## Abstract

Nematic elastic bodies can develop a gradient of response to heat, light and other stimuli. They then bend and develop curvature in a complex manner. Using the results for a general weak response derived in the preceding paper, we solve for strong spontaneous distortion where bend in one direction causes stretch in another direction if that too is bending, and vice versa. Since stretch is elastically expensive, it can cause suppression of one of the bends (we determine which), thus eliminating Gaussian curvature. This is the spontaneous distortion equivalent of the classical Lamb calculation of the anti-clastic suppression when large distortions are imposed in classical elastica. In practice, spontaneously deforming nematic solids, e.g. in actuation, are in this strong bend limit.

## 1. Introduction

Nematic solids develop large, anisotropic, mechanical deformation in response to a variety of stimuli, achieving contraction along and elongation perpendicular to the director and vice versa on recovery. Bend results from a spatial variation of mechanical response through the thickness of a sheet or cantilever of the active material. In an accompanying paper (Warner *et al.* 2010) (WMCI), we have catalogued the ways in which non-uniform strains can arise, and have calculated what weak bending results.

By weak bending, we mean where the bend in one direction is (apart from Poisson effects) decoupled from the bend in another direction. By contrast, strong bending is where two bending directions are coupled by stretch induced by geometric effects. It is a difficult problem in classical elasticity where deformations are imposed. We show that it also arises when large spontaneous deformations exist in more than one direction, that is, when there is spontaneous Gaussian curvature; see figure 1*a*,*b*, which can be either positive (syn-clastic, spherical) or negative (anti-clastic, saddle-like). As a result, stretches arise that can be large if the body is large—curvature in one direction means that parts of the body can be far removed from the neutral plane associated with curvature in the other direction. Stretch is very expensive compared with bend and the body reacts by flattening in one direction, the choice of which we discuss later (§2*b*). We solve this anti-clastic suppression for the first time for spontaneously deforming systems and find that it obtains even for rather small strains and sample sizes (in a sense that we quantify). Indeed, one can almost never neglect the stretches attendant upon the formation of saddles and spherical caps.

In §1*a*, we summarize weak bending results since the curvatures that emerge set the scale for the strong case we address in §2. In appendix A, we sketch the Föppl–von Kármán (FvK) approach to spontaneous bending, an approach also used by others (Gaididei *et al.* 2009; Liang & Mahadevan 2009) in this context of internally generated deformation. We point out differences between our treatment and the other analyses.

### (a) Summary of weak spontaneous bending results

We require the effective strain at a depth *z*, which is the geometric strain minus the spontaneous strain (denoted by ^{s}), :
1.1
Unlike in weak bending (WMCI), and will become functions of the transverse coordinate *x* in strong bending, as well as the usual function of depth, *z*. The geometric strain in the material mid-plane (initially at *z*=0) is , and we reserve the notation for its value at *x*=0. The geometric strain has a curvature-induced part, *z*/*R*, where *z* is the distance from the mid-plane and 1/*R* is a curvature. The spontaneous strain tensor described in WMCI is characterized by strains *ϵ*_{∥} and *ϵ*_{⊥} parallel and perpendicular to the local ordering direction . We relate these spontaneous strains by a thermal (or optical) Poisson ratio, *ν*_{th}, that is, *ϵ*_{⊥}=−*ν*_{th}*ϵ*_{∥}, where *ν*_{th}=1 gives the cross-over in suppression direction.

Adopting coordinates as in Lamb (1891), whose large deflection method for bending we follow, the relevant in-plane elements of and the stresses are
1.2
and
1.3
where *R*_{xx} and *R*_{yy} are the radii of curvature in the *xx* and *yy* directions. WMCI discusses, in the context of twisted-director cantilevers, why off-diagonal components of and do not play a role in weak bending. In appendix A, we discuss eliminating off-diagonal terms in general. Young’s modulus is *E* and *ν* the Poisson ratio. Our cantilevers are thin and without surface tractions so that *σ*_{zz}(*z*)=*σ*_{xz}(*z*)=*σ*_{yz}(*z*)=0.

For a weak spontaneous bend, no net force or torque acts through any section of the cantilever, in particular through the *xx* and *yy* sections at each *y* or *x*, respectively, (Warner & Mahadevan 2004; Corbett & Warner 2007):
1.4
where *σ* is *σ*_{xx} or *σ*_{yy} and the thickness is 2*h*. For instance, the *y* force per unit *x* length acting normally to a *y* section (figure 1*d*) is
1.5
In weak bending, *P* vanishes identically at each *x* across the cantilever. For strong bend, this will not be true—the curvature is suppressed by hoop stresses from forces *P* acting in a curved body, but of a differing sign across the cantilever. In the weak limit, the four conditions from equation (1.4) are, respectively, from torques and forces,
1.6
and
1.7
and give rise to the weak curvatures and mid-plane strains
1.8
A bar denotes an average and the superscript ^{o} on the radii of curvature denotes weak bend forms generated by . Figure 1*a*,*b* shows a cartoon of the response one obtains, either in the classical case of imposed torques, or when the spontaneous distortions in the two directions are of opposite signs. A flat surface deforms into a saddle. The small strain results will be radically altered by a stretch, but the results are still only functions of this underlying response (1.8). Classical analysis (Timoshenko 1925) of curvature of bimetallic strips estimates weak bend analysis to be valid for deflections less than about half the cantilever thickness. We will be ultimately concerned with the other extreme because in the highly responsive systems of interest, the deflections can be hundreds or thousands of times the thickness, see examples in WMCI and a full analysis of weak bend. We now solve the large deflection case in detail and quantify this estimate of validity for equations (1.8) in terms of thickness 2*h*, width 2*b* and curvature 1/*R*_{yy} in the long direction.

## 2. Anti-clastic suppression in strong spontaneous bend

The mechanism whereby transverse curvature (generally anti-clastic, saddle response) is suppressed arises from tension and compression forces in one direction (*y* in figure 1*d*) owing to *x* curvature displacing by a distance *w*(*x*) elements of the *yy* section far from any possible neutral surface. The effect of suppression is to eliminate expensive stretch deformations that arise when there are displacements far from a neutral surface. Our arguments also hold for suppression of syn-clasticity (positive Gaussian curvature, encountered for instance when poly-domain photo-responsive nematic solids are irradiated with unpolarized light). Since there is a *yy* curvature 1/*R*_{yy}, the *y* in-plane forces, *P*(*x*), have perpendicular resultants—hoop stresses, see figure 1*d*—the gradient in the *x* direction of which are effectively torques that unbend the *xx* curvature. The vital point is that the *y* forces per unit length of the *x* direction, *P*(*x*) (see equation (1.5)), are not independent of *x*, but vary between compression around *x*=0 to extension for *x*∼±*b*. Accordingly, the hoop stresses act outward (along *z*) near *x*=0 and conversely inward near *x*=±*b*, these forces undoing the *xx* bend. We follow Lamb’s analysis (Lamb 1891) for saddle suppression when external torques are applied. Later authors (Love 1952) have given accounts of the Lamb approach, but in the context of sufficiently difficult problems (the bend of shells with intrinsic curvature) that their derivations are hard to follow and from which it is hard to glean what the underlying mechanisms of suppression are. It is therefore difficult to see how their methods can be adopted for spontaneous distortions. Shield gives a difficult, sophisticated analysis of the Lamb problem and obtains the same result (Shield 1992). We sketch Lamb’s argument as we give our extension of it. The reader not interested in the technicalities of suppression can find the curvature results at the end of §2*c*(i)–(iii), and be assured that saddle suppression occurs quickly when bend grows. An appendix A sketches how spontaneous curvature form is treated in an FvK analysis.

### (a) Scaling of suppression effects

Given the natural curvatures as above in the *x* and *y* directions of extent approximately *b* and approximately *L*, see figure 1, the bend energy benefit on attaining is approximately . We explore the stretch cost of double curvature, anticipating that it is in the *x* direction that curvature is suppressed. The *xx* curvature displaces the mid-plane a *z*-distance approximately , which, because of *yy* curvature, gives a *yy* stretch of and hence a stretch energy of approximately on integrating across the *x* dimension of the cantilever. Overall, one has
Stretch induced by the *xx* bend in the presence of the *yy* bend is more expensive than the advantage of achieving the *xx* bend for . This geometric combination dictates curvature suppression and delineates weak from strong bending.

### (b) Selection of suppression direction

The direction for curvature suppression is selected by comparing the elastic energy of final states with suppression of either the *xx* curvature (energy *E*_{yy} associated with bending in the *yy* direction) or suppression of the *yy* curvature (with energy *E*_{xx}). Consider first *E*_{yy}. We have the conditions of §1*a* associated with the forces and torques arising from *σ*_{yy}, namely . Likewise, the *x* force must still vanish, , but we simply take the *xx* curvature to have been suppressed, 1/*R*_{xx}=0, by the torques discussed above and therefore do not have a fourth condition relating to torque. Now we only have the first of equations (1.6), with *ν*/*R*_{xx}=0 in it, and still have both of (1.7). The solutions are , and . Recall that and are of opposite signs and hence the curvature 1/*R*_{yy} is thereby less than .

Calculating the energy density from the squares of the effective strains in equation (1.2) and integrating through the thickness *z*, one obtains the energy *E*_{yy}. Likewise, one could suppress 1/*R*_{yy} and retain curvature in the *xx* sense. Repeating the above analysis with *x* and *y* interchanged, one can reduce the condition for *yy* curvature at the expense of *xx* curvature, *E*_{yy}<*E*_{xx}, to
This is an obvious conclusion. Curvature is suppressed in the direction where the natural curvature is lower since the cost of undoing in that direction is less. Since , the curvature in the *y* direction () survives if *ν*_{th}<1 and perpendicular to for *ν*_{th}>1. These conclusions are for an field uniform through the thickness, bend arising because of gradient of stimulating field. For non-uniform , some cases are obvious, for instance, that of splay-bend where there is no saddle formation anyhow. Other cases such as twist have the two perpendicular directions equivalent and the geometrical conditions are more subtle.

### (c) Analysis of curvature suppression

We now analyse the suppression of anti- and syn-clastic response more precisely and take cantilevers longer than they are wide. Since the cantilever is translationally invariant along the (long) *y* direction (except for end regions ∼*b*), then the *x* force per unit *y* length, , is *y* independent as in the small strain limit above. The symmetry between the *y* and *x* directions is now broken in our approach, figure 1. The effective strains (equation (1.1)) are now more complicated and develop an *x* dependence too because the *z* displacement of the cantilever’s mid-plane by *w*(*x*) is *x* dependent. Thus,
2.1and
2.2
where *R*_{yy} is the radius of curvature of the medial line at *z*=0, *x*=0 and 1/*R*_{xx}=−*d*^{2}*w*/*dx*^{2}≡*w*′′(*x*) is the *xx* curvature (ignoring terms like *w*′′*w*′^{2} and smaller) and in general varies with *x*. Imagine the cantilever initially without the *xx* curvature, wrapped on to a cylinder of radius *R*_{yy}−*h* and with the *x*-axis of figure 1*c*,*d* as its generator. Displacing the mid-plane (initially at *z*=0) out radially by *w*(*x*) causes the circumference to increase, and the *yy* strain induced by the outward displacement is *w*/*R*_{yy}. This increase is with respect to that generated by the mean strain at *x*=0, that is, with respect to *e*_{yy}. (Recall, we denote the mid-plane strain at *x*=0 simply by *e*_{yy}.) The combination *e*_{yy}(*x*)=*w*(*x*)/*R*_{yy}+*e*_{yy} is the *yy* strain of the mid-plane a distance *x* away from the centre of the cantilever. Thus, equations (2.1) and (2.2) generalize equation (1.2). The modified radius of curvature is now *R*_{yy}+*w*(*x*) at *x*. Hence, across the thickness, one has an additional *yy* strain of *z*/(*R*_{yy}+*w*). The net strain is with respect to .

Considering small sections of cantilever with normal along , the net *x*-force per unit (*y*) length of section is *dQ*/*dx* and is zero. Since there are no surface tractions *σ*_{xx} at *x*=±*b*, then *Q*=0 there, and the vanishing of the derivative means that *Q*=0 everywhere; see also the note below on the force *Z*. Using the definition of *Q* and the second of equations (1.3), one accordingly has , and thus for the *y* force per unit *x* length *P*(*x*), one has
2.3

The moment per unit *x* length acting on *y* sections to determine the *yy* bend, *G*_{x}, and likewise *H*_{y} governing the *xx* bend are
2.4
and
2.5
Let be the shearing force per unit *y* length acting on a section with its normal along . Then, *dZ*/*dx*−*P*(*x*)/(*R*_{yy}+*w*)=0 since the hoop forces are balanced by the net shearing force on a *yy* element.^{1} The shearing forces are related to the net torques by *dH*_{y}/*dx*−*Z*=0, and hence
2.6
Substituting *H*_{y} and *P* into equation (2.6) and neglecting terms of order *w*/*R*_{yy} (which are of order *h*/*R*_{yy} smaller than surviving terms; Lamb 1891; Shield 1992)
2.7
The final combination in is the mean *yy* strain with respect to the mean spontaneous strain at position *x* across the cantilever which generates *P*(*x*) (equation (2.3)) and which we denote by *u*(*x*)/*R*_{yy}. Thus, we have the definition
2.8
whence equation (2.7) becomes (the curvatures in the first term vanish on *d*/*dx*):
2.9
where *m*^{4}=(3/4)(1−*ν*^{2})/(*hR*_{yy})^{2}, that is is an inverse length. Denoting and etc., the symmetric solution is
2.10
The boundary conditions fixing *A* and *B* are (i) no torques are applied to surfaces *x*=±*b*; thus *H*_{y}(*x*=±*b*)=0:
2.11
and (ii) no surface tractions in the *z* direction at *x*=±*b*, thus *dH*_{y}/*dx*=0, whence
2.12
Clearly, using the latter condition, the total longitudinal (*y*) force vanishes:
as required if we have curvature arising purely from spontaneous processes (Warner & Mahadevan 2004). Likewise, we require there be no net torque involved in the *yy* bend, that is . The spontaneous *yy* curvature is then
2.13
with equation (2.10) giving us
The boundary conditions (2.11) and (2.12) give us
2.14
and
2.15
The functions *p* and *q* (figure 2*a*) are
2.16
and
2.17

Both *p*(*mb*) and *q*(*mb*) tend rapidly to zero at large *mb*, and for small *mb*, one has *p*(*mb*)∼−1/6+(29/1260)(*mb*)^{4}+⋯ and *q*(*mb*)∼(1/2(*mb*)^{2}) (1−(*mb*)^{4}/24+⋯ ), which we exploit below. They each have one imperceptible oscillation about zero for *mb*∼6. Using *A* and *B* in the curvature equation (2.13) gives the final result
2.18
where the function *f*(*mb*) (figure 2*a*) varies between 1 and 0 as *mb* increases:

Expression (2.18) is a transcendental equation for 1/*R*_{yy} or for *mb* since 1/*R*_{yy}=(4/3(1−*ν*^{2}))^{1/2}(*h*/*b*^{2})(*mb*)^{2}. However, equation (2.18) is easy to explore in its limits, which will turn out to be the only regions of interest in practice. Returning 1/*R*_{yy} to equation (2.18) gives an equation for *mb*:
2.19
The solution for *mb* is a function of *ν*, and .

One can model the connection between the underlying radii of curvature, that is between the thermal responses parallel and perpendicular to . For instance, one can take , that is take the thermal Poisson ratio equal to the elastic one, and hence the curvatures in equation (1.8) are related. Then equation (2.19) simplifies to
2.20
*ξ* being the natural *yy* curvature response scaled by *b*^{2}/*h* from the cantilever geometry and also by a Poisson ratio-dependent factor. Returning to equation (2.18), we can thus plot against this natural response, see figure 2*b*. For example, with *ν*=1/2, the curvature goes from the unconstrained value, , to the plate value (with saddle suppression) of at large spontaneous curvatures *ξ*≫1, that is .

#### (i) Weak and strong bending limits

For a weak curvature *mb*≪1, that is 1/*R*_{yy}≪*h*/*b*^{2}, one has from equation (2.18) , as expected. Self-consistency requires, on returning to the condition on *mb*, that
2.21
Recalling that *h*/*b*∼1/10—1/100 is not unusual, then the condition of a small natural curvature is difficult to satisfy—see §3. The transverse curvature is
2.22
which in this limit we can expand in *mx*≤*mb*≪1 and retain only *B* terms to give an overall *w*′′=2*m*^{2}*B*. Likewise expanding *B* in *mb*, one obtains, as expected
2.23

Strong *yy* curvature, *mb*≫1 and , yields in equation (2.18)
2.24
This is the limit of extreme *yy* stretch were there to be any remaining transverse curvature, *w*′′≠0. Transverse sections flatten from the action of effective torques arising from spatially varying hoop forces *P*(*x*)/*R*_{yy}. The result (2.24) is reminiscent of the transition from the cantilever to the plate result in conventional materials with imposed curvature; in that case and thence —curvature is reduced from the weak curvature value by the usual factor (1−*ν*^{2}).

The condition *mb*≫1 implies for consistency:
2.25

The inequality is mostly satisfied and one rarely sees saddles in spontaneously deforming sheets. ‘Photo swimmers’ (Camacho-Lopez *et al.* 2004) rely on saddles for their motion; see WMCI for calculations and photographs.

#### (ii) Mid-plane strain associated with bending

The geometric strain would be equal to the spontaneous strain were there to be no gradient of the strain—there would be elastic match at all points in the solid. Gradients introduce bending and hence mismatches in strain (unless the spontaneous strain were to vary linearly through the thickness). Thus, it is natural that the mid-plane strain at the middle of the cantilever (*x*=0), *e*_{yy}, varies with the reduced natural curvature *ξ*, which is the appropriate measure of spontaneous strain gradient, see equation (2.20). We determine *e*_{yy} by requiring *w*(*x*=0)=0, whence . But *u*(0)=*A* (see equation (2.10)), hence
2.26

Thus, the limiting values for *e*_{yy} are as follows:

— weak bending with

*mb*≪1 and 2.27— strong bending

*mb*≫1 2.28

The former limit is that of §1*a* on weak bending, equation (1.8), with small modifications as (*b*/*R*^{o})^{2} increases (taking a typical weak curvature of the body). The strong limit gives a mid-plane strain also very close to the mean spontaneous strain . It turns out that *e*_{yy} remains close to for all spontaneous distortions as we see in figure 3*a*. Equation (2.26) simplifies when again is assumed, and when is replaced by (*mb*)^{4} and other factors:
2.29
The deviation of the mean *yy* strain at *x*=0 from the mean spontaneous strain scales with (*h*/*b*)^{2} (which is typically very small) and depends on *mb*(*ξ*), equation (2.19) (hence in figure 3*a*, the deviation appears greatly amplified). The mean *yy* strain at general *x* (with respect to the mean spontaneous strain) is *u*(*x*)/*R*_{yy}; see equation (2.8) and the discussion above it. The mean *xx* strains are also *x* dependent and connected with the *yy* strains by the vanishing of *Q*, see equation (2.3).

Although small, the deviation is important because multiplied by *R*_{yy} it enters the cross-sectional shape and this product can be significant.

#### (iii) Form of transverse sections

The solution we have for *u*(*x*) gives us the sectional shape from (see the discussion of boundary conditions above equation (2.26)). Thus,
2.30
The second form arises on inserting *A* and *B* and making the inessential simplifying connection used above, that is . The coefficient involving *ν* takes the value 4/9 when *ν*=1/2, the rubber value used in illustrations.

The scale of *w*, namely the thickness *h*, means that deflections are small. We have seen that for weak bending there is uniform transverse curvature determined by the first moment of the spontaneous strain in the *x* direction, see equation (1.8). The transverse section is thus simply . From the analysis of equation (2.30), we see that this result can only hold if , whence spontaneous deformations must be small enough that .

For a strong bend, both the coefficients and become exponentially small like ∼*e*^{−mb}. Thus, *u*(*x*) vanishes, except for *x*∼±*b* where the *c*_{mx}*ch*_{mx} and *s*_{mx}*sh*_{mx} terms in *u*(*x*), equation (2.10), become exponentially large. From the argument leading to equation (2.26), boundary conditions determine that and thus *w*(*x*)=*u*(*x*)−*A*. In the strong bending limit, *A* is so small that we can take *w*(*x*)≃*u*(*x*). Taking the dominant terms from *A*, *B*, *c*_{mx}*ch*_{mx} and *s*_{mx}*sh*_{mx}, one obtains for
2.31
where we have used equation (2.24) to eliminate 1/*R*_{yy}. The oscillations are a consequence of the equation governing *u* but would presumably be hard to see, as in the classical case (Lamb 1891); see also below.

To plot *w*(*x*) from equation (2.30), we scale transverse lengths *x* by *b* to give and have the scaled length . We choose a natural scaled response *ξ* that fixes *mb* from the solution of equation (2.19) and the related discussion. Figure 3*b* shows against for various *ξ*. From these curves, one can see the extreme effect of saddle suppression. The naive profile would give a reduced deflection of , which is roughly a factor of *ξ* larger than the true values. Figure 3*b* has *ξ*s of 50–100 and so the reductions are considerable. One also sees the negative excursion predicted in the asymptotic form equation (2.31), the amplitude of which is less than 4 per cent of the thickness 2*h*.

## 3. Summary and application of theory

We have calculated the strong (curvature suppressing) bending of cantilevers, weak bending results serving as the input. For spontaneous distortions, the strong limit is quickly approached according to the bending radius *R*, the width 2*b* and thickness 2*h*, in the combination *ξ*∼*b*^{2}/(*hR*^{o}). Typical values of thickness and width are (Mol *et al.* 2005; van Oosten *et al.* 2007), for instance 2*L*=7 mm, 2*b*=4 mm, 2*h*=(2∼4)×10^{−2} mm. Width to thickness ratios in the region of *b*/*h*∼4×10^{2} give (strong bend) already for m.

The above conclusions are borne out by the experience that nematic photo and thermal cantilevers typically bend in a suppressed manner. A particularly rich example is that of the *π*/2-twisted nematic cantilever, where spontaneous strains involve shears. In fact the anti-clastic response should be maximized since the two directions of bend are equivalent and opposite in sign. Accordingly, the criterion established in §2*b* for selecting the direction of suppression, namely the direction of more expensive bend is suppressed, is not relevant. Instead one can argue that the curvature in a principal direction associated with the smaller lineal dimension in the cantilever surface is suppressed and the surviving curvature is in the direction of the longer principal axis, thus minimizing end regions where the energy reduction is not optimized. Figure 4*a*,*b* shows representations of initially flat twist cantilevers that are mirror images of each other, the longer principal sections being shown dark, the shorter ones (being the director on the opposite side) dashed. The strip’s curved principal direction is then wrapped around the surface of a cylinder of the same curvature. The other, flattened, direction is along a generator of the cylinder. Thus, the strip forms a helix. For *ϕ*_{0}<*π*/4, the director (solid lines) defines a direction of greater extent in the strip than that of and is hence curved—for instance convex with respect to the upper surface when cooling. The dotted direction is shorter, with suppressed curvature, and is aligned with the (imaginary) cylinder’s long axis. The strip accordingly wraps as a right-handed helix, with the converse for *ϕ*_{0}>*π*/4, thus explaining the chiral symmetry breaking associated with strongly distorting strips. For *ϕ*_{0}=*π*/4, this symmetry is broken spontaneously.

The experiments of Harris *et al.* (2005) on twisted-director cantilevers with twist off-sets are of this character; see figure 5. Bend and its suppression depends critically on the geometry of the cantilever and its nematic distribution.

## Acknowledgements

We thank Professors Broer and Bastiaansen and Dr van Oosten for the illustration of figure 5, Professor Mahadevan for his preprint on non-uniform deformation, and the EPSRC for funding under grants EP/F013787/1 (D.C.) and EP/E051251/1 (M.W. and C.D.M.).

## Appendix A. FvK analysis for bend resulting from differential spontaneous strains

We here present a brief reformulation of the FvK equations for thin plates when spontaneous deformations are important. Optically inspired bending has been addressed in these terms (Gaididei *et al.* 2009), though not with emphasis on suppression. A similar treatment for thermal stresses (Boley & Weiner 1997, ch. 12) does not consider hoop stresses. Additionally, spatially dependent in-plane growth leads to bending out of plane (Liang & Mahadevan 2009) and this too has been analysed using FvK methods, though we find we do not need some stretch terms found by these authors. The reason for this difference is that the equilibrium state for our spontaneously deformed sample is not flat, rather in the equilibrium case we have a curved plate or shell. For flat plates, the stretching energy for deviations from the equilibrium state is a second-order effect when compared with the bending energy and thus one needs to make large deflections before stretch becomes important. The situation for shells is entirely different, here the stretching is a first-order effect and it is then important even for small deflections (Landau & Lifshitz 1986, p. 50–57). As a simple example, consider a sphere of radius *R* uniformly stretched by a radial displacement *ψ*, the length of the equator increases by 2*πψ* and thus the strain is 2*πψ*/2*πR*=*ψ*/*R*. In contrast, if we take a flat plate initially in the *xy* plane and displace it in the *z* direction by *ψ*, we will generate an *xx* strain proportional to (∂*ψ*/∂*x*)^{2}. Since we are dealing with shells, we neglect these second-order contributions while Liang & Mahadevan (2009), who were concerned with flat-plates, retained them.

Figure 6 shows the forces and moments per unit length acting on an element of a thin plate of thickness 2*h*. The forces and moments per unit length defined in figure 6 are given by
A1
Force balance in the *xy* plane gives
A2
These equations are generally satisfied by introducing the Airy stress function *F*, which satisfies *Q*=∂^{2}*F*/∂*y*^{2}, *S*=−∂^{2}*F*/∂*x*∂*y* and *P*=∂^{2}*F*/∂*x*^{2}. The equation determining the Airy stress function *F*(*x*,*y*) is given in equation (A12). Moment balance gives
A3
We generalized the notation introduced just after equation (2.5); *Z* there (the *z*-shearing force per unit length in the *y* direction) becomes *Z*_{x} here. Analogously *Z*_{y} is the *z*-shearing force per unit length in the *x* direction. Balancing *z* forces gives
A4
*p*(*x*,*y*) is the distributed loading per unit area. 1/*R*_{pq} is the *pq* coefficient of the second fundamental form, that is the second derivative ∂^{2}*ψ*/∂*x*_{p}∂*x*_{q} of the plate’s height function *ψ*(*x*,*y*) (the displacement along the *z* direction of the point initially at (*x*,*y*,0)).

Combining equations (A3) and (A4) gives
A5
In this balance of out-of-plane forces, there is an *S*/*R*_{xy} contribution to hoop forces deriving from in-plane shears (Timoshenko & Gere 2009, fig. 8.12).

We take the stresses to be given by
A6
A7and
A8
Integrating w.r.t. *z* and inverting to find the mid-plane strain components, we find
A9
A10and
A11
Elastic compatibility requires *e*_{xx,yy}+*e*_{yy,xx}−2*e*_{xy,xy}=0, and thus
A12
Using the Airy stress function for *P*, *Q* and *S* gave the second expression.

Solving equations (A9)–(A11) for *P*,*Q*,*S*, we obtain
A13
A14and
A15
Similarly for the moments, writing as before as etc., we have
A16
A17and
A18
Substituting these expressions for the moments into equation (A5) gives
A19
In order to completely describe the spontaneous deformation of the plate, we must therefore solve equations (A12) and (A19) subject to appropriate boundary conditions. Here, we content ourselves with showing that the present formulation admits the solution found in the main text in §2*c* and presented in equation (2.7).

For simple cases, for instance twist antisymmetrically disposed about *z*=0, the principal axes of bend are the *x* and *y* directions of figure 1; otherwise, they are rotated as we discuss for spirals and again below for weak bending. As discussed in §2*c*, translational invariance in the long (*y*) direction coupled with boundary conditions give *Q*=0. One sees most clearly in equation (A5) the effect of such invariance: ∂^{2}*H*_{y}/∂*y*^{2}=∂^{2}*J*_{xy}/∂*x*∂*y*=0. Since 1/*R*_{pq} is symmetric by construction, we can diagonalize it and also eliminate the *S*/*R*_{xy} term in equation (A5). The corresponding terms deriving from *H*, *J* and *S* on the left- and right-hand sides of equation (A19) therefore vanish. We then identify the height function as *ψ*(*x*,*y*)=*w*(*x*)+*y*^{2}/(2*R*_{yy}), and also assume the loading *p*=0. Since *Q*=0, we have . The term *P*(∂^{2}*ψ*/∂*y*^{2}) becomes *P*/*R*_{yy} as a result of the *y*^{2}/(2*R*_{yy}) part of *ψ*, giving the hoop forces of the main text and the second term of equation (2.7). Writing *e*_{yy}(*x*)=*w*(*x*)/*R*_{yy}+*e*_{yy}(*x*=0) gives the form of equation (2.7).

At this point, one can return to the question of shear stresses in weak bending, discussed in the context of twisted nematic cantilevers in WMCI. Since we can diagonalize 1/*R*_{pq}, we need only worry about the bends 1/*R*_{x′x′} and 1/*R*_{y′y′} and their concomitant forces and torques vanishing—the four equilibrium conditions of weak bending in WMCI and in §1*a*. The principal directions are denoted by *a*′ and in general do not coincide with *x*,*y* of the strip. We discussed in WMCI how *σ*_{x′y′}, and the equal quantity *σ*_{y′x′}, did not give a *z*-torque.^{2}

## Footnotes

↵1 One might have expected

*Z*=0 since with*σ*_{xz}=0 since the*z*dimension is small and the surface tractions from*σ*_{xz}vanish on the free surfaces. The finite hoop forces invalidate this thin elastica argument. For the analogous shearing force per unit*x*length acting on a section with its normal along*y*, one has . Translational invariance in*y*gives and hence*Q*=0, a fact used above equation (2.3). One can compare*Z*and*P*roughly. Taking*P*∼*Ehw*(*x*)/*R*_{yy}and*w*(*x*)∼*x*^{2}/*R*_{xx}, one can integrate the expression for*dZ*/*dx*to get (connecting the*xx*and*yy*curvatures by a Poisson ratio). Given that , then*Z*<*P*until*b*/*R*_{yy}∼1, a rather sharp bend.↵2 For strong bend and thus suppression, the analysis we have developed which depended on the

*x*force per*y*length,*Q*, vanishing can be used, provided the obliquity*θ*of the*x*′-,*y*′-axes to the strip’s*x*,*y*is small. Then, the*Q*′ in this frame is translationally invariant along*y*′, and thus zero since no net force through this section exists, for all positions except close to the free edges—end effects are somewhat more complicated than before.

- Received March 8, 2010.
- Accepted March 29, 2010.

- © 2010 The Royal Society