Monotropic Material

33 A plane of elastic symmetry exists at a point where the elastic constants have the same values for every pair of coordinate systems which are the reflected images of one another with respect to the plane. The axes of such coordinate systems are referred to as "equivalent elastic directions".

34 If we assume Xi = xi, X2 = X2 and X3 = — X3, then the transformation Xj — aj Xj is defined through

where the negative sign reflects the symmetry of the mirror image with respect to the x3 plane.

35 We next substitute in Eq.7.33, and as an example we consider c1123 = aia\a2aUcrstu = aiaia2a3c1123 = (1)(1)(1)( — 1)cii23 = —c1123, obviously, this is not possible, and the only way the relation can remanin valid is if C1123 = 0. We note that all terms in c^jui with the index 3 occurring an odd number of times will be equal to zero. Upon substitution, we obtain

C1111

cij km

C1111

C1122

C1133

C1112

0

0

c2222

C2233

C2212

0

0

C3333

C3312

0

0

C1212

0

0

SYM.

c2323

we now have 13 nonzero coefficients.

0 0

Post a comment