The stress field in a pulled cork and some subtle points in the semi-inverse method of nonlinear elasticity
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec The stress field in a pulled cork andsome subtle points in thesemi-inverse method of nonlinear elasticity
Riccardo De Pascalis, Michel Destrade, Giuseppe Saccomandi2006
Abstract
In an attempt to describe cork-pulling, we model a cork as anincompressible rubber-like material and consider that it is subject toa helical shear deformation superimposed onto a shrink fit and a simpletorsion. It turns out that this deformation field provides an insightinto the possible appearance of secondary deformation fields for specialclasses of materials. We also find that these latent deformation fieldsare woken up by normal stress differences. We present some explicitexamples based on the neo-Hookean, the generalized neo-Hookean,and the Mooney–Rivlin forms of the strain-energy density. Using thesimple exact solution found in the neo-Hookean case, we conjecturethat it is advantageous to accompany the usual vertical axial forceby a twisting moment, in order to extrude a cork from the neck of abottle efficiently. Then we analyse departures from the neo-Hookeanbehaviour by exact and by asymptotic analyses. In that process we areable to give an elegant and analytic example of secondary (or latent)deformations in the framework of nonlinear elasticity.
Rubbers and elastomers are highly deformable solids which have the remark-able property of preserving their volume through any deformation. Thispermanent isochoricity can be incorporated into the equations of continuummechanics through the concept of an internal constraint , here the constraintof incompressibility . Mathematically, the formulation of the constraint of in-compressibility has led to the discovery of several exact solutions in isotropicfinite elasticity, most notably to the controllable or universal solutions of1ivlin and co-workers (see for example Rivlin (1948)). Subsequently, Er-icksen (1954) examined the problem of finding all such solutions. He foundthat there are no controllable finite deformations in isotropic compressible elasticity, except for homogeneous deformations (Ericksen 1955). The im-pact of that result on the theory of nonlinear elasticity was quite importantand long-lasting, and for many years a palpable pessimism reigned about thepossibility of finding exact solutions at all for compressible elastic materials.Then Currie & Hayes (1981) showed that one could obtain interesting classesof exact solutions, beyond the homogeneous universal deformations, if onerestricted their attention to certain special classes of compressible materials.A string of results about the search for exact solutions in nonlinear elastic-ity followed. Now a long list exists of classes of exact solutions which areuniversal only relative to some special strain-energy functions (for a recentpresentation of such classes see Fu and Ogden (2001).) These solutions canhelp us to understand the structure of the theory of nonlinear elasticity andto complement the celebrated solutions of Rivlin.In the same vein, some recent efforts focused on determining the maximalstrain energy for which a certain deformation field, fixed a priori, is admis-sible. This is a sort of inverse problem : find the elastic materials (that is,the functional form of the strain-energy function) for which a given defor-mation field is controllable (that is, for which the deformation is a solutionto the equilibrium equations in the absence of body forces). A classical ex-ample illustrating such an approach is obtained by considering deformationsof anti-plane shear type. Knowles (1977) shows that a non-trivial (non-homogeneous) equilibrium state of anti-plane shear is not always (univer-sally) admissible, not only for compressible solids (as expected from Erick-sen’s result) but also for incompressible solids (Horgan (1995) gives a surveyof anti-plane shear deformations in nonlinear elasticity). Only for a spe-cial class of incompressible materials (inclusive of the so-called ‘generalizedneo-Hookean materials’) is an anti-plane shear deformation controllable.Let us consider for example the case of an elastic material filling the annu-lar region between two coaxial cylinders, with the following boundary valueproblem: hold fixed the outer cylinder and pull the inner cylinder by apply-ing a tension in the axial direction. It is well established that a solution tothis problem, valid for every isotropic incompressible elastic solid, is obtainedby assuming a priori that the deformation field is a pure axial shear. Nowconsider the corresponding problem for non-coaxial cylinders, thereby losingthe axial symmetry. Then it is clear that we cannot expect the material todeform as prescribed by a pure axial shear deformation. Knowles’s resulttells us that now the boundary value problem can be solved with a generalanti-plane deformation (not axially symmetric) only for a subclass of incom-2ressible isotropic elastic materials. Of course this restriction does not meanthat for a generic material it is not possible to deform the annular materialas prescribed by our boundary conditions, but rather that in general, theselead to a deformation field which is more complex than an anti-plane shear.And so, we expect secondary in-plane deformations.The theory of non-Newtonian fluid dynamics has generated a substantialliterature about secondary flows, see for example Fosdick & Serrin (1973).In solid mechanics it seems that only Fosdick & Kao (1978) and Mollica& Rajagopal (1997) produced some significant and beautiful examples ofsecondary deformations fields for the non-coaxial cylinders problem, althoughthis topic is clearly of fundamental importance, not only from a theoreticalpoint of view but also for technical applications.In this paper we consider a complex deformation field in isotropic incom-pressible elasticity, to point out by an explicit example the situations justevoked and to elaborate on their possible impact on solid mechanics. Ourdeformation field takes advantage of the radial symmetry and therefore wefind it convenient to visualize it by considering an elastic cylinder.Let us imagine that a corkscrew has been driven through a cork (thecylinder) in a bottle. The inside of the bottleneck is the outer rigid cylinderand the idealization of the gallery carved out by the corkscrew constitutes theinner coaxial rigid cylinder. Our first deformation is purely radial, originatedfrom the introduction of the cork into the bottleneck and then completedwhen the corkscrew penetrates the cork (a so-called shrink fit problem , whichis a source of elastic residual stresses here). We call A , B the respective innerand outer radii of the cork in the reference configuration and r > A , r < B their current counterparts. Then we follow with a simple torsion combinedto a helical shear , in order to model pulling the cork out of the bottleneck inthe presence of a contact force. Figure 1 sketches this deformation.Of course we are aware of the shortcomings of our modelling with re-spect to the description of a ‘real’ cork-pulling problem, because no cork isan infinitely long cylinder, nor is a corkscrew perfectly straight. In addition,traditional corks made from bark are anisotropic (honeycomb mesoscopicstructure) and possess the remarkable (and little-known) property of havingan infinitesimal Poisson ratio equal to zero, see the review article by Gibson et al. (1981). However we note that polymer corks have appeared on theworld wine market; they are made of elastomers, for which incompressible,isotropic elasticity seems like a reasonable framework (indeed the documen-tation of these synthetic wine stoppers indicates that they lengthen duringthe sealing process). We hope that this study provides a first step towarda nonlinear alternative to the linear elasticity testing protocols presented inthe international standard ISO 9727. We also note that low-cost shock ab- sorbers often consist of a moving metal cylinder, glued to the inner face ofan elastomeric tube, whose outer face is glued to a fixed metal cylinder (Hill1975).The plan of the paper is the following. Section 2 is devoted to the deriva-tion of the governing equations and to a detailed description of the boundaryvalue problem. In § neo-Hookean strain-energy density , and find the corresponding exact solution. We use it to showthat it is advantageous to add a twisting moment to an axial force when ex-truding a cork from a bottle. The neo-Hookean strain-energy density is linearwith respect to the first principal invariant of the Cauchy-Green strain ten-sor. It is much used in Finite Elasticity theory, although it captures poorlythe basic features of rubber behaviour (Saccomandi 2004). We thus inves-tigate the consequence of departing from that strain-energy density. First,in § generalized neo-Hookean strain-energy density — non-linear with respect to the first principal invariant of the Cauchy-Green straintensor — to show that in this case torsion is explicitly present in the solutionfor the axial shear displacement, but it is a second-order dependence. Nextwe consider in § Mooney–Rivlin strain-energy density — linear with re-spect to the first and second principal invariants of the Cauchy-Green straintensor — and find that it is then also possible to obtain an exact solution4o our boundary value problem. Its expression is too cumbersome to ma-nipulate and we resort to a small parameter asymptotic expansion from theneo-Hookean case. Section 6 concludes the paper with some remarks on thelimitations of the semi-inverse method.
Consider a long hollow cylindrical tube, composed of an isotropic incom-pressible nonlinearly elastic material. At rest the tube is in the region A ≤ R ≤ B, ≤ Θ ≤ π, −∞ ≤ Z ≤ ∞ , (2.1)where ( R, Θ , Z ) are the cylindrical coordinates associated with the unde-formed configuration, and A and B are the inner and outer radii of the tube,respectively. Consider the general deformation obtained by the combination of radial di-latation, helical shear, and torsion, as r = r ( R ) , θ = Θ + g ( R ) + τ Z, z = λZ + w ( R ) , (2.2)where ( r, θ, z ) are the cylindrical coordinates in the deformed configuration, τ is the amount of torsion and λ is the stretch ratio in the Z direction. Here, g and w are yet unknown functions of R only (The classical case of puretorsion corresponds to w = g = 0, see the textbooks by Ogden (1997) or byAtkin and Fox (2007), for instance.)Hidden inside (2.2) is the shrink fit deformation r = r ( R ) , θ = Θ , z = λZ, (2.3)which is (2.2) without any torsion nor helical shear ( τ = g = w ≡ F and of its inverse F − are then r ′ rg ′ r/R rτw ′ λ , rλ/R rw ′ τ − rg ′ λ r ′ λ − rr ′ τ − rw ′ /R rr ′ /R , (2.4)respectively. Note that we used the incompressibility constraint in order tocompute F − ; it states that det F = 1, so that r ′ = Rλr . (2.5)5n our first deformation, the cylindrical tube is pressed into a cylindricalcavity with inner radius r > A and outer radius r < B . It follows byintegration of the equation above that r ( R ) = r R λ + α, (2.6)where now α = B r − A r B − A , λ = B − A r − r . (2.7)We compute the physical components of the left Cauchy-Green straintensor B ≡ F F t from (2.4) and find its first three principal invariants I ≡ tr B , I ≡ (det B )tr ( B − ), and I ≡ det B as I = ( r ′ ) + ( rg ′ ) + ( r/R ) + ( rτ ) + λ + ( w ′ ) ,I = ( rλ/R ) + ( rw ′ τ − rg ′ λ ) + ( rw ′ /R ) + ( R/r ) + (1 /λ ) + ( Rτ /λ ) , (2.8)and of course, I = 1.For a general incompressible hyperelastic solid, the Cauchy stress tensor T is related to the strain through T = − p I + 2 W B − W B − , (2.9)where p is the Lagrange multiplier introduced by the incompressibility con-straint, W = W ( I , I ) is the strain energy density, and W i ≡ ∂W/∂I i . Hav-ing computed B − ≡ ( F t ) − F − from (2.4), we find that the components of T are T rr = − p + 2 W ( r ′ ) − W (cid:2) ( rλ/R ) + ( rw ′ τ − rg ′ λ ) + ( rw ′ /R ) (cid:3) ,T θθ = − p + 2 W (cid:2) ( rg ′ ) + ( r/R ) + ( rτ ) (cid:3) − W ( R/r ) ,T zz = − p + 2 W [ λ + ( w ′ ) ] − W (cid:2) (1 /λ ) + ( Rτ /λ ) (cid:3) ,T rθ = 2 W ( rr ′ g ′ ) − W ( w ′ τ − g ′ λ ) R,T rz = 2 W ( r ′ w ′ ) − W (cid:2) rRg ′ τ − rRw ′ τ /λ − rw ′ / ( λR ) (cid:3) ,T θz = 2 W ( rw ′ g ′ + rλτ ) + 2 W ( r ′ Rτ ) . (2.10)Finally the equilibrium equations, in the absence of body forces, are:div T = ; for fields depending only on the radial coordinate as here, theyreduce tod T rr d r + T rr − T θθ r = 0 , d T rθ d r + 2 r T rθ = 0 , d T rz d r + 1 r T rz = 0 . (2.11)6 .2 Boundary conditions Now consider the inner face of the tube: we assume that it is subject to avertical pull, T rz ( A ) = T A , T rθ ( A ) = 0 , (2.12)say. Then we can integrate the second and third equations of equilibrium(2.11) , ; we find that T rz ( r ) = r r T A , T rθ ( r ) = 0 . (2.13)The outer face of the tube (in contact with the glass in the cork/bottleproblem) remains fixed, so that w ( B ) = 0 , g ( B ) = 0 , T rr ( B ) = T , (2.14)say. In addition to the axial traction applied on its inner face, the tube issubject to a resultant axial force N (say) and a resultant moment M (say), N = Z π Z r r T zz r d r d θ, M = Z π Z r r T θz r d r d θ. (2.15)Note that the traction T of (2.14) is not arbitrary but is instead deter-mined by the shrink fit pre-deformation (2.3), by requiring that N = 0 when T A = τ = g = w ≡ T is connected with the stress fieldexperienced by the cork when it is introduced in the bottleneck.In the rest of the paper we aim at presenting results in dimensionlessform. To this end we normalize the strain-energy density W and the Cauchystress tensor T with respect to µ , the infinitesimal shear modulus; hence weintroduce W and T defined by W = Wµ , T = T µ . (2.16)Similarly we introduce the following non-dimensional variables, η = AB , R = RB , r i = r i B , w = wB , α = αB , τ = Bτ, (2.17)so that η ≤ R ≤
1. Also, we find from (2.7) that α = r − η r − η , λ = 1 − η r − r . (2.18)7urning to our cork or shock absorber problems, we imagine that the innermetal cylinder is introduced into a pre-existing cylindrical cavity (this pre-caution ensures a one-to-one correspondence of the material points betweenthe reference and the current configurations). In our upcoming numericalsimulations, we take A = B/
10 so that η = 0 .
1; we consider that the outerradius is shrunk by 10%: r = 0 . B , and that the inner radius has doubled: r = 2 A ; finally we apply a traction whose magnitude is half the infinitesimalshear modulus: | T A | = µ/
2. This gives α ≃ . × − , λ ≃ . , T A = − . . (2.19)At this point it is possible to state clearly our main observation. A firstglance at the boundary conditions, in particular at the requirements that g be zero on the outer face of the tube, gives the expectation that g ≡ g ≡ τ or not. However if the solid is not neo-Hookean, then it isnecessary that g = 0 when τ = 0, and the picture becomes more complex.For this reason, we classify as ‘purely academic’ the question: which is themost general strain-energy density for which it is possible to solve the aboveboundary value problem with g ≡ exactly by that strain-energydensity (supposing it exists). Instead a more pertinent issue to raise for ‘realword applications’ is whether we are able to evaluate the importance of latent(secondary) stress fields, because they are bound to be woken up (triggered)by the deformation. First we consider the special strain energy density which generates the classof neo-Hookean materials, namely W = ( I − / , so that 2 W = 1 , W = 0 . (3.1)Note that here and hereafter, we use the non-dimensional quantities intro-duced previously, from which we drop the overbar for convenience. Hence thecomponents of the (non-dimensional) stress field in a neo-Hookean material8 Z R z r
Figure 2: Pulling on the inside face of a neo-Hookean tube. Here the verticalaxis is the symmetry axis of the tube.reduce to T rr = − p + ( r ′ ) , T θθ = − p + ( rg ′ ) + ( r/R ) + ( rτ ) ,T zz = − p + λ + ( w ′ ) , T rθ = rr ′ g ′ ,T rz = r ′ w ′ , T θz = rg ′ w ′ + rλτ. (3.2)Substituting into (2.13) we find that w ′ = λr T A /R, g ′ = 0 , (3.3)and by integration, using (2.14), that w = λr T A ln R, g = 0 . (3.4)In Figure 2a, we present a rectangle in the tube at rest. It is delimitedby 0 . ≤ R ≤ . . ≤ Z ≤ .
0. Then it is subject to the deformationcorresponding to the numerical values of (2.19). To generate Figure 2b, wecomputed the resulting shape for a neo-Hookean tube, using (2.2), (2.6), and(3.4).Now that we know the ful deformation field, see (2.2) and (3.4), we cancompute T rr − T θθ from (3.2) and deduce T rr by integration of (2.11) , withinitial condition (2.14) . Then the other field quantities follow from the rest9f (3.2). In the end we find in turn that T rr = 12 λ (cid:26) ln λr R R + αλ + ( R − (cid:20) αr ( R + αλ ) + τ (cid:21)(cid:27) + T ,T θθ = T rr + (cid:18) R λ + α (cid:19) (cid:18) R + τ (cid:19) − R λ ( R + αλ ) ,T zz = T rr + λ (cid:18) r ( T A ) R (cid:19) − R λ ( R + αλ ) , (3.5)(where we used the identity 1 + αλ = λr , see (2.6) with R = 1), and that T rθ = 0 , T rz = r r R λ + α T A , T θz = λτ r R λ + α. (3.6)The constant T is fixed by the shrink fit pre-deformation (2.3), imposingthat N = 0 when τ = g = w = T A ≡
0, or( T + λ )(1 − η ) + 1 λ Z η (cid:20) ln λr R R + αλ + α ( R − r ( R + αλ ) − R R + αλ (cid:21) R d R = 0 . (3.7)Using this, and (2.15), (3.5), (3.6), we find the following expressions for theresultant moment, M = π ( r − r ) λτ / , (3.8)and for the axial force, N = 2 πλr | ln η | (cid:0) T A (cid:1) − π r − r ) τ . (3.9)We now have a clear picture of the response of a neo-Hookean solid to thedeformation (2.2), with the boundary conditions of § b . First we saw thathere the contribution g ( R ) is not required for the azimuthal displacement,whether there is a torsion τ or not. Also, if a moment M = 0 is applied, thenthe tube suffers an amount of torsion τ = 0 proportional to M . On the otherhand, if the tube is pulled by the application of an axial force only ( N = 0)and no moment ( M = 0), then τ = 0 and no azimuthal shear occurs at all.When we try to apply our results to the extrusion of a cork from the neckof a bottle, the following remarks seem to be relevant. From the elementarytheory of Coulomb friction, it is known that the pulled cork starts to movewhen, in modulus, the friction force exerted on the neck surface is equal to10he normal force times the coefficient of static friction. In our case this meansthat q | T rz (1) | + | T rθ (1) | = f S | T rr (1) | = f S | T | , (3.10)where f S is the coefficient of static friction. Using (2.12) and (2.13), we findthat the elements of the left handside of this inequality are T rz (1) = ( r /r ) T A , T rθ (1) = 0 . (3.11)Now, our main concern is to understand if it is better to apply a moment M =0 when uncorking a bottle, than to pull only. To address this question wenote that the left handside of inequality (3.10) increases when | T A | increases;on the other hand, combining (3.8) and (3.9), we have( T A ) = (cid:20) N + 1 πλ ( r + r ) M (cid:21) / (2 πλr | ln η | ) , (3.12)It is now clear, that for a fixed value of T A , in the case M = 0, it is necessaryto apply an axial force whose intensity is less than the one in the case M = 0.Moreover, the equation above shows that (cid:0) T A (cid:1) grows linearly with N butquadratically with M . With respect to efficient cork-pulling, the conclusionis that adding a twisting moment to a given pure axial force is more advan-tageous than solely increasing the vertical pull. Moreover, we observe that amoment is applied by using a lever and this is always more convenient froman energetic point of view.Recall that we made several simplifying assumptions to reach these re-sults: not only infinite axial length, incompressibility, and isotropy, but alsothe choice of a truly special strain energy density. In the next two sectionswe depart from the neo-Hookean model. As a first broadening of the neo-Hookean strain-energy density (3.1), we con-sider generalized neo-Hookean materials , for which the strain-energy densityis a nonlinear function of the first invariant I only, W = c W ( I ) , (4.1)say. To gain access to the Cauchy stress components in this context, itsuffices to take W = 0 and W = c W ′ in equations (2.10). In particular, T rθ = 2 rr ′ g ′ c W ′ , and the integrated equation of equilibrium (2.13) gives g ′ = 0. Integrating, with (2.12) as an initial value, we find that g ≡ . (4.2)11ence, just as in the neo-Hookean case, azimuthal shear can be avoidedaltogether, whether there is a torsion τ or not. We are left with an equationfor the axial shear, namely (2.13) , which here reads2 c W ′ ( I ) w ′ ( R ) = λr R T A . (4.3)Obviously the same steps as those taken for neo-Hookean solids may befollowed here for any given strain energy density (4.1), but now by resortingto a numerical treatment. Horgan & Saccomandi (2003 a ) show, through somespecific examples of hardening generalized neo-Hookean solids, how rapidlyinvolved the analysis becomes, even when there is only helical shear and noshrink fit.Instead we simply point out some striking differences between our presentsituation and the neo-Hookean case. We remark that I is of the form (2.8) at g ≡ I = λ + R λ ( R + αλ ) + (cid:18) R λ + α (cid:19) (cid:18) R + τ (cid:19) + [ w ′ ( R )] . (4.4)It follows that (4.3) is a nonlinear differential equation for w ′ , in contrastwith the neo-Hookean case. Another contrast is that the axial shear w is nowintimately coupled to the torsion parameter τ , and that this dependence isa second-order effect ( τ appears above as τ ).A similar problem where the azimuthal shear has not been ignored, butthe axial shear has been considered null i.e. w ≡ In this section we specialize the general equations of § W = I − m ( I − m ) , so that 2 W = 11 + m W = m m , (5.1)where m > I , in contrast to the generalizedneo-Hookean solids of the previous section.Then the integrated equations of equilibrium (2.13) read (cid:0) R + mτ r R + mr /R (cid:1) w ′ − ( mτ λr R ) g ′ = (1 + m ) λr T A , ( mτ λ ) w ′ − (1 + mλ ) g ′ = 0 . (5.2)12irst we ask ourselves if it is possible to avoid torsion during the pullingof the inner face. Taking τ = 0 above gives( R + mr /R ) w ′ = (1 + m ) λr T A , g ′ = 0 . (5.3)It follows that here it is indeed possible to solve our boundary value problem.We find w = λr T A λ (1 + m )2( λ + m ) ln (cid:20) mαλ + ( λ + m ) R mαλ + ( λ + m ) (cid:21) , g = 0 . (5.4)However if τ = 0, then it is necessary that g = 0, otherwise (5.2) gives w ′ = 0 while (5.2) gives w ′ = 0, a contradiction. This constitutes the firstdeparture from the neo-Hookean and generalized neo-Hookean behaviours: torsion ( τ = 0 ) is necessarily accompanied by azimuthal shear ( g = 0 ) .In the case τ = 0, we introduce the function Λ = Λ( R ) defined asΛ( R ) = ( R + mr /R )(1 + mλ ) + mτ r R, (5.5)(recall that r = r ( R ) is given explicitly in (2.6).) We then solve the system(5.2) for w ′ and g ′ as w ′ = (1 + m )(1 + mλ ) λ T A Λ( R ) r , g ′ = m (1 + m ) λ T A Λ( R ) τ r , (5.6)making clear the link between g and τ . Thus for the Mooney–Rivlin mate-rial, the azimuthal shear g is a latent mode of deformation; it is woken up byany amount of torsion τ . Recall that at first sight, the azimuthal shear com-ponent of the deformation (2.2) seemed inessential to satisfy the boundaryconditions, especially in view of the boundary condition g (1) = 0. However,a non-zero W term in the constitutive equation clearly couples the effects ofa torsion and of an azimuthal shear, as displayed explicitly by the presenceof τ in the expression for g ′ above.It is perfectly possible to integrate equations (5.6) in the general case, butto save space we do not reproduce the resulting long expressions. With them,we generated the deformation field picture of Figure 3a and Figure 3b. Therewe took the numerical values of (2.19) for α , λ , T A ; we took a Mooney–Rivlinsolid with m = 5 .
0; we imposed a torsion of amount τ = 0 .
5; and we lookedat the deformation field in the plane Z = 1 (reference configuration) and z = λ (current configuration).Although the secondary fields appear to be slight in the picture, they arenonetheless truly present and cannot be neglected. To show this we considera perturbation method to obtain simpler solutions and to understand the13 –1–0.500.51–1 –0.5 0.5 1 Figure 3: Pulling on the inside face of a Mooney–Rivlin tube, with a clockwisetorsion.effect of the coupling, by taking m small. Then integrating (5.6), we find atfirst order that wr T A ≃ (1 + m ) λ ln R − m (cid:2) τ R + 2(1 + τ αλ ) ln R − αλ/R − τ + αλ (cid:3) ,gr T A ≃ λ τ m ln R. (5.7)Hence, the secondary field g exists even for a nearly neo-Hookean solid (if m is small, then g of order m .) Interestingly we also note that the azimuthalshear g in (5.7) varies in a homogeneous and linear manner with respect tothe torsion parameter τ and in a quadratic manner with respect to the axialstretch λ , showing that that the presence of this secondary deformation fieldcannot be neglected when the effects of the prestress and of the torsion areboth taken into account. To complete the picture, we use the first-orderapproximations 2 W ≃ − m, W ≃ m, (5.8)14o obtain the stress field as T rr ≃ − p + (1 − m ) ( r ′ ) − m n ( rλ/R ) + (cid:2) ( rτ ) + ( r/R ) (cid:3) (cid:0) λr T A (cid:1) /R o ,T θθ ≃ − p + (1 − m ) (cid:2) ( r/R ) + ( rτ ) (cid:3) − m ( R/r ) ,T zz ≃ − p + (cid:0) λT A r (cid:1) (cid:20)(cid:0) mλ (cid:1) R − R (cid:18) τ r R + r R + λ R − R (cid:19) m (cid:21) + (1 − m ) λ − m (cid:2) (1 /λ ) + ( Rτ /λ ) (cid:3) ,T rθ ≃ rr ′ g ′ − mλr T A τ,T rz ≃ (1 − m ) ( r ′ w ′ ) + mλr T A (cid:2) rRτ /λ + r/ ( λR ) (cid:3) /R,T θz ≃ (1 − m ) rλτ + λrr T A g ′ /R + m ( r ′ Rτ ) . (5.9)Using this stress field it is straightforward, but long and cumbersome,to derive the analogue for a Mooney–Rivlin solid with a small m of relation(3.12) (which was established for neo-Hookean solids.) However nothingtruly new is gained from these complex formulas with respect to the simpleneo-Hookean case, and we do not pursue this aspect any further. In non-Newtonian fluid mechanics and in turbulence theory, the existence ofshear-induced normal stresses on planes transverse to the direction of shearis at the root of some important phenomena occurring in the flow of fluiddown pipes of non-circular cross section (Fosdick & Serrin 1973). In otherwords, pure parallel flows in tubes without axial symmetry are possible onlywhen we consider the classical theory of Navier-Stokes equations or the lineartheory of turbulence or tubes of circular cross section.In nonlinear elasticity theory, similar phenomena are reported. HenceFosdick & Kao (1978) and Mollica & Rajagopal (1997) show that for gen-eral isotropic incompressible materials, an anti-plane shear deformation of acylinder with non axial-symmetric cross section causes a secondary in-planedeformation field, because of normal stress differences. Horgan & Saccomandi(2003 b ) give a detailed discussion of how the anti-plane shear deformationfield couples with the in-plane deformation field in a generalized neo-Hookeansolid.The appearance of what we called here latent deformations is quite gen-eral and common. For example it is known in compressible nonlinear elas-ticity that pure torsion is possible only in a special class of materials, butwe know that torsion plus a radial displacement is possible in all compress-ible isotropic elastic materials (Polignone & Horgan 1991) (Here we signal15hat ‘possible in all materials’ is not equivalent to ‘universal’, because thecorresponding radial deformation differs from one material to another.)In this paper we give an example where axial symmetry holds, where theboundary conditions suggest that an axial shear deformation field is sufficientto solve the boundary value problem, and where nevertheless, the normalstress difference wakes up a latent azimuthal shear deformation. Moreover,because we are able to find some explicit exact solutions by some perturbationtechniques, we are able to evaluate the importance of the latent deformation.Indeed, we show that if a certain constitutive parameter m (distinguishinga neo-Hookean solid from a Mooney–Rivlin solid) is zero or if the torsionparameter τ is zero, then the solution to the boundary value problem canbe found only in terms of the axial shear deformation field; if these twoparameters are not zero — even if they are small — then the latent mode ofdeformation is quantitatively appreciable.In conclusion we suggest that it is not crucial to determine the class ofmaterials for which a given deformation field is possible. Rather it is crucial toclassify all the latent deformations associated with a given deformation fieldin such a way that this field is controllable for the entire class of materials.Indeed, no real material, even when we accept that its mechanical behaviouris purely elastic, is ever going to be described exactly by a special choiceof strain-energy. Looking for special classes of materials for which specialdeformations fields are admissible may mislead us in our understanding ofthe nonlinear mechanical behaviour of materials.To finish the paper on a light note, we evoke a classic wine party dilemma:which kind of corkscrew system requires the least effort to uncork a bottle?Figure 4 sketches the two working principles commonly found in commercialcorkscrews. The most common (on the left) relies on pulling only (directlyor through levers) and the other type (on the right) relies on a combinationof pulling and twisting. Notwithstanding the shortcomings of this paper’smodelling with respect to an actual uncorking, the authors are confidentthat they have provided a scientific argument to those wine amateurs whofavour the second type of corkscrews over the first. References [1] Currie, P. K., Hayes, M. 1981 On non-universal finite elastic deforma-tions. In
Proc. IUTAM Symp. on Finite Elasticity (ed. D. E. Carlson &R. T. Shield). Martinus Nijhoff, pp. 143–150.16igure 4: There are two main types of corkscrews: one that relies on pullingonly (left) and one that adds a twist to the cork-pulling action (right). Theanalysis developed in this paper indicates that the second type is more effi-cient.[2] Ericksen, J. L. 1954 Deformations possible in every isotropic incompress-ible perfectly elastic body.
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