Topological defects and metric anomalies as sources of incompatibility for piecewise smooth strain fields
The incompatibility of linearized piecewise smooth strain field, arising out of volumetric and surface densities of topological defects and metric anomalies, is investigated. First, general forms of compatibility equations are derived for a piecewise smooth strain field, defined over a simply connected domain, with either a perfectly bonded or an imperfectly bonded interface. Several special cases are considered and discussed in the context of existing results in the literature. Next, defects, representing dislocations and disclinations, and metric anomalies, representing extra matter, interstitials, thermal, and growth strains, etc., are introduced in a unified framework which allows for incorporation of their bulk and surface densities, as well as for surface densities of defect dipoles. Finally, strain incompatibility relations are derived both on the singular interface, and away from it, with sources in terms of defect and metric anomaly densities. With appropriate choice of constitutive equations, the incompatibility relations can be used to determine the state of internal stress within a body in response to the given prescription of defects and metric anomalies.
Keywords: piecewise smooth strain; strain concentration; strain compatibility; strain incompatibility; topological defects; metric anomalies.
Mathematics Subject Classification (2010): 74E05; 74K15; 74K20; 74K25; 53Z05.
A central problem of micromechanics of defects in solids, in the context of linear elasticity, is to determine the internal stress field for a given inhomogeneity field [13, 14, 6, 18]. The latter can be considered in terms of a density of topological defects, such as dislocations and disclinations, or metric anomalies, such as those engendered in problems of thermoelasticity, biological growth, interstitials, extra matter, etc. [14, 17]. The inhomogeneity field appears as a source in strain incompatibility relations, which when written in terms of stress, and combined with equilibrium equations and boundary conditions, yields the complete boundary value problem for the determination of internal stress field . This classical problem of linear elasticity has been formulated, and solved, in the literature assuming the strain (and therefore stress) to be a smooth tensor field over the body. The defects densities have been also assumed, in general, to be smooth fields. The concern of the present paper is to generalize the problems of both strain compatibility and incompatibility with the consideration of piecewise smooth strain and inhomogeneity fields. The bulk fields are therefore allowed to be discontinuous across a surface within the body. The developed framework, in addition, allows us to consider surface concentration of strain and inhomogeneity fields; it is also amenable to situations when these fields are concentrated on a curve within the body.
In the strain compatibility problem, we seek necessary and sufficient conditions on a piecewise smooth symmetric tensor field (strain), defined over a simply connected domain, for there to a exist a piecewise smooth, but continuous (perfectly bonded interface), vector field (displacement) whose symmetric gradient is equal to the tensor field. The conditions consist of the well known compatibility condition on the strain field, away from the singular interface, and the jump conditions on strain and its gradients across the interface. The conditions are also sought for the case when the displacement field is no longer required to be continuous (imperfectly bonded interface). This, however, necessarily requires us to consider a concentration of surface strain field on the interface. The general forms of compatibility conditions, obtained in both the cases, are novel to the best of our knowledge. They are reduced to several specific situations discussed previously in the literature. We recover the interfacial jump conditions obtained by Markenscoff  and Wheeler and Luo . Whereas the former work was restricted to plane strain, the latter was concerned only with perfectly bonded interfaces and expressing the jump conditions in terms of strain components with respect to a specific curvilinear basis. We also use our framework to obtain the compatibility conditions on smooth strain fields over a domain, on a part of whose boundary displacements are specified, as discussed recently by Ciarlet and Mardare .
A strain field is termed incompatible if it does not satisfy the compatibility conditions. There can then no longer exist a displacement field whose symmetric gradient will be equal to the strain field, and hence the strain can not correspond to a physical deformation. The loss of compatibility is attributed to inhomogeneity fields in terms of defects and metric anomalies [14, 6]. In our work we consider piecewise smooth bulk densities, and smooth surface densities (or surface concentrations), of dislocations and disclinations. We also allow for smooth surface densities of defect dipoles. In addition we consider piecewise smooth bulk density, and smooth surface density, of metric anomalies. Beginning with writing these densities in terms of kinematical quantities, such as strain and bend-twist field, we first obtain the conservations laws they should necessarily satisfy. Our formulation is then led towards relating incompatibility of the strain field with densities of defects and metric anomalies. The strain incompatibility relations thus derived, with weaker regularity in the strain and inhomogeneity fields, as compared to the existing literature, are the central results of this paper. The incompatibility itself is described in terms of a piecewise smooth bulk field and smooth surface concentrations.
A brief outline of the paper is as follows. In Section 2, the required mathematical infrastructure is developed. Several elements of the theory of distributions, which forms the backbone of our work, are discussed. The results, already available in the literature, are given without proof but otherwise self-contained proofs are provided within the section and in the appendix. The strain compatibility problem, first for a perfectly bonded and then for an imperfectly bonded interface, is addressed in Section 3. Several remarks are provided in order to connect our results with the existing literature as well as to gain further insights. In Section 4, the central problem of strain incompatibility arising in response to the given inhomogeneity fields is formulated. Various aspects of the theory are simplified and discussed in the context of defect conservation laws, dislocation loops, plane strain simplification, and nilpotent defect densities. The paper concludes in Section 5.
2 Mathematical Preliminaries
Let be a bounded, connected, open set, with a smooth boundary . For two sets and , denotes the difference between the sets, whereas represents the empty set. The Greek indices range over and the Latin indices range over . Let be a fixed orthonormal right-handed basis in . For , the inner product is given by , where , etc; here, and elsewhere, summation is implied over repeated indices, unless stated otherwise. The cross product is such that , where is the alternating symbol. We use to represent the space of second order tensors (or, in other words, the linear transformations from to itself) and , the space of symmetric and skew symmetric second order tensors, respectively. The identity tensor in is denoted by . The dyadic product is defined such that , where . For , , , and represent the transpose, the symmetric part, and the skew part of , respectively. The axial vector of is such that, for any , . For , the inner product is given by with , etc. The trace of is defined as . For and , we define such that . For and , we define , a linear map from to , such that, for any , .
Let be a regular oriented surface with unit normal and boundary . If , then is either a closed surface or its boundary is completely contained within the boundary of . In either case, will divide into mutually exclusive open sets and such that and . The set is the one into which points.
We use , and ( is a positive integer), to represent spaces of continuous, smooth, and -times differentiable functions on , respectively. The spaces of vector valued and tensor valued smooth functions on are represented by and , respectively. Similar notations are used for functions defined over surface . For a function on and a subset , is the restriction of to the subset .
Let be the space of compactly supported smooth functions on . The dual space of is the space of distributions, . Any distribution defines a linear functional which is continuous for an appropriately defined topology on [12, Chapter 1].111A sequence of smooth functions converges to 0 if , for all , are supported in a fixed compact support and and its derivatives to every order converge uniformly to 0. A functional is continuous if, for any sequence of smooth functions converging to 0, converges to 0. For the purpose of this article, we will be interested in certain types of distributions contained in . For , we say that a distribution if it is of the form
where is a piecewise smooth function, possibly discontinuous across with , and is the volume measure on . The discontinuity in is assumed to be a smooth function on . For , , where are limiting values of at on from , represents the discontinuity in . We say that a distribution if it is of the form,
where , the surface density of , is assumed to be a smooth function on and is the area measure on the surface. We say that a distribution if it is of the form
where is assumed to be a smooth function on and represents the partial derivative along , i.e., (here denotes the gradient of ). We say that a distribution if it is of the form
where is assumed to be a smooth function on a smooth oriented curve and is the length measure on . That the above defined functionals are indeed distributions can be verified by first noting that all of them are linear functionals on . We now establish their continuity on . From converging to it is implied that for there exist positive integers , such that for and for . For , , where is the volume of . Hence, converges to 0. Similar arguments hold for , , and .
We use to denote be the space of compactly supported vector valued smooth functions on . The corresponding dual space is the space of vector valued distributions, . For , with each component , and , we define (summation is implied over repeated indices). Analogously, the space of compactly supported tensor valued function on and its dual are represented by and , respectively. For , with each component , and , we define .
2.3 Derivatives of Distributions
The partial derivative of a distribution is a distribution defined as
for all with .222Any locally integrable function can be associated with a distribution such that, for all ,
which implies . The gradient of a scalar distribution is a vector valued distribution such that . The gradient of a vector valued distribution is a tensor valued distribution such that . The divergence of a vector valued distribution is a scalar valued distribution such that . The divergence of a tensor valued distribution is a vector valued distribution such that . The curl of a vector valued distribution is a vector valued distribution such that . The curl of a tensor valued distribution is a tensor valued distribution such that . In particular, for , we have a tensor valued distribution such that .
2.4 Derivatives of Smooth Fields
The gradients of a smooth scalar field and a smooth vector field are denoted by and , respectively. The divergence of is a smooth scalar field defined as . The divergence of a smooth tensor field is a smooth vector field defined by , for any fixed . The curl of is a smooth vector field defined as , for any fixed . The curl of is a smooth tensor field defined as , for any fixed . The gradient of a scalar distribution can be therefore be equivalently defined as , for all . Similarly, the divergence of a vector valued distribution can be equivalently defined as , for all . Furthermore, we can define the curl of a tensor valued distribution as , for all .
The surface gradient of a smooth field , with a smooth extension , i.e., on , is a smooth vector field obtained by projecting onto the tangent plane of the surface. The surface gradient of a smooth vector field is a smooth tensor field such that , where is a smooth extension of (i.e., on ). The surface divergence of is a smooth scalar field defined as . In terms of the extension , it is given by . In particular, the scalar field is twice the mean curvature of surface . The surface divergence of a tensor field is a vector field defined by . In terms of a smooth extension , it is given by . Finally, if is a linear map from to (third order tensor), the surface divergence is given by .
Motivated by the definition of curl of vector fields on , we introduce, for , a vector valued smooth field such that, for any fixed , . Analogous to its bulk counterpart, gives the axial vector of . If has no tangential component, i.e., with , then we obtain . On the other hand, if we consider to be tangential and to be planar, i.e., and , then we have , where is a smooth extension of over . More generally, the following relationship holds:
For , we introduce a tensor valued smooth field such that, for any fixed , . In terms of a smooth extension of , such that on ,
Indeed, for fixed vectors and , we can use the identity to obtain
Consequent to writing the divergence term above in terms of a surface divergence, and proceeding with straightforward manipulations, we obtain the desired result. Equation (9) can be established along similar lines. It is clear that these relationships are independent of the choice of an extension.
Given a smooth oriented curve , with tangent , consider a surface passing through point such that is the normal to at . For a smooth bulk vector field , we define a vector valued smooth field such that, at any , , which is equal to by Equation (9), where is the derivative along . It is immediate that this definition is independent of the choice of the surface as long as the normal to at is .
2.5 Useful Identities
In this section we collect several identities which relate derivatives of distributions to derivatives of smooth functions. These identities will be central to the rest of our work. The proofs of these identities are collected in Appendix A.
(Gradient of distributions) For ,
(a) If , as defined in Equation (1), then
(b) If , as defined in Equation (2), then
where is the in plane normal to .
(c) If , as defined in Equation (3), then
(d) If , as defined in Equation (4), then
where is the unit tangent along . The last term above evaluates the function at the end points of (excluding those which lie on ) and should appropriately take into consideration the orientation of the curve at the evaluation point.
The following two sets of identities are used to calculate divergence and curl of vector valued distributions , , , and such that, for ,
where is a piecewise smooth vector valued function on , possibly discontinuous across with , and are smooth vector valued functions on , and is a smooth vector valued function on . The divergence and curl of a tensor valued distribution can be obtained from the results for vector valued distributions using the identities and for any fixed vector .
(Divergence of distributions) For ,
(a) If then
(b) If then
(c) If then
(d) If then
(Curl of distributions) For ,
(a) If then
(b) If then
(c) If then
(d) If then
The above identities will be used, in particular, to deduce the consequences of vanishing of the left hand sides in terms of derivatives of smooth functions. For instance, arbitrariness of can be exploited in Equation (12) to show the equivalence of with in and on . Similarly, Equation (17) implies the equivalence of with in and on , and (21) implies the equivalence of with in and on .333Given a distribution such that is piecewise smooth (smooth in ) and is a smooth function on . Also, for any . At , if , there exists a connected set with non zero volume such that in . There also exists a connected set such that has a finite volume with and for all . We choose such that for all , for all , and for . Then ( and do not change signs) which gives us a contradiction. So for all . The assumed sign of is clearly of no consequence. A similar argument can be constructed to argue that . To establish similar results from other identities we need the following two results. First, if is such that, for any ,
where , , are smooth functions on the oriented regular surface with normal , then is equivalent to , , and . Indeed, let be a local orthogonal coordinate system with as basis vectors such that defines (locally) with . Let be a smooth extension of to such that . Then . Let be an arbitrary smooth function on with a compact support . Let be the minimum distance of from . Let such that if and only if , where . There always exist a such that for . Then for , and on , and hence for an arbitrary local smooth function . This implies . Similarly, use to conclude that and consequently . Second, if is such that, for any ,
where and are smooth functions on a smooth oriented curve with tangent . Then is equivalent to and . Indeed, let be a local orthogonal coordinate system with as basis vectors such that is locally parameterized by , i.e. , , and on . By considering in terms of an arbitrary smooth function, with local compact support on , in addition to being linear in and , we can use arguments analogous to the previous paragraph to derive the required results.
A direct application of the above results, in conjunction with Equation (18) is the equivalence of with and in and on . Similarly, Equation (22) implies the equivalence of with and in and on . Furthermore, Equation (19) would imply the equivalence of with , , and in , and , on . Analogous consequences can be deduced from other identities.
2.6 Poincaré’s lemma
Given any and ,
These follow immediately by writing and and recalling Equation (8). The converse of these results is less straightforward. The following theorem, stated by Mardare  in this form, establishes that the converse of (27) holds true for a simply connected domain in the case of curl free vector valued distributions. For a proof, we refer the reader to the original paper.
(Mardare, 2008 ) If is a simply connected open subset of and , such that , then there exist a such that .
An immediate corollary of Theorem 2.1 is to establish an analogous result for symmetric tensor valued distributions.
If is a simply connected open subset of and , then is equivalent to existence of a such that .
It should be noted that both Theorem 2.1 and Corollary 2.1 do not establish any regularity on distributions and , respectively, if we were to start with assuming certain regularity on distributions and . For instance, if we start with an in then what distribution space should belong to? We will answer several such questions in Section 2.7.
The next theorem proves the converse of (27) for divergence free vector valued distributions on a contractible domain. Our proof, whose major part appears in Appendix B, is adapted from a more general proof given by Demailly [7, p. 20] within the framework of currents. Currents on open sets in correspond to vector valued distributions, in a manner similar to the correspondence of smooth forms to smooth vector fields  .
If be a contractible open set of and , such that , then there exist a such that .
The above results are well known in the context of smooth fields. In particular, in the language of differential forms , for any smooth form , , where denotes the exterior derivative. For differential forms of degree 0, 1 and 2, the exterior derivative corresponds to gradient, curl, and divergence operator, respectively. Moreover, for any smooth p-form on a contractible domain such that , there exist a (p-1)-form such that . For a 1-form, this result holds even for simply connected domains. Our assertions extend these results to a more general situation where the components of the vector fields are distributions instead of smooth functions.
2.7 Regularity Results
In this section, we collect several results of the kind mentioned in Theorem 2.1 and Corollary 2.1, but restrict ourselves to specific subsets of distributions. In Lemma 2.1 below, we start with curl free vector valued distributions, defined in terms of elements from , , and , and determine the precise form of distributions whose gradients are equal to the vector valued distributions.
Let be a simply connected region and be a regular oriented surface such that . Then, for and ,
(a) The condition , with , is equivalent to existence of a such that .
(b) The condition , with and , where and , is equivalent to existence of a such that .
(c) The condition , with and , where , , and , is equivalent to existence of a such that , where and , with .
The existence of a is guaranteed in all the above cases by Theorem 2.1. Our goal is to however establish a stricter regularity on for the given conditions. That implies that divides into mutually exclusive open sets and such that and .
(a) According to Identity (22), is equivalent to and . Hence , for a fixed . Then such that , where in and in , satisfies .