A Critical Study of Efrati et al.'s Elastic Theory of Unconstrained non-Euclidean Plates
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n A Critical Study of Efrati et al. ’s Elastic Theory ofUnconstrained non-Euclidean Plates
Kavinda Jayawardana ∗ Abstract
In our analysis, we show that Efrati et al. ’s publication [1] is inconsistentwith the mathematics of plate theory. However it is more consistent withthe mathematics of shell theory, but with an incorrect strain tensor. Thus,the authors’ numerical results imply that a thin object can be stretched sub-stantially with very little force, which is physically unrealistic and mathe-matically disprovable. All the theoretical work of the authors, i.e. nonlinearplate equations in curvilinear coordinates, can easily be rectified with theinclusion of both a sufficiently differentiable diffeomorphism and a set ofexternal loadings, such as an external strain field.
Keywords:
Finite Deformation, Mathematical Elasticity, Plate Theory,Shell Theory
1. Introduction A plate is a structural element with planar dimensions that are largecompared to its thickness. Thus, plate theories are derived from the three-dimensional elastic theory by making suitable assumptions concerning thekinematics of deformation or the state stress through the thickness of thelamina, thereby reducing three-dimensional elasticity problem into a two-dimensional one.Efrati et al. [1] present a model, defined as the elastic theory of uncon-strained non-Euclidean plates , for modelling deformation of thin objects. Themain application of the authors’ work is in the study of natural growth of ∗ Corresponding author
Email address: [email protected] (Kavinda Jayawardana)
January 13, 2021 issue such as growth of leaves and other natural slender bodies. Some nu-merical results are present, which is based on an example of a hemisphericalplate, and they imply the occurrence of buckling transition, from a stretching-dominated configuration to a bending-dominated configuration, under vari-ation of the plate thickness.However, we show that what the authors present is not a plate theorymodel; It is, in fact, a shell theory model, but with an incorrect strain ten-sor. Thus, the authors numerical results imply that a thin object can bestretched substantially with very little force, which is physically unrealisticand mathematically disprovable. All the theoretical work of the authors, i.e.nonlinear plate equations in curvilinear coordinates, can easily be rectifiedwith the inclusion of both a sufficiently differentiable R → E diffeomor-phism and a set of external loadings, such as an external strain field.
2. Efrati et al. ’s Work
Given that a growing leaf can be modelled by a plate, Efrati et al. [1] focus on the elastic response of the plate after its planar (i.e. rest) configu-ration is modified, either by growth or by a plastic deformation. Thus, thegoal is to derive a thin plate theory, as a generalisation of existing elasticplate theories, that it is valid for large displacements and small strains inarbitrary intrinsic geometries. The authors ignore the thermodynamic lim-itations on plastic deformations as they are considered to be not relevantwhen modelling naturally growing tissue, and further assume that the ref-erence configuration is a known quantity. Their main postulate is that anon-Euclidean plate cannot assume a rest configuration, i.e. no stress-freeconfiguration can exist, and thus, one faces a nontrivial problem that alwaysexhibits residual stress. Note that by non-Euclidean the authors mean ‘theinternal geometry of the plate is not immersible in 3D Euclidean space’ [1].Also, the authors define ‘a metric is immersible in E ’ if Ricci curvaturetensor with respect to the implicit coordinate system is identically zero, i.e.for a given immersion ϕ : R n → E , where 1 ≤ n ≤
3, the metric ¯ g on R n induced by the immersion ϕ results in Ric = in R n , where R n and E n arecurvilinear and Euclidean spaces respectively. ∼ razk/iWeb/My Site/Publications files/ESK08.pdf x = ( x , x , x ) , in which the ‘reference metric’,¯ g ij = ¯ g ij ( x , x ), takes the form ¯ g αβ = ¯ g βα , ¯ g α = 0, ¯ g = 1. The referencemetric is a symmetric positive-definite tensor and considered to be a knownquantity. The plate is considered to be ‘even’, i.e. the domain D ⊂ R ofcurvilinear coordinates can be decomposed into D = S × [ − h , h ], where S ⊂ R and h is the thickness of the plate. Thus, it is given that an evenplate is fully characterised by the metric of its mid-surface, i.e. at where x = 0.Although thin plates are three-dimensional bodies, the authors took ad-vantage of the large aspect ratio by modelling the plates as two-dimensionalsurfaces, and thus, reducing the dimensionality of the problem. To achievethis the authors assume ‘Kirchhoff-Love assumptions:’ (i) ‘the body is in astate of plane-stress (the stress is parallel to the deformed mid-surface)’, and(ii) ‘points which are located in the undeformed configuration on the normalto the mid-surface at a point p , remain in the deformed state on the normalto the mid-surface at p , and their distance to p remains unchanged’ [1].Now, consider the deformed plate in the Euclidean space, which is de-fined as a compact domain Ω ⊂ R endowed with a regular set of materialcurvilinear coordinates. Define the mapping r : D ⊂ R Ω, from thedomain of parameterisation D into Ω, as the configuration of the body en-dowed with the metric tensor g , which is defined as g ij = ∂ i r · ∂ j r , where · is the Euclidean dot-product. It is given that Kirchhoff-Love second as-sumption implies that g α = 0. Thus, when defined more precisely, one findsthat r ( x , x , x ) = R ( x , x ) + x ˆ N ( x , x ), g αβ = a αβ − x b αβ + ( x ) c αβ , g α = 0 and g = 1, where R is the mid-surface, ˆ N is the unit normal to themid-surface, and a αβ , b αβ and c αβ are the first, the second and the third fun-damental form tensors respectively. With further inspection one finds that a αβ = ∂ α R · ∂ β R , b αβ = ( ∂ α ∂ β R ) · ˆ N and c αβ = ( a − ) γδ b αγ b βδ . The ultimategoal is to find the metric tensor g , and the authors state that the metrictensor g is immersed in R , and thus, the metric tensor uniquely defines thephysical configuration of a three-dimensional body. It is also the case thatone needs to find equations to six unknowns which make up the metric tensorfor the general case, where ¯ g is not defined by r . For the general case, theauthors describe one approach to this problem via the use of ‘the modified3ersion of the hyper-elasticity principle ... the elastic energy stored within adeformed elastic body can be written as a volume integral of a local elasticenergy density, which depends only on (i) the local value of the metric tensorand (ii) local material properties that are independent of the configuration’[1]. It is unclear what the authors mean by this definition; thus, for a moreprecise definition of hyperelasticity, we refer the reader to Ball [2] or Ciarlet[3]. The authors define the strain tensor as follows, ǫ ij = 12 ( g ij − ¯ g ij ) , (1)and thus, the energy functional is expressed as follows, E ( g ) = Z D w ( g ) √ ¯ g dx dx dx , (2)where w = A ijkl ǫ ij ǫ kl is the energy density and A ijkl = λ ¯ g ij ¯ g kl + µ (¯ g ik ¯ g jl +¯ g il ¯ g jk ) is the elasticity tensor. With the use of the energy functional (2), theconfiguration r is varied to find the three constraints that g αβ must satisfy,i.e. ‘the fundamental model for three-dimensional elasticity’ [1]. Note that Einstein’s summation notation is assumed throughout, and we regard theindices i, j, k, l ∈ { , , } and α, β, γ, δ ∈ { , } , unless it is strictly statesotherwise. Also note that the authors define the symmetric Ricci curvaturetensor of the metric g as follows, Ric li = 12 (cid:0) g − ) kj ( ∂ k ∂ i g lj − ∂ k ∂ j g li + ∂ j ∂ l g ki − ∂ i ∂ l g kj (cid:1) +( g − ) kj g pq (Γ plj Γ qki − Γ pkj Γ qli ) . As the elastic body is immersed in R , the variational principle impliesthat the six independent components of the symmetric Ricci curvature tensormust all vanish, i.e. Ric ij = 0. However, Ric ij = 0 and the three equationsobtained by varying the configuration r in equation (2) imply that the systemis over-determined. Thus, the authors postulate that there are two possibleways to resolve this ‘seemingly over-determination’. The first is by noticingthat the six independent components of Ricci curvature tensor’s derivativesare related through second Bianchi identity. The second way of resolving thisissue is by identifying the immersion r as the three unknown functions (as4efined previously), in which case the six equations that form the Ricci tensorare the solvability conditions for the partial differential equation. However,as the equations in r are of the higher order, one needs to supply additionalconditions, namely to set the position and the orientation of the body, inorder to obtain a unique solution for r .To find the reduced energy density the authors integrate the energy den-sity (2) over the thin dimension as w D = R h − h w dx to obtain the equation, w D = hw S + h w B , where w S = Y ν ) (cid:18) ν − ν ¯ g αβ ¯ g γδ + ¯ g αγ ¯ g βδ (cid:19) ( a αβ − ¯ g αβ )( a γδ − ¯ g γδ ) and w B = Y ν ) (cid:18) ν − ν ¯ g αβ ¯ g γδ + ¯ g αγ ¯ g βδ (cid:19) b αβ b γδ , (3)which are defined as stretching and bending densities respectively. Note that Y is the Young’s modulus and ν is the Poisson’s ratio of the plate.It is stated that with the use of Cayley-Hamilton theorem, the density ofthe bending content can be written in the following form, w B = Y ν ) (cid:0) − ν (¯ g αβ b αβ ) − | b || ¯ g | (cid:1) . It is also stated that, if a αβ = ¯ g αβ (i.e. the two-dimensional configurationhas zero-stretching energy), then the density of the bending content can beexpressed as the density of Willmore functional [4] as follows, w W = Y ν ) (cid:18) H − ν − K (cid:19) , where K and H are Gaussian and the mean curvatures of the mid-surfacerespectively.With the vanishing of the Ricci tensor the authors obtain Gaussian curva-ture and Gauss-Mainardi-Peterson-Codazzi equations, which are respectively5efined as K = | b || a | = 12 ( a − ) αβ ( ∂ γ Γ γαβ − ∂ β Γ γαγ + Γ γγδ Γ δαβ − Γ γβδ Γ δαγ ) and (4) ∂ b α + Γ βα b β = ∂ b α + Γ βα b β , (5)where equations (4) and (5) given to provide sufficient conditions for im-mersiblility of the metric tensor g in R .It may appear to the uninitiated in the study mathematical elasticity thatEfrati et al. ’s publication [1] is a coherent piece of work, but it is, in fact,flawed. To illustrate this matter in detail, we direct the reader’s attentionto section 3.4 of Efrati et al. [1]. Upon examining the governing equationsand the boundary conditions, one can see that the governing equations aredefined for zero-external loadings, and the boundary conditions are definedfor zero-tractions, zero-boundary moments and there are no descriptions ofany Dirichlet conditions. Thus, it is mathematically impossible to obtain anon-zero solution (excluding any rigid motions). Furthermore, there is noevidence in the authors’ publication for proof of the existence of solutions,either via rigorous mathematics (Γ-limit or otherwise) or via numerical anal-ysis (only some conjectures regarding solvability conditions are given by theauthors, as we previously discussed). (a) (b) (c) Figure 1: ‘A schematic illustration of an unconstrained plate exhibiting resid-ual stress. (a) The two elements composing the plate are shows side by side.(b) As the red trapezoid is too large to fit into the square opening, it is com-pressed. (c) For a plate sufficiently thin, the induced compression exceedsthe buckling threshold, and the trapezoid buckles out of plane. Note thatthere are many shapes that preserve all lengths along the faces of the plate,yet they cannot be planar’ [1].Efrati et al. ’s [1] erroneous work arises from not fully understanding howto model the given problem. Consider figure 1 (c) (see figure 1 from Efrati6 t al. [1]): the very reason the red trapezoid is deformed is because it iscompressed at the boundary, i.e. it is deformed as it is subjected to a Dirich-let boundary condition. Thus, if one attempts to model this problem withmathematical rigour, then one can derive the actual energy functional forthis problem. To do so, consider the map r ( u ) = R ( u ) + z ∂ x R ( u ) × ∂ y R ( u ) || ∂ x R ( u ) × ∂ y R ( u ) || , (6)which is assumed to be a sufficiently differentiable E → E diffeomorphismfor an appropriate u ∈ E , where R ( u ) = ( x, y, ) E + ( u ¯1 ( x, y ) , u ¯2 ( x, y ) , u ¯3 ( x, y ) ) E is the deformed mid-surface of the plate, u is the displacement field thatdescribes a vector displacement in the three-dimensional Euclidean spaceand × is the Euclidean cross-product. The metric with respect to map (6)in the Euclidean space is g ¯ α ¯ β ( u ) = ∂ ¯ α r ¯ i ( u ) ∂ ¯ β r ¯ i ( u ), where the over-bar in theindices highlights the fact that one is using Euclidean coordinates. Now, thestrain tensor of a plate can be expressed as follows ǫ ¯ α ¯ β ( u ) = 12 (cid:0) g ¯ α ¯ β ( u ) − δ ¯ α ¯ β (cid:1) , where x ¯1 = x and x ¯2 = y . Thus, the energy functional can be expressed asfollows, J ( u ) = Z S " hA ¯ α ¯ β ¯ γ ¯ δ (cid:16) ∂ ¯ α R ¯ i ( u ) ∂ ¯ β R ¯ i ( u ) − δ ¯ α ¯ β (cid:17) (cid:16) ∂ ¯ γ R ¯ k ( u ) ∂ ¯ δ R ¯ k ( u ) − δ ¯ γ ¯ δ (cid:17) + 124 h A ¯ α ¯ β ¯ γ ¯ δ ∂ ¯ α ¯ β R ¯ i ( u ) ( ∂ x R ( u ) × ∂ y R ( u )) ¯ i || ∂ x R ( u ) × ∂ y R ( u ) || ! ∂ ¯ γ ¯ δ R ¯ k ( u ) ( ∂ x R ( u ) × ∂ y R ( u )) ¯ k || ∂ x R ( u ) × ∂ y R ( u ) || ! dxdy , (7) u ∈{ v ∈ W ( S ) | v | ∂ S = u } , (8)where S ⊂ E is the mid-plane of the unstrained plate, W ( S ) is an appro-priate Sobolev space, u ∈ L ( ∂ S ) is a Dirichlet boundary condition, and7 ¯ αβ ¯ γ ¯ δ = (1 + ν ) − Y (2(1 − ν ) − νδ ¯ α ¯ β δ ¯ γ ¯ δ + δ ¯ α ¯ γ δ ¯ β ¯ δ + δ ¯ α ¯ δ δ ¯ β ¯ γ ) is the elasticitytensor of a plate. As R ( u ) describes a surface, one may argue that with anappropriate coordinate transform, one can express equation (7) in the sameform as the authors’ energy functional (see equation 3.7 of Efrati et al. [1]),which is true if one is using an appropriate ( x, y ) E : R → E coordinatetransform (the authors’ erroneous coordinate transform leads to a referencemetric of a shell at x = 0; please see equation 4.1 and see the definition ofΦ( r ) from section 4.2 of Efrati et al. [1]). However, without the Dirichletboundary condition from equation (8) or some other external loading, whichis exactly what the authors are considering (see respectively equation 2.5 andsection 3.2 of Efrati et al. [1]), one gets the trivial zero-displacement solu-tion, i.e. u = in S . The fact that the authors are claiming that nonzerosolutions are possible without tractions, Dirichlet boundary conditions, ex-ternal loadings or boundary moments imply that they are doing somethingfundamentally flawed.Now, consider the sufficiently differentiable diffeomorphism ϕ : ω ⊂ R → ϕ ( ω ) ⊂ E with the property det( J ) >
0, where ( x, y ) E = ( ϕ ¯1 ( x , x ) ,ϕ ¯2 ( x , x ) ) E and J ¯ βα = ∂ α ϕ ¯ β is the Jacobian matrix of the map ϕ . Notethat we reserve the vector brackets ( · ) E for vectors in the Euclidean spaceand ( · ) for vectors in the curvilinear space. Now, consider the mapping R ( u ) ◦ ϕ : ω ⊂ R → R ( u )( S ) ⊂ E . As ϕ is a diffeomorphism, R ( u ) ◦ ϕ is a well defined surface for a suitable displacement field u ∈ W ( S ), and asdet( J ) >
0, the unit normal to the surface R ( u ) is equal to the unit normalto the surface R ( u ) ◦ ϕ . Thus, with respect to ϕ , equation (7) reduces tothe following, J ( u ) = Z ω " A αβγδ (cid:18) ha αβ ( u ) a γδ ( u ) + 112 h b αβ ( u ) b γδ ( u ) (cid:19)p det(¯ g ) dx dx , (9) u ∈ { v ∈ W ( S ) | v | ∂ S = u } , a αβ ( u ) = 12 (cid:16) ( ∂ α R ¯ i ( u ) ◦ ϕ )( ∂ β R ¯ i ( u ) ◦ ϕ ) − ¯ g αβ (cid:17) ,b αβ ( u ) = ( ∂ αβ R ¯ i ( u ) ◦ ϕ ) (( ∂ R ( u ) ◦ ϕ ) × ( ∂ R ( u ) ◦ ϕ )) ¯ i || ( ∂ R ( u ) ◦ ϕ ) × ( ∂ R ( u ) ◦ ϕ ) || , ¯ g αβ = ∂ α ϕ ¯ γ ∂ β ϕ ¯ γ ,A αβγδ = Y ν ) (cid:18) ν − ν ¯ g αβ ¯ g γδ + ¯ g αγ ¯ g βδ + ¯ g αδ ¯ g βγ (cid:19) . Now, equation (9) is exactly the same form as the nonlinear plates equa-tions in curvilinear coordinates put forward by the authors, excluding theDirichlet boundary condition. However, equation (9) is derived from theplate equations in Euclidean coordinates (7) with the use of the map ϕ which is a R → E diffeomorphism, and thus, the reference metric ¯ g isimmersible in E . To be more precise, consider Ricci curvature tensor inthe two-dimensional Euclidean space, Ric
Euclidean¯ α ¯ β = Ric γδ ( J − ) γ ¯ α ( J − ) δ ¯ β . AsRicci curvature tensor is identically zero in the Euclidean space (clearly!)and the map ϕ is a diffeomorphism, we have the vanishing of Ricci curva-ture tensor in ω ⊂ R , i.e. Ric = . Now, consider the map ϕ × ( z ) E : { ω × [ − , ] } ⊂ R × E → { ϕ ( ω ) × [ − , ] } ⊂ E . As ϕ is a diffeo-morphism, the map ϕ × ( z ) E is also a diffeomorphism, and thus, the met-ric generated by the map ϕ × ( z ) E is immersible in E . Furthermore, as r ( u ) : S × [ − , ] ⊂ E → r ( u )( S × [ − , ]) ⊂ E is a diffeomor-phism, the metric on E induced by r ( u ) (i.e. the metric of deformation)is immersible in E . Thus, the metric on R × E generated by the map r ( u ) ◦ ( ϕ × ( z ) E ) : { ω × [ − , ] } ⊂ R × E → r ( u )( S × [ − , ]) ⊂ E (i.e. the metric g ) is also immersible in E . However, the authors assertthat their reference metric is not immersible in E . But it is mathemati-cally impossible to derive equation (9) from equation (7) without the use ofa sufficiently differentiable R → E diffeomorphism; thus, the fact that theauthors claiming that their reference metric is not immersible in the three-dimensional Euclidean space (while their metric g is immersible in E ) meansthat the authors are attempting something fundamentally flawed.Note that, if equation (7) linearised and along with the Dirichlet boundary9ondition, then one gets the following, J ( u ) = Z S A ¯ α ¯ β ¯ γ ¯ δ (cid:18) ha ¯ α ¯ β ( u ) a ¯ γ ¯ δ ( u ) + 112 h b ¯ α ¯ β ( u ) b ¯ γ ¯ δ ( u ) (cid:19) dxdy , u ∈ { v ∈ H ( S ) × H ( S ) × H ( S ) | v | ∂ S = ,n ¯ α ∂ ¯ α v | ∂ S = 0 , v | { ∂ S \ ∂ S } = u } , where a ¯ α ¯ β ( u ) = ( ∂ ¯ α u ¯ β + ∂ ¯ β u ¯ α ), b ¯ α ¯ β ( u ) = ∂ ¯ α ¯ β u ¯3 , n is the unit outwardnormal to the boundary ∂ S and ∂ S ⊂ ∂ S with meas( ∂ S ; R ) >
0, wheremeas( · ; R n ) is the standard Lebesgue measure in R n and H n ( · ) are the stan-dard W n, ( · )-Sobolev spaces (see section 5.2.1 of Evans [5]). Such problemscan be solved by consulting the literature that are specialised in the studyof linear plate theory (see Ciarlet [6] and Reddy [7]).For numerical results, instead of finding the unknown metric tensor g (which is the goal of the publication), the authors attempt to analyse thestretching and the bending densities, i.e. w s and w b respectively, for a prede-termined reference metric ¯ g and a predetermined deformed mid-surface R ,and thus, a predetermined deformed metric g (see section 4 of Efrati et al. [1]). The authors give numerical results for an ‘annular hemispherical plate’,i.e. annular plate deformed in to a hemispherical shape, and state that nu-merical results demonstrate that in the general case there is no ‘equipartition’between bending and stretching energies. The authors conclude by sayingtheir numerical findings support treating very thin bodies as inextensible,and ‘it also shows that not only in the equilibrium 3D configuration domi-nated by the minimisation of the bending energy term, but the total elasticenergy is dominated by it also’ [1]. The reader must understand that theauthors’ numerical results do not imply the existence of a solutions (i.e. theexistence of the deformation metric g not proven), as the numerical resultsare obtained for a predetermined metric g .The authors’ numerical analysis implies that a thin object can be stretchedsubstantially with very little force. To examine this in more detail, considerthe following simple example in accordance with the authors’ numerical anal-ysis. Consider two circular plates: plate c and plate s , with same Young’smodulus Y , Poisson’s ratio ν , thickness h and radius r , and assume that h/r = ε ≪
1. Now, take plate c and deform it into the shape of a semi-cylinder with a radius π r (an area preserving deformation). Following the10uthors’ publication, one finds that the mid-surface can be express by thefollowing map, R ( x , x ) = ( x , π r sin( x ) , π r cos( x ) ) E , and as one knows the deformed configuration in advance, one finds that thereference metric has the form ¯ g ( x , x ) = diag(1 , ( π r ) , E c = 12 Y − ν Z h − h Z Z { ( x ) +( rπ x ) ≤ r } (cid:18) πx r (cid:19) (cid:18) rπ (cid:19) dx dx dx = 196 π Y − ν h . (10)Now, take plate s and deform it into a shape of a hemisphere with a radius √ r (an area preserving deformation). Following the authors’ publication,one finds that the mid-surface can be express by the following map, R ( x , x ) = 12 √ r ( sin( x ) cos( x ) , sin( x ) sin( x ) , cos( x ) ) E (see the definition of R ( r, θ ) from section 4.1 of Efrati et al. [1]), and as oneknows the deformed configuration in advance, one finds that the referencemetric has the form ¯ g ( x , x ) = diag( r , r sin ( x ) ,
1) (see equation 4.1and the definition of Φ( r ) from section 4.2 of Efrati et al. [1]), where thisconfiguration is defined as the ‘stretch-free configuration’ (see section 4.1 ofEfrati et al. [1]). Thus, the stored energy of a circular plate that is beingdeformed into a hemisphere can be expressed as follows, E s = Y − ν Z h − h Z π Z π − π √ x r ! √ r ! sin( x ) dx dx dx = 13 π Y − ν h . (11)Equations (10) and (11), therefore, imply that, if one deforms a circularplate into a semi-cylinder with a radius π r and deform a circular plate intoa hemisphere with radius √ r , then one gets the very similar respective11nergy densities π (1 − ν ) − Y ε Jm − and Y (1 − ν ) − ε Jm − , i.e. bothdeformations’ internal energies are of O ( ε )Jm − . Which in turn impliesthat both deformations require force of O ( ε )N, given that one is applyingthe forces to the boundaries of the each respective plates. Thus, the authors’work asserts that it take approximately the same amount of force to bend aplate into a semi-cylindrical shape or to stretch a plate into a hemisphericalshape with a similar radius. The reader may try this one’s self: find a pieceof aluminium foil (i.e. kitchen foil) and try to bend it over one’s water bot-tle. This is a very simple process and the reader will able to accomplish thiswith a minimum of effort. In fact, the force of gravity is alone may even besufficient to deform the piece of aluminium foil over the bottle without muchinterference. Now, try to stretch that same piece of aluminium foil smoothlyover a rigid sphere with a similar radius, e.g. over a cricket ball. Can thereader do this without tearing or crumpling, and with the same force as oneapplied in the previous case?To attempt this problem with mathematical regiour, consider the set S = { ( x, y, ) E ∈ E | x + y ≤ r } , which describes the mid-plane ofthe unstrained plates c and s . Now, if one deforms plate c is into a semi-cylindrical shape with a radius π r , then one finds that the map of the de-formed mid-surface has the following form, R ( x, y ) = ( x, π r sin (cid:16) π r y (cid:17) , π r cos (cid:16) π r y (cid:17) ) E , and thus, the total stored energy of a circular plate of radius r that is beingdeformed into a semi-cylindrical shape with a radius π r can be expressed asfollows, E c = 12 Y − ν r Z h − h Z Z S (cid:16) πz r (cid:17) dxdydz = 196 π Y − ν h . (12)Now, if one deforms plate s is in to a hemisphere with a radius √ r , thenone finds that the map of the deformed mid-surface has the following form, R ( x, y ) = 12 √ ( x sin (cid:16) π r p x + y (cid:17) , y sin (cid:16) π r p x + y (cid:17) ,r cos (cid:16) π r p x + y (cid:17) ) E , r that isbeing deformed into a hemisphere with a radius √ r can be expressed asfollows E s = 12 Z h − h Z Z S A αβγδ ǫ αβ ( x, y ) ǫ γδ ( x, y ) dxdydz + O ( h )= ChY r + O ( h ) , (13)where C is an order-one positive constant that is independent of h , Y and r ,and ǫ ( x, y,
0) = (cid:16) r (cid:17) sin (cid:16) π r p x + y (cid:17) ( x + y ) (cid:18) y − xy − xy x (cid:19) + (cid:16) π (cid:17) x + y (cid:18) x xyxy y (cid:19) − (cid:18) (cid:19) , is the strain tensor of the plate at z = 0. As the reader can see from equa-tions (12) and (13) that if one deforms a circular plate into a semi-cylinderwith a radius π r and deform a circular plate into a hemisphere with a radius √ r , then one get the respective energy densities O ( ε )Jm − and O (1)Jm − .Thus, one can see that it takes significantly higher amount of energy to de-form plate in to a hemisphere than to simply bend it in to a semi-cylinder,as the former deformation requires a significant amount of stretching andcompression, while the latter requires no such in-plane deformations, whichis far more realistic than results obtained by Efrati et al. ’s approach [1]. Notethat the both deformations conserve area.As further analysis, consider the deformed plate s in curvilinear co-ordinates ( x , x ) , where 0 ≤ x ≤ π and | x | ≤ π . Now, the firstand the second fundamental form tensors of the deformed configurationcan be expressed respectively as F [I] ( x , x ) = r diag(1 , sin ( x )) and F [II] ( x , x ) = − √ r diag(1 , sin ( x )). If one follows the authors’ publi-cation, then one finds that the reference metric tensor has the form¯ g ( x , x ) = diag( 12 r , r sin ( x ) , , and this can only be derived by doing the following,¯ g ij ( x , x ) = ∂ i r ¯ k ( x , x , x ) ∂ j r ¯ k ( x , x , x ) | x =0 , r ( x , x ) = ( √ r + x ) ( sin( x ) cos( x ) , sin( x ) sin( x ) , cos( x ) ) E .This implies that ¯ g is the reference metric of a shell at x = 0, and thus, ¯ g is clearly not immersible in E as Ricci tensor is not identically zero, i.e. Ric = 12 r diag(1 , sin ( x )) = , for x = 0 . Thus, the authors’ erroneous reference metric implies that a αβ − ¯ g αβ = 0, ∀ α, β ∈ { , } , i.e. zero-planar strain, which in turn implies the existence ofa ‘stretch-free configuration’ for a substantially deformed plate.Now, if one attempts this same problem with mathematical precision,then one finds that the reference metric tensor can be expressed as follows,¯ g ( x , x ) = 4 (cid:16) rπ (cid:17) diag(1 , ( x ) ) , where ¯ g αβ ( x , x ) = ∂ α x∂ β x + ∂ α y∂ β y with x = r − π p x + y and x =arctan( y/x ). The coordinate transform ( x ( x , x ) , y ( x , x ) ) E : R → E is a diffeomorphism (except at x = x = 0) and Ricci tensor is identicallyzero, i.e. Ric = . Furthermore, det( ∂ x, ∂ x ; ∂ y, ∂ y ) >
0, and thus, thedefinition of the unit normal to the deformed surface is not violated (again,except at x = x = 0). Thus, half of change in the first fundamental formtensor (i.e. planar strain) can be expressed as follows, a ( x , x ) = 14 r diag(1 − π , sin ( x ) − π ( x ) )and the change in second fundamental form tensor (i.e. bending) can beexpressed as follows, b ( x , x ) = − √ r diag(1 , sin ( x )) . Now, with this coordinate transform no such ‘stretch-free configuration’ canexist for a plate with a radius r that is being deformed into a hemispherewith a √ r , unless the radius of the plate is zero.Above analysis shows that Efrati et al. [1] are not studying plates, butthey are studying nonlinear Koiter shells with an erroneous strain tensor. Theauthors’ definition of the strain tensor leads to an incorrect change in second14undamental form tensor, and thus, an overestimation of the bending energydensity of the shell per h (see equation (3)). To attempt this problem withmathematical precision, let ˜ g ij ( x ) = ∂ i X ¯ k ∂ j X ¯ k be the metric of the referenceconfiguration X ( x ) = σ ( x , x ) + x ( σ , × σ , ) || σ , × σ , || with respect to the curvilinear coordinate system x = ( x , x , x ) , where σ : R → E is a sufficiently differentiable immersion (Efrati et al. ’s refer-ence metric [1] is derived by ¯ g = ˜ g | x =0 ). Thus, in nonlinear shell theory,one defines the strain tensor as ǫ αβ ( u ) = ( g αβ ( u ) − ˜ g αβ ), where g αβ ( u ) = ∂ α r ¯ i ( u ) ∂ β r ¯ i ( u ), r ( u ) = R ( u ) + x ( ∂ R ( u ) × ∂ R ( u )) || ∂ R ( u ) × ∂ R ( u ) || , R ( u ) = σ + u α ∂ α σ + u N and u ( x ) is the displacement field in curvilinearcoordinates. For more on nonlinear Koiter’s shells, please consult Ciarlet [3],Koiter [8], and Libai and Simmonds [9].Even if Efrati et al. [1] obtain the correct form of the strain tensor forshells, they are still unjustified in using the shell strain tensor to model plates.To explain this matter with mathematical rigour, let S be a two-dimensionalplane and let S ′ be a two-dimensional surface. What the authors fail to graspis that an arbitrary mapping from S to S ′ (i.e. σ : S ⊂ R → S ′ ⊂ E ) isnot the same as deforming the plane S into the surface S ′ (i.e. { S ∪ { u ∈ E } ⊂ E } = { S ′ ⊂ E } ). The former is a simple coordinate transform(which may or may not be related to deforming the body), while the latter isa unique vector displacement (unique up to a rigid motion). To understandthe distinction between a coordinate transform and a vector displacement,please consult section 1 and section 2 of Morassi and Paroni [10].
3. Conclusions