On structured surfaces with defects: geometry, strain incompatibility, internal stress, and natural shapes
Given a distribution of defects on a structured surface, such as those represented by 2-dimensional crystalline materials, liquid crystalline surfaces, and thin sandwiched shells, what is the resulting stress field and the deformed shape? Motivated by this concern, we first classify, and quantify, the translational, rotational, and metrical defects allowable over a broad class of structured surfaces. With an appropriate notion of strain, the defect densities are then shown to appear as sources of strain incompatibility. The strain incompatibility relations, with appropriate kinematical assumptions on the decomposition of strain into elastic and plastic parts, and the stress equilibrium relations, with a suitable choice of material response, provide the necessary equations for determining both the internal stress field and the deformed shape. We demonstrate this by applying our theory to Kirchhoff-Love shells with a kinematics which allows for small in-surface strains but moderately large rotations.
Keywords: 2-dimensional materials; thin structures; geometry of defects; surface dislocations; surface disclinations; non-metricity; strain incompatibility; residual stress.
Mathematics Subject Classification (2010): 74E05; 74K15; 74K20; 74K25; 53Z05.
The aim of this article is to study geometry and mechanics of defects in structured surfaces. The term structured surface is used to represent a variety of 2-dimensional material surfaces such as 2-dimensional crystals, with intrinsic translational, rotational, and metrical order (colloidosomes, carbon nanotubes, graphene etc.); thin sandwiched structures; liquid crystalline membranes and shells, with intrinsic crystalline order (single-layer viral capsids) or without (nematic membranes, single layers in smectics and cholesterics); and Cosserat surfaces, which are used to model a hierarchy of plate and shell theories for thin elastic structures abundant in structural engineering applications. The defects are anomalies within the local arrangement of entities in an ordered structure where the order is usually defined in terms of translational, rotational, and metrical symmetries of the underlying material. These anomalies are omnipresent in nature, e.g., 2-dimensional materials such as graphene are known to contain edge dislocations (translational anomalies), wedge disclinations (rotational anomalies), and point imperfections (metric anomalies) such as vacancies and self-interstitials; on the other hand, twist disclinations are commonly observed in lipid membranes. More examples are given in the following section as well as in an extensive review of the subject in . The phenomena of thermal deformation and biological growth can also be categorized as those leading to metric anomalies. Many of the superior physicochemical properties of the 2-dimensional defective structures can be attributed to the internal stress fields resulting from the distribution of defects , and also, unlike 3-dimensional bodies, due to their lower dimensionality, to their ability to relax by acquiring a variety of natural (stress-free) shapes, for instance, the wavy edges of growing leaves , the topological corrugations present on human brain , helical strands of DNA , among other examples. The present work is concerned with the central problem of formulating a general theory that takes under its ambit the geometric characterization of these multifarious 2-dimensional defective structures and also the determination of their internal stress fields and deformed shapes.
Non-Euclidean differential geometry has been established to provide the necessary mathematical infrastructure to describe the geometric nature of defects in 3-dimensional solids, as well as to provide a rightful setting to discuss the related issues of strain incompatibility and residual stress distribution [38, 6, 2, 3, 52, 40, 16, 14]. Despite this success in 3-dimensions, the problem in lower dimensional structures is relatively less developed, primarily due to the complex interplay between the embedding geometries in the physical space, and the unavoidable non-linearities involved in the deformation as well as the constitutive response of 2-dimensional matter. We note the initial attempts made by Eshelby [28, 27] where analytical solutions for internal stress were derived for linearly elastic plates containing isolated screw and edge dislocations. This work was extended in several directions by Chernykh  and Nabarro [49, 50], among others [61, 48, 63]. A theory of continuous distribution of defects over thin structures was first developed by Povstenko [54, 55] by drawing analogies from the non-Euclidean description of continuous distribution of defects in 3-dimensional elastic bodies. Povstenko introduced the notions of in-surface dislocations (characterized by the in-surface torsion tensor), disclinations (characterized by the in-surface curvature tensor), and metric anomalies (characterized by the in-surface non-metricity tensor). He also provided the non-linear conservation laws for all the in-surface defect density fields as direct consequences of the Bianchi-Padova relations in two dimensions.
The compatibility conditions for strain fields in geometrically linear and non-linear shells, and Cosserat surfaces, are extensively discussed in existing works [37, 44, 56, 22]. The presence of defects, however, introduces incompatibility in strain fields. The strain incompatibility equations for both linear elastic plates and von-Kármán plates, with in-surface anomalies, have been derived by Zubov and Derezin [77, 78, 17, 18, 75]; these also include solutions of certain special boundary-value-problems for determining stress and natural shape of the defective plate. The strain incompatibility equations have also appeared in the recent works on non-Euclidean elastic plates [21, 34], and growth and morpho-elasticity of thin biological materials [46, 47, 43, 42, 19]. Without explicitly incorporating defect densities, these works consider a non-Euclidean metric, representing the distribution of plastic/growth strain field, and use the Riemannian curvature of this metric, which is the measure of strain incompatibility, along with the strain decomposition, to pose boundary-value-problems primarily for determining natural shapes.
The concepts of material uniformity, material symmetry, and inhomogeneity in elastic Cosserat surfaces, following the pioneering works of Noll  and Wang , are also firmly established [26, 73, 74, 23, 24, 25], although these works have neither attempted to describe the inhomogeneity distribution in terms of the curvature and non-metricity (the notion of torsion does appear in some of these works), nor have they discussed the relevant issue of strain incompatibility. A theory of materially uniform, inhomogeneous (dislocated) thin elastic films, derived from a 3-dimensional uniform, inhomogeneous (dislocated) elastic body, has been recently proposed by Steigmann , and applied to determining the natural shapes of plastically deformed thin sheets . Finally, we mention, only in passing, the extensive work on mechanics of topologically defective (‘geometrically frustrated’) liquid crystalline surfaces [58, 7, 8, 10], which, in contrast to the local theories mentioned above, have taken a distinguished local-global (geometrical-topological) standpoint in describing the nature of defects.
There is a clear lack of a complete non-Euclidean geometric characterization of continuously distributed material defects in 2-dimensional structured continua. While these certainly have analogous descriptions in the 3-dimensional theory, there is a considerable richness in the description of the allowable defects as well as their geometrical properties for the 2-dimensional structure. Additionally, there are no derivations of strain incompatibility relations for sufficiently general kinematic and constitutive response as afforded by most of the known 2-dimensional materials. With this in mind, we present a theory, within the natural setting of non-Euclidean differential geometry, that on one hand unifies the several seemingly different streams of research discussed above, and also provides a rigorously constructed, sufficiently general, framework for studying a large range of problems associated with geometry and mechanics of defective structured surfaces. In particular, we give a complete non-Euclidean characterization of all the translational, rotational, and metric anomalies in structured surfaces, derive the imposed restrictions from Bianchi-Padova relations, and establish general strain incompatibility relations. To illustrate our theory, we consider the specific case of Kirchhoff-Love shells and provide a framework, involving kinematics, additive decomposition, incompatibility relations, and balance laws, for posing complete boundary-value-problems for determining internal stress and deformed shapes for a class of 2-dimensional continuously defective structures undergoing small stretch but moderately large deformation.
A brief overview of the paper is as follows. In Section 2, we provide several illustrative examples to demonstrate the non-Euclidean character of local material defects in structured surfaces. Motivated from Section 2, we begin Section 3 by introducing the notion of material space, which includes a 2-dimensional body manifold, a non-Riemannian material connection, and a material metric, as our prototype to characterize continuously defective structured surfaces. Both in-surface and out-of-surface material anomalies are taken into account, and are identified with the components of the tensors of non-metricity, torsion, and Riemann-Christoffel curvature of the material connection (see Table 1). Subsequently, by exploiting Bianchi-Padova relations, we obtain several restrictions, both as algebraic relations and differential equations, on these components while emphasizing their interdependence (see Table 2). We conclude the section by introducing a Riemannian structure on the material space induced by the material metric. We obtain geometric relations which connect the curvatures of the Riemannian and non-Riemannian spaces. These relations are central to our formulation of strain incompatibility equations in Section 4. A generalized notion of strain is introduced which defines the kinematical nature of our structured surface. The strain incompatibility relations lead us to pose complete boundary-value-problems for the determination of internal stress fields and deformed shapes of defective structured surfaces. This is illustrated in Section 5 by restricting our attention to Kirchhoff-Love shells. We also postulate an additive decomposition for the in-surface and bending strain fields into elastic and plastic parts, while arguing that this separation of order is sufficiently general to accommodate small in-surface strains with moderately large rotations. The plastic strain fields are to be solved using the incompatibility relations with a given distribution of defects. We show that several existing formulations follow as special cases, in particular the Föppl-von-Kármán equations for continuously defective thin elastic isotropic plates and the shape equations for continuously defective thin isotropic fluid films. We conclude our study in Section 6.
2 Nature of surface defects
In this section, we provide several illustrative examples of defects in structured surfaces. The defects are understood as anomalies within the local arrangement of entities in an ordered structure where the order is usually defined in terms of rotational, translational, and metrical symmetries of the underlying material. Defects can also appear as global anomalies which affect the topology of the surface, such as those present in multiply connected and non-orientable surfaces [31, 32, 7]; these are however not discussed in the present work. The following examples are presented with an intent to emphasize the non-Euclidean geometric nature of the defects as is incorporated in the subsequent sections. In particular, the central idea of our work of embedding the structured surface within a 3-dimensional non-Riemannian geometric space emerges naturally as we proceed through these rudimentary illustrations.
The rotational anomalies in a structured surface appear in the form of disclinations. Depending on the material nature of the surface, rotational order can be present due to intrinsic crystallinity of the surface (such as in colloidosomes, single-layer viral capsids, carbon nanotubes, and graphene) or due to an extrinsic orientation field (such as in nematic membranes, single layers in smectics, and cholesterics) . As a result, we distinguish between rotational order, or lack thereof, appearing intrinsically and extrinsically in a surface. We also note that unlike disclinations in 3-dimensional crystalline solids, which have large formation energy and hence are rarely observed , disclinations in 2-dimensional crystals are omnipresent since the surface can now relax the energy by escaping into the third dimension. Isolated disclinations in structured surfaces without intrinsic crystalline order are shown in Figures 1(a,b). The rotational order is here present due to a director field distribution, denoted by , over a planar domain parametrized by Cartesian coordinates . The director field in Figure 1(a) is restricted to lie strictly in the -plane; it may represent a deformed configuration of a nematic membrane or a single layer in the cholesteric phase of some liquid crystalline material. In contrast, the directors in Figure 1(b) are allowed to orient themselves transversely to the plane; this can model either a lipid monolayer where the director orientation represents the orientation of individual lipid molecules, or a single layer of molecules in the smectic A or C phase . In nematics, smectics, and cholesterics, is identifiable with due to the mirror symmetry about the mid-orthogonal plane of the director axis. The lack of intrinsic crystalline order (translational and rotational), within the plane, in these examples can be primarily attributed to viscous relaxation . Disclinations in such structured surfaces can be characterized by the signed angle through which the director rotates upon circumnavigating along a loop over the surface. The Frank vector of the disclination is a precise measure of this signed angle. A disclination is of wedge or twist type depending on whether is transverse or tangential, respectively, to the surface. The disclination in Figure 1(a) is of wedge type with Frank vector and the one in Figure 1(b) is of twist type with Frank vector . Here, the triple denote the standard basis of the Cartesian coordinate system (. Note that the wedge disclination line in Figure 1(a) and the twist disclination line in Figure 1(b) are both along the -axis. Disclinations can also appear in surfaces with intrinsic crystalline order, e.g., an ordered arrangement of lattice sites where the directors are attached in viral capsids or hexagonal lattice structure of the carbon atoms in graphene sheets. As illustrated in Figure 1(c), circumnavigating along a loop encircling the disclination, a lattice vector rotates through an angle which is an integral multiple of one of the rotational symmetry angles of the lattice. The wedge disclination located at , in the 2-dimensional hexagonal lattice in Figure 1(c), is characterized by its Frank vector . Material surfaces can also possess twist disclinations in the form of local intrinsic orientational anomalies, which correspond to breaking of the reflectional symmetries of the 2-dimensional material with the local tangent plane of the surface as the mirror plane, e.g., hemitropic plates [65, 25]. They are represented mathematically as ill-defined (multi-valued) local orientation field over the surface.111An example of a global intrinsic orientational anomaly would be the global orientational anomaly present in a surface Möbius crystal due to its non-orientability. Note that, in order to quantify the disclinations discussed so far, the loop of circumnavigation is restricted always within the surface. All the disclinations shown in Figures 1, as well as the intrinsic orientational anomalies discussed above, are quantified using an in-surface loop . The case otherwise can appear in 2-dimensional homogenized models of thin 3-dimensional multi-layered structures, e.g., a stack of few monolayers of smectics or cholesterics, thin multi-walled nanotubes, or a thin slice of some 3-dimensional oriented media. In these structures, disclinations may appear over the representative base surface (often the ‘mid-surface’ of the layered structure) as the homogenized or effective rotational anomaly of all the distributed disclinations across the thickness of the thin structure. In describing these disclinations, the loop of circumnavigation must be taken transversely to the base surface, see Figure 2. Depending on the direction of the resulting vector of angular mismatch, these disclinations may either be of wedge or twist type.
The translational anomalies are represented by dislocations. The nature of dislocations in 2-dimensional matter is analogous to that in 3-dimensional materials. Isolated edge and screw surface dislocations are shown in Figures 3(a) and 3(b), respectively, within a 2-dimensional cubic lattice along with the Burgers parallelograms. The Burgers vector, defined as the closure failure of the Burgers parallelogram, is tangential to the surface of the lattice in the former case and transverse in the latter. In these examples, the dislocations appear essentially due to the breaking of the intrinsic translational symmetries of the 2-dimensional matter. On the other hand, in thin multi-layered structures or thin slices of oriented media, dislocations may be present, irrespective of the crystallinity of the material, as a result of either an order-mismatch of individual layers within the stack or as a homogenized or effective limit of all the distributed dislocations within the 3-dimensional slice. The Burgers parallelogram is, naturally, transverse to the representative mid-surface of the stack, in contrast to the examples shown in Figures 3(a,b). The precise type of these dislocations, edge or screw, can be determined from the direction of the Burgers vector. An edge dislocation in a layered medium is shown in Figure 3(c), arising due to the presence of a sandwiched semi-infinite layer between two infinite layers of material [41, Ch. VI].
The metrical anomalies bring about ambiguity in the (local) notion of “length” and “angle” over the surface. Metric anomalies are generated due to intrinsic point imperfections such as vacancies and self-interstitials, see Figure 4(a), as well as a result of in-surface thermal deformation and biological growth, see Figure 4(b). Note that foreign interstitials fall within the realm of materially non-uniform bodies (e.g., functionally graded materials) where the material constitution changes from point to point; their consideration is outside the scope of the present work. If the distance between the constituent entities in a lattice structure is measured by counting lattice steps, the presence of point defects, such as a vacancy or a self-interstitial, clearly introduces ambiguity in this step counting . Apart from these pure in-surface metric anomalies, differential growth (or thermal deformation) across the thickness direction within a thin multi-layered structure may result in transverse metric anomalies within an appropriately homogenized 2-dimensional theory, see Figure 4(c).
The simple examples described above are sufficient to motivate the non-Euclidean nature of the defects. Recall that, in order to quantify disclinations, we required circumnavigation of a vector along a loop and rotational mismatch between the initial and the final orientation of the vector. These notions correspond, respectively, to parallelly transporting a vector with respect to an affine connection and to the Riemann-Christoffel curvature associated with the affine connection. The Frank vector uniquely characterizes the Riemann-Christoffel curvature tensor . The affine connection has to be necessarily non-Euclidean, since the director fields leading to disclinations are clearly not parallel in the Euclidean sense. Moreover, as the directors may point outside the surface, a differential geometric description of disclinations in structured surfaces would necessarily require embedding the surface into a 3-dimensional space with a specific non-Euclidean connection. In the case of dislocations, the closure failure of the Burgers parallelogram is analogous to the notion of torsion of an affine connection over a manifold which characterizes closure failures of infinitesimal parallelograms [38, 6]. Finally, the metric anomalies are characterized by the non-metricity tensor, which quantifies the non-uniformity of the metric tensor with respect to an appropriate affine connection [3, 39]. Motivated with these geometric analogies, we are now in a position to pursue a systematic study of geometry of defects in a structured surface.
3 Geometry of surface defects
The mathematical prototype for structured surfaces is a connected, compact 2-dimensional manifold , possibly with boundary, which is embeddable (as a topological submanifold) in . Examples of such manifolds, in the orientable category, are sphere, sphere with a finite number of handles added, twisted bands with twists for integers etc., and in non-orientable category, twisted bands with twists for integers , e.g., a Möbius band for which is zero. We can add boundaries to these manifolds by removing finite number of open discs. The condition of embeddability in precludes Klein bottle like surfaces and real projective planes. Our prototype manifold is topologically characterized by its orientability, twistedness, Euler characteristic, the number of open discs removed, i.e., the boundaries, and other topological invariants. We will call the body manifold. A fundamental theorem in differential topology (Tubular Neighbourhood Theorem [9, Theorem 11.4]) guarantees the existence of a tubular neighbourhood of in , for sufficiently small . Here, denotes the minimum Euclidean distance of from . As a bounded open set in , naturally admits a manifold structure, with as an embedded submanifold. Existence of induces a vector bundle (the normal bundle) structure over , which entails a vector field defined over . Our choice of , naturally endowed with a director field , is therefore appropriate for modelling structured surfaces. The differential structure, and all the fields to be defined over and , including , is assumed to be as smooth as the context demands.
Our strategy for characterising material defects on a structured surface is to first equip with a geometrical structure by associating with it a metric and an affine connection. This is then used to induce an appropriate non-Riemannian geometrical structure over , where various fundamental geometric objects, such as non-metricity, torsion, and curvature, are interpreted as defect density measures. The induced metric and connection on is sufficient to encode all the information about the material structure of the structured surface. The Binachi-Padova relations are used to obtain several restrictions on defect density fields. With these relations, it is emphasized that the various defect densities are in fact dependent on each other. The metric associated with is also used to induce a Riemannian structure over . The relationship of the curvature tensor, associated with the affine connection, with Riemann-Christoffel curvature tensor, obtained from the metric, is derived. These relations will provide the starting point for deducing strain incompatibility equations in the following section. They also lead to the well known local conditions under which is isometrically embeddable into , a notion that is related to compatibility of the strain fields.
In rest of the paper, lowercase Greek indices , , etc. take values from the set and lowercase Roman indices , , etc., from the set . Einstein’s summation convention hold over repeated indices unless specified otherwise. Round and square brackets enclosing indices indicate symmetrization and anti-symmetrization, respectively, with respect to them. The superscript is used to denote the inverse of an invertible matrix, whereas the superscript is used to denote the transpose.
3.1 Geometry on induced from the non-Riemannian structure on : the material space
Let the 3-dimensional embedding manifold be equipped with an affine connection and a metric . Consider a chart of with such that the coordinates defined over lie along with at . Such a coordinate system is called adapted to . The restriction of the natural basis vector fields over to will be denoted by , i.e., , hence is transverse to . The coefficients of and the covariant components of are denoted by and , respectively, with respect to . The covariant derivative of a sufficiently smooth vector field , , with respect to , is denoted by
The notation is used for the surface covariant derivative of a tangent vector field , , with respect to the projection of on , i.e., a connection with coefficients ,
Here, the subscript denotes ordinary partial derivative with respect to . A vector field along a curve over is called parallel with respect to if, and only if, its covariant derivative along the curve vanishes identically.
The body manifold , equipped with connection and metric from the embedding space , forms the material space of the structured surface. We will call the material connection and the material metric. The “material” nature of these mathematical objects is due to the fact that the geometric quantities derived from and , when restricted to , represent various material inhomogeneities or defects within the material structure of the structured surface. As we will see immediately below, the non-metricity tensor is a measure of distributed metric anomalies, the torsion tensor is a measure of distributed translational anomalies (dislocations), and the Riemann-Christoffel curvature tensor is a measure of distributed rotational anomalies (disclinations). Most importantly, we assume to be such that the non-metricity, torsion, and curvature tensors associated with are uniform in the coordinate and equal to their respective values at , i.e., at . This assumption alludes to the applicability of our model to thin multi-layered structures, or thin slices of defective media, represented as homogenized 2-dimensional surfaces. It should also be noted that we are only looking at local defects and not the ones which could arise out of various topological anomalies for multiply connected and non-orientable surfaces.
3.1.1 Non-metricity of the material connection: metric anomalies
The third order non-metricity tensor of the material space, measuring non-uniformity of the metric with respect to the connection , has covariant components defined as
The negative sign in the definition is purely conventional. We assume that . The pure in-surface components provide measure for the distributed surface metric anomalies, whereas components , with either of , or taking the value 3, indicate the presence of out-of-surface metric anomalies, e.g., thickness-wise growth. A non-zero lead to variation in angle between tangent vectors during parallel transport with respect to the projected connection , see Figure 5(a). Indeed, the inner product of two tangent vectors and , where , changes under parallel transport with respect to from the initial point to any generic point , along some parametrized curve lying over , by the amount
Here, we have used, and throughout , as they are parallelly transported fields along , where denotes the ordinary derivative of with respect to its argument. In structured surfaces, as we have earlier discussed in Section 2, this variation in inner product, characterized above in terms of a non-trivial , may arise from a distribution of point imperfections in the arrangement of molecules or atoms over the surface, e.g., vacancies and self-interstitials in 2-dimensional crystals, inserting (or removing) a lipid molecule into (or out of) a crystalline arrangement of identical molecules over a monolayer, thermal deformation of the surface, biological growth of cell membranes, leaves etc. The remaining components and measure the non-uniformity of the material metric in the -direction, i.e., along the thickness of the structured surface, and the change in length of transverse vectors along the surface, respectively, see Figures 5(b) and 5(c). These provide faithful representations for differential growth along the thickness in thin multi-layered structures discussed in Section 2 and illustrated in Figure 4(c).
3.1.2 Torsion of the material connection: dislocations
Consider two tangent vectors at some point on . Translating parallelly along and along with respect to , we obtain the vectors
respectively. The closure failure of the parallelogram is given by (see Figure 6)
where the functions
constitute the components of the third-order torsion tensor (anti-symmetric in the lower indices) over . Let . We assume that , which in turn is same as . Associated with the torsion tensor, we have the second-order axial tensor
Here, , where is the 3-dimensional permutation symbol and . For later use, we define . The components provide measures for a variety of dislocation distributions over the structured surface. Taking (i.e., and tangential to , see Figure 6(a)), and comparing with Figures 3(a,b), it is immediate that
represent a distribution of in-surface edge dislocations and
a distribution of in-surface screw dislocations (cf. [54, 55]). Next, taking (i.e., tangential and transverse to , see Figure 6(b)), and comparing with Figure 3(c), it is evident that the components represent the out-of-surface dislocations in thin multi-layered oriented media such as those discussed in Section 2.
3.1.3 Curvature of the material connection: disclinations
The components of the fourth order Riemann-Christoffel curvature tensor of the material connection are given by
The functions measure, in the linear approximation, the change that a vector, , , suffers under parallel transport with respect to along an infinitesimal loop based at and lying within :
where are the components of the initial vector with respect to the basis ; the integral represents the infinitesimal area bounded by the loop . The above formula in fact holds true for any general loop (not necessarily infinitesimal) in . We define the purely covariant components by lowering the fourth index with the material metric as Clearly, and . Moreover, as we did for non-metricity and torsion tensors, we assume .
It is useful to decompose the components into skew and symmetric parts 
are components of the second-order tensor field and the third-order tensor field . They represent, respectively, the skew part and the symmetric part of with respect to the last two indices. A geometric interpretation of these two fundamental tensors is as follows (see Figure 7). Let the infinitesimal loop in (12) be based at . Then the change that a vector undergoes when parallelly transported along , in the linear approximation, can be characterized by a second-order tensor , i.e., , where
Here, is a measure of the infinitesimal area bounded by and its unit normal. The first term in the above expression is skew with axial vector . It represents the rotation that has experienced under parallel transport about the axis , for each fixed , probed by the three Euler angles . Thus, is the measure of the rotation of about the axis . The second term , on the other hand, is symmetric; it represents a stretching, with the three principal values of the tensor as measures of the stretch along their respective (linearly independent) principal directions. The tensor can be shown to be related to the metrical properties of as it gives rise to a smeared out anomaly within the material structure which causes elongation or shortening of material vectors under parallel transport along loops (see  for details), as shown in Figure 7(a). We will assume in rest of the paper since, at present, we do not know of any defects in 2-dimensional materials which they would otherwise represent. Some consequences of this assumption will be discussed in the next section. The curvature tensor is then fully characterized in terms of the non-trivial independent components , i.e., the second-order tensor .
We distinguish between two families of local rotational anomalies characterized by . Consider, first, the infinitesimal loop completely lying within , see Figure 7(b). Then, the and indices in can assume only values 1 and 2, and the resulting angular mismatch after parallel transport of arbitrary vectors is characterized by three fields
These provide a measure for the distributed rotational anomalies within the material structure of the base manifold . Drawing analogy with Figure 1, it is clear that the out-of-surface component provides a measure for the density of distributed wedge disclinations over the structured surface, see Figures 1(a,c), irrespective of its crystallinity, whereas the in-surface components characterize either the distributed intrinsic orientational anomalies, in case of intrinsically crystalline surfaces, or distributed twist disclinations, in case of directed surfaces (as shown in Figure 1(b)). Next, we consider , based at , to lie transversely to , see Figure 7(c). Then one of the indices and in will take the value 3, and the resulting angular mismatch after parallel transport of arbitrary vectors is characterized by the remaining six independent components of :
Recalling our discussion in Section 2 on disclinations in thin multi-layered structures of oriented media, see also Figure 2, we conclude that these components provide a measure for a variety of homogenized/effective rotational anomalies of the distributed disclinations across the thickness of the multi-layered structured surface. Out of these six functions, and are of wedge type, and , , and are of twist type. As we will see shortly, these functions are in fact dependent on each other in very thin monolayer structures where the dislocation densities vanish altogether.
|Geometric objects||Defect densities|
|In-surface metric anomalies; Figures 4(a,b) and 5(a)|
|and||Out-of-surface metric anomalies; Figures 4(c) and 5(b,c)|
|In-surface edge dislocations; Figures 3(a) and 6(a)|
|In-surface screw dislocations; Figures 3(b) and 6(a)|
|Out-of-surface dislocations; Figures 3(c) and 6(b)|
|In-surface wedge disclinations; Figures 1(a,c) and 7(b)|
|Disclinations associated with transverse loops; Figures 2 and 7(c)|
|Metrical disclinations; Figure 7(a)|
We have summarized the set of all defect densities in Table 1.
3.1.4 Bianchi-Padova relations
The tensors of non-metricity, torsion, and curvature of a non-Riemannian space cannot be arbitrary due to geometric restrictions. Besides the restrictions , , and , which follow from their definitions, they satisfy the following system of differential relations, known as the Bianchi-Padova relations [62, p. 144]:
In the above expressions, anti-symmetrization with respect to three indices is defined as
The enclosed indices within two vertical bars in the subscript are to be exempted from anti-symmetrization. Clearly, and (no summation on ). Additionally, there is a fourth Bianchi-Padova relation [62, p. 145], purely algebraic in nature, based on the following identity satisfied by the components of any fourth-order tensor with :
After substituting relations (18a) and (18c) into (20), it boils down to an expression for in terms of , , , and . For a torsion-free, metric-compatible connection (i.e., a Levi-Civita connection), this implies the familiar symmetry . However, as shown below, this particular symmetry is achieved in very thin structured surfaces under much less restrictive conditions. The first three Bianchi-Padova relations, restricted to a surface, have been considered previously by Povstenko , but without studying any of the implications, some of which are noted below.
Consequences of the first Bianchi-Padova relation: Equation (18a) is non-trivial only when at least one of the indices , and assume the value 3, since otherwise . Recalling our assumption that that is uniform with respect to the coordinate, (18a) reduces to
Furthermore, if we assume that the structured surface is sufficiently thin with no dislocations associated with the transverse Burgers parallelograms, i.e., (the in-surface dislocations can still be present), then (21) simplifies into a system of algebraic equations:
For , we obtain , since (from ). This is equivalent to , or
For , (22) can be rewritten as , or equivalently
Combining the above two relations we can therefore infer that, for vanishing , the disclination density tensor is symmetric. Moreover, due to (16), , i.e., the pure in-surface disclination densities (which may either characterize densities of twist disclinations in directed surfaces or intrinsic orientational anomalies in hemitropic surfaces) should be identical to the wedge disclination densities associated with transverse loops, e.g., in multi-layered surfaces as discussed in Section 2; in particular, they should vanish in sufficiently thin structured surfaces, e.g., in 2-dimensional crystals, where both and will be absent. We note that, in contrast, for 3-dimensional solids, the symmetry of the disclination density tensor is implied only under vanishing of the full torsion and the non-metricity tensor. It is worthwhile to reemphasize that the assumption is realistic only in sufficiently thin structures (biological membranes, graphene sheets, etc.), which, otherwise, can support only surface edge and screw dislocations (characterized by ).
On the other hand, if we consider multi-layered or moderately thin structures of oriented media, where the assumption of vanishing is no longer physical, and assume that they do not contain any disclinations and metric anomalies, and also that and are small (of the same order), then (21) yields
This is a conservation law for the -type dislocations enforcing that they must always form loops or leave the surface. In either case, whether the -dislocations are absent or not, there is no restriction on the distribution of in-surface dislocations . This again is in contrast to 3-dimensional solids, where the first Bianchi-Padova relation provides a conservation law for all dislocation densities [55, 60].
Consequences of the second Bianchi-Padova relation: Equation (18b), in the absence of both -type dislocations and metric anomalies (), in addition to , reduces to a simple conservation law
to be satisfied by disclinations characterized by , as well as owing to the symmetries (Equations (23) and (24)), and surface edge dislocations. Assuming that and are small, and of the same order, we obtain
These are linear conservation laws for the respective disclinations, requiring their lines to either form loops or leave the surface. Note that there is no restriction on (wedge disclinations), in contrast to what one would expect for 3-dimensional solids.
It can be shown by direct substitution that a non-trivial solution of (28) is given by
where are arbitrary symmetric functions over . It is a consequence of the fundamental existence theorem of linear differential systems that in absence of disclinations (i.e., ) over a simply connected (hence ), if the matrix field is positive-definite for symmetric functions , then is the only solution to (28) over . This result is proved in . As the density of metric anomalies is assumed to be uniform with respect to the coordinate, we will interpret this representation of the metric anomalies in absence of disclinations over simply connected patches over as
The symmetric matrix field is known as quasi-plastic strain . In absence of disclinations, the positive-definite symmetric matrix field can be used to define an auxiliary material space , equipped with the original material connection but a metric . The non-metricity of the auxiliary material space vanishes identically by definition. The second-order tensor field , where , characterizing pure in-surface metric anomalies in the absence of disclinations, has the unique decomposition
where is the trace of and is the deviatoric part of (i.e., ). The first term represents isotropic metric anomalies and the second represents anisotropic metric anomalies . When is purely isotropic, i.e., , it is straightforward to obtain , where . The surface metric of the auxiliary material space for isotropic metric anomalies is, hence, conformal to the surface metric of the original material space, . This formulation is readily applicable to model various real-life surface metric anomalies such as 2-dimensional anisotropic biological growth, thermal expansion, distributed point defects, etc.
Consequences of the fourth Bianchi-Padova relation: The fourth Bianchi-Padova relation imposes interdependence on the disclination density measures , out of which the interdependence between the two distinct families of disclinations characterized by and , derived in the following, are in particular interesting. Assuming , the in-surface components of (20) require , since , which is the trivial relation . Next, if we also assume that the metric anomalies are absent, i.e., , then (20), together with (18a), yields
Here, . After substituting , as per our assumption on , and , or equivalently , the right hand side of the above relation vanishes identically, thereby enforcing the symmetries
In terms of disclination densities, these are, respectively, and . Interestingly, we reached the same conclusion from the first Bianchi-Padova relation. We will of course obtain a non-trivial consequence of the fourth Bianchi-Padova identity whenever .
The results of this section are summarized in Table 2.
|Implications on defect densities|
3.2 The induced Riemannian structure
The coefficients of any general affine connection of a manifold , with non-trivial torsion and non-metricity , can be decomposed as [62, p. 141]
where the functions are coefficients of the Levi-Civita connection (torsion-free, metric-compatible) induced by the metric :
with , and
The functions form the components of the contortion tensor, whereas the tensor is an equivalent measure of non-metricity. The covariant components
of the Riemann-Christoffel curvature tensor of the Levi-Civita connection and components of the material curvature are related as [62, p. 141]
where and denotes covariant differentiation with respect to the Levi-Civita connection . From the general symmetry relations, , of the Riemannian curvature induced by a metric, it is evident that it has only six independent components characterized by , and . The only non-trivial relations out of (38), when restricted to , are
Here, , , , and denotes covariant differentiation with respect to the projected Levi-Civita connection on , consisting of components . The relations (39) are central to the theory of mechanics of defects as they are directly related to the strain incompatibility equations which we discuss next. Indeed, once we have identified the material metric in terms of strain fields on the structured surface, (39) constitute a system of PDEs for the strain fields, with defect densities as source terms. It should be noted that the components do not appear in any of the equations (39). This is because, according to the fourth Bianchi-Padova relation (32), they can be written in terms of and other defect measures, and hence are not independent quantities.
4 Strain incompatibility relations for structured surfaces
In this section, we begin by introducing the notion of strain for a structured surface. The complete set of strains represent essentially the kinematical nature of shell theory that is being employed to describe structured surfaces. The strain fields also provide us with fundamental variables for construction of constitutive responses of the continuum. Once the strain fields are fixed, we look for the necessary and sufficient (compatibility) conditions for the existence of a local isometric embedding of the surface. Finally, we discuss how various defect densities become sources of strain incompatibility precluding the existence of the local isometric embedding. This will then set the stage for posing complete boundary-value-problems for internal stress distribution and natural shapes of defective structured surfaces, as will be discussed subsequently.
4.1 Strain measures and strain compatibility
Let us assume that there exist a local isometric embedding of into . Let and . The first and second fundamental forms associated with this embedding (over a local patch) are therefore and , respectively. We consider the following sufficiently smooth fields, defined over , as descriptors of strain on the structured surface: (i) a symmetric tensor