Numerical Implementation of Continuum Dislocation Dynamics with the Discontinuous-Galerkin Method.
- PDF / 811,378 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 97 Downloads / 209 Views
Numerical Implementation of Continuum Dislocation Dynamics with the Discontinuous-Galerkin Method. Alireza Ebrahimi ¹ , Mehran Monavari ², Thomas Hochrainer ¹ 1 2
Universität Bremen, Am Biologischen Garten 2, 28359 Bremen, Germany Friedrich-Alexander-Universität Erlangen-Nürnberg, Dr.-Mack-Str. 77, Fürth, Germany
Contact e-mail: [email protected]
ABSTRACT
In the current paper we modify the evolution equations of the simplified continuum dislocation dynamics theory presented in [T. Hochrainer, S. Sandfeld, M. Zaiser, P. Gumbsch, Continuum dislocation dynamics: Towards a physical theory of crystal plasticity. J. Mech. Phys. Solids. (in print)] to account for the nature of the so-called curvature density as a conserved quantity. The derived evolution equations define a dislocation flux based crystal plasticity law, which we present in a fully three-dimensional form. Because the total curvature is a conserved quantity in the theory the time integration of the equations benefit from using conservative numerical schemes. We present a discontinuous Galerkin implementation for integrating the time evolution of the dislocation state and show that this allows simulating the evolution of a single dislocation loop as well as of a distributed loop density on different slip systems.
INTRODUCTION Crystal plasticity is the result of the motion of dislocations which produce a permanent shear of the crystal without altering the crystal structure. Dislocation density based models of crystal plasticity, however, typically assume the dislocation density evolution to be local and driven by strain evolution. With the term continuum dislocation dynamics (CDD) we denominate theories of crystal plasticity which derive strain solely from the motion of dislocations and which consider dislocation fluxes and line length changes. For a pseudo-continuum description of dislocations or for smoothly aligned distributions of only so-called geometrically necessary dislocations (GND) such a theory is given through the evolution of the dislocation density tensor (Kröner-Nye-tensor) (see for example [1]). For averaged descriptions containing also so-called statistically stored dislocations such a theory was first developed in a higher dimensional configuration space [2], from which we recently derived a simplified version without extradimensions [3]. The simplified theory is based on three internal state variables, that are the total dislocation density , the classical dislocation density tensor α and a new variable called the curvature density q . It was argued in [4] that the integral of the curvature density corresponds to
the total number of dislocations in the system up to a factor of 2π . As a result, the curvature density should be a conserved quantity. This means that the time rate of change of q needs to be a divergence of an appropriate flux vector. This is not the case for the evolution equation for q as provided in [3]. In the current paper we derive a conservative version of the evolution of the curvature density. Because of the
Data Loading...
