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Positive disclination in a thin elastic sheet with boundary

An isolated positive wedge disclination deforms an initially flat elastic sheet into a perfect cone when the sheet is of infinite extent and is elastically inextensible. The latter requires the elastic stretching strains to be vanishingly small. In this paper, rigorous analytical and numerical results are obtained for the disclination induced deformed shape and stress field of a bounded F{\"o}ppl-von K{\'a}rm{\'a}n elastic sheet with finite extensibility, while emphasising the deviations from the perfect cone solution. In particular, the Gaussian curvature field is no longer localised as a Dirac singularity at the defect location whenever elastic extensibility is allowed and is necessarily negative in large regions away from the defect. The stress field, similarly, has no Dirac singularity in the presence of elastic extensibility. However, with increasing Young's modulus of the sheet, while keeping the bending modulus and the domain size fixed, both of these fields tend to develop a Dirac singularity. Noticeably, in this limiting behaviour, inextensibility eludes the bounded elastic sheet due to persisting regions of non-trivial Gaussian curvature away from the defect. Other results in the paper include studying the effect of specific boundary conditions (free, simply supported, or partially clamped) on the Gaussian curvature field away from the defect and on the buckling transition from the flat to a conical solution.


Introduction
Isolated conical singularities due to positive wedge disclinations appear ubiquitously as point defects in thin elastic sheets [1][2][3] and liquid crystal polymer films [1,4,5]. Such a disclination can be introduced in a thin sheet of paper by first removing one or more wedges (all sharing a common apex) and then gluing together the exposed edges, see Figure 1. The resulting conical deformation and the singular stress field are a consequence of the concentration in the disclination induced strain incompatibility without any external forces [6][7][8]. The incompatibility of the strain field is a direct implication of the multivaluedness of the in-plane deformation field (which in turn arises due to the gluing operation in Figure 1) [9]. Conical singularities can also appear due to strain incompatibility arising due to non-metricity [10], such as those observed recently in shape morphing elastomers [11]. This is in contrast with the developable cones (dcones) which are formed in response to external forces while maintaining compatibility of the strain field; they appear commonly in crumpled sheets [12]. The solution to the disclination problem is analytically tractable when we idealise the thin elastic sheet as a Föppl-von Kármán plate of infinite extent with elastic inextensibility (i.e., with a vanishing elastic stretching strain field). The deformed shape then is a perfect cone with a Dirac concentration (at the defect point) in both the Gaussian curvature and the stress field. The Gaussian curvature field vanishes elsewhere while the stress field decays as the inverse squared distance from the defect, see Section 3.2 for details. The purpose of this paper is to study the deviations from the perfect cone solution when the Föppl-von Kármán elastic plate is bounded and has finite extensional elasticity. We do so by combining tools from measure theory and distribution theory with finite element based numerical simulations. The central contributions of our work are summarized next.
The finite extensibility and boundedness (including the type of boundary conditions) of a disclinated sheet significantly alters its Gaussian curvature field. It is no longer localised at the defect point (as a Dirac concentration or otherwise), although it remains unbounded therein. The regularised solution, as obtained from the Föppl-von Kármán equations with finite extensibility, is not merely a smoothening of the cone tip [4]. The elastic extensibility regularises the total curvature (bending strain) to yield a finite bending energy with the curvature field remaining unbounded in the vicinity of the defect and the Gaussian curvature taking non-trivial values away from the defect. The Gaussian curvature field in fact takes a negative value over finite regions of the plate domain away from the defect. This follows immediately for the clamped boundary condition where the average Gaussian curvature (over the entire plate) is necessarily zero, irrespective of material parameters and the shape of the boundary. The appearance of negative regions is to balance the substantial positive Gaussian curvature in the neighbourhood of the defect. The same conclusion holds for simply supported plates with polygonal shapes. For plates with a free boundary, such a restriction does not hold but the Gaussian curvature and its slope are necessarily negative at the boundary.
As the two-dimensional (2D) Young's modulus of the bounded sheet increases towards large values, keeping the bending modulus and the plate size fixed, both the Gaussian curvature and the stress field tend to develop a Dirac singularity. This limiting behaviour, however, does not lead to inextensibility, i.e., to the vanishing of the elastic stretching strain field, of the elastic sheet since the Gaussian curvature field remains non-zero in finite regions outside the defect location irrespective of the value of material constants (for reasons mentioned in the previous paragraph). In other words, no solution is possible (within Föppl-von Kármán theory) for a disclination in an inextensible elastic sheet with a finite domain size.
The choice of boundary conditions (free, simply supported, or clamped) affects the buckling transition from flat to conical solutions. A conical solution appears when the flat plate solution becomes energetically unfavourable. The buckling transition, characterised by a dimensionless number, has been shown to depend on the Poisson's ratio for a finite elastic sheet with free boundary by Seung and Nelson [6]. A similar behaviour is observed for the simply supported plate. The buckling of a clamped plate is however independent of the Poisson's ratio. Moreover, the critical buckling elastic modulus (while keeping all other parameters fixed) is lowest for plates with free boundaries and highest for plates with clamped boundaries.
There are several implications of our work. First, it provides a physically relevant, and rigorously justified, regularisation to the perfect cone solution which has infinite bending and stretching elastic energies. The regularisation yields finite energies despite allowing for unbounded bending strain and stress fields. Second, our solutions provide a quantitatively complete picture of the micromechanical response of a disclination in an elastic sheet. Much of the previous work related to this widely applicable problem relies on the perfect cone solution obtained for an unbounded and inextensible sheet [1,2,13]. The difference in the two solutions is considerable as has been highlighted throughout this paper. Third, the role of boundary conditions in affecting the solution of the disclination problem, previously unexplored, is central to the design and fabrication of systems involving disclinated thin sheets. The choice of the boundary condition strongly influences the morphology of the elastic sheet away from the defect. Finally, our work establishes quantitative benchmarks against which the applicability of the Föppl-von Kármán model can be justified with respect to direct experimental observations (of sheet morphology, for instance).
We provide a brief outline of the paper. In Section 2 we pose the boundary value problem of our interest including the Föppl-von Kármán plate equations (with a disclination) and various possibilities for the boundary conditions. In Section 3 we provide several analytical results including the well known flat plate solution, the perfect cone solution (with several novel insights), and the impossibility of an inextensible bounded sheet with a disclination. In Section 4 the numerical framework is outlined and implemented to discuss a typical numerical solution. The latter is used to motivate the specific concerns that are addressed in the rest of the paper. In Section 5 the nature of the Gaussian curvature and the stress field is investigated in a close vicinity of the defect both for finite extensional elasticity and in the limit of increasing Young's modulus values (for fixed bending modulus and plate size). In Section 6 we discuss the effect of the boundary conditions on the Gaussian curvature field (away from the defect) and the buckling transition. The paper concludes in Section 7.  ), where det represents the determinant. The operators ∇ and div represent the gradient and the divergence, respectively. In particular, the second gradient ∇ 2 f = ∇(∇f ) of a scalar field is a tensor valued field having components f ,αβ . The inner product, cross product, and tensor product, in 2D Euclidean vector spaces are denoted by ·, · , ×, and ⊗, respectively. The 2D permutation symbol e αβ is such that e 12 = −e 21 = 1 and e 11 = e 22 = 0. The 2D Kronecker delta symbol δ αβ is such that δ 11 = δ 22 = 1 and δ 12 = δ 21 = 0.

The Föppl-von Kármán equations
We consider a positive disclination of strength s located at a point o within the 2D simply-connected plate domain ω with a piecewise smooth boundary ∂ω. The equilibrium equations for a Föppl-von Kármán plate, in the absence of body forces, are written in terms of the in-plane stress σ (with components σ αβ ) and moment m (with components m αβ ) tensors, both symmetric, as [14] σ αβ ,β = 0 and m αβ ,αβ − w ,αβ σ αβ = 0, where w represents the transverse displacement field of the plate. The stress and moment components are assumed to be related to the stretching and bending strains (given by ε αβ and w ,αβ , respectively) through the isotropic, materially uniform, linear elastic constitutive relations [14] where E is the Young's modulus of the 2D sheet, D is the bending modulus, and ν is the Poisson's ratio.
The equilibrium equation (1) 1 is identically satisfied if the stress components are expressed in terms of the scalar Airy stress function Φ such that σ αβ = e αµ e βν Φ ,µν . Due to the wedge disclination at o ∈ ω, the strain field ε (with components ε αβ ) cannot be written in terms of a single-valued displacement field.
More precisely, we can write ε αβ = γ αβ + 1 2 w ,α w ,β , where γ αβ are the components of the infinitesimal stretching strain tensor which is not expressible in terms of a well defined in-plane displacement field [6,9].
This incompatibility of the strain field is expressed in terms of the following equation [6,7]: where δ o is the Dirac measure supported at point o. Note that 1 2 [w, w] = det(w ,αβ ) is the Gaussian curvature of the deformed plate. If s = 0, i.e., there is no disclination, then there exists a single-valued in-plane displacement field u α such that ε αβ = 1 2 (u α,β + u β,α ) + 1 2 w ,α w ,β ; the converse is also true. The incompatibility associated with the positive disclination can be understood in terms of removal of wedges (of net angle s) centred at o, see Figure 1. Consider, for instance, removing of four symmetrical wedges (of angle s/4) from a square domain of side length L 0 . The domain ω is then given by the square plate of side length L = L 0 (1 − tan(s/8)) obtained from the plate of size L 0 after removing the four wedges and identifying each pair of the newly exposed edges with a straight line.
The Föppl-von Kármán equations with a disclination can be obtained by writing strain and moment components in terms of stress (and hence stress function) and w, using (2a) and (2b), respectively, and then substituting them into Equations (3) and (1) 2 . We obtain in ω. These equations, combined with appropriate boundary conditions, can be used to obtain the stress field and the deformed shape of the plate due to the presence of a disclination of strength s at o ∈ ω. More detailed derivations of these equations are available elsewhere [6,7]. In writing (4), and in interpreting Gaussian curvature as det(w ,αβ ), we assume both Φ and w to be sufficiently smooth over ω. Strictly speaking, this is overly restrictive and one alternative is to interpret these equations in the sense of distributions. This is however not immediate due to the nonlinear terms in the equations. In Appendix A.2 we have given the assumptions on Φ and w such that both (4) and the Gaussian curvature can be interpreted reasonably in a distributional form.
Considering a length parameter L (representing the size of the finite plate), we can write the nondimensional form of Föppl-von Kármán equations (4) as whereΦ = Φ/D,ŵ = w/L,x i = x i /L, and Γ = EL 2 /D. The dimensionless constant Γ is known as the Föppl-von Kármán number [13]. The condition for inextensibility is ε = 0 in ω, which corresponds to (1/Γ)∆ 2Φ = 0. We note that Γ → ∞, or E → ∞ with D and L fixed, does not necessarily imply

Boundary conditions
We state three types of boundary conditions that are most commonly used with Equations (4) to yield a well posed boundary value problem [15]. All of these can be derived as part of the stationarity conditions from the functional (17) Figure 2. We will be discussing the solution to the disclination problem (4) subjected to either (6), (7), or (8). It is evident that if w is a solution to any of these boundary value problems, then so is −w. The problems are, in general, analytically intractable and have to be attended numerically. There are however two scenarios, as discussed next, when we are able to obtain exact closed form solutions.

The flat plate solution
For any of the three boundary value problems stated above, the flat plate solution, with w = 0 in ω, always holds true. All three problems are reduced to whose unique solution for a circular plate of radius R is [16] Φ = Es 8π with the corresponding stress field where e r and e θ are the orthonormal basis vectors in the polar coordinate system (r, θ). The flat plate solution is not well defined for an unbounded plate. The solution is, in any case, unstable beyond a critical value of R (for fixed Es/D) giving way to buckled solutions with w = 0 [6]. In this article we will always be working with parametric values where the buckled solution is the stable solution.

The buckled solution to the inextensible problem with an unbounded domain
The simplest buckled solution is obtained assuming the plate to be elastically inextensible and with an unbounded domain. The former is tantamount to a priori imposing ε = 0 in ω. Substituting this into (3), we can derive the reformulated governing equations as with the stress and moment fields vanishing identically as r → ∞. The stress field, and hence Φ, appears here as a Lagrange multiplier associated with the inextensibility constraint. The minimum energy solution to this problem is given by w = s π r, which represents a perfect cone, and Φ = −D ln r, whence we calculate A rigorous verification of the claim, that w = s π r and Φ = −D ln r indeed solves the problem at hand, is not straightforward. We use distribution theory to establish the result in Appendix A.2, wherein we also derive stress field (13) from the stress function. Note that both stress and Gaussian curvature fields develop a Dirac singularity at the location of the defect in the plate. This should be compared with (11), where the stress is unbounded at o but has no Dirac singularity. More importantly, the stress field in (13) is independent of the defect strength s. For s = 0, and considering w = 0 as a solution, any stress function field (including Φ = −D ln r) which yields a stress field vanishing at infinity is a solution. With s = 0, irrespective of the magnitude of s, the extent of non-uniqueness in stress is significantly reduced. We show, in Appendix A.3, that given w = s π r the most general form of the solution for the stress function is Φ = −D ln r + g 0 (θ) + rg 1 (θ), where g 0 (θ) and g 1 (θ) are arbitrary periodic functions (with period 2π) which satisfy

The inextensional problem with boundary
A somewhat surprising result is that if we consider the inextensional equations (12) for a bounded plate, with boundary conditions of any form given in Section 2.2, then the ensuing boundary value problem has no solution. To establish this, we start by writing a loop condition where e 3 is the unit vector such that {e r , e θ , e 3 } form a right-handed orthonormal triad in R 3 and dL is an infinitesimal line element in ω. Equation (14) can be derived by integrating Equation (37) from Appendix A.2, which is the distributional counterpart of (12) 1 , over the plate domain ω and using the Stokes' theorem. The loop condition in fact holds for any arbitrary loop enclosing the defect point o.
Under certain regularity conditions on w, the loop condition over arbitrary loops is equivalent to (12) 1 .
According to (14), ∇ 2 w cannot vanish everywhere on ∂ω. We first consider free and simply supported boundary conditions. The boundary condition m, n ⊗ n = 0, on using the constitutive relationship, Using this we can calculate the Gaussian curvature on the boundary as Therefore, with m, n ⊗ n = 0 on ∂ω, ∇ 2 w = 0 implies [w, w] = 0. Since ∇ 2 w cannot vanish everywhere on ∂ω, the same would follow for [w, w]. This is inconsistent with (12) 1 which requires [w, w] = 0 at each In the case of clamped boundary condition, we have w = 0 and ∇w, n = 0 on ∂ω, which together imply that ∇w = 0 on ∂ω. This, however, would trivialise the loop integral (14) and hence render it unequal to the right-hand side constant term.

A variational formulation
We solve the boundary value problems using a finite element methodology. We have developed our own code based on a mixed variational principle, according to which the governing equations appear as the stationary conditions of the functional [17, p. 165] where dA is an infinitesimal area measure on ω. The square plate domain is discretised using nonconforming C 1 -continuous rectangular elements and the weak form of the variational principle is used to

A typical numerical solution
We use the numerical framework to solve a typical problem. We will use the results to motivate the central concerns of this work. We consider a square plate with free boundary condition and a positive disclination of strength s = π/3 at the centre of the plate. There are no external loads acting on the plate. We take L = 2, E/D = 8000, D = 0.01, ν = 0.3, and a mesh of 48×48 square elements. The plate axes are denoted as X and Y (both taking values from the interval [0, 2]) with origin at one corner. The simulation results are given in Figure 3. In solving for w we fix three corners of the plate to avoid any rigid body motions.
The plate appears to deform into a conical shape with a rounded vertex [18]. The smoothening of the cone tip is due to extensional elasticity; the fourth-order derivative term ( 1 E ∆ 2 Φ) acts as a regulariser for the nonlinear Monge-Ampere bracket term. Both the Gaussian curvature field and the scaled biharmonic of show singular behaviour at the defect location. However, unlike the inextensional case, it is not clear how the Dirac singularity in Equation (4a) is distributed between the two terms.
The biharmonic plot also reveals an interesting cusp-like feature with the function decreasing sharply to a negative value, as one moves away from the defect, before rising again to a near zero magnitude.
This feature is neither a numerical artefact nor a consequence of the boundary conditions, as has been checked rigorously through numerical experiments. The stresses again are singular but whether they have a Dirac singularity, or not, is unclear. The behaviour of the deformation and Gaussian curvature, away from the defect, seems uninteresting from the plots in Figure 3. This is however not so. Indeed, a simple conical solution for w away from the defect will not work at the boundary. It will violate all three sets of boundary conditions mentioned in Section 2.2.

The questions
Motivated by the discussion so far, we enumerate the questions that will be addressed in the rest of this article: 1. What is the nature of solution close to the defect? More precisely, a) whether the Gaussian curvature field and the stress fields have a Dirac singularity at the defect location?, b) how the Dirac source term in (4a) is shared between the biharmonic and the Gaussian curvature terms?, c) are solution fields, in the close vicinity of the defect, invariant with respect to the type of boundary conditions considered? d) how do these solutions behave as E/D is increased for a fixed L?
2. What is the nature of solution away from the defect? We study this question with an emphasis on the behaviour of the Gaussian curvature field away from the defect location. In particular, a) how the field behaves for varying E/D (keeping L fixed) and varying plate sizes (keeping E/D fixed)?
and b) how the three boundary conditions affect the Gaussian curvature field away from the defect point?
3. To what extent buckling is dependent on the three boundary conditions? In this we extend the previous work of Mitchell and Head [19] and Seung and Nelson [6].
In the rest of the paper the domain ω is taken to be a square plate of side length L with the disclination of strength s located at its centre (position denoted as o). We will fix D = 0.01, s = π/3, and ν = 0.3, unless stated otherwise. Let µ be a measure such that

Solution near the defect
where g is an integrable function and a µ ∈ R is a constant. For any measurable subset Ω ⊂ ω, we have where ξ = 1 if o ∈ Ω and ξ = 0 otherwise. Let Ω n ⊂ ω be a sequence of measurable subsets such that, for each n, o ∈ Ω n and Ωn dA → 0 as n → ∞. Then Ωn dµ = Ωn gdA + a µ , which yields Ωn dµ → a µ as n → ∞.

Gaussian curvature near the defect
We assume both 1 E ∆ 2 Φ and 1 2 [w, w] to be measures like µ, i.e., where G 1 and G 2 are integrable functions and a 1 , a 2 are constants. In other words, we posit both the For establishing that the solution close to o is not significantly affected by our choice of the boundary condition as well as the plate size, we introduce an error for a given size (L > 1) and boundary condition, where R is a domain centred around o of a size less than that of a unit square, w 1 is the deformation field corresponding to a plate of size L = 1, w L is the deformation field for a plate of size L, k 1 is the Gaussian curvature field for a plate of size L = 1, and k L is the Gaussian curvature field for a plate of size L. For a chosen boundary condition, and for a fixed region R, error e measures the deviation of the solution for a plate of size L from that for a plate of size L = 1.
The results are reported in Figure 5, where each plot corresponds to a different boundary condition.
Within each plot, we have reported errors for four plate sizes (L = 1.33, 1.5, 1.67, 2) and four choices of    Our results should not be seen contradictory, to those discussed in Section 3.3, for we are dealing with the solution only in the neighbourhood of the defect. As we shall see in a following section, the value of the Gaussian curvature, away from the defect closer to the boundary, indeed does not become vanishing small even for large values of E/D.
We now combine the arguments presented in the last two paragraphs. We showed that, for any finite E (keeping D and L fixed), the Gaussian curvature behaves like an integrable function G 2 and the scaled

Stresses near the defect
We now study the singular nature of the stress field around the defect. The stress distribution is observed to remain invariant with respect to the choice of the boundary conditions and plate size, while keeping all other parameters fixed. We establish the nature of singularity in the stress field for a fixed E/D. Following the framework developed in the preceding section, we assume all the three Cartesian components of the stress (σ 11 , σ 22 , and σ 12 ) to be measures like µ, i.e., each of them is given in terms of an integrable . For each instance of the mesh refinement we calculate three numbers: V n σ 11 = Ωn σ 11 dA, V n σ 22 = Ωn σ 22 dA, and V n σ 12 = Ωn σ 12 dA. We observe that V n σ 11 = V n σ 22 and V n σ 12 = 0, irrespective of the mesh element size. Moreover, with increasing mesh refinement V n σ 11 = V n σ 22 tend towards 0. Therefore, like the Gaussian curvature field, the stresses are unbounded at o but without developing a Dirac concentration. This is again contrary to what was derived in the inextensible case (for an unbounded plate).

The role of boundary conditions
The concerns raised in the second and the third question of Section 4.3 are now addressed. First, we discuss the nature of the Gaussian curvature and its slope close to the boundary points for various boundary conditions. In doing so we are able to obtain definite analytical insights and their confirmation from our numerical results. In particular, we establish that regions of negative Gaussian curvature are inevitable in a finite plate even when we are placing a positive disclination at the centre. Next, we investigate the role of ν and the choice of boundary condition in affecting the buckling transition.

Gaussian curvature away from the defect
We begin by determining the sign of the Gaussian curvature at the boundary points of the plate domain for various boundary conditions. We assume that the curvature ∇ 2 w remains non-zero in the considered regions. This is reasonable since we do not expect the solution to deviate far from the perfect cone. Recall, for the free boundary condition, that we require m, n ⊗ n = 0 for all points on ∂ω, which on using the constitutive relation can be rewritten as ∇ 2 w, n ⊗ n = −ν ∇ 2 w, t ⊗ t , assuming D = 0. If ν > 0 then ∇ 2 w, n ⊗ n and ∇ 2 w, t ⊗ t are of opposite sign and if ν < 0 (auxetic materials) then they are of same sign on ∂ω. The Gaussian curvature 1 2 [w, w] = ∇ 2 w, n ⊗ n ∇ 2 w, t ⊗ t − ∇ 2 w, t ⊗ n 2 on ∂ω is therefore negative for plates with ν > 0 while its sign is undecided when ν < 0. If ν = 0 then ∇ 2 w, n ⊗ n = 0 yielding a negative Gaussian curvature on ∂ω. For the simply supported boundary condition, w = 0 and m, n ⊗ n = 0 on ∂ω. If in addition the boundary is piecewise straight, as is the case for the square plate, we have ∇t = 0 and ∇n = 0 almost everywhere on ∂ω. Under this simplification, the boundary condition w = 0 can be differentiated twice to yield ∇ 2 w, t ⊗ t = 0 which, on using the other boundary condition, gives ∇ 2 w, n ⊗ n = 0 almost everywhere on ∂ω, regardless of the value of ν. Therefore, the Gaussian curvature for almost all the boundary points of a simply supported square plate is necessarily negative. For the clamped boundary condition, w = 0 and ∇w, n = 0 on ∂ω. For a square plate these conditions yield ∇ 2 w, t ⊗ t = 0 and ∇ 2 w, t ⊗ n = 0 almost everywhere on ∂ω. Consequently the Gaussian curvature is identically zero at almost all boundary points of the square plate with a clamped boundary. Following similar arguments we can show that the derivative of the Gaussian curvature, along n, also vanishes at almost all boundary points for the square plate with clamped boundary condition.
The results about the sign of Gaussian curvature for free and simply supported boundary conditions, and We can obtain some further analytical understanding of the nature of the Gaussian curvature, and its slope, at the boundary if we restrict our attention to a circular plate (of radius r 0 ) with the free boundary condition. This allows us to consider a smooth axisymmetric solution for w, away from the defect, of the form w = f (r). Hence ∇ 2 w = f ′′ e r ⊗ e r + (f ′ /r)e θ ⊗ e θ , where the superscript prime denotes the derivative with respect to r. The Gaussian curvature and its slope along the radial direction can then be written as respectively. The moment tensor takes the form The boundary condition (6) 3 then implies It is reasonable to assume that f (r) is close to a cone like solution in the sense that f ′ ≈ s/π, as is clear from our numerical simulations. Consequently, f ′′ ≈ − s/π(ν/r 0 ) and f ′′′ ≈ s/π(1 + ν)/r 0 2 .
Substituting these into (23) we obtain The Gaussian curvature and its slope at the boundary therefore scale as −(1/r 0 2 ) and (1/r 0 3 ), respectively, with the size of the domain. Taking the effective radius of the square plate with side L as r 0 = L/ √ π, we superpose the analytically predicted behaviour of the end-values with those obtained from numerical solutions in Figure 9(c). The two solutions are in very good agreement except for the slope value at L = 1.
The Gaussian curvature field oscillates as it moves away from the defect towards the boundary, irrespective of the material parameters, plate size, and the type of boundary condition, see Figures 9(a) and 9(b). In doing so, the curvature values become negative over large regions in the plate, see Figure 10.
The existence of non-positive Gaussian curvature values at the boundary has been argued analytically in the preceding discussion for all the boundary conditions. In fact, as we demonstrate below, the average Gaussian curvature (over the plate) is necessarily zero both for the clamped and the simply supported case irrespective of the material and geometric parameters. Therefore the sheet will necessarily have regions of negative Gaussian curvature, large enough in their extent and in the magnitude of the curvature, so as to balance the substantial positive Gaussian curvature in the neighbourhood of the defect. Using identity (31) from Appendix A, we can write For a clamped boundary ∇w = 0 on ∂ω. The net Gaussian curvature over ω, given by the left-hand side of (26), is therefore zero. This is strictly a topological requirement for sheets with clamped boundary condition over boundaries of arbitrary shape. Indeed, clamping the sheet forces the integrated geodesic curvature on the boundary to be 2π which, on using the Gauss-Bonnet theorem, implies the vanishing of the net Gaussian curvature in the domain. For a simply supported boundary we can reach the same conclusion for piece-wise linear boundaries (like that of a square plate). Over the boundary, away from the corner points, ∇ 2 w, t ⊗ t = 0 and ∇w = ∇w, n n, as is immediate from (7) 4 and the constancy of t. This leads to the vanishing of the integrand on the right-hand side of (26) over the boundary except at the corner points. At a corner point, a non-zero value of ∇w would necessarily lead to a non-trivial jump in the value of ∇w and therefore to a concentration in ∇ 2 w, which is energetically unfavourable. Hence ∇w will necessarily vanish at the corner points, leading to our assertion. This result, again topological in nature, will hold for any simply supported plate with a polygonal shape. We have verified these claims, for the vanishing of the average Gaussian curvature, from our numerical simulations. Keeping them in mind, and recalling Figure 9(a), it is difficult to argue for the formation of a boundary layer as we move towards large E/D values, contradictory to what one would intuitively expect in such a limiting solution.

Buckling
The total strain energy U stored in the plate due to a positive disclination is given in terms of stretching and bending energies, respectively. The identity (31) from Appendix A, when used for Φ, yields For any of the boundary conditions given in Section 2.2, ∇Φ = 0. The contribution from stretching energy therefore is limited to only the first term of the integral in (27a). Similarly, using (26), we note that the second term in the bending energy integral (27b) is identically zero for clamped boundary condition where ∇w = 0 on ∂ω. In fact, the total energy for the clamped problem is independent of ν. Indeed, ν does not enter either the boundary conditions or the energy expression for a clamped boundary value problem. On the contrary, there is a ν dependence in the free boundary and the simply supported boundary problems through the boundary conditions as well as the second term in the bending energy integral (27b). For the flat solution, irrespective of the boundary condition, U = U s = (1/2E) ω (∆Φ) 2 dA with Φ determined from solving the system of Equations (9). For a circular plate of radius R, the total energy for the flat solution is Es 2 R 2 /32π (which increases unboundedly with the size of the plate). The flat solution remains the stable solution to our problem prior to the buckling transition to the non-flat (buckled, w = 0) solutions [6,19].
According to Seung and Nelson [6], for a plate with free boundary condition, the buckling transition is given in terms of a dimensionless number y c = R Es/D, where y c depends only on ν. For a fixed plate size (R), disclination strength (s), and bending modulus (D) this formula can also be used to calculate the critical value of the stretching modulus E and its variations with respect to ν. For our present discussion, we fix L = 2, s = π/3, D = 0.01, and a mesh size of 64 × 64 elements. The effective radius is calculated The calculated values for the critical E for a range of ν values (corresponding to stable isotropic elastic materials) and for various choices of boundary conditions are given in Figure 11. The variation of critical E with ν for the free boundary condition is consistent with the prediction of Seung and Nelson [6]. On the other hand, as expected, the critical E for the clamped boundary does not vary with ν. The trend in the variation of critical E with ν for the simply supported boundary condition is similar to that for the free boundary, however with higher magnitudes. For any given ν, the critical E is always highest for the clamped boundary and lowest for the free boundary. More importantly, it is clear that the buckling transitions are significantly dependent on the nature of boundary conditions.

Conclusion
We combined methods from measure theory and distribution theory with finite element based numerical respectively, for all ψ ∈ D(ω). The curl and the divergence of smooth fields is denoted using curl and div, respectively. The distributional gradient and the gradient of smooth fields are both represented by ∇; its appropriate usage will be clear from the context at hand. We note the following identities: (i) For smooth functions f : ω → R, g : ω → R we have the equivalence Using , and the Stokes' theorem, we can then obtain This identity can be shown to hold true even under milder regularity assumptions on f (as in cases when f is singular only at an interior point of ω) as long as f is integrable [20].
for all ψ ∈ D(ω, R 2 ), where B ǫ represent a disc of radius ǫ > 0 centred at o ∈ ω. Then, using the definition of distributional curl, we can calculate Curl V (ψ) = 2π 0 g(θ)dθ ψ(o), which implies (iv) Consider a distribution V ∈ D ′ (ω, Lin), where Lin is the space of linear transformations, such that for all ψ ∈ D(ω, Lin). Then, using the definition of distributional divergence, we can calculate Div

A.2 Perfect cone solution
To rigorously discuss the singular solutions of the inextensional problem (in an unbounded domain) we would need to restate the problem statement (12) in the sense of distributions. However, care is needed in doing so due to the nonlinear terms. We consider w ∈ D ′ (ω) and Φ ∈ D ′ (ω) such that (i) w| For w and Φ satisfying these assumptions, we define ∇w ⊗ ∇w ∈ D ′ (ω, Lin) as the unique extension of In fact, the stated assumptions on w and Φ are sufficient to describe the more general problem (4) in the sense of distributions. For smooth w and Φ, the left hand sides of the above equations reduce to those in (12). For w = cr, where c is a constant, ∇w = ce r . Moreover, Curl c 2 e r ⊗ e r = −(c 2 /r)e θ . Thereupon, using (34), we obtain Curl Curl c 2 e r ⊗ e r = −2πc 2 δ o . Accordingly, w = s π r satisfies Equation (37). On the other hand, we can use a generalised form of identity (32) to rewrite (38) as D Div Div(∇ 2 w) + Div Div ((e 3 × ∇Φ) ⊗ (e 3 × ∇w)) = 0.
A little loosely, we write the stress field as σ = −D 1 r 2 (e r ⊗ e r − e θ ⊗ e θ ) + πδ o 1 .
In ω − o, Φ 0 is smooth allowing us to calculate Curl Curl(e r ⊗ ∇Φ 0 ) = − 1 r ∂ 2 Φ 0 ∂r 2 . Thereupon, we can integrate ∂ 2 Φ 0 /∂r 2 = 0 to obtain the general solution for Φ 0 in ω − o as where g 0 and g 1 are smooth functions. The smoothness of Φ 0 in ω − o imposes periodicity on g 0 , g 1 , and their derivatives, i.e., g 0 (θ) = g 0 (θ + 2π), g 1 (θ) = g 1 (θ + 2π), etc. Given the degree of divergence of Φ 0 , we can use (48) to evaluate e r ⊗ ∇Φ 0 as a well defined unique distribution on ω. This allows us to calculate Curl Curl(e r ⊗ ∇Φ 0 ) on ω. We use the identities from Section A.1 to obtain Curl Curl(e r ⊗ ∇Φ 0 ) = − where the superscript prime denotes the derivative with respect to θ. Substituting this in (46) we obtain the following restrictions on g 0 and g 1 : The stress field corresponding to Φ 0 in ω is e r ⊗ e r + g ′ 0 r 2 (e r ⊗ e θ + e θ ⊗ e r ).
Clearly, as expected, σ 0 → 0 with r → ∞ for any choice of g 0 and g 1 with bounded values and derivatives.
The stress σ 0 should be appended to (13) to obtain the general expression for stress field in the plate due to a positive wedge disclination in an inextensional plate of infinite extent.

A.4 Biharmonic of φ in the inextensional limit
Consider a sequence of integrable functions G 1n ≤ 0 on ω and a sequence of real numbers E n > 0 such that G 1n → −sδ o and E n → ∞ as n → ∞. Furthermore, we make the following assumptions: (i) G 1n is axisymmetric that is G 1n (re r ) = G 1n (r), (ii) lim n→∞ E n Ω G 1n dA = 0 for any Ω ⊂ ω such that o / ∈ Ω and lim n→∞ E n Ω G 1n dA + s = 0 for any Ω ⊂ ω such that o ∈ Ω, and (iii) lim n→∞ E n ω r 2 G 1n dA = c 0 , which implies lim n→∞ E n R 0 r 3 G 1n dr = c 0 /2π, where R > 0 is arbitrary and c 0 ∈ R is a constant. For each G 1n and E n there corresponds a Φ n ∈ D ′ (ω) such that which implies lim n→∞ (1/E n ) ∆ 2 Φ n = 0. Our aim, however, is to calculate Towards this end, we consider a small disc B ro (of radius r o around o) and use the first part of assumption