Geometrically necessary dislocation and size-dependent plasticity
Introduction
The background for our discussion is quickly set below. During the past five years, we have been actively engaging in the development of a mechanism-based theory of strain gradient (MSG) plasticity [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Our effort in modeling size-dependent plasticity was initially inspired by the work of Fleck and Hutchinson [13], [14], [15] who eloquently pointed out the importance of geometrically necessary dislocations and the need to develop a strain gradient plasticity theory with an intrinsic material length scale. The Fleck–Hutchinson theory fits nicely with the mathematical framework of Cosserat–Koiter–Mindlin theories of higher order elasticity [16], [17], [18]. We quickly learned that there are also other possibilities of formulating continuum theories with a length scale, such as gradient theories without higher-order stress [19], [20], gradient-dependent hardening modulus [21], and non-local theories of continuum mechanics [22], [23].
The existence of a material length scale for plasticity is now firmly supported by direct dislocation simulations [24], [25] and by four kinds of laboratory experiments: micro-torsion [13], micro-bending [26], particle-reinforced metal-matrix composites [27] and micro-indentation hardness tests [28], [29], [30], [31], [32]. These experiments have repeatedly shown that metallic materials display significant size effects when the characteristic length scale of non-uniform plastic deformation is close to a micron: The smaller the size, the harder the material. Conventional plasticity lacks an intrinsic length scale and hence cannot predict the size effects observed in experiments.
Section snippets
What is a geometrically necessary dislocation?
The concept of geometrically necessary dislocations has been a subject of extensive discussion in the literature [33], [34], [35], [36], [37], [38]. Our view is that the geometrically necessary dislocations represent an extra storage of dislocations required to accommodate the lattice curvature that arises whenever there is a non-uniform plastic deformation. This concept can be best explained using two examples. Fig. 1(a) depicts a plastically bent metal beam. In such a beam, certain number of
What does the concept of geometrically necessary dislocations tell us that we would not otherwise know?
In the classical continuum mechanics, the stress-strain laws, or so-called material constitutive relations, are largely obtained from bulk mechanical tests such as uniaxial tension performed on an MTS machine with bulk specimens. The concept of geometrically necessary dislocations tells us that there is an extra storage of defects associated with non-uniform plastic deformation and that the effects of such defects cannot be captured by macroscopic tests alone. For example, uniaxial tension
Can geometrically necessary dislocations be identified experimentally, i.e. what is testable about geometrically necessary dislocations?
Geometrically necessary dislocations do not necessarily have different atomic structures from statistically stored dislocations. They can be just ordinary dislocations, and it would be impossible to identify an individual dislocation as a geometrically necessary one or otherwise. This is a group concept. They correspond to extra storage of defects in the presence of a plastic strain gradient. As in the case of any useful physical concepts, geometrically necessary dislocations should of course
Which size effects can be modelled in terms of geometrically necessary dislocations, and which cannot?
The size effects due to non-uniform plastic deformation in metals can be modelled in terms of geometrically necessary dislocations. It may be possible to generalize this concept to other materials such as polymers or amorphous solids if we understand geometrically necessary dislocations as extra storage of defects (not necessarily dislocations) when the plastic deformation is non-uniform. However, there are also size effects which could not be modelled in terms of geometrically necessary
Geometrically necessary dislocations are often said to be associated with gradients––But any dislocations give rise to inhomogeneous stress and deformation fields––What characterizes the gradients associated with geometrically necessary dislocations from those that are not?
It is important to consider the length scale of the phenomena involved. Continuum plasticity differs from atomistic modelling or discrete modelling of dislocations in that only the collective behaviours or effects of dislocations are of interest. It is true that any dislocations give rise to inhomogeneous stress and deformation fields on an individual level, but we still find it useful to define a state of homogeneous plastic deformation on the continuum plasticity scale. This state of
How should geometrically necessary dislocations be incorporated into a phenomenological continuum theory of plasticity? What does this imply for the formulation of boundary value problems?
We strongly believe in the concept of geometrically necessary dislocations and advocate for incorporating geometrically necessary dislocations into a phenomenological continuum theory of plasticity. This is a significant challenge and should be collectively tackled by the whole mechanics and materials community. We ourselves have been trying to take a first step in this direction [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] by establishing mesoscale plasticity theories based on
What about surfaces and interfaces? Can anything we know about geometrically necessary dislocations tell us what boundary conditions should be imposed?
In our point of view, the concept of geometrically necessary dislocations, at least in its current understanding, do not distinguish between the states of matter near surfaces or interfaces and those in the bulk. Hence we do not expect that the concept of geometrically necessary dislocations could provide a guideline on how the boundary conditions should or should not be imposed. The issue of boundary conditions should be left to the specific theoretical framework one is trying to fit the
How many materials properties are needed in a continuum plasticity theory to represent the effects of geometrically necessary dislocations? How can these be measured?
An obvious material property in a continuum plasticity theory designed to represent the effects of geometrically necessary dislocations is an intrinsic material length scale. From a dimensional consideration, it seems that the most basic length scale is the Burgers vector which is the most fundamental quantity characterizing plastic deformation in a crystalline material. It is in fact surprising that most of the plasticity theories, which are supposed to describe collective behaviour of
What critical experiments are needed?
We believe that the most critical experiments needed at this point of time are those which allow one to directly quantify geometrically necessary dislocations. One example is the direct measurement of dislocation density underneath a micro- or nano-indenter, as has already been discussed in Section 4 above. It would be very desirable to directly measure the change in dislocation density as the plastic strain gradient is increased in a controlled fashion while the strain is maintained at a
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