Consider the case of a crack problem in an anisotropic monoclinic material under anti-plane deformation as described
Question:
Consider the case of a crack problem in an anisotropic monoclinic material under anti-plane deformation as described in Exercise 11.16. Following relation (11.6.6), choose the complex potential form as F(z∗) = A = √z∗, where A =–√2 K3μ/(C55 + μC45) and K3 is a real constant. Using this form, show that the nonzero displacement and stresses in the vicinity of the crack tip (see Fig. 11.11) are given by:
Note that the parameter K3 will be related to the stress intensity factor for this case. Verify that shear stress τ yz vanishes on each side of the crack face, θ =± π These results can be compared to the corresponding solution for the isotropic case given in Exercise 8.44.
Data from exercise 11.16
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
Equation 11.6.6
Fig 11.11
Fig 11.2
Data from exercise 8.44
Consider the crack problem shown for the anti-plane strain case with u = v = 0, w = w(x,y).
From Section 7.4, the governing equation for the unknown displacement component with
zero body force was given by Laplace’s equation, which in polar coordinates reads.
Step by Step Answer:
Elasticity Theory Applications And Numerics
ISBN: 9780128159873
4th Edition
Authors: Martin H. Sadd Ph.D.