A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials

Abstract

We present a practical algorithm for partially relaxing multiwell energy densities such as pertain to materials undergoing martensitic phase transitions. The algorithm is based on sequential lamination, but the evolution of the microstructure during a deformation process is required to satisfy a continuity constraint, in the sense that the new microstructure should be reachable from the preceding one by a combination of branching and pruning operations. All microstructures generated by the algorithm are in static and configurational equilibrium. Owing to the continuity constrained imposed upon the microstructural evolution, the predicted material behavior may be path-dependent and exhibit hysteresis. In cases in which there is a strict separation of micro and macrostructural lengthscales, the proposed relaxation algorithm may effectively be integrated into macroscopic finite-element calculations at the subgrid level. We demonstrate this aspect of the algorithm by means of a numerical example concerned with the indentation of an Cu-Al-Ni shape memory alloy by a spherical indenter.

Importance of Shear in the bcc-to-hcp Transformation in Iron

“…The compatibility constraint precludes direct optimization of the variant volume fractions so that the simple Gibbs construction does not apply [25]. The specific algorithm used equilibrate the laminate and optimize its layout is given in [26].…”

“…Moreover, the approach is entirely predictive, since all input data come from first principles, with no calibration or fitting of the model to experiment. Lastly, the model currently does not yet account for nucleation, interfacial energies and kinetics, finite temperature, fcc variants, or plasticity, but these effects can [26] and will be included in future work.…”

On certain convex compactifications for relaxation in evolution problems

“…In fact, such type of compactification has been used for relaxation of various evolutionary phase-change problems e.g. in [2,4,6,7,18,19,22,23,24,29,32,39] and many others. Up to an equivalence introduced in Section 3, we can obtain this convex compactification in the above framework, too.…”

“…e.g. [2,6,18,22,23,24,29,32,39]. This is a relatively standardly used approach, especially if a correlation of microstructures between particular time instances is rather "macroscopical" only.…”

Modeling of shape-memory alloys on the mesoscopic level

“…The microstructure (described by the gradient Young measure) is approximated by a laminate (as already in e.g. [2,13,14,28], some theoretical insight is given in [15]); here of the second order although theoretically a laminate of any order can be used in the code developed by the author. Figure 3 on element zero (the same for which the reconstruction was shown also in 3) performed on a fine 180-element mesh.…”