A method for linking thermally activated dislocation mechanisms of yielding with continuum plasticity theory

Abstract

This work combines classical continuum mechanics, the continuously dislocated continuum theory as developed by Kröner and Bilby with discrete dislocation theory to develop quantities that permit models involving interactions between individual dislocations to be incorporated into a description of multiaxial yielding of a material. Two quantities distinguish this approach from earlier efforts: firstly, a dislocation mobility tensor relating the velocity of a dislocation configuration to the net Peach–Koehler force on the configuration and, secondly, a vector quantity representing the dislocation content of the materials. The theory of thermally activated motion of dislocations past obstacles is employed to relate the dislocation velocity to stress by a stress-dependent mobility tensor whose components are determined by the nature of the interaction of the moving dislocation with the obstacle. An example is presented in which the obstacle is a forest dislocation that affects a gliding dislocation through mutual interaction of their stress fields. The development leads to a quantity that can be used as a plastic potential for the construction of an associated flow law.

Acknowledgements

Research facilities were provided by the US Air Force Office of Scientific Research. The author acknowledges helpful suggestions from Dean Mook, Nasr Ghoniem and the referees for improvements to the clarity and rigor of the presentation.

Notes

† Email: [email protected].

‡ A compatible deformation is one in which positions of material points in the deformed state are related to their positions in the undeformed state by a continuous one-to-one mapping.

† Bilby (1960) referred to these quantities as ‘lattice correspondence functions’.

‡ Adams and Olson (1998) referred to this state as ‘local crystal’.

§ For brevity in the following discussion, we shall consider deformations from the natural state at constant temperature Θ₀ and drop this parameter from the descriptions of the various states.

† This definition differs from that of Kröner and Teodosiu (1974) who removed all constraints and considered the state before defects have moved to adjust to the traction-free boundary conditions. However, their assumption places N(X) out of static equilibrium. A traction-free boundary can be maintained without dislocation motion if the dislocations are pinned in place by a suitable distribution of body forces, such as those produced by point defects introduced by irradiation.

‡ Tensor components and operations referred to lattice coordinates will be indicated by lower-case Greek letters with subscripts, while the corresponding quantities referred to spatial coordinates will be indicated by italic letters with subscripts. This convention is the reverse of that used by Bilby et al. (1955) but the same as that used by Teodosiu (1970). Summation with respect to repeated subscripts is implied unless otherwise indicated.

§ Capital italic subscripts on the spatial coordinates indicate the natural state, while lower-case italic subscripts indicate the deformed state.

† The distortions are also called ‘deformation gradients’ (Malvern 1969).

‡ The symmetric part of the dislocation distortion is the same as Eshelby's (1956) ‘transformation strain’, which is also stress free.

† In this and the subsequent discussion, all quantities are evaluated at the general material point x in the deformed state unless otherwise noted, and time derivatives are evaluated at t = t.

† We use the term ‘deformation system’ instead of ‘slip system’ to distinguish situations in which dislocation motion may not be conservative. In this event, the deformation system is defined by the Burgers vector and the normal to the plane of motion of the dislocations.

† In the subsequent discussion we drop the superscript referring to a specific deformation system, with the understanding that the treatment applies to each deformation system separately, the collective behaviour of all systems being obtained by superposition.

† We employ Greek letters for vectors with a roman italic letter as a subscript to emphasize that the quantity is a lattice vector with components referred to the shape coordinates.

† Although all field quantities are evaluated at the material point X, the notation (X) will be omitted for economy.

† In an earlier treatment (Hartley 2001) the pre-exponential v ₀ was approximated by the product of b and ν*, the lattice vibration frequency.

† This expression corrects an error in the power of b _e in equation (21) of an earlier treatment (Hartley 2001).

Additional information

Notes on contributors

Craig S. Hartley†

† Email: [email protected].