Interfaces within strain gradient plasticity: Theory and experiments

Abstract

In this paper, it is shown that the occurrence of dislocation pileups across grain boundaries, as well as subsequent emission to the adjacent grains, is captured theoretically by gradient plasticity and confirmed experimentally by nanoindentation. From a theoretical point of view, this is accomplished (within a deformation theory framework applicable to continued loading) by accounting for a specific interfacial term in the overall potential of the material, in terms of which its response, taken to conform to strain gradient plasticity, is defined. The main features that result from the addition of this interfacial term are (i) significant size effects of Hall-Petch type in the overall stress-strain response of polycrystals and (ii) the determination of an analytical expression for the stress corresponding to the onset of dislocation transfer across interfaces. From an experimental point of view, the effective stress at which dislocation transfer takes place across an interface can be obtained from nanoindentations performed in close proximity to an Fe-2.2 wt.% Si grain boundary, since they exhibit a distinct strain burst that is related to the presence of the boundary. It is possible, therefore, to fit the theoretically determined analytical expression for the interfacial yield stress to the experimental data. From this fit, first estimates are obtained for key material parameters, namely the interfacial term and the internal length, that are required for the theoretical formulation. Dislocation mechanics are employed to provide physical insight of these parameters.

The effect of interfaces on the plastic behavior of periodic composites

“…This result avoids the complication of having to differentiate expression (20) numerically. In addition to estimating the effective response, it is of interest to estimate the statistics of the plastic strain field.…”

“…An estimate for the effective response of the composite can be obtained by differentiating (20), in accordance with (12). By following similar arguments to those of [23], it can be shown that…”

“…However, their computation becomes straightforward in the case of isotropic potentials such as those considered in Section 4. In any event, the functions that result from the optimizations (21)- (22) are convex in the tensors M (r) , N (r) , S (i) , so that the minimization in (20) can be carried out using any standard method for convex optimization.…”

“…The presence of plastic strain gradients in the constitutive law accounts phenomenologically for the effect of geometrically necessary dislocations on bulk material hardening (see, for instance, Fleck and Hutchinson [19]), while the introduction of an interfacial 'energy' term, penalizing the accumulation of plastic strain there, follows from the accepted notion that interfaces represent an obstruction to the motion of dislocations. Recent attempts to determine the interface potential φ experimentally include nanoindentation tests in the vicinity of grain boundaries [20], and tensile tests on bicrystals [21]. Additionally, atomistic simulations of dislocation/grain-boundary interactions [22] should prove useful in determining this potential.…”

“…In view of the 'particulate' character of the microstructure considered, refined estimates are obtained here by making use of the nonlinear variational estimate (20) in conjuction with the linear Hashin-Shtrikman estimate (49). Motivated by the particular form of the potentials (51) and (52), we restrict the potentials (18) and (19) in the LCC to the form…”