5. Continuum Mechanics of the Interaction of Phase Boundaries and Dislocations in Solids

However, a dislocation-only defect model does not require any consideration of torque balance or couple stresses, as shown in [Ach11, AF12] and in Sect, 5.5.3.

As an aside, this observation also shows why the typical assumptions made in deriving transport relations for various types of control volumes containing a shock surface do not hold when the discontinuity in question is of the ‘terminating jump’ being considered here.

\({{\varvec{W}}}{{\varvec{V}}}\) is to be interpreted as the name for a single field.

Here it is understood that if \(n = 0\) then the symbol \(i_1 \cdots i_n\) correspond to the absence of any indices and the \(\textit{curl}\) of the higher-order tensor field is understood as the natural analog of the second-order case defined in Sect. 5.2.

It is to be noted that the decomposition (5.3) is merely a means to understand the definitions (5.2), (5.4), the latter being fundamental to the theory.

Note that the choice of C affects the \({{\varvec{W}}}\) field at most by a superposed spatio-temporally uniform rotation field.

An important feature of conservation statements for signed ‘topological charge’ as here is that even without explicit source terms nucleation (of loops) is allowed. This fact, along with the coupling of \(\varvec{\varPi }\) to the material velocity field through the convected derivative provides an avenue for predicting homogeneous nucleation of line defects. In the dislocation-only theory, some success has been achieved with this idea in ongoing work.

In the classical disclination-dislocation case, the corresponding question to what we have considered would be to ask for the existence, on a cut-induced simply-connected domain, of a vector field \({{\varvec{y}}}\) and the characterization of its jump field across the cut-surface, subject to \((\mathrm {grad}\,{{\varvec{y}}})^T\mathop {\mathrm {grad}}\nolimits \,{{\varvec{y}}}= {{\varvec{C}}}\) and the Riemann-Christoffel curvature tensor field of (twice continuously differentiable) \({{\varvec{C}}}\) (see [Shi73] for definition) vanishing on the original multiply-connected domain. Existence of a global smooth solution can be shown (cf. [Sok51] using the result of [Tho34] and the property of preservation of inner-product of two vector fields under parallel transport in Riemannian geometry). The corresponding result iswhere \(\textit{grad}\,{{\varvec{y}}}= {{\varvec{R}}}{{\varvec{U}}}\) on the cut-induced simply-connected domain, and \({{\varvec{R}}}\) is a proper-orthogonal, and \({{\varvec{U}}}= \sqrt{{{\varvec{C}}}}\) is a symmetric, positive-definite, 2nd-order tensor field. \({{\varvec{U}}}\) cannot have a jump across any cut-surface and the jump \(\llbracket {{\varvec{R}}}\rrbracket \) takes the same value regardless of the cut-surface invoked to define it, as can be inferred from the results of [Shi73]. By rearranging the independent-of-\({{\varvec{x}}}\) term in the above expression, the result can be shown to be identical to that in [Cas04]. Of course, for the purpose of understanding the properties of the Burgers vector of a general defect curve, it is important to observe the dependence of the ‘constant’ translational term on the cut-surface. An explicit characterization of the jump in \(\textit{grad}\, {{\varvec{y}}}\) in terms of the strength of the disclination is given in [DZ11].

Note that such a tensor field is not \({{\varvec{F}}}^p\) of classical elastoplasticity theory; for instance, its invariance under superposed rigid body motions of the current configuration is entirely different from that of \({{\varvec{F}}}^p\).