there also exists the concept of rheological elements. The kinematics is described by the deformation gradient which gets multiplicatively decomposed into several sub deformation gradients. For the definition of the material behaviour, a common approach is to formulate a free energy function as a sum of sub free energy functions attributed to the defined sub deformation gradients. By the evaluation of the Clausius-Duheminequality and under consideration of constitutive assumptions a system of equations describing a thermodynamically consistent material behaviour can be derived. Depending on particular definitions, those materials can include a mixture of elastic, viscous and plastic material behaviour. An alternative approach has been presented by Ihlemann (2006). It is based on the additive decomposition of the stress power density and leads to a system of equations for the derivation of the stress equilibrium on defined intermediate configurations as well as the derivation of the total stresses. Special attention has to be given to the definition of accurate stress measures on the intermediate configurations. By this approach, a formal procedure for the direct connection of single elements at large deformation processes can be obtained. The procedure itself is independent of the concrete material behaviour. The concept of direct connection of rheological elements in nonlinear continuum mechanics and some examples for its application will be presented. Furthermore, a numerical strategy for the direct implementation of this concept will be demonstrated. [1] J. Ihlemann (2006), Beobachterkonzepte und Darstellungsformen der nichtlinearen Kontinuumsmechanik, Habilitation, Universität Hannover A stochastic model for the direct and the inverse problem of adhesive materials N. Nörenberg, R. Mahnken (Universität Paderborn) This work deals with the generation of artificial data [1] based on experimental data for adhesive materials and the application of this data to the inverse and the direct problem. In reality there are only a very limited number of experimental data available. Therefore, the prediction of material behaviour is difficult and a statistical analysis with a stochastic proved thesis is nearly impossible. In order to increase the number of tests a method of stochastic simulation based on time series analysis [2] is applied. With artificial data an arbitrary number of data is available and the process of the parameter identification can be statistically analysed. Additionally, two examples are shown, which adapt the analysed material parameter to the direct problem. The stochastic finite element method [3] is used to take into account the distribution and deviation of the fracture strain. [1] S. Schwan, Identifikation der Parameter inelastischer Werkstoffmodelle: Statistische Analyse und Versuchsplanung. Diss. Shaker, 2000. [2] P. J. Brockwell and R. A. Davis, Time series: theory and methods. Springer, 2009 [3] I. Babuška, R. Tempone and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM Journal on Numerical Analysis 42(2) (2005), 800 – 825
Section 6: Material modelling in solid mechanics 31 Analysis of nanoindentation experiments by means of rheological models Holger Worrack, Wolfgang H. Müller (TU Berlin) The nanoindentation technique, which is similar to the instrumented indentation hardness according to MARTENS, is established in the field of material characterization at small dimensions. It is daily practice to analyze nanoindentation data with an almost classical formula based on the publications by Oliver and Pharr and Fischer-Cripps. In this formula the gradient at the beginning of the unloading curve is used to determine Youngs modulus of the tested material, which is one of the material parameters of interest. The procedure works well for elastic time-independent plastic material behavior, for example copper and the calibration material fused silica, even at higher test temperatures. However, low melting solder materials are susceptible to creep behavior, especially at the higher indentation temperatures where the homologous temperature is > 0.5. As a result of the creep effects the beginning of the unloading curve often shows a bulge. This discrepancy from the ideal unloading curve complicates the correct determination of the unloading stiffness and finally yields to incorrect results for Youngs modulus. For this reason, additional analysis procedures are required to determine the material parameters more precisely. In this paper the authors want to give an introduction to an enhanced analysis of nanoindentation data based on rheological models, which are often used to describe the time-dependence of material response. Two examples of such models are the MAXWELL- and the KELVIN-body. In these models springs and dashpots connected in series and/or in parallel are used to describe the time-dependent material response. The viscous parameters, determined from the recorded data during the nanoindentation experiment, can be used to identify the mechanical material parameters. The authors present miscellaneous viscoeleastic and viscoplastic rheological models in connection with the corresponding equations which are used to extract the material properties from the recorded data. Results of the analysis are presented and discussed in context with the results from the classical Oliver and Pharr procedure and with the material parameters published in the literature. Finally, the models are assessed by their applicability of modeling the time-dependent material response of low melting solder materials. S6.12: Special Material Behavior Thu, 16:00–18:00 Chair: Lurie Sergey, Jaan-Willem Simon S1|01–A1 Carbon Fibre Prepregs: Simulation of a Thermo-Mechanical-Chemical Coupled Problem F. Hankeln, R. Mahnken (Universität Paderborn) In automotive industry research is done to replace high strength steel by combinations of steel and carbon-fibre prepregs (pre-impregnated fibres). It is planned to form both steel and uncured prepregs in one step followed by the curing process under pressure in the forming die [1]. The ability to simulate the mechanical behaviour during forming and curing would allow more economical processes. The simulation of prepregs must regard highly anisotropic, viscoelastic and thermal- chemical properties. For this the model is split into an anisotropic elastic part, which represents the fibre fraction and an isotropic, viscoelastic part, representing the matrix. This part also contains curing, causing a dependency on time and temperature. During deep-drawing large deformations are occurring, so a large strain model regarding anisotropy[2], viscoelasticity [3] and curing [4] has been developed. Also experiments were made to validate this model. Current progress is the identification of material parameters.
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