Consider an anisotropic monoclinic material symmetric about the x,y-plane (see Fig. 11.2) and subject to an anti-plane deformation specified by u = v = 0, w = w(x,y). Show that in the absence of body forces, the out-of-plane displacement must satisfy the Navier equation:

where z^∗= x + μy and μ are the roots of the equation C44μ² +2C₄₅μ + C₅₅ = 0. Note that for this case, positive definite strain energy implies that C₄₄C₅₅ > C² ₄₅; therefore, the roots will occur in complex conjugate pairs.

Fig 11.2

Transcribed Image Text:

3w aw +44 ay xy Next looking for solutions that are of the form w = F(x + uy), show that this problem is solved by aw C55 +2C45 x = 0 w = 2Re {F(z*)] Txz = 2Re{(uC45+C55) F'(z*)] Tyz = 2Re{(C44 +C45) F'(z*)]