Efficient modeling of microstructure evolution in magnesium by energy minimization

Abstract

The description of the complex interplay between deformation-induced twinning and dislocation slip, typical for metals showing an hcp structure such as magnesium, is of utmost importance for understanding their deformation behavior. In the present paper, an incremental energy principle is presented for that purpose. Within this principle, dislocation slip is modeled by crystal plasticity theory, while the phase decomposition associated with twinning is considered by a mixture theory. This mixture theory naturally avoids the decomposition of the twinning process into so-called pseudo-dislocations followed by a reorientation of the total crystal. By way of contrast, the proposed model captures the transformation of the crystal lattice due to twinning in a continuous fashion by simultaneously taking dislocation slip within both, possibly co-existent, phases into account. The shear strain induced by twinning as well as the deformation history are consistently included within the twinned domain by an enhanced multiplicative decomposition of the deformation gradient. Kinematic compatibility between the different phases is enforced by a Hadamard-type compatibility condition, while compatibility with respect to the boundary conditions requires the introduction of a boundary layer. The evolution of all state variables such as the twinning volume and the plastic strains associated with dislocation slip follow jointly and conveniently from minimizing the stress power of the total crystal. This canonical variational principle is closely related to the postulate of maximum dissipation and guarantees thermodynamical consistency of the resulting model. Particularly, the second law of thermodynamics is fulfilled. In sharp contrast to previous models suitable for the analysis of the deformation systems in magnesium, the Helmholtz energy of the twinning interfaces and that of the aforementioned boundary layer are considered. Analogously, the energy due to twinning nucleation and that related to twinning growth are accounted for by suitable dissipation functionals. By doing so, the number of twinning laminates becomes an additional unknown within the minimization principle and thus, the thickness of the lamellas can be computed. Interestingly, by interpreting this thickness as the mean free path of dislocations, a size effect of Hall-Petch-type can naturally be included within the novel model. The predictive capabilities of the resulting approach are finally demonstrated by analyzing the channel die test. For that purpose, a certain rank-two laminate structure is considered. However, it bears emphasis that the proposed framework is very general and consequently, it can also be applied to other materials.

A variational constitutive model for slip-twinning interactions in hcp metals: Application to single- and polycrystalline magnesium

“…This is different than e.g. the recent model of Homayonifar and Mosler (2012) who assumed all dislocations are consumed by the moving twinning front and no dislocations are transferred from the untwinned to the twinned orientation.…”

“…Both slip and twinning contribute to the plastic deformation and the specific order of these two mechanisms is not obvious. Therefore, we adopt the approach of Kalidindi (2001) and work with a single inelastic deformation gradient F in to incorporate both relaxation schemes rather than further splitting F in multiplicatively into plastic slip and twinning components (Homayonifar and Mosler, 2011Mosler, , 2012. This further implies that no kinematic interactions between slip an twinning are built into this decomposition but will be accounted for by the specific constitutive model presented below.…”

Incremental work minimization algorithm for rate‐independent plasticity of single crystals

“…More promising here is the incremental energy minimization extended to second-order terms with due account of geometric nonlinearity, developed for plastic instability problems by Petryk [36][37][38], in view of its effectiveness in selecting exact post-bifurcation modes among nonunique possibilities [39,40]. A similar idea of incremental energy minimization but extended to a finite time step has frequently appeared in more recent papers by other authors, e.g., [41][42][43][44][45][46][47]. For a piecewise regular solution path so determined to provide an exact solution in the first-order rates in the limit as the magnitude of all increments tends to zero, the rate problem must be of potential type with respect to unknown rates of internal variables and hence must satisfy the related symmetry restriction [48].…”

On the Large Strain Torsion of HCP Polycrystals

“…Tremendous experimental and theoretical works have been very recently undertaken to better understand and more accurately describe plastic deformation in HCP materials under various deformation processes [see e.g., Abdolvand et al, 2011;Beyerlein et al, 2011;Chapuis and Driver, 2011;Choi et al, 2011;Chun and Davies, 2011;Ghaderi and Barnett, 2011;Guillemer et al, 2011;Homayonifar and Mosler, 2012;Izadbakhsh et al, 2011;Jonas et al, 2011;Khan et al, 2011;Ma et al, 2012;Mayama et al, 2011;Ostapovets et al, 2012;Park et al, 2012;Raeisinia et al, 2011;Segurado et al, 2012;Stanford et al, 2011;Wang et al, 2012a;Xin et al, 2012;Yu et al, 2011;Zhang et al, 2011].…”