A discrete dislocation analysis of residual stresses in a composite material

Abstract

A bstractResidual stresses and strains in a two-dimensional model composite consisting of elastic reinforcements in a crystalline matrix are analysed. The composite is subject to macroscopic shear and then unloaded. Plane-strain conditions and single slip on slip planes parallel to the shear direction are assumed. The dislocations are modelled as line defects in a linear elastic medium. A t each stage of loading, superposition is used to represent the solution in terms of the in® nite medium solution for the discrete dislocations and an image solution that enforces the boundary conditions, which is non-singular and obtained f rom a linear elastic ® nite-element solution. The lattice resistance to dislocation motion, dislocation nucleation and dislocation annihilation are incorporated into the f ormulation through a set of constitutive rules. Obstacles leading to possible dislocation pile-ups are also accounted for. Considerable reverse plasticity is found when the reinforcement arrangement is such that all slip planes are cut by particles and when the unloading rate is equal to the loading rate. When unloading takes place at a very high rate, the unloading slope is essentially elastic but relaxation of the dislocation structure occurs in the unloaded state. Predictions of the discrete dislocation formulation f or residual stresses, residual strains and the strain variance are compared with corresponding predictions obtained using conventional continuum slip crystal plasticity. The e ect of particle size, as predicted by the discrete dislocation description, is also addressed.

A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations

“…The resulting theory contains a length scale through a set of coupled transport equations for two dislocation density ÿelds: one is the total dislocation density and the other is a net-Burgers vector density. After a summary of the theory for single slip, we proceed to numerical implementation of the theory and to a comparison with direct simulations of discrete dislocation plasticity in a model composite material based on the work of Cleveringa et al (1997Cleveringa et al ( , 1998Cleveringa et al ( , 1999a. Similar comparisons have been carried out by Bassani et al (2001) and Bittencourt et al (2003) with the nonlocal theories of Acharya and Bassani (2000) and Gurtin (2002), respectively.…”

“…Groma (1997) and Groma and Balogh (1999). Let us ÿrst look at the nucleation and annihilation mechanism incorporated in the discrete dislocation simulations of Cleveringa et al (1997Cleveringa et al ( , 1998Cleveringa et al ( , 1999a which will serve as "numerical experiments" for veriÿcation in a subsequent section.…”

“…In two dimensions, a Frank-Read source is emulated by a point source on the slip plane which generates a dislocation dipole when the magnitude of the resolved shear stress at the source exceeds the source strength nuc during a period of time t nuc . Annihilation of two dislocations on the same slip plane with opposite Burgers vectors occurs when they are within a material-dependent, critical annihilation distance L e , which is taken to be L e = 6b (Cleveringa et al, 1997(Cleveringa et al, , 1998(Cleveringa et al, , 1999a). As we are here aiming at a comparison with the discrete dislocation results in these references, we choose a form of the source term in (7) that mimics the above-mentioned nucleation and annihilation rules:…”

Incompatibility Induced Nonlocal Crystal Plasticity

“…Finally the discrete dislocation results of CLEVERINGA ET AL. [4] are fitted in figure 1 c) employing the extended local formulation. Obviously a corresponding size effect can be picked up and (not shown here) the GND distribution coincides qualitatively with the microforce distribution in BITTENCOURT ET AL.…”

“…In a second example a composite material (CLEVERINGA ET AL. [4]) of dimensions h × √ 3h with an aluminium matrix and stiff inclusions is subjected to shear. First the global FE-formulation is compared with the extended local formulation for h = 8µm and h → ∞ in figure 1 b).…”