Incompatibility of strains and its application to mesoscopic studies of plasticity

Abstract

Structural transitions are invariably affected by lattice distortions. If the body is to remain crackfree, the strain field cannot be arbitrary but has to satisfy the Saint-Venant compatibility constraint. Equivalently, an incompatibility constraint consistent with the actual dislocation network has to be satisfied in media with dislocations. This constraint can be incorporated into strain-based free energy functionals to study the influence of dislocations on phase stability. We provide a systematic analysis of this constraint in 3D and show how three incompatibility equations accommodate an arbitrary dislocation density. This approach allows the internal stress field to be calculated for an anisotropic material with spatially inhomogeneous microstructure and distribution of dislocations by minimizing the free energy. This is illustrated by calculating the stress field of an edge dislocation and comparing it with that of an edge dislocation in an infinite isotropic medium. We outline how this procedure can be utilized to study the interaction of plasticity with polarization and magnetization.

Coupled Nonlinear Ginzburg-Landau and Mechanics Model for Martensitic Transformations in Polycrystals

“…Figure 6.3(c) and (d) show the morphology of martensites represented by the deviatoric strain e 2 and the deviatoric stress σ 2 undergoing a square-to-rectangular MT. The characteristic deviatoric strain e 2 of the lattice caused by an edge dislocation in the middle of the block is similar to the strain based TDGL results of [GLS08,GLS10]. However, our simulated stress field shows the internal stress is only accumulated around the dislocation, while a stress-free state is found in the interior parts of the perfectly formed martensite variants.…”

“…The high similarity inspires the initiatives of integrating these two theories to shed light on the physical mechanisms of the interplaying of dislocations and phase transformations, which gives rise to the formation of microstructure. Recently, Gröger utilized Kröner's continuum theory to deal with the formation of distinct microstructure in a MT affected by the dislocations [GLS08,GLS10]. The presence and evolution of dislocations were depicted as the response to the nonlocal coupling of the incompatibility field and the OP strains.…”

“…Following the strategy of [GLS08], only the motion of crystal dislocations by glide in their corresponding slip planes is considered. The evolution of the scalar dislocation density is In comparison is the strain-based method developed by Gröger [GLS08,GLS10], in which the introduction of incompatibilities allows the coupling to dislocation behavior via a Nye tensor, and then the incompatibilities enter the long-range kernel of OP strains and evolve with OP strains to represent the microstructure of martensites and dislocations. It has advantages of clear physical picture but fails to describe the stress free state inside the martensite after the coupled transformation.…”

Modelling Avalanches in Martensites

“…Arbitrary distributions of crystal dislocations within snap-springs may in general lead to a non-zero Burgers vector at mesoscopic scales and this translates into a lack of elastic compatibility between snap-springs. The non-compatibility effects were analysed in [112,113] using Landau theories for weak phase transitions. In principle, such methods could be extended to be included in the model described here.…”

Continuum Mechanics of the Interaction of Phase Boundaries and Dislocations in Solids

“…We note here the work of[SB98] that discusses kinetics of junctions of phase-boundaries under the assumption that the deformation is compatible while[GLS10],[PBSR08] discuss coupling of strain incompatibilities with phasetransformation without explicit kinematic consideration of strain-gradient incompatibilities due to terminating phase boundaries.2 However, a dislocation-only defect model does not require any consideration of torque balance or couple stresses, as shown in[Ach11,AF12] and in Sec 5.3.…”