4.5 Material Anisotropy

Transcription

1 . Mteril Anisoopy.. Mteril Symmey he isoopic mteril ws defined s one whose mteril response ws unffected y rigid ody rottions of the reference configurtion. Other mteril symmeies re possile; to generlise the notion insted of considering orthogonl nsformtions consider n riry deformtion F of the reference configurtion S ringing it to new configurtion S Fig... compre with the isoopic cse Fig F S F S X x X S reference configurtion F Figure..: deformtion of the reference configurtion onsidering the uchy-elstic mteril if the deformtion F hs no effect on the response of the mteril then σ F = σ F σ FF = σ F.. hen F = Q one hs the isoopic mteril. Setting the most usul form: G = F.. cn e cst in σ F = σ FG.. Note tht the resiction det G = ± is ssumed since otherwise riry dilttions could occur with no chnge in mteril response which seems physiclly unresonle. Note tht the set of ll tensors G which stisfy.. forms group see the Appendix to this hpter.a. nd hence is clled the symmey group of the mteril with respect to the configurtion S. Aprt from isoopy the two most importnt prcticl cses of mteril symmey re nsverse isoopy nd orthoopy. 86

2 .. rnsverse soopy onsider first the nsversely isoopic mteril. Such mteril hs single preferred direction defined y unit vector in the reference configurtion. Such vector is illusted in Fig... showing lso the unit vectors n ˆ ˆ n completing n orthonorml set. he symmey group of the nsversely isoopic mteril is the set of orthogonl tensors Q which nsform the set { n n } into the new orthonorml ± n n. n prticulr set { } Q = ±.. n order to ensure tht the sense of Q is immteril it is est to inoduce the sucturl tensor which nsforms s the xes chnge ccording to Q ±.. Q = ± or Q Q =.. n n or n n Q n or n Figure..: n orthonorml set of vectors he sin energy cn now e tken to e function of s in the isoopic cse nd which chrcterises the sucture of the mteril: =..6 Allowing for nsformtions of the undeformed configurtion = S QQ Q..7 Q with Q here resicted to the symmey group defined y... hen is n isoopic sclr function of two symmeic tensors nd so from le.a. tkes the form 87

3 =..8 Since { Prolem } = = = = =..9 one rrives t the representtion =.. where the fourth nd fifth sclr pseudo- invrints re defined y = =.. Note lso tht from the definition of the setch Eqn...7 = = λ.. where λ is the setch of the unit line element. f the preferred direction is e then the fourth nd fifth invrints in terms of components re = = = = in which cse the five invrints cn e tken s { }... Using the reltions { Prolem } = =.. the PK sesses for hyperelstic mteril re then S = i= = i i.. 88

4 89 Let e unit vector in the current configurtion in the direction of F tht is F = λ..6 hen using Eqn...7 FSF σ = J with FF = FF = see Eqn... F F = nd noting tht nd hve the sme principl invrints.. ecomes = σ J..7 Using the yley-hmilton theorem llows one to re-write the uchy sess s = σ J..8 Expressing the sclr invrints in terms of rther thn the coefficients of the tensors in..7-8 re functions of the set { } λ λ..9 rnsversely soopic Mterils with onsints For n incompressile mteril = nd nlogous to.. the sin energy tkes the form p.. For mteril which is inextensile in the direction of from.. = nd the sin energy tkes the form q..

5 .. Orthoopy onsider now mteril which is dependent on two chrcteristic directions nd ; gin the sense of these directions is immteril. he sin energy is now of the form =.. As isoopic sclr function of three symmeic tensors depends on the following ces see le.a. Using..9 see Eqn..9.e this reduces to the set of nine invrints.... with =. he term is the cosine of the ngle etween the two chrcteristic directions; this does not chnge during the deformtion nd so this term cn e omitted leving.. An orthoopic mteril is one for which nd re perpendiculr = mking the lst term here zero. his lso then defines third preferred direction c orthogonl to oth nd which inoduces ex terms c c c c. But = = c c c c..6 so tht c c c c re redundnt. Finlly the sin energy is of the form =..7 9

6 As efore the sesses in hyperelstic mteril cn now e otined y differentition... Prolems. Show tht for unit vector i = ii = iii = iv =. Show tht. Show tht = = i F F = λ ii F F = λ For ii it might help to note the following reltions for vector nd second-order tensors A B: A B A B AB = A B A A = AA 9