## Stress Strain Relations in Generalized Elasticity Anisotropic

28 From Eq. 7.22 and 7.23 we obtain the stress-strain relation for homogeneous anisotropic material which is Hooke's law for small strain in linear elasticity.

29 We also observe that for symmetric cij we retrieve Clapeyron formula

 Tu c1111 c1112 c1133 C1112 c1123 c1131 E11 T22 c2222 c2233 c2212 c2223 c2231 E22 T33 c3333 c3312 c3323 c3331 E33 S T12 / — C1212 c1223 c1231 S 2^12(712) ( T23 SYM. c2323 c2331 2^23(723) 1 T31 J _ c3131 [ 2E3i(731) J T- -T13 k m Ekm 30 In general the elastic moduli cij relating the cartesian components of stress and strain depend on the orientation of the coordinate system with respect to the body. If the form of elastic potential function W and the values cij are independent of the orientation, the material is said to be isotropic, if not it is anisotropic. 31 cijkm is a fourth order tensor resulting with 34 81 terms. ci,1,3,1 C2,1,1,1 C2,1,2,1 c2,1,3,1 C3,1,1,1 c3,1,2,1 c3,1,3,1 c2,1,1,2 c2,1,2,2 c2,1,3,2 c3,1,1,2 c3,1,2,2 c3,1,3,2 c1,1,1,3 c1,1,2,3 c1,1,3,3 c2,1,1,3 c2,1,2,3 c2,1,3,3 c3,1,1,3 c3,1,2,3 c3,1,3,3 c2,2,1,1 c2,2,2,1 c2,2,3,1 c3,2,1,1 c3,2,2,1 c3,2,3,1 c1,2,1,2 c1,2,2,2 c1,2,3,2 c2,2,1,2 c2,2,2,2 c2,2,3,2 c3,2,1,2 c3,2,2,2 c3,2,3,2 c1,2,1,3 c1,2,2,3 c1,2,3,3 c2,2,1,3 c2,2,2,3 c2,2,3,3 c3,2,1,3 c3,2,2,3 c3,2,3,3 c1,3,3,1 c2,3,1,1 c2,3,2,1 c2,3,3,1 c3,3,1,1 c3,3,2,1 c1,3,3,2 c2,3,1,2 c2,3,2,2 c2,3,3,2 c3,3,1,2 c3,3,2,2 c1,3,1,3 c1,3,2,3 c1,3,3,3 c2,3,1,3 c2,3,2,3 c2,3,3,3 c3,3,1,3 c3,3,2,3 But the matrix must be symmetric thanks to Cauchy's second law of motion (i.e symmetry of both the stress and the strain), and thus for anisotropic material we will have a symmetric 6 by 6 matrix with (6)(6+1) =21 independent coefficients. 32 By means of coordinate transformation we can relate the material properties in one coordinate system (old) Xi, to a new one Xi, thus from Eq. 1.27 (Vj = apvp) we can rewrite cjcrstu ErsEtu i-,crstuai ajak amEij E km r)cijk,mE ij E km thus we deduce cijkm ajakamcrstu that is the fourth order tensor of material constants in old coordinates may be transformed into a new a j a\am coordinate system through an eighth-order tensor a¡ aSa^ am
0 0