Soft Matter PAPER. Buckling pathways in spherical shells with soft spots. Introduction. Jayson Paulose* a and David R. Nelson b

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1 PAPER Buckling athways in sherical shells with soft sots Cite this: Soft Matter, 013, 9, 87 Received 13th March 013 Acceted 13th May 013 DOI: /c3sm50719j Jayson Paulose* a and David R. Nelson b Thin elastic sherical shells are known to exhibit a buckling instability at a finite external ressure. We study how this buckling is influenced by a weak region in an otherwise uniform shell, focusing on the case of a single small circular soft sot that is thinner than the rest of the shell. Using numerical simulations and theoretical arguments, we show that the soft region fundamentally alters the buckling behavior of the shell. The soft sot influences both the ressure at which the shell buckles, and the ostbuckling shae of the shell. Deending on the roerties of the soft sot, we find either a single buckling transition or two searate transitions as the external ressure is increased. We analyze the deendence of these buckling transitions on the size and thickness of the soft region. Besides contributing to our fundamental understanding of buckling transitions in inhomogeneous shells, our results can be alied to designing casules with tunable shaes. I Introduction Buckling is a ubiquitous henomenon in thin-walled elastic structures, such as lates and shells. 1, A rototyical examle is the buckling transition of a thin sherical shell under ressure. Under an increasing external ressure, idealized shells without imerfections remain sherical until a threshold ressure is reached at which the shell buckles: at this oint, a very small increment in ressure triggers a large deformation in the form of one or more indentations that signi cantly reduce the enclosed volume. Signi cant hysteresis is associated with this transition. In this sense, the buckling of shells in three dimensions resembles a rst order hase transition, and the critical buckling ressure reresents a limit of metastability. The situation is quite different for ressurized rings in two dimensions (or ressurized cylinders), where the behavior at buckling resembles a continuous hase transition (see, e.g. ref. 3). Understanding and mitigating buckling has long been imortant to engineers to revent the catastrohic failure of macroscoic shell structures such as water tanks and submarines. 4 More recently, the buckling of thin sherical shells has been exloited for various functions, such as the measurement of shell elastic constants, 5 controlled release of encasulated chemicals, 6 and mass-roduction of colloids with non-sherical, functional shaes. 7,8 Whether buckling is desirable or avoidable, inhomogeneities in the shell, i.e. variations in the shae, thickness or material roerties, strongly in uence the buckling behavior. For instance, even small deviations from the erfect sherical shae induce shells to buckle at signi cantly reduced ressures, requiring engineers to build shells several times stiffer than a Harvard School of Engineering and Alied Sciences, Cambridge, MA 0138, USA. jaulose@ost.harvard.edu b Deartment of Physics, Harvard University, Cambridge, MA 0138, USA their theoretical design seci cation to withstand a articular load. 9,10 Thermally excited shae uctuations, which may be signi cant for shells with nanoscale thicknesses, also markedly reduce the buckling ressure. 11 The ollen grains of certain lant secies, which can be modeled as elastic shells, have evolved one or more carefully shaed so sectors in the otherwise stiff shell that guide their shae change as they dry out to evade the hysteresis associated with sudden buckling behavior. 1 Placing circular so regions in seci c geometric con gurations on sherical shells also fundamentally alters their buckling under ressure, allowing them to shrink by about half their volume while retaining their sherical shae. 13 A recent study of sherical microcasules whose thickness varied with latitude showed that the thickness inhomogeneity affects the buckling ressure, dynamics and ostbuckling shaes. 14 These studies are only beginning to exlore the rich variety of shell inhomogeneities that may be exloited to control and extend buckling behavior, ranging from naturally occurring ra s and domains in multicomonent vesicles 15,16,17 to carefully tuned variations in shell shae, thickness and comosition via micro- uidic 18 and raid rototying 13,19 techniques. Understanding the effect of inhomogeneity on the buckling transition, besides being a challenging roblem due to the highly nonlinear nature of large de ections in shells, has the otential to uncover new ways of using buckling to rovide form and function. In this work, we study the effect of a reviously unexlored inhomogeneity that is simle, yet general: a so sot with a circular boundary in an otherwise uniform sherical shell. We use numerical simulations and theoretical arguments to understand the buckling and ostbuckling behavior as the sot size and stiffness are varied. The resonse of the shell may be broken down into the resonse of the so region and the resonse of the remainder of the shell; the interlay of these resonses lays a signi cant role in exlaining the observed behavior. This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

2 Our numerical model is similar to ones used in other studies of the buckling of both uniform 0 and nonuniform 1,14 shells under ressure. Our theoretical analysis builds on revious work both on uniform and slightly imerfect sherical shells, 10,1 and on the buckling of sherical cas i.e. shallow sections of sheres, with various boundary conditions.,3 The rest of this aer is structured as follows: we introduce the elastic energy used to describe deformations in thin sherical shells, and identify the dimensionless quantities that are relevant for our descrition. We then resent our numerical results from simulations of buckling shells under external ressure, and describe different buckling athways that arise as the arameters are changed. To exlain our numerical observations, we analyze the underlying continuum shell theory equations for a so sot embedded in the rest of the shell. To rovide context, we reca the buckling behavior of uniform shells, which rovides much insight on the buckling of shells with so sots. We then consider searately the deformations of the ca, and of the remainder of the shell, using scaling arguments and aroximations to understand the underlying nonlinear differential equations that describe stresses and deformations in the shell. Finally we describe the hysteresis roerties of the inhomogeneous shells when the ressure is cyclically varied, which is strongly in uenced by the so region. We conclude with otential alications and ossible avenues for further research. II Elastic theory of thin sherical shells Tyically, the elastic energy of deforming a three-dimensional (3D) object made u of a uniform elastically isotroic material deends on the detailed elastic strains at each oint within the object and the 3D elastic Young's modulus E that relates strains to stresses. However, a simli cation can be made for elastic lates and shells, structures which have a very small extent in one of the three satial dimensions. 1, Then, it is sufficient to consider the deformation of the middle surface, a two-dimensional manifold. The elastic energy can be described using in-surface strains, which carry an energy enalty set by a two-dimensional (D) Young's modulus Y, and changes in curvature of the middle surface, enalized by a bending rigidity k. In terms of the 3D elastic modulus E and Poisson ratio n of the material making u the shell, the D moduli are Y ¼ Eh, (1) Eh 3 k ¼ 1ð1 n Þ ; () where h is the shell thickness. As a result, the thickness of the shell strongly affects its elastic resonse. In articular, for extremely thin lates and shells, the h 3 -deendence of k shows that bending is highly favoured over stretching, as demonstrated by the ease of bending a iece of aer comared to stretching it. Whereas thin, at elastic lates can exhibit ure bending deformations with no associated in-lane stretching and Paer consequently a very low energy cost (the aforementioned sheet of aer being an examle), the same is not always true of shells which are curved in their undeformed state. In at lates, the contribution of normal (out-of-lane) deformations of the midsurface to the in-lane strain is quadratic in the deformation, whereas the contribution to the bending tensor is linear. As a result, the bending energy dominates the elastic enalty of normal deformations of at lates. In contrast, the underlying curvature of a shell mediates a linear couling between out-oflane dislacements and in-lane strains: any transverse deformation z introduces a strain of order z/r, where R is the radius of curvature of the shell in its ground state. 1 Thus, bending and stretching are intimately couled in curved shells, unlike elastic lates and membranes which are at in the ground state. An imortant consequence of this couling is that normal deformations tend to be localized to narrow widths in sherical shells. A scaling analysis of the energy density reveals a length scale that sets the extent of characteristic deformations, which is small comared to the shell radius. Consider a deformation of maximum deth z localized over a length l in the sherical shell, which has a corresonding curvature change of aroximately z/l and strain z/r. The bending energy er unit area of the deformation is kz /l 4 while the stretching energy er unit area is Yz /R. Minimizing the sum of the bending and stretching energies with resect to l gives rise to a tyical elastic length scale of localization of deformations for which the total elastic energy is minimized, kr 1=4 l ¼ h h R ; (3) Y g1=4 where we have introduced the dimensionless Föl von Kármán number, gh YR R k z10 : (4) h The Föl von Kármán number quanti es the relative imortance of stretching to bending for sherical shells. It is large for thin elastic shells. Tyically, a shell is considered thin, i.e. the abstraction to a two-dimensional surface is valid and shell theory can be used, when h ( 0.1R; i.e., g T Very large Föl von Kármán numbers are common in both natural and arti cial systems. For instance, the radius of a ing-ong ball is about 100 times its wall thickness, leading to g z Tyical values for the elastic moduli and radius of curvature of a red blood cell 4 lead to estimates for g in the range 10 4 to Therefore, tyical deformations vary over a length scale l that is much smaller than the radius of the shell. This searation of scales simli es the theoretical analysis of deformations of sherical shells. A Elastic energy of thin shells We use the standard Kirchhoff Love descrition of thin elastic shells, in which the elastic energy density of the surface is quadratic in two-dimensional strain and curvature tensors Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

3 Paer The elastic energy is the sum of a stretching and a bending comonent, ð Y E ¼ ds ðu ð1 n 11 þ u Þ ð1 nþ u 11 u u 1 Þ þ k ðk 11 þ k Þ ð1 nþ k 11 k k i 1 (5) where the indices i, j label the local coordinate directions tangent to the shell mid-surface, u ij is the strain tensor and k ij ¼ K ij Kij 0 the change in the curvature tensor K ij from its initial value Kij. 0 The integration is carried out over the midsurface of the shell. The stress and curvature tensors are de ned to vanish when the external ressure vanishes the equilibrium shae is a shere of radius R. The initial curvature tensor Kij 0 of the shell mid-surface is that of a shere with radius R: Kij 0 ¼ d ij /R at every oint for an orthonormal basis set u in the tangent lane to the shere at that oint. This exression forms the basis of a discretized model of a thin shell aroriate for numerical simulation (see Aendix A). Uon icking a suitable coordinate system, rewriting the elastic energy exressions in terms of dislacement elds and stress functions, and erforming a functional extremization with resect to these elds, we obtain the equations of shell theory, which are nonlinear but amenable to analysis using numerical and aroximate analytical methods. We use both numerical simulations and a theoretical analysis of the shell theory equations to understand the behavior of inhomogeneous shells. B Shells with so sots In our study, we restrict ourselves to a seci c inhomogeneity: a thin region with a circular boundary in an otherwise uniform sherical shell. By considering a small region that is so er than the rest of the shell, we oen u the ossibility of triggering an instability in the so region before the shell as a whole collases. (A stiffer region in the shell would affect the ressure at which the shell collases, but would not show any additional instabilities before this collase; we do not consider stiffer regions here.) The thin region, or ca, is assumed to be made of the same material as the rest of the shell (the remainder), but with a thickness h ~ < h where h is the uniform thickness of the remainder. The elastic moduli in the ca and the remainder are obtained using eqn (1) and () with the aroriate thicknesses; the ca is thus easier to bend and stretch than the shell. We treat the ca and the remainder as materially couled to each other; i.e. the dislacements, stresses and bending moments are continuous across the boundary between the ca and the shell. In what follows, we will use ~Y, ~k,. to refer to the roerties of the ca and Y, k,. for the remainder. Without loss of generality, we choose the so ca to be centered at the north ole, and its size is then de ned by the azimuthal angle a (in radians) subtended by its rim. We consider cas small comared to the rest of the shell (a ( /4). The dimensionless arameters characterizing the shell are the Föl von Kármán number g (comuted using the elastic roerties of the remainder), the ca remainder thickness ratio s h h/h ~ < 1, and the dimensionless size l h Ra/l ¼ ag 1/4 (which exresses the radial size of the ca in units of the elastic length scale l of the remainder). Alternatively, one could use the Föl von Kármán number and elastic length scale of the ca, which are simly related to the remainder quantities via ~g ¼ g/s > g, ffiffi ~ ¼ s \. III Numerical results Soft Matter We numerically simulate the buckling of shells with so sots under hydrostatic ressure. We restrict ourselves to a single inhomogeneity in the form of a region with a circular boundary that is thinner than the rest of the shell. In buckling simulations, the ressure is measured in units of the classical buckling ressure 0 ¼ 4 ffiffiffiffiffiffi ky =R of a uniform shell with the same thickness as the remainder (see Section V for more details on the buckling transition for uniform sherical shells). In a tyical simulation, a uniform ressure is exerted on the shell by adding a volume integral Ð dv to eqn (5), and the local equilibrium con guration of the shell is comuted. More generally, would be the ressure difference between the inside and the outside of the shell; if the shell contains a uid such as air or water, we assume that it is ermeable to the enveloed uid, thus allowing reductions in volume in resonse to the ressure. (An exerimental realization is the osmotic crushing of susensions of drolet-temlated microcasules that are ermeable to the inner and outer uid but not to solute ions; introducing a solute into the outer hase exerts an osmotic ressure on the casules which can shrink in resonse by releasing their enclosed uid. 14 ) The ressure is initially set to zero, and incremented by a small amount between successive equilibrations. Fig. 1 shows the outcome of a simulation on a shell with Föl von Kármán number g ¼ 10 4, thickness ratio s ¼ 0.4, and dimensionless sot size l ¼ 5.. At early stages (low ressures) the shell resonds via a near-uniform contraction while maintaining a roughly sherical shae. At a critical value of the ressure, which we call c1, the ca exeriences a snathrough transition to an inverted shae, although the rest of the shell remains sherical. This shae ersists until a higher Fig. 1 Results of a buckling simulation on a shell with g ¼ 10 4, s ¼ 0.4 and ca size a ¼ 0.5 (where a is the azimuthal angle subtended by the sot at the center of the undeformed sherical shae), corresonding to l ¼ Ra/l ¼ 5.. Equilibrium configurations at different ressures, measured in units of the classical buckling ressure of a uniform shell without the soft sot, are shown. The mesh oints in the soft region are colored differentlyfrom the remainder. The shae of the shell is sherical u to the first buckling ressure c1 ¼ , when the soft ca snas through to an inverted shae. This shae is largely unchanged u to the second, more catastrohic buckling event at c ¼ This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

4 critical ressure c is reached, at which oint the rest of the shell exeriences a catastrohic collase. In real shells, this collase would be terminated by self-contact of the shell; in our simulations, we introduce a minimum volume restriction of roughly 10% of the original volume to stabilize the collased shae, as it is comutationally simler to imlement than selfavoidance. For this articular set of arameters, c1 ¼ and c ¼ This buckling athway, with two distinct buckling events as the ressure is ramed u, is observed over a wide range of values of s and l. For future reference, we will call this athway I. However, for other arameter ranges, different behavior is observed. For modestly thinned cas such that s T 0.6, ractically all shells only dislay one buckling transition, a catastrohic collase from a sherical shae to a fully de ated shae similar to the nal shae in Fig. 1 without any intermediate range of ressures where only the ca has snaed through. We refer to this as athway II. There is also a narrow range of arameters, for small values of s and l, in which a deformation is observed in the ca before the rest of the shell buckles, but the deformation is a smooth and continuous inversion of the shell as the ressure rises, rather than a sudden sna-through at a articular ressure. For these shells, the collase of the rest of the shell still haens when a articular ressure is reached. This is termed athway III in what follows. For both athway II and athway III, c1 refers to the critical ressure at which the lone buckling event occurs, and c is unde ned. Fig. summarizes the various buckling athways for shells with different thickness ratios s as a function of the dimensionless size l ¼ ag 1/4 of the so sot. These results are indeendent of g for small sots, as will be shown in the Paer theoretical analysis below. We observe that when the so sot is extremely thin comared to the rest of the shell (s < 0.5), athway I is almost always observed: two distinct buckling events occur at ressures c1 and c, and there is a range of ressures c1 < < c over which the ca has exerienced snathrough buckling but the shell is stable against further collase. The excetion is for extremely small sots (l ( 1.5) where buckling occurs via athway III with a continuous deformation of the ca from its initial shae to a fully inverted shae, followed by a single catastrohic buckling event at a ressure c1. For less drastic thickness changes (s $ 0.6), athway II is almost exclusively observed, i.e. a single catastrohic buckling event induces collase of the entire shell, with no signi cant deviation from the sherical shae rior to buckling. For s ¼ 0.5, athway I is observed in a narrow range of sot sizes 1.5 < l < 3.5, and athway II is observed outside this range. Note that, even when the ca size shrinks to zero (l / 0) the critical buckling ressure remains smaller than 0, the buckling ressure of an ideal shell, due to inevitable small scale inhomogeneities introduced by the amorhous mesh that describes the shell (see Aendix A). As exected, the sna-through buckling of the ca haens at higher ressures for thicker (i.e. stiffer) cas. However, for shells following athway I (0. # s # 0.5), the ressure c at which the whole shell collases has a very weak deendence on the thickness ratio s. Excet for some small deviations at small l, the ressure of bulk collase for these thickness ratios seems to follow a near-universal curve of / 0 as a function of l. This trend is in contrast to shells following athway II (s > 0.5), for which the ressure associated with large-scale collase of the shell does deend strongly on s. We also notice an interesting common structure in the c1 vs. l curves for different thickness ratios, all of which have a distinctive concave-uwards shae with strong variations u to a sot size l 3 4 beyond which they level off to a roughly constant value that deends on s. To exlain these features of the critical ressure function de ned by ¼ c1 (l,s) and ¼ c (l,s), we analyze the various mechanisms that bring about the elastic buckling instability in the following sections. IV Governing equations of shell theory Fig. Buckling ressures c1 and c of the first and second buckling events as the ressure is ramed u in numerical simulations, as a function of rescaled ca size for a variety of thickness ratios s ¼ h/h. ~ The ressure is rescaled by the classical buckling ressure 0 of a shere with the same roerties as the remainder. In all cases, g ¼ 10 4, although the results when exressed with rescaled variables are largely insensitive to g. Emty symbols signify a sna-through transition localized to the ca, whereas filled symbols signify catastrohic collase of the entire shell. Some shells exhibit two distinct buckling events sna-through of the ca at the lower ressure c1 and collase of the rest of the shell at the higher ressure c. Other shells dislay only a single transition, which is always a catastrohic collase of the whole shell, at the first buckling ressure c1. The second buckling ressure c is not defined for these shells. We begin by summarizing the equations governing the stresses and strains of elastic sherical shells. Initially, we consider comletely uniform shells. For theoretical analysis, we use shallow shell theory,,5 where we consider a section of the shell small enough so that sloes relative to a tangent lane at the basal oint are small. Then, we set u a Cartesian coordinate system (x,y) in this tangent lane, and describe deformations of the shell in terms of tangential dislacement elds U(x,y), V(x,y) and the normal dislacement W(x,y). The nonlinear strain tensor is related to the dislacements via,5 u xx ¼ U ;x þ 1 W ;x W R ; (6) u yy ¼ V ;y þ 1 W ;y W R ; (7) 830 Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

5 Paer u xy ¼ 1 U;y þ V ;x þ W ;x W ;y ; (8) where we use the notation f,a h v a f for satial derivatives. The curvature tensor is aroximately k ij ¼ W,ij. (9) The total energy, including the elastic contribution [eqn (5)] as well as the contribution due to a nite external ressure, reduces to E tot ¼ ð dxdy Y ð1 n Þ uxx þ u yy ð1 nþ u xx u yy u xy þ k W;xx þ W ;yy ð1 nþ W ;xx W ;yy W ;xy W : (10) Equilibrium shaes of the shell section corresond to elds U, V and W for which the energy functional, eqn (10), is stationary. Extremizing E tot with resect to variations in U, V and W gives rise to three nonlinear differential equations: kv 4 W 1 sxx þ s yy sxx W ;x þ s xy W ;y R ;x (11) s xy W ;x þ s yy W ¼ ; ;y ;y s xx,x + s xy,y ¼ 0, (1) s xy,x + s yy,y ¼ 0, (13) where V f ¼ f,xx + f,yy, V 4 f ¼ V (V f ), and s ij is the D stress tensor, with comonents s xx ¼ s yy ¼ Y 1 n uxx þ nu yy ; (14) Y 1 n uyy þ nu xx ; (15) s xy ¼ s yx ¼ Y 1 þ n u xy: (16) Eqn (1) and (13) are automatically satis ed by introducing the Airy stress function c, derivatives of which give the stress tensor: 1 s xx ¼ c,yy ; s yy ¼ c,xx ; s xy ¼ c,xy. (17) When s ij is written in terms of the Airy function, eqn (11) becomes kv 4 W 1 R V c N ðc; WÞ ¼; (18) dislacement elds U, V and W via eqn (14) (16). Eliminating the elds U and V from these three equations reduces them to a single comatibility condition for c: 1 Y V4 c þ 1 R V W þ 1 N ðw; WÞ ¼0: (0) Eqn (18) and (0), together with aroriate boundary conditions, govern the equilibrium de ections and stresses of a uniform shallow shell. The resence of the nonlinear oerator makes solving the shallow shell equations subtle and challenging, but they are amenable to analysis through scaling arguments and numerical methods. A Equations in olar coordinates When dealing with shells with so cas, it is advantageous to use olar coordinates r,q rather than x,y, with the origin at the center of the so region (so that x ¼ rcos q, y ¼ rsin q). In olar coordinates, the strain tensor reads u rr ¼ U ;r W R þ 1 W ;r ; (1) u qq ¼ U r þ V ;q r W R þ 1 W;q r ; () u rq ¼ 1 U;q r þ V ;r V r þ W ;rw ;q ; (3) r the curvature tensor is k rr ¼ W,rr, (4) k qq ¼ W ;r r þ W ;qq ; (5) r k rq ¼ W ;rq W ;q r r ; (6) and stresses are given by s rr ¼ c ;r r þ c ;qq r ; s qq ¼ c ;rr ; s rq ¼ c ;q r c ;rq r : (7) The governing equations remain unchanged (eqn (18) and (0)) but with the differential oerators V f ¼ f ;rr þ f ;r r þ f ;qq r ; V4 f ¼ V V f ; (8) g;r N ð f ; gþ ¼f ;rr r þ g ;qq f;r þ g r ;rr r þ f ;qq r f;rq r f g;rq ;q r r g ;q : r Soft Matter (9) where N ( f,g) ¼ f,xx g,yy + f,yy g,xx f,xy g,xy (19) V Buckling of uniform sherical shells under ressure de nes a second-order nonlinear differential oerator. Although any choice of the function c(x,y) automatically satis es eqn (1) and (13), c must also be related to real In rearation for our analysis of nonuniform shells, we review here the buckling of uniform sherical shells under external ressure, following the aroach in ref. 10. The classical This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

6 buckling of sherical shells can be understood using a linearized analysis of the shell equations. Prior to buckling, a consideration of the forces on a section of the shell shows that the shell contracts uniformly by an amount W 0 ¼ R (1 n)/et and is in a uniform state of comressive stress s xx ¼ s yy ¼ R/, corresonding to an Airy stress function c 0 ¼ R(x + y )/4. The buckling mode can be obtained by linearizing the nonlinear shell equations around this re-stressed state. Uon substituting W ¼ W 0 + W 1 and c ¼ c 0 + c 1 in eqn (40) and keeing terms linear in the normal dislacement W 1 and Airy function c 1,we nd: kv 4 W 1 1 R V c 1 þ R V W 1 ¼ 0; (30) V 4 c 1 þ Y R V W 1 ¼ 0: (31) The nonlinear couling between the re-stress and the normal de ection via the term N (c 0,W 1 ) is resonsible for the last term on the le -hand side of eqn (30). We search for oscillatory solutions of the form W 1 ¼ Ae iq$x, c 1 ¼ Be iq$x with some two-dimensional wavevector q ¼ (q x,q y ). Such a solution to eqn (30) and (31) can only exist rovided kq 4 R q þ Y ¼ 0; (3) R with the requirement for oscillatory solutions being that the wavevector magnitude q h ffiffiffiffiffiffiffiffi q$q is real and ositive. The smallest value of at which solutions with q > 0 exist is ¼ 4 ffiffiffiffiffiffi ky =R h 0, and the corresonding wavevector magnitude of the solution that arises is q ¼ 1/l ¼ g 1/4 /R. This ressure is identi ed as the buckling ressure of the shell. Note that the instability would not exist without the couling between c 0 and W 1 mentioned above, and is a nonlinear effect even though it is catured by a linearized analysis. The linearized buckling analysis shows that a uniform shell which contracts isotroically under low external ressures can release this comressive stress, at the cost of bending energy, by taking on an oscillatory transverse de ection. Such a de ection becomes energetically favorable at the ressure 0 ¼ 4 ffiffiffiffiffiffi ky =R. In fact, there are many degenerate de ection modes that arise at this critical ressure, since all wavevectors q such that q ¼ q ¼ 1/ l are allowed. The wavelength associated with these modes is l ¼ R/g 1/4 R since g [ 1 for thin shells. Thus, the buckling of sherical shells is triggered by one of many degenerate modes with wavelength much smaller than the shell radius. We have not established here that the oscillatory modes that arise at the buckling ressure are unstable toward further growth. This can be done by evaluating higher order terms in the elastic energy associated with the modes (see e.g. ref. ). We do not reroduce that calculation here, but instead resent a qualitative argument for the ultimate fate of the shell at the buckling ressure, from ref. 1, which considers the energetics of inversions of sections of the shell. Thin sherical shells under ressure nd it favorable to reduce their volume using nearly isometric inversions, which leave the metric within most of the inverted region unchanged and localize the elastic energy to the narrow rim of the inversion (which has a width of order l ). The energy associated with an inversion of deth d is characterized by a quantity we denote the Pogorelov energy, 1,6 E el ¼ ckg 1/4 (d/r) 3/, (33) where c is a dimensionless constant. If the shell is under external ressure, the net energy including the work done by the ressure is E ¼ E el + DV, where DV Rd is the volume change due to the inversion. We immediately see that at large values of inversion deth d, the second term dominates, and inversions can reduce their net energy by growing even deeer. The total energy E(d) has a maximum when kg 1=4d1= R Rd0 d max 3= R k R 3 ffiffiffi g : (34) Therefore, for a articular ressure, inversions of deth d > d max ( ) grow uncontrollably, until some other constraint such as self-contact revents further growth of the inversion. At the buckling ressure, d max ð 0 ÞR= ffiffiffi g h; thus, even an inversion of very small deth, comarable to the thickness h of the shell, is unstable to growth. Pogorelov's icture can be reconciled with the linear stability analysis sketched above as follows: the oscillatory buckling modes that arise at the buckling ressure can be thought of as many tiny inversions covering the entire shere; any one of these inversions can grow uncontrollably, giving rise to the characteristic inverted shae of a buckled sherical shell. Thus the buckling mode is a transient, and the nal buckled shae of the shell dislays one or more large inversions that signi cantly reduce the enclosed volume of the shell, very similar to the nal con guration in Fig. 1. These large inversions with deth comarable to the shell radius have additional features, such as the shar oints of stress focusing, that deviate from the Pogorelov isometric inversion, but the associated scaling of the elastic energy with deth is hardly changed. 7,8 The transient buckling mode can only be observed in real shells by arresting its uncontrolled growth. This has been done for metallic thin shells by buckling a shell that encloses a solid ball of slightly smaller radius: the deformed shell contacts the inner ball when the buckling mode arises, arresting it so it can be visualized. 9 We can relicate this remarkable exeriment in our simulations by including a stee reulsive otential within the mesh and raming u the external ressure until the shell buckles, at which oint the internal reulsive otential revents the unstable mode from growing uncontrollably. The result is shown in Fig. 3, showing the small-wavelength buckling attern that extends over the shell. VI Sna-through of soft cas Paer We now turn to the shells with so sots, numerically investigated in Section III. The radius of the so sot, measured along a shere geodesic, is r 0 h Ra. We exect large changes in the buckling due to external ressure whenever r 0 > l ¼ R/g 1/4, the elastic length; i.e. l ¼ (r 0 /R)g 1/4 > 1. Indeed, unless l 1, the so sot will substantially in uence the mechanics of the shell and, in articular, be the rst region resonding to 83 Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

7 Paer Soft Matter Fig. 3 Simulation of a uniform shell with large Föl von Kármán number (g ¼ 10 6 ) under an external ressure above the buckling transition, and an inner ball of radius R inner ¼ 0.99R that reels all mesh oints with a stee reulsive otential. The reulsive otential revents vertices from moving closer than a small amount to the center of the shere, thus arresting the oscillatory buckling mode. The mesh is colored by the distance of each oint from the centre in units of R, to highlight small differences. Note the eriodic array of indentations with amlitudes that vary slowly over the surface. the external ressure by either deforming continuously or by sna-through buckling when a threshold ressure is reached. The in uence of the ca on the remainder is only exected to extend over a narrow region of width l into the area occuied by the remainder. As a result, we exect the equations of shallow shell theory to also accurately describe the stresses in the remainder for small so cas. We use olar coordinates, locating the origin at the center of the ca. We then have two sets of equations, one for the ca (with elds ~W, ~c and elastic moduli ~Y, ~k) when r < Ra and one for the remainder (with elds W, c and elastic moduli Y, k) when r > Ra. The elds are related by matching dislacements, stresses and moments at the boundary, r ¼ Ra. A Mechanism for sna-through buckling In the limit of an in nitely stiff remainder (s ¼ h/h ~ / 0), all dislacements along the ca boundary are restricted, and by continuity of sloes the sloe of the ca shae at the boundary is xed to be a. This limit corresonds to the roblem of buckling of an elastic ca, clamed along the entire edge to restrict all deformations, under hydrostatic ressure. The buckling of clamed cas is a well-studied roblem in shell theory, with a de nitive numerical analysis conducted by Huang. We exect that the sna-through buckling of cas embedded in thin shells with a stiff remainder follows a similar mechanism, which we summarize here. At each value of ressure, one or more equilibrium solutions may exist for the elds within the ca. The signature of loss of stability under ressure is seen in the ressure de ection curve associated with these solutions, qualitatively illustrated in Fig. 4 (a er ref. ). The curve OAB in Fig. 4 shows a reresentative ressure de ection relation for axisymmetric de ections. At low enough ressures, an analysis of the linearized equations shows that the ca deforms axisymmetrically and the de ection averaged over the ca, Fig. 4 Reresentative ressure deflection curve for a clamed shallow sherical ca (after ref. ). The deflection is averaged over the area of the ca (see eqn (35)). Uon increasing the ressure, the average deflection jums from A to B at the ressure. The green dashed curve CD that branches off at * corresonds to a nonaxisymmetric deformation. ~W ¼ ð r 0 0 r ~WðrÞdr ð r 0 0 rdr ; (35) grows with ressure (r 0 ¼ Ra is the radius of the so sot boundary). At higher ressures, however, nonlinear effects become imortant, and the governing equations of the ca admit multile solutions that satisfy the boundary conditions. In the range 1 < <, for instance, three solutions exist at each ressure. When this haens, the ressure de ection relation is no longer single-valued. With increasing ressure, at the maximum oint A wheretwoofthesolutionsmerge,thecade ection jums directly to oint B, signifying axisymmetric sna-though at ressure. A subsequent slow decrease in ressure leads to a snaback in the shae at the ressure 1. We study this mechanism for cas embedded in sherical shells in Section VI E. However, for cas beyond a certain size, a non-axisymmetric de ection mode may aear at a lower ressure *, withasa ressure de ection relation of the form of the dashed line in Fig. 4. Unlike the axisymmetric mode, the nonaxisymmetric de ection modes are highly degenerate. A stability analysis of such modes in shallow shells shows that there is no stable shae in the vicinity of such a state; i.e. the sloe of the curve CD is indeed negative at C. Therefore,the aearance of a nonaxisymmetric mode at * < triggers sna-through buckling of the ca to an inverted shae via the ath OCD. InFig.4,weshow* < 1 for clarity, although this inequality need not hold in general. We consider the nonaxisymmetric buckling mode for our cas in Section VI D. In our analysis, we assume that the aearance of a nonaxisymmetric de ection in the ca immediately triggers snathrough buckling of the ca, whichisthecaseforclamedcas. B Non-dimensionalizing the equations Since the sna-through of the ca is rimarily determined by the de ections and stresses in the ca itself, we rescale all quantities with hysical arameters that deend on the elastic roerties of the ca. We de ne the nondimensional radial coordinate and ressure: This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

8 Paer s ¼ r ~ ; (36) h ¼ ¼ R ~ 0 4 ~k ffiffiffiffiffiffi ; (37) ~Y where ~ ¼ R/~g 1/4 and ~ 0 h4 ~k~y ffiffiffiffiffiffi =R is the classical buckling ressure associated with a uniform sherical shell with the elastic roerties of the ca. We also de ne the rescaled elds in both the ca and the remainder ~w ¼ R W; ~ ~ ~f ¼ R ~Y ~ ~c; (38) 4 w ¼ R ~ W; f ¼ R ~Y ~ c: (39) 4 Uon substituting these quantities into the shell equations [eqn (18) and (0)], we obtain the following nondimensional equations for the ca: V 4 ~w V ~f N (~f, ~w) ¼ 4h, (40) V 4 ~f þ V ~w þ 1 N ~w; ~w ¼ 0; (41) and for the remainder: 1 w V f N s 3V4 ðf; wþ ¼4h; (4) sv 4 f þ V w þ 1 N ðw; wþ ¼ 0; (43) where the differential oerators are now de ned using the variables (s,q) rather than (r,q). From this analysis, we see that when ca size is rescaled by the elastic length scale ~ and ressure by the classical buckling ressure ~ 0, the deendence on elastic moduli dros out, and cas of vastly different roerties but the same thickness ratio s relative to the remainder of the shell can be described by just two dimensionless quantities ~ l ¼ Ra/ ~ and h ¼ /~ 0 (as long as they are small enough that shallow shell theory alies). Fig. 5 lots the ressure of the rst buckling transition h c1 ¼ c1 /~ 0 (whether it corresonds to a sna-through transition of the ca as for s < 0.5, or a collase of the whole shell as for s > 0.5), as a function of the arameters s and ~ l. When comared to Fig., we see that the buckling ressures, when rescaled using ~ 0, vary over a much smaller range and dislay similar behavior as s is changed. This observation suggests that the rst buckling transition of the shell is always a sna-through transition of the ca, and that the mechanism of the sna-through does not vary signi cantly as s is changed. Indeed, all of the h c1 vs. ~ l curves for the buckling ressure share similar characteristics for the deendence of buckling ressure on ca size: there is an aarent kink in the range 5 < ~ l < 6 that searates a convex deendence for smaller cas from a roughly constant deendence for larger cas. A similar feature is seen in the buckling ressure curve for clamed cas, i.e. shallow sections of sheres with clamed circular edges under ressure. In that case, the kink searates values of ~ l for which the buckling mode is axisymmetric i.e. with no circumferential angular deendence Fig. 5 Reduced ressure c1 /~ 0 of the first buckling event, whether a snathrough of the ca (for shells following buckling athway I, emty symbols), or a catastrohic collase of the whole shell (buckling athways II and III, filled symbols), as a function of the dimensionless sot size ~ l ¼ a~g 1/4 in buckling simulations on shells with g ¼ The data shown is the same as for the first buckling event in Fig., but is rescaled using the ca arameters ~ 0 and ~ l rather than the shell arameters 0 and l. about the z axis, from ~ l values for which the buckling is nonaxisymmetric and has some circumferential variation. Furthermore, no sna-through buckling is observed for clamed cas with ~ l < 3.3; below this size the ca deforms continuously as the ressure is increased. This is similar to the resent case with shells that follow buckling athway III, where no sna-through buckling of the ca is observed. We rst use our numerical results to test whether the structure of the h c1 vs. ~ l curves is linked to the buckling mode. We investigate the nature of the buckling mode that drives the snathrough of the ca by arresting its growth with an internal reulsive otential, as was done for the uniform shere in Section V. Fig. 6 illustrates the buckling mode that drives sna-through of the ca, as the ca size is varied for a shell with g ¼ 10 4 and s ¼ 0.4. For small cas, the buckling mode is axisymmetric, whereas for large cas, it has a eriodic structure in the circumferential direction, similar to the situation for clamed cas. The nature of the buckling mode changes from axisymmetric to nonaxisymmetric around ~ l ¼ 5, which corresonds to the kink in the h c1 vs. ~ l curve for s ¼ 0.4 (see Fig. 5). The number of nodes in the circumferential direction then increases with ca size, although the wavelength of the buckling mode itself is hardly changed from an intrinsic size of order ~. The deendence of the buckling ressure on ca size ~ l in the nonaxisymmetric region is also weak. This structure is retained for other values of s, although the ~ l-value of the transition from axisymmetric to nonaxisymmetric buckling falls as s aroaches the uniform shell limit, s ¼ 1. The roughly constant value of h c1 for nonaxisymmetric buckling also aroaches 1 as s / 1, which is exected because buckling of a uniform shell with the thickness of the ca would haen at h ¼ 1. C Analysis of linearized equations The buckling of the ca is a nonlinear henomenon, that arises due to the couling terms in the shell equations [eqn (40) (43)], 834 Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

9 Paer Soft Matter Fig. 6 (a) Mode for sna-through buckling of a soft ca, for shells with g ¼ 10 4, s ¼ 0.4, and various dimensionless ca sizes ~ l ¼ a~g 1/4. The buckling mode was arrested in numerical simulations by including an inner ball of radius R inner ¼ 0.99R whose surface reels all mesh oints with a stee reulsive otential. The mesh is colored by the distance of each oint from the center of the shere on a linear color scale, to highlight small differences. The soft region coincides with the region of large variations in deflection at the to of each shell as an illustration, the dashed line shows the boundary of the soft region for the shell with ~ l ¼ 6.0. which uts an exact analytical study out of reach. However, we can gain some insight about the behavior under ressure and the aroach to buckling by linearizing the equations about some aroriate state of stress to get the so-called Reissner equations of shallow-shell theory. 30,31 We saw in Section V that a uniform shell contracts uniformly u to the oint of buckling; we can exand de ections and stresses about this state of uniform re-stress. In the ca, the rescaled uniform radial de ection ~w 0 ¼ h(1 n) and Airy function ~f 0 ¼ s h solve the shell equations exactly. Uon substituting ~w ¼ ~w 0 + ~w 1 and ~f ¼ ~f 0 + ~f 1 in eqn (40) and (41) and keeing terms linear in ~w 1 and ~f 1, we get the Reissner equations within the ca: V 4 ~w 1 +hv ~w 1 V ~f 1 ¼ 0, (44) V 4 ~f 1 + V ~w 1 ¼ 0. (45) General axisymmetric solutions to the differential equations that satisfy the requirement of continuous and nite stresses at the origin are written in terms of real and imaginary arts of J 0, the Bessel function of the rst kind: # h ~w 1 ðsþ ¼c 1 "J R ðsþþffiffiffiffiffiffiffiffiffiffiffiffiffi J I ðsþ 1 h # h þ c "J I ðsþ ffiffiffiffiffiffiffiffiffiffiffiffiffi J R ðsþ þ c 3 ; (46) 1 h 1 ~f 1 ðsþ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi ½c 1 J I ðsþ c J R ðsþš; (47) 1 h where h i h i ffiffiffiffiffiffiffiffiffiffiffiffiffi 1=: J R ðsþhre J 0 ~rs ; J I ðsþhim J 0 ~rs ; ~rh h þ i 1 h (48) Similarly, substituting w ¼ w 0 + w 1 and f ¼ f 0 + f 1, where w 0 ¼ h(1 n)s and f 0 ¼ s h, into the nonlinear equations for the remainder (eqn (4) and (43)), and keeing terms linear in w 1 and f 1, gives the Reissner equations for the remainder: 1 s 3 V4 w 1 þ h V w 1 V f 1 ¼ 0; (49) sv 4 f 1 + V w 1 ¼ 0. (50) In the remainder, we look for solutions to the de ection and stress that decay to zero as s / 0. These can be written in terms of H (1) 0, the Hankel function of the rst kind: # hs w 1 ðsþ ¼c 4 "H R ðsþþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H I ðsþ 1 h s 4 # hs þ c 5 "H I ðsþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H R ðsþ ; 1 h s 4 f 1 ðsþ ¼ s (51) 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½c 4 H I ðsþ c 5 H R ðsþš; (5) 1 h s 4 This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

10 where h h H R ðsþhre H ð1þ 0 i; ðrsþ H I ðsþhim H ð1þ 0 i; ðrsþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=: (53) rh s 3 h þ is 1 h s 4 At the boundary we require continuity of the transverse de ection W and its rst derivative W,r, the radial dislacement U ¼ r(u qq + W/R) [or equivalently the strain comonent u qq ¼ (c,rr nc,r /r)/y], the radial stress s rr ¼ c,r /r, and the bending moment M rr ¼ k(w,rr + nw,r /r). The resulting dimensionless equations, to be satis ed at the ca edge s ¼ ~ l, are: ~w ¼ w, ~w,s ¼ w,s, ~f,s ¼ f,s ; (54) f ~f ;ss n ~ ;s ¼ s f ~ ;ss n f ;s l ~ l! ; (55) s 3 ~w ;ss þ n ~w ;s ~ l!! ¼ w ;ss þ n w ;s : (56) ~ l Note that the non-constant comonents ~w 1, w 1, ~f 1, f 1 must be nonzero for h > 0 since ~w 0 s w 0, ~f 0,s s f 0,s. For each air of {h, ~ l} values, the boundary conditions give rise to ve linear equations which can be solved for the ve constants c 1.c 5.We will call the result the Reissner solution for the de ection and stresses in the ca and the remainder. Fig. 7 comares the solution of the Reissner equations to the transverse de ection from simulations for a few shells, with h values below the buckling threshold. The linearized solutions cature many features of the full solution, articularly in the vicinity of the boundary between the ca and the remainder and the eriod of the decaying oscillations in the ca. Paer D Nonaxisymmetric buckling The solution of the linearized equations is sufficient to obtain a qualitative understanding of the nonaxisymmetric buckling of large cas, which we have seen is largely indeendent of the ca size ~ l for each value of s. Fig. 8 shows the behavior of the axisymmetric Reissner solution for h ¼ 0.8, close to the snathrough ressure, for cas of various sizes and thickness ratios relative to the remainder. This is exected to aroximate the axisymmetric de ection of the shell before the nonaxisymmetric mode aears and causes the ca to sna through. From Fig. 8(a), we see that the solution develos a boundary layer which does not change signi cantly when ~ l > 10; the solutions for large ca size are related to each other by a simle shi along the r axis. (Clamed cas dislay a similar behavior. ) Since the stresses are also largest in this region close to the ca shell boundary, the behavior here dominates the buckling transition. It is likely that the relative insensitivity of the buckling ressure to the ca size for large cas is a consequence of boundary layer dominance. To quantify this effect, we estimate the couling between the excess stresses due to the boundary and any nonaxisymmetric mode that may arise. Recall that for uniform shells, the nontrivial buckling mode arose due to the couling between gradients of the transverse dislacement W 1 and the uniform re-stress s xx ¼ s yy ¼ R/ (Section V). Now we have an axisymmetric re-stress {~w a,~f a } that varies in the r direction, which coules to nonaxisymmetric elds {~w n,~f n } which we shall assume small. We substitute ~w ¼ ~w a + ~w n and ~f ¼ ~f a + ~f n into the full nonlinear shell equations [eqn (40) and (41)] and kee terms linear in the nonaxisymmetric elds. For simlicity, we work in scaled Cartesian coordinates at a osition on the x axis near the ca shell boundary, so that ~w a (s) z ~w a (x), and x z ~ l [ 1 which means that the q direction coincides aroximately with the y direction. Then the nonlinear couling gives rise to terms of the form ~w a,xx ~f n,yy, ~f a,xx ~w a,yy and ~w a,xx ~w n,yy. The equations for ~w n and ~f n now read V 4 ~w n V ~f n ~w a,xx ~f n,yy ~f a,xx ~w a,yy ¼ 0, (57) V 4 ~f n + V ~w n + ~w a,xx ~w n,yy ¼ 0. (58) For the uniform shell, only ~f a,xx ¼ h was nonzero, but the resence of the ca remainder boundary gives rise to a satially varying ~w n and thus a nonzero ~w a,xx as well. In fact, ~f a,xx z h remains a good aroximation to the re-buckling Airy stress function in the ca, so the additional effects which bring down Fig. 7 Comarison of the scaled normal deflection WR/ ~ as a function of radial osition r/ ~ measured from simulations with an amorhous mesh (symbols), to the solution of the linearized shell equations (lines), for different values of the reduced ressure h. The values from simulations are obtained by averaging the ositions of all mesh oints sharing each r value (i.e. over all values of q for each value of r). The three subfigures (a c) corresond to three different shells, all with g ¼ 10 4 and ca angle a ¼ cos 1 (0.85), but with different ca-to-shell thickness ratios s which changes the elastic length ~ and therefore the dimensionless ca size ~ l ¼ Ra/. ~ The vertical arrows indicate the r osition of the boundary between the soft ca and the remainder. All three shells buckle around h ¼ Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

11 Paer Soft Matter Fig. 8 Reissner solution for the scaled normal deflection WR/ ~ for cas with (a) increasing ca size ~ l at constant ressure h and thickness ratio s, and (b) increasing thickness ratio at constant ressure and ca size. The vertical arrows indicate the r osition of the boundary between the soft ca and the remainder. the buckling ressure below h ¼ 1 are mainly due to the new couling with ~w a,xx. Let x denote a tyical value of ~w a,xx in the boundary layer region. A rough estimate of its effect can be obtained from the equations V 4 ~w n V ~f n x~f n,yy +h ~w a,yy ¼ 0, (59) V 4 ~f n + V ~w n + x ~w n,yy ¼ 0. (60) Similar in sirit to the stability calculation for uniform shallow shells, we look for solutions that oscillate in the y direction of the form ~w n, ~f n f e iqy, which leads to the eigenvalue equation q 4 hq +(1+x) ¼ 0. (61) Therefore, oscillatory solutions with q > 0 exist only if h >1+x. (6) According to this argument, regions of the axisymmetric de ection with negative curvature in the radial direction, x ~w a,ss <0, cause nonaxisymmetric modes to aear at h z 1 + x < 1. The Reissner solution, as well as the full nonlinear solution, dislay these regions rominently in the boundary region near the ca shell boundary (see Fig. 7 and 8). Therefore, the couling of the nonaxisymmetric deformation with the re-buckling axisymmetric solution can in fact give rise to a nonaxisymmetric solution at ressures below ~ 0, the buckling ressure of a comlete shell with the same roerties as the ca. Fig. 7 and 8 show that the most negative value of the curvature in the axisymmetric de ection, ~w a,xx, occurs at the shar eak in ~w a in the boundary layer. The magnitude of the curvature deends on the ressure as well as the ca shell thickness ratio. To estimate when a nonaxisymmetric mode may rst arise, we take the value of ~w a,xx at this eak to be the scale x. We use the Reissner solution to nd x(h); the value h Reissner that solves the self-consistent equation h ¼ 1+x(h) isa rough estimate of the buckling ressure. Fig. 9 shows the result for large cas with different values of s. Although our estimate ignores the s-deendence of ~w a,xx and the boundary conditions on ~w a,xx and ~w n, it successfully reroduces the features associated with the nonaxisymmetric buckling ressure: h Reissner aroaches an aroximately constant value for ~ l T 6, and this value gets closer to 1 as s / 1. Our rough calculation also redicts the inverse wavelength of the circumferential buckling mode. The solution to eqn (61) at the buckling ressure h ¼ 1+xis q ¼ ffiffiffiffiffiffiffiffiffiffiffi 1 þ x ¼ ffiffiffi h. Thus, the length scale of the circumferential oscillation is exected to be unchanged in the nonaxisymmetric buckling region, in agreement with Fig. 6. E Axisymmetric buckling In the small ~ l region, to the le of the kink in the buckling ressure curves (Fig. 5), we have seen that the sna-through is axisymmetric. The axisymmetric Reissner solution to the linearized equations, eqn (44), (45), (49), and (50), shows no discontinuity in the de ection as the ressure is increased; thus, the discontinuous sna-through of the ca haens only when the nonlinear terms are signi cant in the governing equations of the ca. To simlify our analysis, we assume that de ections in the remainder are small, so that the linearized aroach Fig. 9 Estimate of the reduced ressure h ¼ /~ 0 at which a nonaxisymmetric mode arises in the ca for various thickness ratios, using the criterion of eqn (6) with the largest value of second derivative ~w a,xx of the Reissner solution for the ca as the estimate for x. This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

12 introduced in Section VI C still alies in the remainder and the Reissner solution is sufficient to describe its resonse. This aroximation is accurate for small thickness ratios s, when the sna-through of the ca haens at small ressures comared to the buckling ressure of the remainder (Fig. ). In the axisymmetric region, ~w and ~f are functions solely of s and the nonlinear governing equations of the ca [eqn (40) and (41)] simlify to V 4 ~w V ~f 1 ~f s ;s ~w ;ss þ ~w ;s ~f ;ss ¼ 4h; (63) V 4 ~f þ V ~w þ 1 s ~w ;s ~w ;ss ¼ 0; (64) exressions which can be integrated once to give equations for the derivative functions Q h ~w,s and F h ~f,s :,3 sq ;ss þ Q ;s Q s þ sf QF þ hs ¼ 0; (65) sf ;ss þ F ;s F Q sq þ ¼ 0: (66) s In addition to these equations and the linearized equations [eqn (49) and (50)] in the remainder, the continuity equations [eqn (54) (56)] must be satis ed at the boundary between ca and remainder. 1 Qualitative analysis. An aroximate analytical solution to the nonlinear equations, comuted using the Galerkin method 33 recovers many asects of the sna-through buckling transition. We rst consider the reviously studied clamed ca limit, which corresonds to an in nitely stiff remainder, i.e. s / 0. For a clamed ca, the tangent sloe and all deformations at the boundary are xed, whereas the shae and stresses at the origin must be continuous, which leads to the following boundary conditions for Q and F: Q(0) ¼ F(0) ¼ 0, (67) Q( ~ l) ¼ 0, (68) ~ lf,s ( ~ l) nf( ~ l) ¼ 0. (69) For small cas that exhibit axisymmetric buckling ( ~ l ( 5), the size of the ca is smaller than the length scale of oscillatory variations in ~w, which is set by ~. As a result, the inward de ection ~w has a maximum at the centre of the ca and decays monotonically to zero at the ca boundary. Corresondingly, Q(s) is zero at the origin and the ca boundary, and has a single extremal oint in between. Many characteristics of the full solution to the nonlinear equations can be obtained by considering an aroximate solution which reroduces these main features. We choose as our trial solution QðsÞ ¼As 1 s ~ l!; (70) which automatically satis es the boundary conditions. We obtain the value of A that best aroximates the true solution using the Galerkin rocedure: the trial solution, substituted into Paer eqn (66), rovides a linear differential equation for F which can be solved with the corresonding boundary conditions to obtain a trial solution for F(s) that also deends on A. These solutions are substituted into the le hand side of eqn (65), which is then multilied by Q(s)ds and integrated from s ¼ 0tos ¼ l to get a third order algebraic equation for the de ection amlitude A: ~ l 4 (194A A + 95A) A ~ l h ¼ 0. (71) Roots of this equation rovide the best estimate of the solution for the ca shae at any ressure. The buckling mechanism is evident from the nature of these roots. For small values of ~ l, eqn (71) has a single real solution for all values of the ressure, and the average de ection increases continuously as a function of h. This corresonds to buckling athway III for small cas, in which the ca deforms smoothly to its inverted shae without snaing. However, for ~ l > 3.5, there is a range of ressures for which A, and consequently the ca-averaged de ection ~u, can take on three real values. As a result, the ressure de ection curves, shown in Fig. 10(a), take on the characteristic form of curve OAB in Fig. 4, and sna-through buckling occurs at the maximal value of h at which three real solutions exist (i.e., at oint A in Fig. 4). This sna-through ressure is obtained in the Galerkin solution by setting the discriminant of eqn (71) to zero, and the solution is shown as the bottom curve in Fig. 10(b). The aroximate solution reroduces many features of the buckling ressure of clamed cas in the axisymmetric region, such as the absence of a snathrough for cas below a certain size, and a non-monotonic deendence of buckling ressure on ca size. Embedding the ca in a more exible shell requires modifying the boundary conditions from the clamed limit. The full effect of the remainder on the ca at the boundary is comlicated, and cannot be exressed succinctly. Requiring continuity of stresses and deformations across the boundary relaces eqn (68) and (69) with equations that relate the dislacement and rotation angle at the boundary to the corresonding stresses and bending moments, themselves imlicitly deendent on the solution to Q and F. However, the qualitative effect of the remainder on the boundary conditions can be gauged from the Reissner solutions, which satisfy the same continuity conditions, albeit for linearized versions of the governing equations. The Reissner solutions suggest that as s is changed from zero, Q( ~ l) will be ositive but remains small, whereas F,s ( ~ l) nf( ~ l)/ ~ l, which is roortional to the circumferential strain u qq, will be negative because of the comression of the remainder under external ressure. As the thicknesses in the ca and the remainder become comarable, this circumferential strain becomes larger in magnitude. The qualitative effect of this modi ed boundary condition on our aroximate solution can be assessed by relacing the boundary condition on F, eqn (69), with ~ lf,s ( ~ l) nf( ~ l) ¼ 3, (7) where 3 < 0 becomes increasingly negative as s / 1, and recomuting the aroximate Galerkin solution. The Reissner solutions show that r3r increases over a range 0 ( r3r ( ass is 838 Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

13 Paer Soft Matter Fig. 10 (a) Deflection ~w, averaged over the entire ca, as a function of the reduced ressure h, for the aroximate solutions for the clamed ca obtained using the Galerkin method for different values of the ca size, ~ l. (b) Estimate of the buckling ressure using the Galerkin method, for different values of the circumferential strain 3 at the boundary. The clamed ca corresonds to 3 ¼ 0. increased from 0 to 1, for ressures in the range of The variation in the corresonding estimate for the sna-through ressure is shown in Fig. 10(b). As the circumferential strain at the boundary increases, the sna-through buckling ressure curve moves uward and the smallest value of ~ l at which buckling is observed decreases, both in qualitative agreement with the trends in the sna-through ressure as s is increased (see Fig. 5). Although our analysis is a highly simli ed descrition of the true boundary conditions (which in reality deend subtly on ~ l, s and h), it qualitatively catures the effect of the remainder on the sna-through transition. Numerical analysis of governing equations. To make our analysis more quantitative, we use the Reissner solution in the shell remainder to establish the boundary conditions on Q and F at the boundary, and then numerically solve the nonlinear governing equations in the ca. The continuity conditions, eqn (54) (56), together with eqn (49) and (50) for the elds in the remainder, can be reduced to two equations that must be satis ed by Q, F and its derivatives at s ¼ ~ l. These equations deend on the dimensionless quantities ~ l, h and s. Together with eqn (67), they rovide a comlete set of boundary conditions for the nonlinear differential equations of the ca, eqn (65) and (66), which are then solved numerically using the bv4c differential equation solver in the MATLAB (Mathworks Inc.) scienti c comuting ackage. A discontinuity in the ressure de ection curve for the comuted solutions signals the sna-through buckling ressure, which we comare to the sna-through buckling ressure from simulations in Fig. 11. The results quantitatively relicate the observed behavior in simulations at low values of s. Deviations from the results of simulations become larger as s increases, because our assumtion of linear behavior in the remainder becomes less accurate as s / 1. However, these results con rm that snathrough buckling of the ca haens via an axisymmetric mode for small values of s, which arises due to nonlinear coulings between deformations and stresses in the ca. The remainder in uences the boundary conditions on these deformation and stress elds, which shi s the sna-through ressure closer to h ¼ 1ass / 1. Our results show that the deendence of the rst buckling ressure c1 on the sot size and thickness can be exlained by Fig. 11 Sna-through ressure estimate from numerically solving the governing equations of the ca with boundary conditions set by the Reissner solution for the remainder (solid lines), comared to c1,thefirst buckling ressure measured in simulations (symbols). the loss of stability of a sherical ca of dimensionless size ~ l, embedded in a remainder whose relative stiffness deends on the thickness ratio s. Signi cantly, this sna-through loss of stability of the ca determines c1 for both buckling athways I and II, even though the shaes of the shells at c1 differ dramatically between the two athways (a sna-through localized to the ca for shells following athway I, as oosed to a catastrohic collase for shells following athway II). From Fig., we see that the transition from athway I to athway II occurs when this sna-through buckling ressure curve moves above the second buckling ressure c at which an inversion localized to the so ca becomes unstable to catastrohic growth into the remainder of the shell. In contrast to c1, which is determined rimarily by the elastic roerties of the ca, this second loss of stability is set mainly by the elastic roerties of the remainder. We analyze this second loss of stability in the following section. VII Collase of the remainder Whether through a continuous deformation or a sna-through buckling event, the so ca attains an inverted shae, at a This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

14 articular ressure determined largely by the thickness ratio s (the deendence on the ca size itself is weak for large cas). The inversion is an examle of a nearly isometric shae, that reserves the metric everywhere excet for a narrow region at its rim (whose width is aroximately the elastic length within the ca, ). ~ When this inversion has formed, the fate of the remainder of the shell deends on the stability of the inversion against further growth into the remainder. A scaling argument shows why such an inversion is unstable for large enough ressures: suose the inversion has grown so that the rim is entirely within the remainder. Then, as outlined in Section V, there exists an inversion deth d max ð Þ ðk=r 3 Þ R ffiffiffi g that searates stable inversions from unstable ones. Inversions of deth d > d max () can reduce their net energy by growing deeer, leaving the shell unstable to comlete ffiffiffiffiffiffiffiffi collase. Conversely, ffiffiffiffiffiffiffiffi there is a ressure I ðdþ ðkg 1=4 =R 3 Þ R=d 0 g 1=4 R=d above which an inversion of deth d grows uncontrollably leading to collase of the whole shell. But the inversion of the so ca has the effect of seeding an inversion with a deth d 0 =Rh1 cos aza =zl = ffiffiffi g within the remainder (here, d0 is the ca height, an alternative way of measuring the ca size). If the sna-though of the ca haens at c1 < I (d 0 ), the seeded inversion is stable over the range of ressures c1 < < I (d 0 ), and a second collase of the whole shell occurs only when the ressure increases to c ¼ I (d 0 ). This corresonds to athway I as described in Section III. However, if the ca thickness is so high that sna-through haens at c1 > I (d 0 ) (note that c1 is set by the ca elastic arameters while I is set by the remainder) then the inversion of the ca is immediately unstable to further growth into the remainder, and only a single collase of the whole shell is observed at ¼ c1. This scenario corresonds to athway II for collase of the shell. Thus, a consideration of the stability of inversions at different ressures qualitatively exlains the transition from athway I to athway II as the thickness of the ca increases relative to the shell (Fig. ). This aroximate descrition does not take into account the recise evolution of the inversion energy as the rim of the inversion grows out of the ca into the remainder. This transition is crucial in determining the true value of I (d 0 ) and thus the ressure of collase of the remainder. We study it in detail in the rest of this section. Paer change and energy) for just one of the resulting inversions. The equilibrium con guration and corresonding elastic energy of the mesh are obtained by numerical minimization in the resence of the indenters. The deth of the inversion is varied by changing the distance between the indenting hard sheres. To revent the shell from sliding out from between the indenters, we restrict a few mesh vertices in the vicinity of the oles to move only in the z direction. Fig. 1 shows the elastic energy of inversions of deth d for shells with so cas of various sizes [reorted in terms of the ca height d 0 ¼ R(1 cos a)] at zero ressure, keeing the elastic roerties of the shell and the ca unchanged. All energy curves can be divided into three distinct regions: d d 0, when the inversion rim is localized within the so ca; d [ d 0, when the rim is localized in the remainder, and a transition region around d z d 0. This division is a direct consequence of the fact that the inversion rim is localized to a narrow region whose width is set by the elastic length scale which is small in both the ca and the remainder. As a result, the elastic energy aroaches the Pogorelov scaling form 6 with the aroriate elastic constants in the small-d and large-d regions: E el / c~k~g 1/4 (d/r) 3/ ¼ cs 5/ kg 1/4 (d/r) 3/ for h h 0 and E el / ckg 1/4 (d/r) 3/ for d [ d 0. The best t to the numerics is for c ¼ 19, which is consistent with estimates from asymtotic analyses of the nonlinear shell equations governing the elastic energy of the inversion 6,35 and was also indeendently checked via indentation simulations on uniform shells. The volume change associated with the inversion, meanwhile, is wellaroximated by that of a erfectly isometric inversion and does not deend strongly on the ca size (see inset to Fig. 1). The stability of an inverted so ca at nite ressure can now be studied by examining the behavior of the total energy E tot (d) ¼ E el (d) DV(d) as the ressure is ramed u. Fig. 13(a) A Energetics of a nearly isometric inversion in a shell with a so ca To numerically investigate the energetics of inversions centered at the north ole for shells with so cas, we arti cially create inversions by indenting the mesh with hard sheres that are the same radius as the shell itself. (We use sherical indenters to avoid the olygonal inversions that are associated with inversions created by oint indenters. 34 In the resent case, inversions are seeded by the erfectly circular so ca, and thus we are interested in inversions with erfectly circular rims.) To maintain equilibrium of forces, we reare shells with two identical cas one at the north ole and one at the south ole and then squeeze the shere between two indenters at each ole, although we reort hysical quantities (volume Fig. 1 Elastic energy associated with an inversion of deth d, at zero ressure, for a shell with a soft ca of height d 0. In all cases, g ¼ 8000 and s ¼ 0.4. The lines corresond to the Pogorelov form of the elastic energy for a nearly isometric inversion localized in the ca (solid line) and remainder (dotted line) resectively, with dimensionless refactor c ¼ 19. Inset: the volume reduction DV associated with the inversion matches the exectation for a urely isometric inversion, DV iso ¼ d (3R d)/3 z Rd for small cas. 840 Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

15 Paer Soft Matter Fig. 13 (a) Total energy of an inversion in a ressurized shell with g ¼ 8000, s ¼ 0.4 and l ¼ 6.09 (corresonding to a ca height d 0 ¼ 0.R), showing that the shell with inversion confined to the ca becomes unstable for > The arrow indicates the r coordinate of the border between the ca and the remainder. (b) Collase ressure estimated using stability considerations from indentation simulations (symbols) comared to c, the ressure of catastrohic collase of the remainder measured from ressure buckling simulations (line) for shells with g ¼ 8000, s ¼ 0.4, as a function of the dimensionless ca size l. shows the total elastic energy for a ressurized shell with g ¼ 8000, s ¼ 0.4 and ca size d 0 ¼ 0.R (corresonding to l z 6). For this shell, the ca snas through to an inverted shae at c1 ¼ The inversion localized to the ca is stable for ressures u to 0. 0 because of the existence of a local minimum in the energy curve at d/r z 0.. However, at ¼ 0. 0, the energy minimum vanishes and the ressurized shell can reduce its energy inde nitely if the inversion grows into the remainder of the shell. Thus, the energetics of the inversion dictates that the remainder collases comletely at c ¼ 0. 0, which matches the oint of collase observed in numerical collase simulations (Fig. ). The loss of stability of the remainder of the shell, triggered by the merging of two xed oints in the energy curve, is analogous to a limit of metastability at a rst order hase transition, and is classi ed as a saddle-node bifurcation in dynamical systems theory. 36 To test this mechanism for loss of stability of the remainder, we analyzed the total energy E tot (d) from indentation simulations on shells with various ca sizes, to estimate the ressure at which the bifurcation occurs. This was comared to the ressure of collase of the remainder as measured in ressure collase simulations (Section III). The results are reorted in Fig. 13(b). The estimates from indentation simulations agree quantitatively with the ressure collase simulations for ca sizes down to l z 1. The discreancy for smaller ca sizes is to be exected when the ca size is small comared to the elastic length scale of the shell, there is no searation of scales that lets us consider the stability of an inverted ca in the remainder of the shell. In that case, the deformations of the ca and the remainder are closely couled and jointly determine the ressure of collase of the shell. B Deendence of collase ressure on ca size and thickness Having shown that stability considerations quantitatively determine c as a function of sot size for a articular value of the thickness ratio (s ¼ 0.4), we now use scaling arguments to qualitatively understand how c changes with thickness ratio ffiffiffiffiffiffiffiffiffiffi and ca size. Our revious estimate, c ¼ I ðd 0 Þ 0 g 1=4 R=d 0 0 =l was inconsistent with the simulation results (Fig. ) which show c aroaching a constant value for large ca sizes rather than falling with ca size. (Recall that the ca height d 0 and the dimensionless ca size l are related via d 0 zrl = ffiffiffi g.)thereviousestimatefor c did not take into account the jum in the elastic energy E el (d) fromcs 5/ kg 1/4 (d/r) 3/ to ckg 1/4 (d/r) 3/ when the inversion rim transitions from the ca into the remainder, which stabilizes the inversion of deth d z d 0 that arose due to the sna-through of the ca [see Fig. 1 and 13(a)]. This transition haens over some narrow deth range d 0 ( d ( d 0 + d, whered is the deth change that causes the inversion rim radius rz ffiffiffiffiffiffiffiffi dr to increase from r 0 ¼ ffiffiffiffiffiffiffiffiffiffi d 0 R by an amount of order l (i.e. the rim moved away from the ca boundary by an elastic length so that it no longer feels its in uence). The required change in rim radius r corresonds to a deth change of order d ffiffiffiffiffiffiffiffiffiffi d 0 =R ffiffiffiffiffiffiffiffi d 0 R =g 1=4. The inversion of deth d z d 0 loses its stability when the jum in elastic energy is overcome by the ressure contribution, i.e. when E tot (d 0 ) z E tot (d 0 + d). For d 0 [ d, which corresonds to l [ 1 (the limit of large ca size), this criterion reduces to ckg 1/4 (d 0 /R) 3/ (1 s 5/ ) z 4Rd 0 d. Therefore, we exect the second buckling ressure c to scale as c kg1=4 ðd 0 =RÞ 3= 1 s 5= Rd 0 d k ffiffiffi g ðd0 =RÞ 3= 1 s 5= ðrd 0 Þ 3= 0 1 s 5= ; (73) where we have used 0 ¼ 4 ffiffiffiffiffiffi ky =R ¼ 4k ffiffiffi g =R 3 and d ffiffiffiffiffiffiffiffi d 0 R =g 1=4. According to this estimate, the ratio c / 0 is indeendent of ca size for large l, consistent with the simulation results (Fig. ). The redicted s-deendence of the second buckling ressure is also consistent with the simulation results: the limiting value of c for large l falls from c ¼ for s ¼ 0. to c ¼ for s ¼ 0.4, consistent with c / 0 f (1 s 5/ ). By combining this result for the loss of stability of the remainder with the results of the revious sections, we can now fully exlain the buckling athways summarized in Fig.. The ( c1 / 0 ) vs. l curves, set by the sna-through of the ca, move to higher values of / 0 as s increases, whereas the ( c / 0 ) vs. l curves, set by the stability of the inverted ca against further growth into the remainder, move to lower values as s increases [eqn (73)]. For s # 0.4, the former curve lies below the latter for This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,

16 all values of l, and two distinct buckling transitions are observed at c1 and c (athway I) over a wide range of ca sizes. For s $ 0.6, the ca sna-through ressures lie above the stability curve of the remainder, and an inverted ca is immediately unstable to growing into the remainder, leading to collase of the whole shell at c1 (athway II) for all ca sizes. The searate buckling ressure curves cross at thickness ratio s ¼ 0.5, and shells with this thickness ratio buckle via either athway I or athway II, deending on the size of the so region. VIII Hysteresis The buckling of uniform sherical shells is known to be hysteretic under cyclic changes in external ressure. A buckled shell with a large inversion re-in ates only when the ressure is brought down signi cantly below its buckling ressure 0. This is illustrated in Fig. 14 for a numerical simulation on a uniform shell with g ¼ The shell collases close to 0 (in ractice, all simulated uniform shells buckle slightly below the classical buckling ressure because of the nite mesh and the nonuniformity in the mesh, to which the buckling transition is very sensitive), but only re-in ates when the ressure is brought down to about 10% of the buckling ressure. This behavior can be understood from the Pogorelov scaling form [eqn (33)] for the elastic energy of an inversion, which we have already ffiffiffiffiffiffiffiffi seen leads to an in ation ressure I ðdþ g 1=4 R=d 0 to re-in ate an inversion of deth d. In collased shells, the inversion deth is comarable the shell radius, and as a result I g 1/4 0 is very small comared to the buckling ressure for thin shells with g [ 1. In ractice, this makes sherical buckled casules difficult to re-in ate. A large, thin so region (s < 0.5, l T 3) mitigates hysteresis to some extent by bringing the buckling and re-in ation ressures closer to each other. Fig. 14 shows the volume evolution of a shell with s ¼ 0.4 and l ¼ 6.4 ( ~ l ¼ l= ffiffi s ¼ 10:1), corresonding to a ca of height of d 0 ¼ 0.R. The shell follows buckling athway I (a sna-through of the ca followed by buckling of the Fig. 14 Hysteresis loos for cyclical ressure variation on a shell without (dashed) and with (solid) a soft sot with s ¼ 0.4 and l ¼ 6.4. In both cases, g ¼ The curves show the evolution of the enclosed volume V, normalized by the initial volume V 0 ¼ 4R 3 /3, as the ressure is first ramed u to ¼ 0 and then back down to zero in stes of D ¼ remainder at higher ressure, as in Fig. 1). The ressure of catastrohic collase of the shell is brought down to c ¼ 0. 0, and the shell re-in ates when the ressure is about a third of this value. The so region also exhibits some hysteresis of its own, snaing back to its initial con guration at a ressure of which is lower than the sna-through ressure c1 ¼ This ca hysteresis is characteristic of the bistable state with two equilibrium solutions that is resonsible for sna-through buckling (Fig. 4). These features may be exloited for alications that require shae changes of elastic casules in resonse to changes in the ressure. First, the de ation and re-in ation ressures associated with large-scale collase of the shell are brought much closer to each other for shells with so sots (when s < 0.4), reducing the area of the hysteresis loo and making the shells more resonsive to ressure cycling. Second, there is a wide range of arameters for which the so sot can be inverted and re-in ated without collasing the shell as a whole. This rovides a mechanism for making casules with a well-de ned indentation (set by the ca size) that can be triggered or removed by changing the ressure. Such casules could be used to make tunable lock colloids in a lock-and-key colloidal assembly system, 8 which may be induced to associate with, or be indifferent to, smaller key colloids deending on whether the ca is inverted to form a recetacle for the keys. IX Conclusion Paer Our analysis shows that a circular thin region in a sherical shell, although a simle inhomogeneity to describe, strongly in uences the buckling roerties in imortant and subtle ways. By varying the thickness and size of the so region, we can induce the shell to follow three different buckling athways, each of which can be understood by analyzing the governing equations of the ca and the remainder. These athways can be exloited to control the shell shaes swet out by varying the external ressure. A wide range of buckling henomena in inhomogeneous shells remains to be exlored. It is already evident that the shae of the so region 1 and the number of so sots 13 fundamentally changes the buckling behavior; a systematic study of so sots with non-circular shaes and of multile so sots on the same shell could uncover additional ways to control and exloit the buckling transition. We could also consider sectors on a shell with a different 3D elastic modulus E comared to the remainder, allowing us to vary the ratio of D Young's moduli ~Y/Y indeendently from the ratio of bending rigidities ~k/k. Localized variations in the shell curvature, already studied in the context of imerfect shells, 37 may have interesting effects on the buckling roerties as well. Inhomogeneities could likewise affect buckling athways in other shell shaes known to buckle under external ressure, such as ellisoids, 38 toroids, 39 and cylinders. 40 These and many more shell shaes and inhomogeneities can be exlored using numerical simulations of the tye used in this work. Finally, we remark that a shell with a single so region is likely to be less sensitive to imerfections in the shell shae 84 Soft Matter, 013, 9, This journal is ª The Royal Society of Chemistry 013

17 Paer than a uniform shell. The sensitivity of the buckling transition to even minute deviations in the shell shae (of the order of the shell thickness) from the erfect shere has already been mentioned. This sensitivity occurs because the mode resonsible for buckling of a uniform shell (ictured in Fig. 3) is highly degenerate. 10 Since many such modes exist, it is likely that one or more of the modes coules strongly with the satial shae variation, and acquires a non-zero amlitude (thus triggering shell collase) at a ressure lower than the classical buckling ressure 0 for a erfect shell. Deending on the strength of the couling, the instability is triggered at a ressure that may be anywhere from 0 80% of 0 for real-world shells, even if the deviations from the erfect shere are only of the order of the thickness of the shell. 41 As a result, the true ressure at which uniform shells lose stability can be unredictable. In contrast, the collase of shells with thin so sots (s < 0.5) is triggered by loss of stability of the sna-through state. As described in Section VII, this is governed by the energetics of nearly isometric inversions rather than the resence of highly degenerate modes. Since the scaling of the inversion energy with inversion deth is not sensitive to small imerfections, the buckling of the shell is exected to haen reliably at roughly 0. 0 for large sots (l T 4), regardless of small deviations from the erfect shae. Even though the buckling ressure is greatly reduced by the so sot, it can be known much more recisely. Paradoxically, introducing a known inhomogeneity in the shell thickness could make the buckling transition more reliable than for a uniform shell! This roerty could be exloited for alications, such as release of encasulated materials under seci c environmental conditions, that require recise control over the ressure at which buckling haens. Aendix A: details of numerical simulation For our simulations, we discretize the elastic energies on a oating mesh of oints. The initial mesh is disordered, with a distribution of nearest neighbor distances and a nite fraction of ve-fold and seven-fold disclinations distributed throughout the mesh (Fig. 15). The disordered mesh aroximates a shell made u of amorhous material, eliminating the 1 regularly saced ve-fold disclinations that inevitably arise when tiling a sherical surface with equilateral triangles. 4 The elastic stretching energy of deformations from the initial unstrained con guration is aroximated by a harmonic sring energy associated with bonds connecting nearest-neighbor airs: 43 E s ¼ X ffiffi 3 4 Y r ij rij 0 (A1) hiji where r ij and r 0 ij are the lengths in the deformed and initial states of the bond connecting nearest-neighbor mesh oints i and j. This form reroduces the stretching energy comonent of the elastic energy (eqn (5)) in the continuum limit, with n ¼ 1/3. 43 Previously used discretizations of the bending energy in terms of the angles between surface normals of adjacent facets in the mesh 1,43 are not suitable for disordered meshes. 44 Instead, we reconstruct the change in curvature k ij from the Soft Matter Fig. 15 Disordered mesh used in simulations. For clarity, only one octant of the mesh is shown. Mesh oints are colored according to the number of nearest neighbors: blue (five neighbors), yellow (six neighbors) or red (seven neighbors); and nearest-neighbor airs are connected by lines. mean curvature H i and Gaussian curvature K i estimated at each oint i. This requires the discretized surface area A and enclosed volume V of the shell, obtained by treating the mesh as a olyhedron with lanar triangular facets whose edges are the nearest-neighbor bonds. Each facet j contributes an area A j ¼ (r j r j1 ) (r j3 r j1 ) /, where r ji (i {1,,3}) are the osition vectors of the three mesh oints j i that de ne the facet. The tetrahedron de ned by the origin and the three vertices of each facet j contributes a signed volume V j ¼ [r j1 $(r j r j3 )]/6 to the total volume of the olyhedron. The contribution correctly calculates the enclosed volume of the mesh, rovided the facet vertices j 1, j and j 3 are ordered so that the vector (r j r j1 ) (r j3 r j1 ) oints towards the outside of the shell. An estimate of the signed mean curvature at mesh oint i is 45 H i ¼ 1 F i $N i (A) N i $N i where F i ¼ V ri A is the gradient of the discretized surface area A ¼ P ja j with resect to r i, and N i ¼ V ri V is the gradient of the discretized volume V ¼ P jv j. The Gaussian curvature at each oint is estimated using the Gauss Bonnet theorem: K i ¼ 1 X! a j (A3) s i jðiþ where a j is the angle subtended by facet j at oint i, and the sum is over all triangular facets sharing oint i. s i ¼ P jðiþ A j=3 is the area associated with each vertex, comuted as one-third of the sum of areas A j of the facets j(i) sharing oint i. The discretized bending energy is 14 E b ¼ X ks i 4 H i 1 ð1 nþ K i R H i R þ 1 : (A4) R i The ressure is imlemented by adding a term V to the total energy. This net discretized energy E s + E b + V of the mesh oints is then minimized using the BFGS quasi-newton This journal is ª The Royal Society of Chemistry 013 Soft Matter, 013, 9,