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A computational micro-sphere model for the simulation of phase-transformations




In this work, an efficient model for the simulation of polycrystalline materials undergoing solid to solid phase transformations is presented. As a basis, a one-dimensional, thermodynamically consistent phase-transformation model is embedded into a micro-sphere formulation, facilitating to simulate three-dimensional boundary value problems. To solve the underlying evolution equations, an explicit integration algorithm that could be proved to be unconditionally A-stable was developed. In addition to the investigation of homogeneous deformation states, representative finite element examples are discussed.


One-dimensional probabilistic phase transformation model



The one-dimensional phase transformation model implemented in this work can handle an arbitrary amount of material phases. Each material phase is presumed to behave thermo-elastically, thus a Helmholtz free energy function can be assigned to each phase. The overall free energy of the mixture is given by the free energy contributions of the respective constituents, whereby the distortional energy of the phase boundaries is neglected. Based on this, a Legendre-transformation can be carried out in order to obtain a Gibbs potential. The approach of calculating the evolution of the volume fractions is based on statistical physics and makes use of a transformation probability matrix. For simplicity, in the given example a three-phase material with an austenitic parent phase is chosen. Furthermore, two martensite variants, in particular, one tensile variant and one compression variant, are assumed.


Application of the micro-sphere model



In order to generalize the one-dimensional phase-transformation model, a kinematically constrained micro-sphere approach is applied. In the context of kinematically constrained micro-sphere models, the one-dimensional micro-scale strains are obtained using projections of the macro-scale strain tensor with respect to the underlying integration directions. The micro-macro-relations represent the scale bridging equations facilitating to establish connections between micro and macro quantities. Conceptually speaking, the macroscopic state of deformation is transferred to the micro-level by means of projecting the macroscopic strain tensor onto microscopic strains, while the macroscopic stresses---or rather the flux term of the macroscopic balance of linear momentum representation---is determined based on microscopic strain and the related history, respectively microscopic volume fractions. Besides that, the macroscopic stress tensor is assembled from the scalar-valued stress responses of the underlying one-dimensional phase transformation model. In view of the finite element simulations discussed later on, the macroscopic algorithmic tangent operator is additionally computed.


Numerical Example


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To investigate the behaviour of the material under inhomogeneous deformations, the constitutive model is embedded into a finite element formulation. The particular example studied consists of an axisymmetric specimen loaded under tension-compression. The geometry of the axisymmetric rod is depicted in the figure on the left. The bottom of the rod is constrained in axial direction, while on the top an axial displacement is imposed. In the finite element model the displacements are applied to the center of the rod, as for symmetry resons it is sufficient to take into account only one quarter of the cross-section of a double-symmetric rod. The overall height of the specimen is 100 mm while the center cross section has a diameter of 10 mm. Furthermore, the rod initially consists of pure austenite. It is then subjected to cyclic tension and compression. The figure shows the radial stresses occurring within the rod due to the cyclic loads.




R. Ostwald, T. Bartel, A. Menzel, A computational micro-sphere model applied to the simulation of phase-transformations, Z. angew. Math. Mech., 90(7-8):605-622, 2010

Contact/Author Information


Dr.-Ing. Richard Ostwald, Dr.-Ing. Thorsten Bartel, Prof. Dr.-Ing. Andreas Menzel