1. The scalar-valued function of a second order tensor φ ( T) , m an integer, is isotropic. T, m an integer is an isotropic function.
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1 ecton ppendx to Chapter Isotropc Functons he scalar- vector- and tensor-valued functons φ a and of the scalar varable φ vector varable v and second-order tensor varable B are sotropc functons f a φ( v φ( v φ( B φ( B ( φ a( φ a( v a( v a( B a( B ( φ ( φ ( v ( v ( B ( B Isotropc Functons (4.. for all orthoonal tensors. Isotropc functons are also called sotropc nvarants. Here follow some examples. Examples (of Isotropc Functons. he scalar-valued functon of a second order tensor φ ( det s an sotropc functon snce φ ( det( det. he scalar-valued functon of two second order tensors ( B tr( B sotropc functon of ts two tensor varables snce φ φ s an ( B tr( B tr( B tr( B More enerally the functon tr ( m B m m an nteer s sotropc. he vector-valued functon a of a vector v and a second order tensor a ( v v s an sotropc functon snce m m ( v v v a( v a Indeed the functon m a( v v m an nteer s an sotropc functon. 4. he tensor-valued functon of a second order tensor: functon snce ( s an sotropc ( ( ( ( ( Indeed the functon m ( m an nteer s an sotropc functon. old Mechancs Part III 46
2 ecton 4. Restrctons on the form that sotropc functons can tae s next examned. Isotropc calar-valued Functons Consder frst an sotropc scalar-valued functon of a vector u φ ( u so that φ ( u φ( u. nce only the mantude of u s nvarant under an orthoonal tensor transformaton t follows that φ depends on u only throuh u u u so φ( u u Here u u s called the nterty bass of φ. φ. mlarly an sotropc scalar-valued functon of two aruments s defned throuh ( u v φ( u v φ (4.. for every orthoonal and ts nterty bass conssts of the three scalar nvarants u u u v v v (4.. snce only the lenths of the two vectors and the anle between them are preserved under a rotaton. Consder next a scalar-valued sotropc functon φ of a symmetrc second-order tensor. nce s symnmeytrc t has the spectral decomposton representaton s s where { } are the eenvalues and { ˆ nˆ n } s sotropc ˆ n are the eenvectors of. nce ( φ( φ( ( nˆ nˆ φ( nˆ nˆ φ (4..4 hus φ s ndependent of the orentaton of the prncpal drectons of and so must depend only on the three prncpal values ( φ( f (4..5 Note also that f must be a symmetrc functon of the eenvalues. For example tae to be a postve rotaton about ˆn. hen nˆ ˆ n nˆ ˆ n and nˆ ˆ n so ( φ( nˆ nˆ nˆ nˆ nˆ ˆ f ( φ( φ n (4..6 and smlarly the subscrpts on any par of eenvalues n 4..5 can be nterchaned. nce the set{ tr tr tr } the set of three prncpal scalar nvarants { I II III } and the set of eenvalues { } unquely determne one another any of these sets can be rearded as the nterty bass of φ (. old Mechancs Part III 47
3 ecton 4. ome mportant sotropc scalar-valued functons and ther nterty bases are lsted n able 4.. below. he nterty bass conssts of that entry toether wth approprate entres from hher up n the able for example the nterty bass for a tensor and two vectors u and v s u u v v u v uu u u vv tr tr v v tr uv u v calarvalued functons B C are symmetrc tensors Isotropc Functon φ u u u φ uv u v φ ( u ( φ ( u v ( φ ( φ( ( u φ φ( u φ ( B φ( B φ ( u v φ( u v tr tr uu u u trb Interty Bass tr tr B trb uv u v φ ( B C tr BC Four or more tensors redundant able 4..: Isotropc calar Functons and Interty Bases tr B Isotropc Vector-valued Functons Next consder a vector-valued sotropc functon a of a vector v so ( v a( v a. o fnd the dependence of a on v consder the scalar-valued functon φ ven by (note that φ here s lnear n ts frst arument u: It follows that ( u v u a( v φ (4..7 ( u v u a( v u a( v u a( v φ( u v φ (4..8 and so φ s an sotropc functon of ts two vector aruments and must depend only on the three nvarants 4.. and so taes the eneral form Fnally a must tae the form ( u u u v v v u α( v vv φ (4..9 a ( v αv (4.. where the coeffcent α s a functon of the scalar nvarant of v.e. v v. old Mechancs Part III 48
4 ecton 4. Note that the only sotropc vector functon a of a tensor B s the null vector a o. nother mportant sotropc vector-valued functons s that of a vector and symmetrc tensor. hs and ther nterty bases are lsted n able 4.. below. Isotropc Functon Interty Bass Vector-valued a ( v a ( v a( v v functons a ( a ( a( o s a symmetrc a ( v a ( v a( v v v tensor able 4..: Isotropc Vector Functons and Interty Bases Isotropc ensor-valued Functons Consder next a second-order tensor-valued functon of a tensor B. o fnd how depends on B ths tme consder the scalar-valued functon φ ven by (aan note that by defnton φ s lnear n ts frst arument ( B tr[ ( B ] φ (4.. It follows that φ ( B tr[ ( B ] ( B ] tr[ ( B ] tr φ [ ( B ] ( B (4.. hus φ s an sotropc functon of ts two tensor aruments and so f and B are symmetrc s a functon of the ten nvarants lsted n able 4... nce φ s lnear n t can only depend on sx of these ten nvarants namely tr trb trb trb trb trb and so taes the form and so taes the form [ ] [ ( B ] tr ( α I α B α B φ tr (4.. ( B α I α B α B Form for a symmetrc sotropc tensor functon of a symmetrc tensor (4..4 where α α α are scalar functons of the nvarants of B. Equaton 4.. can be rewrtten n varous alternatve forms usn the Cayley-Hamlton theorem old Mechancs Part III 49
5 ecton 4. ome mportant symmetrc sotropc tensor-valued functons are lsted n able 4.. below. Isotropc Functon Interty Bass v v I v v ( v ( ( ensorvalued ( ( ( I functons ( u v ( u v ( u v u v v u B are ( u ( u ( u u u u u u u symmetrc ( u v ( u v u v v u tensors ( u v v u u v ( B B B B BB able 4..: Isotropc (ymmetrc ensor Functons and Interty Bases ome Results for Isotropc Functons Here follow some other mportant results reardn sotropc functons.. he prncpal values of an sotropc tensor functon of a tensor B are scalar nvarants of B. t be the prncpal values of (B prncpal values of ( B. hen o show ths let ( B ( ( B ( B I det and let ( B t det( ( B t ( B I Because of the sotropy and usn the relaton.9.a second of these can be wrtten as det t be the det( B det det B the ( ( B t ( B I det( ( B t ( B I det( ( B t ( B I hs holds for all orthoonal and hence whch s the defnton of an sotropc scalar nvarant of B. t ( B t ( B (4..5. n sotropc tensor functon of a tensor B s coaxal wth B. hs follows drectly from 4..4 snce n B has the same prncpal drectons as B. old Mechancs Part III 4
6 ecton 4.. Let be a symmetrc sotropc tensor functon of the symmetrc tensor B; f n addton the functon s a lnear functon of B then t has the representaton where ( B α ( B I βb α β are arbtrary constants (ndependent of B. tr (4..6 hs follows drectly from 4..4 notn that only the frst nvarant tr B s lnear n B. It wll be noted that ths s the form of the (sotropc lnear elastc materal model Let C be a fourth-order sotropc functon that s C C (4..7 l m jn p lq mnpq wth the mnor symmetres.9.65 C C C. hen t has the representaton Cl l l jl l ( δ δ δ δ δ μ δ (4..8 In terms of the dentty tensors of.9.6 (compare wth Eqn...7 jl ( I I l j C I I μ (4..9 o show ths consder a symmetrc second-order tensor and defne C:. hen the ndex notaton for ( s C l l and s clearly symmetrc. hen ( ( : : C l m ( C m mnl mn l l n jn p p p jq jq jq δ C r rm pqmn ls δ C sn C mn pqrs pqrs m mn mn l n (4.. from whch t can be seen that s a symmetrc sotropc tensor functon of the tensor varable. Further s lnear n and for symmetrc t follows that taes the representaton 4..6 In component form ths s ( ( tr I μ (4.. δ μ δ δ δ l μ l ( j μ( δ δ jl δ l δ j l (4.. old Mechancs Part III 4
7 ecton 4. from whch 4..8 follows. 4.. he ymmetry Group he nonempty set G wth a bnary operaton that s to each par of elements a b G there s assned an element ab G s called a roup f the follown axoms hold:. assocatve law: ( ab c a( bc for any a b c G. dentty element: there exsts an element e G called the dentty element such that ae ea a. nverse: for each a G there exsts an element a G called the nverse of a such that aa a a e Consder the set of tensors G of 4... nce for two tensors G and G n G G G G s G G FG G ( FG ( (4.. ( F. he assocatve law clearly holds the dentty element s I and the nverse of. hus the set of tensors G forms a roup. 4.. hear of an Isotropc quare Bloc Consder a combned stretch and smple shear of an sotropc hyperelastc materal F Relatve to the Cartesan coordnate system x X X x X x X (4..4 hen F (4..5 and so can be consdered to be a homoeneous stretch followed by a smple shear. he left Cauchy-Green stran and nverse are / / b b / / / (4..6 / he compressble and ncompressble sotropc relatons are (4.4.8 and 4.4. respectvely old Mechancs Part III 4
8 ecton 4. ( c ( β I β b β b pi α b α b (4..7 ubsttutn n the Cauchy-Green strans one fnds that and ( c β β α α ( (4..8 Usn ths relaton t can then be seen that (4..9 whch holds for both compressble and ncompressble materals and s the unversal relaton analoous to Here however the stretches can be chosen so as to mae the normal stress-dfference zero. y B N N e o x Fure 4..: bloc under stretch and smple shear Introduce now base vectors alon the edes of the deformed bloc wth correspondn contravarant base vectors and F. 4.. so that e e e e e e e e (4.. he metrc coeffcents are old Mechancs Part III 4
9 ecton 4. old Mechancs Part III 44 (4.. From.9. the unt normals to the bloc surfaces are (see ˆ ˆ e n e e n (4.. he stress components wth respect to the curvlnear system can be obtaned from the transformaton rule n..: [ ] [ ] [ ][ ] j j j n mn m e (4.. leadn to [ ] (4..4 he normal and shear stresses actn on the surfaces of the bloc are (see F. 4.. are N N (4..5 In order that the normal stresses actn on the bloc are zero then one requres (4..6 From 4..9 ths means that ( (4..7 physcal nterpretaton of ths results s that the lenths of the sdes of the deformed bloc are equal ob o n F In ths case 4..4 reduces to usn 4..8 [ ] [ ] ( ( ( ( ( ( α β α β c c (4..8
10 ecton 4. old Mechancs Part III 45 hus a state of pure shear s acheved wth only shear stresses actn on the faces and a square bloc deforms nto a rhombc bloc. Consder now the (ncompressble Neo-Hooean model Eqn for whch b I p c (4..9 he stress components are then b c p (4..9 he metrc components are ven by 4... he contravarant components of the left Cauchy-Green stran can be obtaned from coordnate transformaton equatons smlar to 4.. (wth b b n the Cartesan system leadn to [ ] b (4..4 wth. hen wth the stress tan the representaton 4..8 wth c α α ( ( / / c p c c (4..4 olvn leads to ( ( ( 6 / / / c p (4..4 he soluton shows that > and < and so the bloc deforms as n F. 4...
11 ecton 4. y x Fure 4..: smple shear of a Neo-Hooean bloc Note that n contrast to the decomposton old Mechancs Part III 46
where the coordinates are related to those in the old frame as follows.
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