On barrelling instabilities in finite elasticity

Journal of Elasticity 14 (1984) 103-125 © 1984 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands On barrelling instabilities in finite elasticity H E N R Y C. S I M P S O N * and S C O T T J. S P E C T O R ** • Department of Mathematics, University of Tennessee, Knoxoille, TN 37996-1300, U.S.A. • * Department of Mathematics, Southern Illinois Unioersity, Carbondale, IL 62901, U.S.A. (Received March 31, 1982; in revised form August 17, 1983) Introduction In this paper we investigate the question of stability for a solid circular cylinder, c o m p o s e d of h o m o g e n e o u s isotropic (compressible) nonlinearly elastic material, that is subjected to compressive end forces in the direction of its axis (so as to give fixed end displacements). The initial configuration is assumed to be natural with positive elasticity tensor. Experiments of Beatty and H o o k [4] and Beatty and Dadras [3] have shown that such cylinders retain a cylindrical shape until a certain critical value of the loading is reached. Further loading of the cylinder results in buckling for thin cylinders and axisymmetric bulging or barrelling .for thick cylinders. W e have investigated this barrelling phenomena. The first analysis of the stability of a nonlinearly elastic cylinder t was d o n e by Wilkes [34], who assumed that the cylinder was infinite and c o m p o s e d of an incompressible material. After formulating an appropriate non-linear boundary-value problem, Wilkes proceeded to linearize his equations about a given solution of the non-linear problem. 2 This linearization yielded a system of second order partial differential equations with b o u n d a r y conditions. Separation of variables was then used to obtain a family of functions that satisfied the linear problem, sans the traction free b o u n d a r y condition on the curved surface of the cylinder. Finally, Wilkes considered the problem of satisfying the remaining b o u n d a r y condition for the special case of a n e o - H o o k e a n material. Here he showed numerically that the traction free condition could be satisfied for an appropriate loading value. i Green and Spencer [11] have studied the problem of extension and torsion of an infinite incompressible cylinder. Sensenig [29] has studied the problem considered in this paper, the axial compression of a finite compressible cylinder. Both papers use separation of variables and a numerical verification that appropriate loading values produce instability. Sensenig's analysis is limited to a harmonic material (cf., e.g., John [14,15]) which has a number of physically unrealistic properties (for example, a cylinder composed of harmonic material will collapse to zero height and finite radius under finite compressive loading, cf. Sensenig [29, p. 475]). 2 For an incompressible material the existence of trivial solutions to the nonlinear problem is obvious. 103 Henry C. Simpson and Scott J. Spector 104 I n this p a p e r we i m p r o v e u p o n W i l k e s ' t e c h n i q u e s 3 a n d a p p l y t h e m to o u r p r o b l e m . W e b e g i n o u r a n a l y s i s w i t h a f o r m u l a t i o n o f the r e l e v a n t n o n - l i n e a r b o u n d a r y - v a l u e p r o b l e m a n d a p r o o f o f the e x i s t e n c e o f a trivial h o m o g e n e o u s s o l u t i o n 4 tO this p r o b l e m . T h e c r u c i a l a s s u m p t i o n s u s e d are g r o w t h c o n d i t i o n s 5 o n the s t o r e d e n e r g y or. I n p a r t i c u l a r we a s s u m e o(F)~o¢ asboth detF~0 + and JlFll~oo. I n S e c t i o n 4 we c o n s i d e r t h e s u p e r i m p o s e d l i n e a r ( i n f i n i t e s i m a l ) p r o b l e m a b o u t a g i v e n s o l u t i o n o f t h e n o n l i n e a r p r o b l e m . W e first p r o v e t h a t if t h e l i n e a r p r o b l e m has a n o n t r i v i a l s o l u t i o n (i.e., if t h e e q u a t i o n s are n o t l i n e a r i z a t i o n stable) t h e n t h e u n d e r l y i n g n o n - l i n e a r s o l u t i o n is n o t H a d a m a r d - s t a b l e . W e t h e n p r o v e t h a t all s o l u t i o n s 6 of the linear problem, sans the traction free boundary condition on the curved surface of the cylinder, can be obtained by separation of variables. 7 W e t h e r e f o r e r e d u c e the q u e s t i o n o f s t a b i l i t y to t h a t o f p r o v i n g that a c e r t a i n d e t e r m i n a n t , t h a t c o r r e s p o n d s to the t r a c t i o n f r e e c o n d i t i o n , is z e r o at s o m e critical v a l u e o f the l o a d i n g . F i n a l l y , in S e c t i o n 6, w e c o n s i d e r the special case o f a s t r o n g l y - e l l i p t i c Hadamard-Green m a t e r i a l , 8 i.e., a(F)=-2F.F+b((F.F)2-FFT.FFr)+ xt'(det F ) . H e r e w e p r o v i d e a n a n a l y t i c p r o o f o f the e x i s t e n c e o f a l o a d i n g v a l u e s u c h t h a t the a b o v e m e n t i o n e d d e t e r m i n a n t is z e r o a n d h e n c e c o n c l u d e t h a t every solid circular cylinder composed of Hadamard-Green material will eventually become unstable when compressed in an axial direction. 9 3 Stability analyses of this type have also been done for rectangular solids. For a neo-Hookean or a harmonic (cf. footnote 1) rectangular solid it is fairly straightforward to establish the existence of a loading value that allows the traction free boundary condition to be satisfied (cf. Levinson [19] and Nowinski [21] for the neo-Hookean material and Kerr [16] and Kerr and Tang [17] for the harmonic material). Sawyers and Rivlin [26] and Sawyers [27] consider a class of incompressible materials that includes both the neo-Hookean and Mooney-Rivlin materials. Both papers provide a rigorous analysis that the traction free boundary condition will be satisfied at certain Ioadings. For other stability analyses cf. Fosdick and Shield [10], Sawyers [28], Wesolowski [33], and Zhong-Heng [36]. 4 A solution in which the cylinder is deformed homogeneously and retains a cylindrical shape. For a discussion of assumptions of this type see Ball [2]. The idea for this kind of assumption was developed by Antman in a series of papers concerning non-linear rod and shell theories, cf., e.g., Antman [11. 6 This is a crucial point that (to our knowledge) has not previously been addressed in the literature. If an additional solution not obtainable by separation of variables exists, this solution might satisfy the remaining boundary condition at a smaller loading value than the separation of variables solution. 7 The technique used to establish this result can also be used to prove that all solutions of a number of other linear problems (without one boundary condition) considered in the literature can be obtained by separation of variables. Cf., e.g., Burgess and Levinson [7], Kerr [16], Kerr and Tang [17,18], Nowinsld [211, Sawyers and Rivlin [261, Sensenig [291, Wu and Widera [351. s Theoretical and experimental work on matching constitutive equations to data can be found in Ogden [23,24,25], Blatz and Ko [5], Ciferri and Flory [8] and Murnaghan [20]. Blatz and Ko [5], in particular, have shown that the data of Bridgman [6] can be matched to a particular Hadamard-Green material. 9 We note that we have not considered the problem of buckling instabilities, nor the question of whether buckling or barrelling occurs first. On barrelling instabilities in finite elasticity 105 1. Notation We let Lin -- space of all linear transformations from R 3 into R 3, with inner product and norm G ' H = trace(GHr), IIGII2 = G - G , where H r is the transpose of H. We write Lin + = { H ~ Lin : det H > 0}, O r t h + ~ { Q ~ Lin ÷ : Q r Q = I }, where det is the determinant and I the identity. Any tensor (linear transformation) H admits the unique decomposition H=E+ W into a symmetric tensor E and a skew tensor W: E-½(H+HT), W=-½(H-Hr). W e call E and W, respectively, the symmetricand skew parts of H. Given two vectors a, b e R 3 we write a @ b for the tensor product of a and b; in components (a ® b),j = aibj. We write V and div for the gradient and divergence operators in ii3: for a vector field u, x7u is the tensor field with components ( V u ) i j = 0ui/Oxj; for a tensor field S, div S is the vector field with components ~,jOSiJax j. Finally, given any function ~ ( F ), we denote the Frechet derivative of • by d r ~' ( r ) . 2. The constitutive relation We consider a homogeneous body ~ and identify ~ with the region it occupies in a fixed homogeneous reference configuration. A d e f o r m a t i o n / o f the body is a member of the space D e f - ( f ~ C 1 ( ~ , R 3 ) : d e t V f > 0}. We assume that ~ is hyperelastic with a C 2 response function o : L i n + - , R. o gives the stored energy at any point x ~ ~ when the body is deformed by f. Writing F for V f ( x ) , we assume that o(OF) o(F) = for all F ~ Lin + and Q ~ Orth +. Henry C Simpson and Scott J. Spector 106 T h e (Piola-Kirchhoff) stress S : Lin +--, Lin is given by S(F)-Ao(F), while the linear transformation A ( F ) : Lin ---, Lin defined by A(F)-- ~rs(r) is called the elasticity tensor. We assume that ~ is isotropic so that o(FQ)=o(r) for all F ~ Lin + and Q ~ Orth +. A well-known consequence of this a s s u m p t i o n is that there is a function 0 • R 3 ~ R such that o( F ) = O( ½F. F, ¼FF r. FF r, det F ) . We shall assume that 0 is C 2. We suppose that the reference configuration is natural: S(t)=0, and positive: E.A(1)E> O, (2.1) for all symmetric E 4= 0. We say that the elasticity tensor is strongly-elliptic at a deformation t if, for each xE,~, H-A ( v/(x))H > 0 whenever H = a ® b with a ~ 0, b 4: 0. We note for future reference PROPOSITION 2.1: 3 S ( F ) = E °,, Gi, iffil A ( F ) [ H ] = o l H + o 2 [ H F r F + F H r F + FFrH] + o 3 [(det F ) ( F - r . H ) F - r - ( d e t F ) F - r H r F -r ] 3 + ~ (Gi.H)GJo,ij i,j~l for any H ~ Lin and any F ~ Lin +. Here G 1 ~ F, G 2 = FFTF, G 3 = (det F ) F -r, o.i = o i(½F'F , ¼FFr'FF r, det F ) . (2.2) 107 On barrelling instabilities in finite elasticity PROOF: Equation (2.2)1 follows upon differentiating o and noting that d dF ( F F r . F F r) = 4FFrF, A ( d e t F ) = (det F ) F -r. Equation (2.2)2 follows upon differentiating (2.2)1 and noting that ddF ( F - r ) [ H ] = _ F _ r H r F _ r [] DEFINITION: We say that the body is composed of Hadamard-Green material if there is a C 2 function ~I,: R + ~ R such that (2.3) PROPOSITION 2.2: Suppose that the body is composed of Hadamard-Green Material. Then necessary and sufficient conditions for the reference configuration to be natural and positive are: ~"(1) = - a - 2 b , a+b>0, 3~'"(1) + 2b > a. (2.4) PROOF: We first differentiate (2.3), at F = I, to conclude, with the aid of (2,2)1, that S ( 1 ) = (a + 2b + ~t"(1))l. It is now clear that the reference configuration is natural if and only if the first equation in (2.4) is satisfied. If we next differentiate (2.3) twice and evaluate at F = I, we find, with the aid of (2.2)2 and (2.4), that H.A(I)H = 2(a + b ) E . E + ( I . E)2[ xI'"(1) - a ] , where E = ½(H + H r ) is the symmetric part of H. The orthogonal decomposition E=Eo +½(E'I)I, yields H.A(t)H= 2(a + b ) E o . E o + ½( E - I)213'I'"(1) + 2b - a]. The desired result now follows from (2.1) and the fact that E 0 and E . 1 may be specified arbitrarily. [] Let f ~ Def, x ~ ~ , and F = V f ( x ) . Then F has a polar decomposition F = VR, where R ~ Orth ÷ and V is symmetric and positive definite. By the spectral theorem there is an orthonormal basis {e 1, e 2, e 3 } such that 3 v= E X,e' e'. (2.5) i~l The scalars Xi are called local principal stretches (of the deformation f at the point x) while the vectors e i are principal axes of strain. Henry C Simpson and Scott J. Spector 108 The Cauehy Stress T is given by the relation T( F) = S( F ) F r / ( d e t F ). (2.6) We note that, as a consequence of (2.2)1, T is symmetric. The eigenvalues of T are called local pdneilml stresses. If we combine (2.2)1, (2.5), and (2.6) we find that 3 T(F) = E lid® e', i=l where the principal stresses t i are given by ti(Xl, X 2, X3) = 6.3 + (h2i# l + X4/O.z)/XlA2X3 (2.7) The Baker-Ericksen inequality is the requirement that the principal stresses have the same order as the principal stretches: (ti-tj)(Xi-Xj)>O, Xi*Xj. In view of (2.7) a slightly stronger requirement is that BEi=-6.a + 62(X] + h~ + X23- X ] ) > 0 (2.8) (even if h i = ~j). The tenslon-extension inequality is the requirement that each principal stress is a strictly increasing function of the corresponding principal stretch. Slightly stronger than this is the requirement TEi~ " -1" ~t, (XlX2X3Xi ) ' ~ / > 0. (2.9) REMARk: For a detailed discussion of constitutive inequalities see Truesdell and Noll [32], pp. 153-163. 3. The non-linear problem We consider the body in the reference configuration to be in the shape of a right circular cylinder of height L and radius R. In rectangular coordinates we take ~ = { ( x l , x 2 , x3):x~+x~<~R 2, x 3 ~ [ 0 , L ] ) with lateral surface The remaining two pieces of a ~ , the top and bottom of the cylinder, we denote by cgr and ~s, respectively. Let A ~ (0,1] and consider the boundary-value problem div S ( V / ) = 0 f3=0 f3=•L Sa3(Vf)=S23(Vf)=O S( V f ) n = 0 in ~ , on ~s, on fgr, on'sand"r, one, (3.1) On barrelling instabilities in finite elasticity 109 where n is the o u t w a r d unit normal to the lateral surface and the p a r a m e t e r h, which we refer to as the load modulus, is the ratio of the final to initial height of the cylinder. We note that if a d e f o r m a t i o n I satisfies (3.1) then so does g o ] if g is either a translation perpendicular to the axis of the cylinder or any rotation about this axis, i.e., g ( x ) = Ox + ( a , fl, 0) T, where a, fl ~ R, 0 011] and R r R = I. In order to eliminate this trivial non-uniqueness of the solutions we impose additional constraints u p o n our deformation f. We require f(f,.2 - A . 1 ) --- 0. (3.2) A deformation f that satisfies (3.1) and (3.2) will be called a solution of the non-linear p r o b l e m or simply a non-linear solution. THEOREM 3.1: A s s u m e that lo (i) o ( F ) ~ 00 as det F --, 0+; (ii) o ( F ) - - , oo as IIFII--' ~ Then f o r any )~ ~ (0,1] there is a constant IX = IX(X) such that h(x)=( Ixl/2 IXl/2 ~k/X is a solution to (3.1) and (3.2). PROOF: It is clear that h satisfies (3.1)1. 4 and (3.2) for any IX. By (3.1)5 and (2.2)1 we want to determine IX so that o.1 + IXo.2 + X03 = 0, (3.3) where o., = o., (IX + X2/2, IX2/2 + X4/4, IXX). F o r fixed A we find that ~o(Vh) = 01 + IX02 + Ao.3 . (3.4) Finally, by (i) and (ii) we find that o 1' oo as both IX--, 0 + and IX--, oo. T h e m e a n value theorem and (3.4) yield (3.3) for some IX, and this is the desired result. [] 1o Cf. footnote5. 110 Henry (7. Simpson and Scott J. Spector 4. The linear problem We now investigate the stability of the non-linear solution h by linearizing the boundary-value problem. For a given ~ E (0,1] and h = h a we linearize (3.1) and (3.2) and arrive at the problem of finding a C 2 function u such that div A [ Vu] = 0 in ~ , U3 = 0 o n c~B and cgr, A[Vu]13=A[Vu]23=O onCgBand~r, A[vu]n=0 one', ~l (4.1) = f u2 = O, (4.2) f (u, 2 - u2,) = o, where A = A (XThx) and once again n is the outward unit normal to the lateral surface. A C 2 function u that satisfies (4.1) and (4.2) will be called a solution of the linear problem corresponding to h x or simply a linear solution. Let Var = { u ~ C 1 ( ~ , R3): u satisfies (4.1)2 and (4.2)}. DEFINITION: [12, 31]. A deformation f ~ Def is Hadamard-stable with respect to perturbations in Var if f vu.,a(vf)w, >o for all u ~ Var. THEOREM 4.1 : I f u is a non-trivial solution of (4.1) and (4.2) then h x is not Hadamard-stable with respect to perturbations in Var. PROOF: Let u m 0 satisfy (4.1) and (4.2). If we multiply (4.1)1 by u, integrate by parts, use the divergence theorem, and apply the boundary conditions (4.1)2. 4 we conclude that f v , , - ~ [ vu] = 0. This completes the proof, since (4.1)2 and (4.2) imply u ~ Var. [] We now observe that by (2.2)2, (2.8), (2.9), and (3.3) the tensor B=A(vh~)[V] (4.3) On barrelling instabilities in finite elasticity 111 is given by Bii= 2BE3Uii + ( T E 1 - 2BE3)(U n + U22) + XU33, B33 -= TE3U33 + X(Ull + U22) , B12 = B21 = BE3 (U12 +/-/21 ), (4.4) B,3 = BEi(Ui3 + t1/2U3,), B3i = BEI( U3i + tl/2u/3), for i = 1, 2, where t --- bt/h 2 and + (~/2X4 + ~5/2)O23 + ~3/2~k3022 + ]g3/2~kO"33 "1- ~1/20.3. (4.5) In deriving the above equations we have made use of the fact that BE, = BE 2 and T E 1 = TE 2 when two of the principal stretches are equal. For future reference we define (4.6) N - X + t'/2BEI. For the remainder of the paper we will only be interested in considering specific types of instabilities, those that are axisymmetric. We shall therefore consider linear solutions of the form f+<r,z)x, / u(xi'Xz'X3)= ] d?(r'z)x2 t d(r,z) J' (4.7) ] where r 2 = x~ + x~ and z = x 3. A straightforward computation using (4.4), (4.5), (4.6), and (4.7) shows that (4.1)1 reduces to 0 = (div n ) , = x,[TEi(3gPrlr + qJ.) + BEiq),: + Nfrzlr], 0 = (div B E l ( e r l r + err) + N(2q): + r•rz) + TE3erz , J~)3 = for i = 1, 2. If we let O(r, z) = r 2 e p ( r , z) we find that the linear boundary-value problem reduces to the problem of finding a pair (0, d) of C 2 functions such that r E i ( O r l r ) , + aElOzJr + Ndr, = 0 on (0, R] × [0, L ] , (4.8) BEi(rd~) r + rTE3dz, + NOr2 = 0 d(r, O) = d(r, L ) = 0 Oz(r,O)=Oz(r,L)=O B E i R d ( R, z) + ti/2BEiOz( R, z) = 0 rElOr( , z) + X R d ( R , z) = 2BE30( R, z ) / R for r ~ (0, R], for z E [0, L ], (4.9) (4.10) 112 Henry C. Simpson and Scott J. Spector In addition the requirement u ~ C z yields O(r,z)/r--*O as r--*O (4.11) uniformly in z. Before proceeding further, we first recall a result concerning Fourier series of a function of two variables. PROPOSITION 4.2: Let 0 and dbe C z on [0, R] × [0, L] with ¢(r, o) = d ( r , L ) = O~(r, O) = Oz(r, L ) = O. Then 0 and d have Fourier series O(r, z) = ~ O,(r) cos(p,z), .~-0 (4.12) d(r, z ) = E d , ( r ) sin(p,z), n~l where p, = ncr/L. Moreover the series are uniformly and absolutely convergent and 0,, d, are C z on [0, R] and are given by O,( r ) = ~La( r, z) c o s ( p . z ) d z , (4.13) e.(r) = f0Le(r, z) sin(pnz)dz. PROOF: The only non-trivial part of the proof is the uniform and absolute convergence of the series. If we integrate by parts twice we find that O.(r) = - foLO~z(r, z) cos(p.z)/p2.dz and since 0 is C 2 that 10,(r)l ~ k / n z, where k is independent of z, r, and n. The desired result follows from the Weierstrass M-test. [] THEOREM 4.3: Suppose that the quadratic polynomial BE, TEle 2 + ( T E 1 T E 3 + B E ? - N Z ) e + BE1TE3=O (4.14) has distinct roots. Then any C 2 solution (0, d) of (4.8), (4.9), and (4.11) satisfies the conclusions of the previous proposition with O, and d, a finear combination of O,i(r)=rJl(p,r~) , <i(r) = -D, Jo(pnr~), (4.15) where e i are the roots of (4.14) and D~= ( BE, + e~TEa) / N ~ . (4.16) PROOF: Let (0, d) be a C 2 solution of (4.8) and (4.9). If we multiply (4.8)1 by cos(p,z) and (4.8)2 by sin(p,z), integrate over [0, L] with respect to z, and then integrate by On barrelling instabilities in finite elasticity 113 parts we conclude, with the aid of (4.9), (4.12), and (4.13), that TE 1( O~/r)' + p, N • = p2 B E lOn/r ' (4.17) BE, ( rg')' -- pnNO" = p2nTE3rgn . A straightforward computation shows that (Oni, ~ni) as given in (4.15) and (4.16) satisfies (4.17). It is also clear that the replacement of the Bessel functions J0 and ,/1 by the Bessel functions of the second kind II0 and II1 gives two additional solutions. Hence, we have constructed four linearly independent solutions to (4.17) and any other solution can be written as a linear combintion of these solutions. To complete the proof we must, of course, eliminate the functions Y0 and Y1. By (4.11) and (4.13) we conclude that as On(r)/r ~ 0 r--* O, from which the desired result follows. [] In Section 6, when we consider Hadamard-Green materials, we will find that the quadratic polynomial (4.14) has negative real roots. The following result will therefore be useful. PROPOSITION 4.4: Suppose that the roots of (4.14) are negatioe real and distinct. Then (4.15) can be rewritten Oni(r ) = rll(Pnrfi), ~ni(r) = D, Io(Pnrf~), (4.18) where f~ -- I~,l, D i = ( BE 1 - fi2TEa )/Nf~. (4.19) PROOF: Equation (4.18) may be obtained from (4.15) by noting that lo(r) =Jo(ir), I i ( r ) = i-lJa(ir). [] Although we do not use the following result in this paper, we include it to complete our discussion of (4.8)-(4.11). PROPOSITION 4.5: Suppose that (4.14) has a double root. Then any C 2 solution (0, ~) of (4.8), (4.9), and (4.11) satisfies the conclusions of Proposition 4.3 with 0n and ~n a linear combination of gnl=rJ,(pnrVre), gn2 = r2Jo ( pnrv~ ) , <1 = -DJo(Pnrv/e), ~2 = DrJ, ( p n r ~ ) - FJo ( pnrv~ ) , where e is the unique solution of (4.14) and D = ( B E , + eTE, )/NvCe, F = 2TE,/Non. Henry C. Simpson and Scott J. Spector 114 5. The boundary conditions We would now like to show that there is a X ~ (0, 1] such that our linear problem, in the form (4.8)-(4.11), has a non-trivial solution. Unfortunately, we have been unable to prove this for an arbitrary homogeneous isotropic material. We will however discuss this problem. Clearly, Theorem 4.3 (and the assumption that a certain quadratic has distinct roots) reduces this problem to that of finding a X such that one of the solutions (0, g) to (4.8), (4.9), and (4.11) satisfies the boundary conditions on the lateral surface (4.10). We note that (4.12) and (4.13) give necessary and sufficient conditions for (0, Y) to satisfy (4.10); namely, at r = R, where 1 TE d 1dr 2BE3 --~ o.Xr B"= [_O, tl/2BE BElr f___ ~ Since we desire a non-trivial solution, (0,, g,) must be non-zero for at least one n. We therefore conclude, with the aid of Theorem 4.3, that a necessary and sufficient condition for the existence of a solution to (4.8)-(4.11) (given that (4.14) has distinct roots) is that 0,2 [•"1]lr--R; Bn[e"2] r-R)=0 det{ B, [ 0"1 (5.1) for some n. 6. Instability for Hadamard-Green materials We now make the simplifying assumption that the body is composedof Hadamard-Green material; that is, we assume that there is a C 2 function q : R + ~ R such that o(F)=2 (F.F ) +b ((F.F)2- FFr.FF r) + ff'(det F ) . (6.1) In addition, we will assume that a>0, b>~0, (t~'(t))' >~0 q'(1) = -a- for t ~ (0, 1], (6.2) 2b. REMARK: It follows from Proposition A.2 that Eqn. (6.2)1 is necessary for strong ellipticity and Eqn. (6.2)3 is equivalent to assuming that the reference configuration is On barrelling instabilities in finite elasticity 115 natural. As we shall see from the next two results, Eqn. (6.2) 2 is used to prove strong ellipticity and also to prove that the energy becomes infinite as det F goes to zero. PROPOSITION 6.1: I f 0 is given by (6.1) then (6.2)1_ 3 imply that the elasticity tensor is strongly elliptic at every F ~ Lin + satisfying det F ~< 1. PROOF: It is clear from Proposition A.2 that all we need show is that xI," is non-negative on (0, 1]. If we integrate (6.2)2 over [s, 1] we discover e/'(1) >~ s e l ' ( s ) for s ~ (0, 1]. (6.3) We note that (6.2) 2 also implies that for s~P"(s)+Xt"(s)>~O s ~ (0, 1]. (6.4) If we combine (6.3) and (6.4), we find, with the aid of (6.2)1.3, that for Ol"(s)>~--et'(1)/s2>O s E (0, 1], (6.5) which is the desired result. [] W e now are ready to state our main results for H a d a m a r d - G r e e n materials. THEOREM 6.2: Let o be given by (6.1) and (6.2). Then for every ~ ~ (0, 1] there exists a unique Ix = t~( X ) such that X/~(X) < 1 (if X :~ 1), (6.6) and h(x) = ~1/2 x is a solution to (3.1) and (3.2). Moreover,/.t : (0, 1] ~ R is a C 1 function that satisfies a + b(p.(X) + X2) + Xxl"(X/-t (X)) = O, /~(1) = 1. (6.7) In addition if • " ( t ) >~ 0 for t E [1, oo), then there are no other l~ ~ (0, oo)'such that h is a solution of (3.1) and (3.2). THEOREM 6.3. I f o is given by (6.1) and (6.2) then for any positive integer n there is a h n ~ (0, 1) such that (4.8)-(4.11) has a solution that is a linear combination of(Oni, d~i ) as given in (4.18) and (4.19). A n d the following is immediate. COROLLARY: I f the body is composed of Hadamard-Green material then there is a A ~ (0, 1) such that the cylinder is not H a d a m a r d stable. 116 Henry C. Simpson and Scott J. Spector REMARK 1: Theorem 6.3 will turn out to be a direct consequence of Proposition 6.5. REMARK 2: Although we have not explicity assumed that the reference configuration is positive, this assumption is implicit in Eqn. (6.2). To see this, combine (6.4) (at s = 1), (6.2)1, and (6.2)3 to arrive at (2.4). PROOF OF THEOREM 6.2: It follows from (3.3) and (6.1) that h will be a solution to (3.1) and (3.2) if p satisfies (6.7). Define F: (0, 1] × (0, 1] --* R by F ( a , X) - X [a + b ( a / X - ' + X2) + X ~ ' ( a ) ] . Since xI, is C 2, we find that F is C I with derivative F~( a, X) = b + h2xI'"(a). (6.8) We shall prove that for each ~, ~ (0, 1] there is a unique a ~ (0, 1] satisfying F(a, ?~) = O. Defining a = a ( h ) , a : (0, 1] ~ (0, 1], we shall prove that a is C 1 and the desired result will follow with /~(?~) = a(~,)X -~ . (6.9) Uniqueness. By (6.1)1, (6.5), and (6.8) we find that F,~ is strictly positive on its domain of definition and uniqueness follows immediately. Existence. If we evaluate F at a = 1 we conclude with the aid of (6.2)3, that F(1,)Q=a)~(l_~)+b~[(l_)~)2 + ( 2 - 1 _ 1)], and hence, by (6.2)~, that F(1, 1) = 0 and F(1, )~) > 0 for ~, ~ (0, 1). (6.10) Now, (6.2)3 and (6.3) yield F ( a , X) ~< aX + ab + )~3b - ~ 2 ( a + 2 b ) , Ot and hence we find, with the aid of (6.2) 1, that for fixed 2, ~ (0, 1] Lira F ( a , )~) = - o0. (6.11) a--~0 Existence now follows directly from (6.10) and (6.11) upon application of the intermediate value theorem. Smoothness. If we now define a = a(),) we can apply the implicit function theorem to F to conclude that a is C 1. Finally, we note that (6.10) yields (6.6)1, (6.7)2, and that the nonexistence of additional solutions # ~ ()~-1, oo) will follow from (6.8) and the assumption that q'" is strictly positive. This concludes the proof. [] 117 On barrelling instabilities in finite elasticity To prove Theorem 6.3 we derive a number of intermediate results. We first observe that for a Hadamard-Green material bX2), TEa = a + b ( i t + A 2 ) + q , BE 3 = (a + B E a = a + bit, N = ta/2( bX 2 + q), TE 3 = a + 2 btt+ tq, X= ta/2(-a- where q = itA2q,"(itX), bit + bh 2 + q), t = it X-z, and it = g(X) satisfies Eqn. (6.7)v A simple computation, using (6.12) and (6.13), shows that - 1 quadratic polynomial (4.14) and hence that ea=-l, (6.12) (6.13) is a root of the e2 = - [ a + t q + 2 b i t ] / [ a + q + b ( i t + X 2 ) ] . (6.14) We now prove that the roots of (4.14) are distinct so that we can use Proposition 4.4 and Eqn. (5.1). PROPOSITION 6.4: Let o be given by (6.1) and (6.2). Then e2(X) < - 1 for a ~ (0, 1). PROOF: If we take the derivative of (6.7) with respect to X and solve for d i t / d X we arrive at dit dX itX~"(it~) + ~l/'(itX) + 2bX b + )~2g,,,,(it),) It then follows from (6.2)a, (6.4), and (6.6), that dit dR < 0 for X ~ (0, 1], (6.15) and, with the aid of (6.7)2, we conclude that 1 ~ it(),) for X ~ (0, 1]. (6.16) Next, by (6.13) and (6.14), we find that e 2 + 1 = (1 - t) (q + b~2) a + q + b(it + Xz) " (6.17) If we consider (6.2)1, (6.5), (6.13), and (6.17) we conclude that sign(e z + 1) = sign(1 - t). The desired result now follows from (6.13), (6.16), and the fact that X ~ (0, 1). [] We now return our attention to Eqn. (6.14). By Proposition 6.4 we can apply Proposition 4.4, and Eqns. (6.12) and (6.13) to conclude that 8., = rIa(p.r ), 8.2 = r l , ( p r f ) , d,, = Dalo(p.r), d,,: = D2Io(p.rf) , (6.18) Henry C Simpson and Scott J. Spector 118 where f = + -~--~2, If we (6.19) now substitute (6.18) into (5.1) and use the relations ~r(r],(r))=rlo(r), we /)2= - ~ / t / f . D1 = _7~--1, ff-~Io(r)=Ii(r), obtain, upon evaluation of the determinant (TE, + XD,)( BE, D2f - t'/2SE,)v(p.R ) + 2BE3BE,( D, - fa2) - ( BE, D, - t' ~2Be, )( TE, + X O J f ) v ( PnRf ) -- O, (6.20) where v(r) =--rlo(r)/I,(r ). (6.21) If we use (6.12), (6.13), (6.14), (6.19), and tt)2f2v(p v(pf)-(1 p = pnR, equation (6.20) reduces to 2 ( t - 1 ) f 2a+bh2 +(l+t) 2J a+b/x O, (6.22) f2--1+ ( t - 1 ) ( q + b k 2) (q + b~Z) +(a + bl~) Define Ka = a + b # , r 2 = a + b X 2, R ( P ' K:I' t¢2' K:3' t ) ( 1 K3 = q + b X 2, tt)zf2V(p 2 ( t - 1) •__2f2, ~" q--; 7 K:I (6.23) a ( p , r,, x 2, r 3, t)= v ( p f ) - R(p, x,, x 2, x 3, t), and note that f2 = 1 + ( t - 1 ) r 3 / ( r I + x3). Then, for each fixed n, we see that for a nontrivial solution (0,, f,) necessary and sufficient that there exist a ~ ~ (0, 1) such that H = 0 for fixed p = n~rR/L we seek k ~ (0, 1) such that H = 0 holds. We are functions of k only with/L --/~(k) as defined by Theorem 6.2. reduces to solving (6.24) to satisfy (5.1) it is is satisfied. That is, note that x 1, x 2, x 3 Thus, the problem a ( p , ~,(X), ~2(X), ~3(X), t ( X ) ) = 0 for A in terms of p. PROPOSITION 6.5: Let o be given by (6.1) and (6.2). Then for each p > 0 there exists a k ~ (0, 1) satisfying H(p, g , ( h ) , ~2(k), g3(k), t ( k ) ) = 0. On barrelling instabilities in finite elasticity 119 PROOF: Define /4(p, h) = H ( p , K,(h), r 2 ( h ) , g3(h), t ( h ) ) with bt = # ( h ) as defined by Theorem 6.2. Then the continuity of H will imply the existence of such a h if we show that /4(0, 1) = 0, 0 H ( p , 1) > 0, 0h Lim /4(p, h) = + oo. A--.0 + The first equation follows from (6.7)2, (6.13), and (6.24). The last two are consequences of Lemmas 6.6 and 6.7. LEMMA 6.6: Let gl be given by (6.1) and (6.2). Then oi:i Oh ( p ' 1) > O. (6.25) PROOF: We first note for future reference that v(o) =- OIo(P)/I,(P) satifies v (0) = 2, lira [v(P)/p] = 1, (6.26) p--* oO pv'(p) = p2 + 2 v ( p ) - v 2 ( p ) . By the chain rule 0/t 0h- 0H 0t ,~3 0H 0Ki Or-Oh + L 0~i Oh" i--1 Now at h = 1, # = 1 so that t = 1 (of. (6.7)2 and (6.13)). Noting that H(p, rl, r2, K3, 1) = 0 for all p, ~,, it is clear that OH/OK~ = 0 at h = 1 and thus OH Oh 0H 0t Ot Oh at h=l. If we take the derivative of (6.3), with respect to h, we find, with the aid of (6.15), that dt x= = dg x= - 2 < 0 , and thus it suffices to show that OH ~---~-< 0 at h=l, in order to prove (6.25). (6.27) 120 Henry C. Simpson and Scott J. Spector A straightforward computation, using (6.23) and (6.24) shows that when t = 1 (irrespective of the other variables) OR K3 Ot xI + x3 OH 81 = v(p) K2 2~ 1 ' 1 K3 2 K 1 "4- K 3 pv'(p) -- --~-. OR Thus, (6.26)3 allows us to conclude that 02 - 02(0) -~ x2(rlxlx3 + x3) 8-7 = 2(K, + "3) at t = 1, and hence we discover, with the aid of (6.13) and (6.23), that (6.27) is equivalent to pZ-oZ(p)+ q ' " ( 1 ) + 2b + a xi,,,(1) + b <0. Define, for all p > 0, h(p) = (p~ + c) '/2, c= • "(1) + 2b + a q'"(1) + b (6.28) T h e n (6.25) will follow if we can show that v(p) > h(p) for all p> 0 (6.29) (Note that v ( p ) is positive, since I 0, 11 > 0 for all p > 0.) At p = 0 we find, with the aid of (6.2)1,3, (6.21), (6.26)], and (6.4) (at s = 1) that v(0)-h(0) = 2 - c '/2 > O. We next suppose, for the sake of contradiction, that (6.29) does not hold and define 0 < Pl = inf{ p ~ (0, ~ ) : v(p) = h ( p ) } . It is then clear (since v(p)> h(p) for p ~ [0, Pl)) that v ' ( p , ) ~< h ' ( p l ) . (6.30) However, (putting hi = h ( p l ) and using (6.26)3) , ' ( p , ) - h'(pl)= 1p, [p{ + 2 v ( p l ) _ v 2 ( p , ) ] _~_~11 = 1 [hl(p~+2h,_h,~)_p~,] Plhl - 1 [h~-chl+c] p]hl where c and h are given by (6.28). We therefore conclude that 1 v'(p,)-h'(p])=p---~l [(h 1 -½c)2+ c2(c -1 - ¼ ) ] . On barrelling instabilities in finite elasticity 121 But, (6.2)1.3 and (6.4) (at s = 1) imply c - 1 - 1 / 4 > 0 and hence the last equation contradicts (6.30). Therefore no such Pl can exist and we have proven that (6.29) must hold. This completes the proof of Lemma 6.6. rq LEMMA 6.7: Let xIt be given by (6.1) and (6.2). Then L i m / ~ ( 0 , X) = + o0. ~ 0 (6.31) + PROOF: We demonstrate (6.31) by proving that as ~, ~ 0 + (for fixed p). f - * -t- do, f 2 / ( 1 + t) is bounded, 4w(p)- (6.32) 2 ( t - 1)r2/r 1 is bounded. l+t Equation (6.31) then follows from (6.23)1 and (6.26)2. We first note that (6.32)2 is immediate, since (6.24) implies f2 1 t-1 l+t+l+t l+t 1 l+rl/x3 where t, xl/~ 3 > 0 by (6.2) and (6.23). We next note that (6.32) 3 will follow if we can show that x2/K 1 is bounded. Equation (6.23)1 reduces this to proving that 1 is bounded as h ~ 0 ÷, # (6.33) but this follows immediately from (6.16). Thus we arrive at (6.32)3. Finally, we note that (6.32)1 will follow if as )~ ~ 0 ÷ 1 1 x2 1 t- b + ~ is bounded, /~ q + b)~~ (6.34) since f 2 - 1 is equal to (6.34)1 divided by (6.34)2. Of course (6.33) yields (6.34)1 and we need only show that as h ~ 0 ÷, q is bounded away from zero, to prove (6.34)2. By (6.4), (6.7)1, and (6.13) q = X ( # X ) ~ " ( / x X ) ~ a + b(/~ + ~2), and hence (6.2)1 and (6.33) yield (6.34)2. This completes the proof of Lemma 6.7. [] R~MARK: In our nonlinear problem we have assumed that the cylinder will frictionlessly adhere to the top and bottom surfaces for ~ ~ (0, 1]. Physically, this will only occur if the solution puts the body in compression in the z-direction. 11 We verify that 11 This observation is due to ProfessorE. Sternberg. 122 Henry C Simpson and Scott J. Spector this indeed occurs. First note that (2.2)1 and (6.1) imply $33 ( v h ) = aX + 2bpX + #xI,'(pX), and hence by (6.7)1 we find that XS33( V h ) ----(X2 -- p ) ( a + b~). It is clear from the last, (6.2)1, and (6.16) that S33(vh) < 0 for X ~ (0, 1], and hence that the body is in compression in the z-direction. Appendix. We wish to consider the consequences of strong ellipticity for a Hadamard-Green material. 12 We will do so using the machinery we have constructed in this paper. We first consider the implications of strong ellipticity for a general homogeneous isotropic material. Define .,~= ( H ~ Lin+: H=diag(l~l/2, t~l/2,~),t~,X ~ R + }. THEOREM A.I: Necessary and sufficient conditions for the elasticity tensor A to be strongly-elliptic at H ~ ~:are BE 1 > 0 , BE 3 > 0 , TE 1 > 0 , TE 3 > 0 , BE~ + (TE, TE3) t : > IX + tt/ZBE1[. (m.1) PROOF:If we take the inner product of (4.3) with U ~ Lin we find, with the aid of (4.4) and (4.5), that U.A[U]-- rE,(U1] + U~) + 2(TE, - 2BE3)U,,U:2 + 2tW2BE1 ( U13U31 ÷ U23U32) ÷ TE3U2 + 2 x ( v l , + v22)u33 + Be3(vl2 + ~1) ~ If we let U = a ® b where a = (a*, a) and b = (b*, b) we find that A is strongly-elliptic if and only if BE31a*121b*l 2 + ( TE 1 - BE 3)(a*" b*) 2 + BEl( la*12b 2 + [b*12a2) + 2ab(a* "b*) N + a2b2TE3 > 0 (A.2) for all a*, b* ~ R 2 and a, b ~ R, where N is given by (4.6). 12 For a direct proof of Proposition A.2 cf. Ogden [22]. See also John [15], Hayes [13] and Currie and Hayes [9] for related results. 123 On barrelling instabilities in finite elasticity Case L Suppose a = 0. Then if we let cos 0 = (a* "b*)/(la*llb*l) we find that U . A [ U ] = [(sin20)BE3 + TE 1 cos20] la*121b*l 2 + BElla*12b 2. (A.3) The choice b* = (0, 0) yields BE, > 0 as necessary for strong ellipticity, while the choice b = 0 yields B E 3, TE 1 > 0 as necessary conditions since 0 is arbitrary. These conditions are clearly sufficient for the positivity of (A.3). Case IIa. Suppose a = b = 1 and IN[~< B E 1. Then U . A [ U ] = [la*121b*l 2 - ( a * "b*) 2] B E 3 + TEl(a* "b*) 2 + TE 3 + ( B E l - [NI)(la*l 2 + Ib*l 2) +lNl(la*l 2 + Ib*l 2 + 2 s g n ( N ) ( a * " b*)). (m.4) The choice a * = b* = (0, 0) yields TE 3 > 0 as necessary for strong ellipticity. The conditions BE 3, TEa, T E 3 > 0 are clearly sufficient for the positivity of (A.4). Case lib. Suppose a = b = 1 and INI > BE,. Then u. At u] = [la*l lb*l BE3 + B E , [la*l 2 + Ib*l 2 + 2 s g n ( N ) ( a * "b*)] + Q ( a * .b*), (A.5) where Q ( s ) = TE, s 2 + 2~s + TE3, (A.6) 71 = N - ( s g n N ) B E , . If BE l, BE 3 > 0 then U ' A [ U ] >~Man Q ( s ) . We first show that Min U . A [ U ] = Man Q ( s ) . a*,b* (A.7) Noting that the minimum of Q ( s ) occurs at s --- - T / T E 1 , we choose a*, b* that satisfy a* + (sgn N ) b * = 0 , a* • b* = - ~I/TE 1. (This is possible since sgn(a*, b * ) = - s g n ( N ) = - s g n ( ~ ) = s g n ( - ~ i / T E 1 ) if TE 1 > 0.) A simple computation now shows that (A.7) holds. Thus a necessary condition for strong ellipticity is that Q ( s ) > 0, for all s ~ R. In particular at s = - ~ I / T E l this yields TE, TE 3 > ~q2 or by (A.6)2 and the fact that ISl > BEI (TEITE3) '/z + BE, > INI- It is clear that the last, TEl, BEa, BE3 > 0 are sufficient for the positivity of (A.5). 124 Henry C. Simpson and Scott J. Spector It is now clear that the conditions (A.1) are necessary conditions for strong ellipticity. The sufficiency of these conditions also follows from our proof, since one can always reduce (A.2) to one of the forms (A.3), (A.4), or (A.5). [] And for a Hadamard-Green material we obtain the following result of Hayes [13] and Ogden [22]. PROPOSITION A.2: Suppose that the body is composed of Hadamard-Green material and that the elasticity tensor is strongly-elliptic at every H ~ )[° Then a>0, b>~O or a > ~ O , b>0 (A.8) and xI'" >/0. (A.9) PROOF: By (6.12)2 and (A.1) we find that a+bt~>0. Since /~ > 0 is arbitrary, we arrive at (A.8). Next, by (6.12)4, (6.13), and (A.1) we discover that g(/~, A) = a + 2b# + #2gt"(ttX ) > 0. Suppose, for the sake of contradiction, that there is an a > 0 such that q " ( a ) is strictly negative. Then g(t~,~)=a+2bl~+tx2'~'(a). If we view the last equation as a quadratic polynomial in tt, we conclude that g is not strictly positive and we have a contradiction. Hence we arrive at (A.9). [] Acknowledgement The authors would like to thank Professor J.L. Ericksen for suggesting we pursue this problem and Professors E. Sternberg and M.E. Gurtin for their helpful comments. 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