Abstract
In classical elasticity theory the stress-field of a dislocation is characterized by a 1∕r-type singularity. When such a dislocation is considered together with an Allen–Cahn-type phase-field description for microstructure evolution this leads to singular driving forces for the order parameter, resulting in non-physical (and discretization-dependent) predictions for the interaction between dislocations and phase-, twin- or grain-boundaries. We introduce a framework based on first strain gradient elasticity to regularize the dislocation core. It is shown that the use of strain energy density that is quadratic in the gradient of elastic deformation results in non-singular stresses but may result in singular driving forces, whereas a strain energy, which is quadratic in the gradient of the full deformation tensor, regularizes both stresses and driving forces for the order parameter and is therefore a suitable choice. The applicability of the framework is demonstrated using a comprehensive example.
Highlights
Phase field approaches have proven to be very powerful for the investigation of the formation and evolution of microstructures due to solid-solid phase transformations and twinning
We introduce a scalar product between second order tensors denoted by “:” as Preprint submitted to JMPS
In this paper we developed a framework for coupling a phase-field description of planar defects such as phase or twin-boundaries with a discrete representation of dislocations within first-strain-gradient elasticity
Summary
Phase field approaches have proven to be very powerful for the investigation of the formation and evolution of microstructures due to solid-solid phase transformations and twinning. The first strain gradient approach advocated by Po et al (2018) has the advantage that the obtained regularization is independent of the type of defect in question and does not require any defect-specific information for the determination of model parameters. In principle, these parameters can directly be obtained from atomistic interaction potentials (Admal et al, 2017). The purpose of this work is to follow a micromorphic approach and to derive a framework which consistently couples first strain gradient elasticity to Allen-Cahn-type microstructure evolution ensuring non-singular driving forces on the order parameters in the presence of line defects
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