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

Post a comment