TR/06/83 September 1983 LOCAL AND GLOBAL BIFURCATION PHENOMENA IN PLANE STRAIN FINITE ELASTICITY R. W. OGDEN

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1 TR/06/8 Sepember 98 LOCAL AND GLOBAL BIFURCATION PHENOMENA IN PLANE STRAIN FINITE ELASTICITY by R W OGDEN

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3 Local and global bifurcaion phenomena in plane srain finie elasiciy By RW OGDEN Deparmen of Mahemaics and Saisics, Brunel Universiy Absrac Bifurcaion, global non-uniqueness and sabiliy of soluions o he plane-srain problem of an incompressible isoropic elasic maerial subjec o in-plane dead-load racions are considered In paricular, for loading in equibiaxial ension, bifurcaion from a configuraion in which he in-plane principal sreches are equal is shown o occur a a cerain criical value of he ension (which depends on he form of srain-energy funcion) Resuls concerning he global inveribiliy of he elasic sressdeformaion relaions are obained and hen used o derive an equaion governing he deformaion pahs branching from his criical value The sabiliy of each branch is also examined The analysis is carried hrough for a general form of srain-energy funcion and he resuls are hen illusraed for a paricular class of srain-energy funcions Inroducion Non-uniqueness of soluion o he problem of a cube of incompressible isoropic elasic maerial subjec o hree equal pairs of equal and opposie dead-load racions acing normally o is faces has been examined by Rivlin ( 8, 9 ) in respec of he neo-hookean form of srainenergy funcion Rivlin also deermined he sabiliy of he soluions; Hill ( ) poined ou some deficiencies in Rivlin's ( 8 ) sabiliy analysis which were subsequenly correced ( 9 ) Somewha differen aspecs of he

4 same problem have been considered recenly by Ball and Schaeffer ( ); in paricular, hey have examined he branching of soluions from a configur- aion of pure dilaaion as he applied load increases from zero and he sabiliy of he branches Their work made exensive use of he Mooney form of srain-energy funcion and hey used singulariy heory o examine he local aspecs of bifurcaion The closely-relaed problem of non uniqueness and sabiliy of a cube subjec o wo equal pairs of racions differing from he hird pair has been discussed by Sawyers ( 0 ) wih aenion resriced o he neo-hookean srain-energy funcion In he presen paper we consider a slighly differen problem, namely he plane-srain bifurcaion, non-uniqueness and sabiliy of a body subjec o in-plane equibiaxial dead-load racions The maerial is aken o be incompressible and isoropic, bu no oher resricion is placed on he form of elasic srain-energy funcion We begin by giving a brief general accoun of he governing equaions and heir applicaion o he sudy of bifurcaion and sabiliy in order o place he problem under consideraion wihin a broader framework The basic equaions Consider a maerial body which occupies he region B0 in some reference configuraion, and suppose ha he deformaion X: B B defines he region B occupied by he body in he curren Configur- 0 aion Le X denoe a ypical poin of B 0 and also is posiion vecor relaive o an arbirary choice of origin Similarly, le X denoe he posiion vecor of X in B, so ha X = X (X ) X B () 0 The deformaion gradien ensor A is defined by A = Grad X, ()

5 where Grad denoes he gradien operaor wih respec o X Use will be made of he polar decomposiion A = R U, () where R is proper orhogonal and U is he posiive definie, symmeric righ srech ensor In his paper aenion is resriced o incom- pressible maerials so ha we have he consrain de A = de U = (4) For an (incompressible) elasic maerial he nominal sress ensor S is given by S W = PA, A (5) where p is he arbirary hydrosaic pressure (Lagrange muliplier) arising from he consrain (4) and W( A ) is he elasic srain energy per uni volume The srain-energy is objecive, ie unaffeced by a superposed rigid-body roaion afer deformaion, and i herefore follows from () ha W depends on A only hrough U Thus W( A) W ( U) (6) This leads o an alernaive and very useful form of sress-deformaion relaion, namely where () U () W T = P U T is he symmeric Bio sress ensor, (7) () () The noaion T is used because T is conjugae o U T corres- ponding o m= in he class ( U m I )/m of srain ensors For a full accoun of conjugae sress and srain ensors see ( 7)

6 4 For an isoropic maerial W depends on U only hrough is principal values,,, he principal sreches We wrie his dependence as W,, ) and noe he symmeries ( wih W(,, ) = W(,, ) W(,, ), (8) = (9) = following from (4) () () If i (i =,,) denoe he principal componens of T hen, for an soropic maerial, (7) gives () i W - = Pi i=,, (0) i Also, for an isoropic maerial, we have he connecion W A W = U T R (he proof of his is sraighforward), and hence, by (), (5) and (7), S = T () R T () In erms of he principal componens sress ensor T = A S we have i (i =,,) of he Cauchy i () W = ii = i P i =,, () i Finally in his secion we record ha in he absence of body forces he equilibrium equaion is simply Div S = O in B o, ()

7 5 where Div is he divergence operaor wih respec o X Typical boundary condiions involve he specificaion of X and he racion S T N on com- plemenary pars of he boundary B0 of B 0, N, being he uni ouward normal o B 0 The incremenal equaions Le Xdenoe an incremen in X, ie a small deformaion from he curren configuraion Then, on aking he incremen of (5), we obain he incremenal consiuive law S A = A where A = Grad X, S and - PA + PA A A, p denoe incremens in S and p and A = W A (4) is he (fourh-order) ensor of firs-order elasic moduli associaed wih he conjugae pair ( S, A ) The righ-hand side of (4) is correc o he firs order in incremenal quaniies I is convenien for our purposes o choose he reference configur- aion o coincide wih he curren configuraion, in which case (4) becomes where T S& = A A& PI & + PA& , + PA, 0 (5) is he (second-order) ideniy ensor and he subscrip zero signifies evaluaion in he curren configuraion The ensor A 0 is called he ensor of firs-order insananeous moduli associaed wih ( S, A ), while A 0= A A = grad, V (6) where V (X ) is idenified wih X (X ) hrough () and grad denoes he

8 6 gradien operaor wih respec o X Correc o he firs order in incremenal quaniies he incremenal form of he consrain (4) is r(a ) div v 0 = 0, (7) div represening he divergence wih respec o X For an isoropic maerial we shall require he componens of A 0 referred o he underlying principal axes These are given by A 0iijj W = i j, i j (ie he principal axes of T ) A = 0ijij ( - i - j ) i /( i j ) (A 0iiii W A 0iijj + i ) i i j, i = j, i j i j, (8) in (7) A 0ijji of () is = A 0jiij = A 0ijij W i i i j Derivaions of hese expressions are conained in ( 5 ) and fuller deails Referred o he curren configuraion he incremenal counerpar Div S & = 0 0 in B, (9) and he boundary condiions may specify, for example, Vand he incre- menal racion n ST 0 ouward normal o B on complemenary pars of B, where n is he uni

9 4 Incremenal uniqueness and sabiliy Suppose ha X is a soluion of he underlying problem and ha he incremenal problem has homogeneous boundary condiions I follows ha X = 0 is a soluion of he incremenal problem If X = V 0 is also a soluion hen, by use of (6), (9) and he divergence heorem, we obain r (S A B T ) dv = v (S = 0, B n ) ds where dv, ds denoe volume and surface elemens in B and B respecively Hence, incremenal uniqueness (of boh he problem wih homogeneous daa and ha wih inhomogeneous daa) is guaraneed if B r(s A ) dv 00 for all (wice coninuously differeniable) v saisfying v = on he 0 appropriae par of B and (7) in B We refer o such v as admissible The configuraion x is said o be unsable if he above funcional 0 is negaive for some admissible v and sable if for all admissible v B r(s A ) dv 0 (0) 0 0 If sric equaliy holds in (0) for some admissible v wih grad v / 0 hen x is said o be neurally sable However, his descripion is valid only o he second order in incremenal quaniies and a neurally sable configuraion may, in fac, be unsable when assessed o higher order in A 0 (he lef-hand side of (0), which represens wice he increase in oal energy of he body under dead-loading condiions due o he incremen x, mus hen include such higher-order erms) For full discussion of his we refer o ( ), ( 4 ) and ( ) 7

10 8 When (0) is srenghened o for all admissible r(s) 0 A) dv > 0 () 0 B V wih A 0, he curren configuraion is sable and he incremenal problem has a unique soluion Thus () excludes bifurcaion and is herefore referred o as he exclusion condiion ( 4) Suppose henceforh ha he maerial properies and he deform- aion x are homogeneous Suppose furher ha he body is subjec o all-round dead load Then () is equivalen o r ( S A ) > for all A saisfying (7), or, from (5), 0 r {(A A ) A + P A } > 0, () () independenly of he geomery of he body Along a deformaion pah on which () holds he exclusion condiion firs fails where r{(a A ) A + P A } 0, (4) for all A saisfying (7) wih equaliy holding for some A Exremizing (4) shows ha in a neurally sable configuraion S A A + p A p I = 0, where p is he Lagrange muliplier inroduced in respec of he con- srain (7), ie he nominal sress is saionary wih respec o incremenal deformaions which minimize he quadraic form (4) For an isoropic maerial use of (7), (8), () and a lile algebra shows ha he exclusion condiion () may be arranged in he form (5)

11 W V W V + W W + + W V V i j i j j i + ( v + v ) + V V > 0, (6) i, j i {,,} i j j i i ji j i j i j i j 9 where V are he componens of grad v, on he principal axes of T I is ij an easy maer o obain necessary and sufficien condiions for (6) o hold for all grad v 0; for he plane srain specializaion of (6) such condiions are given in he following secion For an isoropic maerial, neurally sable configuraions can be found eiher by use of (5) or by direc exremizaion of (6) Henceforh we resric aenion o he plane srain problem for isoropic maerials 5 Plane srain bifurcaion crieria Suppose ha he srech is fixed and normal o he plane in quesion and ha he maerial is subjec o a pure homogeneous srain wih in-plane principal sreches and Le he principal axes of he srain define Caresian axes wih coordinaes x,x,x We resric he incremenal deformaion so ha v = 0 and v,v are independen of x Equaion (7) reduces o V + v = 0 (7) V and he exclusion condiion (6) o w v + ( V v ) Where v ij = vi / x j v v > 0 (8)

12 0 Necessary and sufficien condiions for (8) o hold are, wih he help of (), easily seen o be W > 0, (9) () () + 0 (0) > () () ( ) ) /( ) > 0, () joinly, and, by a limiing process, i can also be seen ha () is equivalen o (9) when X = X Corresponding inequaliies for compressible maerials are well known ( 4) bu for incompressible maerials (9) - () are apparenly new alhough (0) and () are idenical in form o heir compressible counerpars Neurally sable configuraions are defined by values of and for which one or more of (9) -() jus fails Such values define bifurcaion poins in he (, )-plane The bifurcaion crierion W = 0 () is associaed wih a pure shear mode of incremenal deformaion (v =- v ) coaxial wih he underlying pure srain, while and () () + = 0 () () ( /( ) = 0 (4) Permi in plane shearing modes such ha v / v = / and /

13 ŵ a ) (, {v + 4 (V + V ) } + (v v ) > 0 (40) Necessary and sufficien condiions for (40) o hold for all v 0 for some pair i, j are simply ij v ij wih Ŵ < < (, ) (4) 0 On a pah of equibiaxial ensile loading wih = bifurcaion becomes possible when reaches he value Ŵ = (, ) (4) and for larger values of he deformaion = is clearly unsable From he resuls of Secion 5 we deduce ha a symmery-breaking mode of deformaion appears a he value of defined by (4) This de- formaion comprises a pure shear v = -v coaxial wih he chosen Caresian axes and a shearing mode wih v = v The combinaion of hese modes represens a pure shear deformaion whose axes have orienaion dependen on v / v (which is arbirary) This arbirariness is a consequence of he fac ha he orienaion of he in-plane principal axes of () () () T is arbirary since in he = considered configuraion This laer poin also arises in connecion wih he global inversion of (5), as we see in he following secion 7 Global resuls According o (), for an isoropic maerial we may decompose he nominal sress as () S = T R T However, for a given S his polar decomposiion, unlike (), is no unique since ( ) T (4) need no be posiive (or negaive) definie When S S T has disinc principal values i is known ( 6, 7 ) ha, for given

14 respecively Equaions () - (4) may be derived direcly from (5), wih () requiring eliminaion of P & The (nominal) sress-deformaion relaion is clearly locally in- verible excep in configuraions such as hose jus described and, more generally, in configuraions where (5) holds for A 0 Given S 0 0 in a configuraion where he exclusion condiion holds, A is uniquely defined by (5), pbeing fixed by he consrain r( A ) = Applicaion o he case of equibiaxial ension A his poin i is convenien o inroduce he auxiliary variable defined by =, = (5) ogeher wih he noaion ŵ(, ) = w(,, ) (6) From he symmery (8) we deduce ha ŵ(, ) = ŵ(, ), (7) a resul which will be required in Secion 7 From () and (6) we obain and ŵ = (8) ŵ w =, (9) he laer expression occurring in () Of paricular ineres is he deformaion for which =, ie = wih = correspondingly, when he exclusion condiion (8) reduces o

15 S, here are jus four disinc polar decomposiions of he form (4) and, moreover, only one of hese saisfies he sabiliy requiremens () i + () j > 0 i j (44) By consras, if wo principal values of S S T are equal and non- () zero, he principal values i (i=,,) of T are again uniquely deermined under he requiremens (44), bu he orienaion of he () principal axes of T in he plane of he equal values is arbirary ( 6, 7 ) In paricular, his is he case for he equibiaxial problem discussed in Secion 6 () Once T, and hence R, has been obained from (4) wih (44) o wihin he arbirariness jus menioned, he righ srech ensor U, and hence A = R U he equibiaxial ension problem () is o be found by invering (7) subjec o (4) For = () () () and he in-plane principal axes of T and hence of U, have arbirary orienaion in he (,)- plane Since is fixed i herefore remains o find and, () subjec o (9), ogeher wih p, if required, in erms of from (0) Eliminaion of p from () gives () w () w () W = = (45) The firs equaion in (45) serves o deermine and for fixed () () and prescribed and while he second equaion hen deermines () N o w s e () = () and in roduce he noaion () () = (46) Clearly, he firs equaion in (45) is saisfied for all () and when = ( = in he noaion of (5)) On use of (5), (6) and

16 4 (46) we see ha a soluion wih is governed by () ŵ = /( ) (47) and ha in he limi his becomes () ŵ () = = (, ) c, (48) wherein he criical value ( c ) is defined Recalling (4) we noe ha his is precisely he criical value a which bifurcaion can occur on he fundamenal deformaion pah = as () increases from zero Thus, equaion (47) governs he global pah of deformaion ( ) emanaing from he poin ( c,) in he ( (), )-plane As we indicaed in Secion 6 he fundamenal pah is unsable for values () ( ) of greaer han c In order o assess he sabiliy of he pah described by (47) we noe ha he exclusion condiion (8) simplifies o Ŵ () v + ( v v ) / ( + ) > 0 (49) However, he lef-hand side of (49) vanishes for shearing modes governed by v = if v = 0 (recall ha (4) holds) I v follows ha he deformaion branch governed by (47) is (neurally) sable provided 0 ( ) Ŵ or, equivalenly, Ŵ Ŵ / ( ) 0 (50) In order o illusrae he consequences of he inequaliy (50)

17 5 we consider he class of srain-energy funcions defined by m m m W (,, ) = μ( + + ) / m, where μ > 0 is he shear modulus From (6) we obain Ŵ (, m m m m m ) = μ ( + + ) / m (5) and hence Ŵ m m m m = μ( ) ) / m (5) so ha he sric form of he righ - hand inequaliy in (50) can be rearranged as f m ( ) { (m ) m+ (m + ) m + (m + ) m+ (m ) m }/ ( ) 0 (5) This is clearly symmeric wih respec o inerchange of and - and reduces o zero in he limi for any m For m =, f m ( ) = 0 while f m ( ) 0 0 for for m > m < wih equaliy if and only if = I follows ha for srain-energy funcions wih m he deformaion branch is (neurally) sable, bu for hose wih m < i is unsable FIG ABOUT HERE The branching of he deformaion pah is illusraed in Figure where is ploed agains () for a ypical value of m in each of ranges m < and m > on he basis of (47) wih (5) In each

18 6 case he curves above and below he line = are obained one from he oher by reversal of he roles of and -, ie of and Thus, for a body of square cross secion, for example, under biaxial deformaion he curves represen geomerically equivalen deformaions I is insrucive o examine he behaviour of he branches in he neighbourhood of he bifurcaion poin and for his purpose we differeniae (47) wih respec o and evaluae he resul for = One differeniaion yields ) ( = 0 for = (54) for arbirary Ŵ and To obain his resul a limiing process mus be used ogeher wih Ŵ (, ) / = 0, which follows from (8), and he connecion Ŵ (, ) w ) = (, ), which is obained by differeniaing (7) hree imes wih respec o Equaion (54) shows ha, as indicaed in Figure, he iniial gradien o each branch is verical irrespecive of he form of Ŵ The iniial curvaure depends on he second derivaive; a second differeniaion of (47) yields (55) () 4 Ŵ Ŵ = (, ) (, ) for =, 4 6 (55) again having been used For he srain-energy funcion (5), equaion (56) simplifies o (56)

19 () = μ (m 7 ) m for = and his reflecs he inermediae role played by m = in Figure The Taylor expansion () () () ( ) = c + + of (47) near = emphasizes ha wo deformaion branches emanae () from ( c,) for arbirary Ŵ since, o he second order in ( -), i describes a parabola in he ( c (), )-plane Thus far, by fixing, we have resriced aenion o he plane srain problem The corresponding problem of all-round dead load wih () () (),,, prescribed is also governed by equaions (45), and we () () conclude wih some remarks abou his If, in paricular, = hen, in he noaion of (5) and (6), equaions (45) become () ( ) ( + () ) = () Ŵ = Ŵ (57) (58) The resuls described in ( 0 ) for he neo-hookean srain-energy funcion (m = in (5)) are recovered by appropriae specializaion of (57) and (58), and generalized by examining he soluions of (57) () () and (58) for prescribed and along he lines described here for he plane-srain problem We do no consider he general case here bu illusrae he problem by aking energy funcion (5) Eliminaion of ha eiher = or is given by m () () = m m = ( ) / ( ) 0 and using he srain- from (57) and (58) shows, (59)

20 8 which has value -m in he limi For (59) o have a posiive soluion for we mus have m< By aking m =, for example, we obain 4 () = ( + ), = μ, which define an unsable pah of deformaion from he bifurcaion poin () = μ on he pah = Finally, on seing () = () in (58) we see ha a possible soluion of (57) and (58) is = = () for all A soluion wih = and is governed by () Ŵ = (, ) / ( ) (60) and bifurcaes from = = a he criical value Ŵ () = (,) as () increases Secondary bifurcaion ino a deformaion pah () wih # may occur subsequenly when reaches he criical value obained from (57) in he limi, namely () Ŵ = (, ), () where and are also conneced by (60) In respec of he Mooney srain-energy funcion such bifurcaions were analyzed in (), bu for () (5) he above criical values of are boh equal o μ (independenly of m) REFERENCES ( ) BALL, JM and SCHAEFFER, DG Bifurcaion and sabiliy of homo- geneous equilibrium configuraions of an elasic body under dead-load racions To appear

21 9 ( ) HILL, R On uniqueness and sabiliy in he heory of finie elasic srain J Mech Phys Solids 5 (957), 9-4 ( ) HILL, R Eigenmodal deformaions in elasic/plasic coninua J Mech Phys Solids 5 (967), 7-86 ( 4) HILL, R Aspecs of invariance in solid mechanics Advances in Applied Mechanics 8 (978), -75 ( 5) OGDEN, RW On sress raes in solid mechanics wih applicaion o elasiciy heory Proc Cambridge Philos Soc 75 (974), 0-9 ( 6) OGDEN, RW Inequaliies associaed wih he inversion of elasic sress-deformaion relaions and heir implicaions Mah Proc Cambridge Philos Soc (8 ) (977), -4 ( 7) OGDEN, RW Nonlinear Elasic Deformaions (Ellis Horwood, 98), o appear ( 8) RIVLIN, RS Large elasic deformaions of isoropic maerials - II Some uniqueness heorems for pure homogeneous deformaion Phil Trans Roy Soc London A 40 (948), ( 9) RIVLIN, RS Sabiliy of pure homogeneous deformaions of an elasic cube under dead loading Quar Appl Mah (974), 65-7 ( 0 ) SAWYERS, KN Sabiliy of an elasic cube under dead loading: wo equal forces In J Non-Linear Mech (976), -

22 0 Capion o Figure Fig Bifurcaion diagram in he ( (), ) -plane showing branches emanaing from bifurcaion poin on he pah = for differen values of m () = ( c )

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