Section 6: Material modelling in solid mechanics - GAMM 2012

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18 Section 6: Material modelling in solid mechanics Phase separation phenomena in alloys, such as spinodal decomposition and Ostwald ripening can be described by phasefield models of Cahn-Hilliard-type. Realistic models based on thermodynamically correct logarithmic free energies contain highly nonlinear and singular terms as well as drastically varying length scales (cf. [1]). We present a globally convergent nonsmooth Schur-Newton Multigrid method for vector-valued Cahn-Hilliard-type equations ([2, 3]) and a numerical study of coarsening in a binary eutectic AgCu alloy taking into account elastic effects. Vector-valued Cahn-Hilliard computations open the perspective to multicomponent simulations. [1] W. Dreyer, W.H. Müller, F. Duderstadt and T. Böhme, “Higher Gradient Theory of Mixtures”, WIAS-Preprint No 1286 (2008) [2] C. Gräser and R. Kornhuber, “Nonsmooth Newton Methods for Set-valued Saddle Point Problems”, SIAM J. Numer. Anal. (2009) 47 (2) [3] C. Gräser, R. Kornhuber, “Schur-Newton Multigrid Methods for Vector-Valued Cahn–Hilliard Equations”, in prep A Phase Field Model for Martensitc Transformations Regina Schmitt, Ralf Müller, Charlotte Kuhn (TU Kaiserslautern) Considering the microscopic level of steel, there are different structures with different mechanical properties. Under mechanical deformation the metastable austenitic face-centered cubic phase transforms into the tetragonal martensitic phase at which transformation induced eigenstrain arises. On the other hand, the mircostructure affects the macroscopic mechanical behavior of the specimen. In order to take the complex interactions into account, a phase field model for martensitic transformation is developed. Within the phase field approach, an order parameter is introduced to indicate different material phases. Its time derivative is assumed to follow the time-dependent Ginzburg-Landau equation. The coupled field equations are solved using finite elements together with an implicit time integration scheme. With the aid of this model, the effects of the elastic strain minimization on the formation of microstructure can be studied so that the evolution of the martensitic phase is predictable. The applicability of the model is illustrated through different numerical examples. Investigation of the strain localization behavior with application of the phase transition approach M. Ievdokymov, H. Altenbach, V.A. Eremeyev (Universität Magdeburg) Metal foams found recently many applications in civil, airspace and mechanical engineering. In particular, foams have a good energy absorption property. This property relates to the phenomenon of localization of strains in foams under loading. In porous materials the localization of strains leads to change of mass density of material and appearance of areas with low and high densities separated by sharp interface. This behavior is similar to the phase transitions in solids with sharp interfaces. In this paper we use the methods of modelling of phase transitions in solids to the description of strain localization in foams. We assume that the foam consist of two phases with different densities and mechanical properties. The results of modelling of one- and two-dimensional problems are discussed. The calculations are performed by Abaqus and script language Python. S6.8: Elasticity, Viscoelasticity, -plasticity II Wed, 16:00–18:00 Chair: Ismail Caylak S1|01–A1
Section 6: Material modelling in solid mechanics 19 A New Continuum Approach to the Coupling of Shear Yielding and Crazing with Fracture in Glassy Polymers Lisa Schänzel, Christian Miehe (Universität Stuttgart) Over the past decades, considerable effort was made to develop constitutive models that account for finite viscoplasticity and failure in glassy polymers. Recently, we developed a new model of ductile thermoviscoplasticity of glassy polymers in the logarithmic strain space [1]. However, depending on thermal and loading rate conditions to which the material is subjected, the response might change from ductile to brittle. This brittle response is characterized by inelastically deformed zones, so-called crazes, having the thickness of micrometers and spanning at some fractions of a millimeter [2]. The crazing is associated with considerable dilatational plasticity, containing a dense array of fibrils interspersed with elongated voids. The shear yielding and crazing are not completely independent excluding each other. In this lecture, we outline an extension of the ductile plasticity model [1] towards the description of (i) volumetric directional plasticity effect due to crazing and (ii) the modeling of the local failure due to fracture. To this end, the first extension accounts for a (i) dilatational plastic deformation mechanism in the direction of the maximum principle tensile stress. The ultimate amount of this volumetric plastic craze strain is bounded by a limiting value, where failure occurs. Then, in a second step, the (ii) modeling of subsequent failure mechanisms is realized by the introduction of a fracture phase field, characterizing via an auxiliary variable the crack topology. Here, we adopt structures of a recently developed continuum phase field model of fracture in brittle solids [3], and modify it for a fracture driving term related to the volumetric plastic deformation of the crazes. We demonstrate the performance of proposed formulation by means of representative boundary value problems. [1] Miehe, C.; Mendez, J.; Göktepe, S.; Schänzel, L. [2011]: Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory. International Journal of Solids and Structures, 48: 1799–1817. [2] Kramer, E. J.[1983]:Microscopic and Molecular Fundamentals of Crazing. Advances in Polymer Science, 52/53: 1–56. [3] Miehe, C; Hofacker, M.; Welschinger, F. [2010]: A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 199: 2765-2778. Anisotropic finite strain hyperelasticity based on the multiplicative decomposition of the deformation gradient Raad Al-Kinani, Kaveh Talebam, Stefan Hartmann (TU Clausthal) Frequently, the case of finite strain anisotropy, particularly, the case of transversal isotropy, is applied to biological applications or to model fiber-reinforced composite materials. In this article the multiplicative decomposition of the deformation gradient into one part constrained in the direction of the axis of anisotropy and one part describing the remaining deformation is proposed. Accordingly, a form of additively decomposed strain-energy function is proposed. This leads to a clear assignment of deformation and stress states in the direction of anisotropy and the remaining part. The decomposition of the case of transversal isotropy is explained. The behavior of the model is investigated at different simple analytical examples, such as uniaxial tension along and perpendicular to the axis of anisotropy, and simple shear. In addition, the model is also studied for the case of a thick-walled tube under internal pressure, where a second order ordinary differential
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