Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor
- PDF / 791,370 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 22 Downloads / 201 Views
Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor
Thomas Hochrainer Universität Bremen, IW3, Am Biologischen Garten 2,28359 Bremen, Germany. Contact e-mail: [email protected]
ABSTRACT In the current paper we present a continuum theory of dislocations based on the second-order alignment tensor in conjunction with the classical dislocation density tensor (Kröner-Nye-tensor) and a scalar dislocation curvature measure. The second-order alignment tensor is a symmetric second order tensor characterizing the orientation distribution of dislocations in elliptic form. It is closely connected to total densities of screw and edge dislocations introduced in the literature. The scalar dislocation curvature density is a conserved quantity the integral of which represents the total number of dislocations in the system. The presented evolution equations of these dislocation density measures partly parallel earlier developed theories based on screw-edge decompositions but handle line length changes and segment reorientation consistently. We demonstrate that the presented equations allow predicting the evolution of a single dislocation loop in a non-trivial velocity field.
INTRODUCTION The (re-)discovery of size effects in micro- and nano-plasticity during the 1990s into the early 2000s revived interest in dislocation density based models of crystal plasticity. Inspired by the successful quasi-two dimensional dislocation theory for straight edge dislocations developed around the last turn of century [1] several authors proposed dislocation flux based models also for general (curved) dislocation densities based on a screw-edge decomposition of the dislocation density [2,3]. Upon fixing positive directions for edge and screw dislocations these models use separate densities of positive and negative screw s and edge dislocations e . From these densities the Kröner-Nye tensor α of the given slip system may be obtained using the net dislocation densities s s s and e e e of screw and edge dislocations, respectively, as α κ b ses eee bee . Here κ denotes the average dislocation density vector, b is the Burgers vector (with modulus b ) and es and e e denote the unit vectors in positive screw and edge dislocation direction, respectively. These models are kinematically self-contained in the sense that this density information suffices for deriving the plastic slip rate and the density evolution on the system in closed form. With the total screw s s s and edge e e e dislocation densities the plastic slip rate is obtained by Orowan’s equation,
vss vee b, where vs and ve are the average velocities of screw and edge dislocations, respectively. In the sequel we will assume equal velocities for screw and edge dislocations and introduce the average velocity v vs ve . Together with the total dislocation density e s we obtain the classical Orowan equation vb . Under the premise of the isotropic velocity and leaving aside
Data Loading...
