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Micromechanical modeling of martensitic phase transformations via energy relaxation

Basics

 

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Martensitic phase transformations are characterised by spontaneous shape changes of the underlying crystalline lattice, for example from body-centered-cubic (bcc) austenite to body-centered-tetragonal (bct) martensite where effects from diffusion are negligible. Such deformations cause inelastic strains which are referred to as Bain strains. The crystal structures of the underlying material's phases furthermore determine the number of possible martensite variants. In the case of bcc to bct transformations, the number of martensite variants is 3.

The cornerstone of the constitutive model is laid by phase energy densities for each martensite variant - the energy wells - from which a representative averaged energy density for the phase mixture is assembled. In this context the assumption of homogeneously distributed strain for the phases leads to a highly non-quasiconvex energy landscape. Stationary states following from such functionals are known to be unstable. Furthermore, the implementation of material models based on non-quasiconvex energies into FE simulations will definitely result in mesh-dependent and thus unphysical results.

 



Relaxation via laminated microstructures

 

In fact, many materials exhibit fine scale periodic microstructures induced by mechanical or thermal loading. In relaxation theory, such microstructures are assumed to regularize the microscopic energy landscape. More precisely, microstructures will form if energetically favorable or, in other terms, energetically necessary in order to avoid non-quasiconvexity. We approximate the materials' microstructure by laminates which yields an upper bound to the quasiconvex hull of the underlying multi-well energy. According to the specifically chosen microstructure an inhomogeneous displacement perturbation field is applied from which (piecewise homogeneous) total strain states for each phase are derived.


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discretised microstructure via laminates       applied roof-like displacement perturbations       example of displacement perturbation field



(Partially) Relaxed energy density

 

The different phase strains depend upon several internal variables which characterise the microstructure of the material, e.g., the volume fraction of each martensite variant and the amplitudes of the displacement perturbation field. In this context we distinguish between elastic variables - which are assumed to evolve with the speed of sound - and dissipative ones. Accordingly, the elastic variables are designed to minimize the averaged energy density which results in relaxed energy density. More precisely, the method we apply along with the energy density we obtain is referred to as rank-one convexification with respect to first and second order laminates.


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Simplified, qualitative 1D example of a relaxed energy density       Corresponding evolution of martensite volume fraction       Material response in terms of stress



Evolution laws and algorithmic implementation

 

The dissipative variables like the phase volume fractions are treated by evolution laws consistently derived from inelastic potentials. Driving forces for the evolution of these variables are derived as thermodynamical conjugates from the relaxed energy density in a standard way. An important aspect of the material routine is the algorithmic treatment of mass restrictions (in terms of martensite volume fractions) and of the monolithic system of differential and algebraic equations. The latter issue requires an unconditionally A- and L-stable integration scheme. For the fulfillment of inequality constraints (exemplified by the aforementioned mass restrictions as well as by the consistency conditions for the rate-independent evolution laws) we use a sophisticated reformulation of Karush-Kuhn-Tucker conditions via Fischer-Burmeister NCP-functions (NCP: Nonlinear Complementarity Problems).




Numerical studies

 

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In [1] the material response of single crystalline shape memory alloys for different martensite systems (tetragonal, orthorhombic, monoclinic) is discussed at length. Furthermore, we focus on the consistent derivation of different types of evolution laws from specifically chosen dissipation potentials. More precisely, the specific choices for the inelastic potentials presented in this work yield von Mises-, multisurface- and viscoplastic-type evolution laws for, e.g., the martensite volume fractions. Among other effects, this material model is capable of simulating the tension-compression asymmetry in terms of the stress response. Furthermore, the transformation induced anisotropy revealed in (single crystal) experiments is as well captured in a natural way.

The numerical examples presented in [2] reveal a significant and interesting result. The underlying problem is characterised by a Representative Volume Element (RVE). It consists of an SMA–matrix (where the aforementioned material model is applied) and a linear isotropic elastic inclusion. This mesostructure is spatially fully resolved and dicretised by Finite Elements. By applying numerical homogenization via a multiscale FE approach (assuming periodic boundary conditions for the introduced displacement fluctuation field), the effective material response of the heterogeneous aggregate subjected to homogeneous macroscopic deformation is calculated. It turns out that, even without presupposing them in this model, heterogeneous distributions of, e.g., volume fractions are energetically favorable. The different phases arrange themselves in locally homogeneous patterns which can be regarded as mesoscopic laminates. In fact, experimental observations confirm that the phase distributions in SMAs are always heterogeneous and that martensite rather forms in such bands than in a continuous manner. In this sense, the applied microscopic material model presented in [1] yields physically well-motivated and mesh independent results without the need of a global regularization.

 



Selected references

 

[1] Bartel, T., Hackl, K. (2009), A micromechanical model for martensitic phase-transformations in shape-memory alloys based on energy-relaxation, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 89: 792–809. doi: 10.1002/zamm.200900244

[2] Bartel, T., Hackl, K. (2010), Multiscale modeling of martensitic phase transformations: on the numerical determination of heterogeneous mesostructures within shape-memory alloys induced by precipitates. Tech. Mech. 30 (4), 324–342.




Contact/Author Information

Dr.-Ing. Thorsten Bartel, Prof. Dr. rer. nat. Klaus Hackl (Ruhr-Universität Bochum)