## Transversely Isotropic Material

38 A material is transversely isotropic if there is a preferential direction normal to all but one of the three axes. If this axis is x3, then rotation about it will require that cos 6 sin 6 0 — sin 6 cos 6 0 0 0 1

substituting Eq. 7.33 into Eq. 7.41, using the above transformation matrix, we obtain

C1111 = (cos4 6)c1111 + (cos2 6 sin2 6)(2cm2 +4cm2) + (sin4 6)c2222 C1122 = (cos2 6 sin2 6)cm1 + (cos4 6)cm2 — 4(cos2 6 sin2 6)cm2 + (sin4 6)c2211 + (sin2 6 cos2 6)c2222 (cos2 6)c1133 + (sin2 6)c2233

(sin4 6)cm1 + (cos2 6 sin2 6)(2cm2 +4cm2) + (cos4 6)c2222

C1133 C2222 C1212

= (cos2 6 sin2 6)c1111 — 2(cos2 6 sin2 0)c1122 — 2(cos2 6 sin2 0)c1212 + (cos4 6)c1212 (7.39-f)

+(sin2 6 cos2 6)c2222 + sin4 6c1212

But in order to respect our initial assumption about symmetry, these results require that c1111 =

c1133 =

c2323 =

c1212 =

c2222 c2233 c3131

yielding cijkm we now have 5 nonzero coefficients.

c1111

c1212 =

c1111

 c1122 c1133 0 0 0 c2222 c2233 0 0 0 c3333 0 0 0 2(c1111 — c1122) 0 0 SYM. c2323 c3131

39 It should be noted that very few natural or man-made materials are truly orthotropic (certain crystals as topaz are), but a number are transversely isotropic (laminates, shist, quartz, roller compacted concrete, etc...).

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