s law also called the elastic constitutive relation may be written in the following form where u03c3 and u03b5 are second rank stress and strain tensors respectively and c is a fourth rank transformation tensor for a transversely isotropic material properties of x u2261 y the characteristic array of elastic coefficients for the middle castellated layer can be defined by the following tensor see a p boresi and k p chong elasticity in engineering mechanics elsevier science publishing co inc new york 1987 pp 250 259 c u03b1 ue89e ue89e u03b2 c 11 c 12 c 13 0 0 0 c 12 c 11 c 13 0 0 0 c 13 c 13 c 33 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 55 0 0 0 0 0 0 c 55 7 where c 11 1 nv zx 2 ue89e e x ab ue89e c 12 v xy nv zx 2 ue89e e x ab ue89e c 13 v zx ue89e e x b c 33 1 v xy ue89e e z b ue89e c 44 1 2 ue89e c 11 c 12 ue89e c 55 g xz g yz e z 2 ue89e 1 v xz a 1 v xy ue89e b 1 v xy 2 ue89e nv zx 2 ue89e and ue89e ue89e n e x e z and where the ratio n is a measure of the degree of an isotropy u03bd xy u03bd yz u03bd xz is poisson 39 s ratio 0 3 for a typical metal and e x e y and e z are the moduli of the respective axes for the special case e x e y lt lt e z and e z e substrate 8 the stresses in the.