The goal of this contribution is to enhance a statitics-based phase-transformation model introduced by Govindjee and Hall (2000) in order to take into account plasticity as well as the interaction between phase-transformation and plasticity effects by introducing a so-called plastic inheritance law. Therefore we extended the Helmholtz free energy functions for each material phase, taking into account plastic strains as new variables for each individual phase. Based on the extended multi-well energy potentials, the probabilistic phase transformation model is then derived. In addition, the differential equations driving the evolution of plasticity as well as the potential-based derivation of the individual plastic driving forces are provided. The coupling of phase-transformation and plasticity effects is incorporated by means of a staggered algorithm. To this end, an inheritance algorithm for the inheritance of plastic strains resulting from a propagating phase front is introduced. Therefore two physically reasonable exponential-type inheritance probability functions are presented. Besides shape memory alloys, the model is also applied to TRIP steel material parameters.

We restrict ourselves to two material phases in this work, namely austenite and martensite. The according transformation probability matrix then reduces to two independent components, which are given by the Gibbs energy barriers required for transformation from austenite to martensite and vice versa. The Gibbs energy barriers are obtained by carrying out a Legendre-transformation of the Helmholtz free energy function. Based on the Gibbs energy functions, the transformation energy barriers are given in terms of the minima and intersections of the respective Gibbs potential parabolas in strain-space. For the time-integration of the resulting coupled non-linear system of evolution equations for the volume fractions, we make use of an A-stable explicit algorithm.

For the incorporation of plasticity effects, we - for conceptual simplicity - assume von Mises-type plasticity with linear proportional hardening. Based on the overall free energy potential, the plastic driving force is derived for each phase. With the driving force and the current yield stress at hand, the yield function determining the admissible elastic domain in the respective material phase is computed. Here, the current yield stress is given by the initial yield stress being modified due to accumulated plastic strains present in the respective material phase, where the hardening modulus is considered in addition. Due to the underlying Voigt assumption, an initial stress acts in each phase and is therefore considered as a back stress in the yield function.

When the phase front of a phase evolves throughout a crystal during one discrete time step, the question arises, whether plastic strains present in the decreasing phase are inherited by the phase front of the increasing phase or not. Conceptually speaking, one has to specify to which amount a positive volume fraction increment of one phase inherits plastic strain from the accordingly decreasing phase. It is shown that it is physically reasonable to assume that the inheritance probability is not constant, but rather a function depending on the remaining volume fraction and plastic strain of the decreasing phase, as well as on further material parameters characterizing the actual functional dependency. To this end, two reasonable approaches for introducing exponential-type inheritance probability functions, namely a convex and a concave one, are introduced and elaborated.

Numerous numerical examples were studied in order to investigate the behavior of the coupled model in terms of stress-strain response and evolution of volume fractions. Furthermore, different types of inheritance laws were elaborated. The material model is not only applied to SMA, but also to TRIP steel material parameters. The results obtained for TRIP steel parameters are promising, but not yet completely satisfactory when compared to experimentally observed material behavior. However, we expect the accurracy of the model to increase further by taking into account thermal coupling effects, a specific martensitic compression phase, and the extension to three dimensions, which is a part of ongoing research work.

R. Ostwald, T. Bartel, A. Menzel (2011), A one-dimensional computational model for the interaction of phase-transformations and plasticity, Int. J. of Structural Changes in Solids, 3(1):63-82, 2011

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

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