Wu tensors and single inclusion problems

Eshelby (1957) calculated the strain field in an elastic ellipsoidal inclusion imbedded in an elastic host material subjected to a displacement with uniform strain field at infinity. The important general result of Eshelby is that for all ellipsoidal inclusions the resulting strain in the inclusion is homogeneous. Host and inclusion strains are therefore related algebraically. For arbitrary spheroidal inclusions, Wu (1966) subsequently found the components of the tensor $\matT^{hi}$ relating the strain in the host e_h to the strain in the inclusion e_i according to

e_i = ^hie_h.  

If the composite of interest is isotropic, no particular orientation of the spheroidal inclusions is preferred. Then, it is not actually that is of most interest but rather its isotropic average. This average is obtained by considering

P^hi 13(^hi)_mmpp   and   Q^hi 15[(^hi)_mnmn -13(^hi)_mmpp],   where the summation convention is implied for repeated subscripts. I treat from this point on in the paper as an isotropic tensor of the form

()_mnpq = 13(P-Q)_mn_pq + 12Q(_mp_nq+_mq_np).   Specific results on P and Q for inclusions shaped like spheres, needles, disks, and penny-shaped cracks are given in Table 1. Formulas for more general prolate and oblate spheroidal inclusions were presented by Berryman (1980), based on the work of Wu (1966) and Kuster and Toksöz (1974).

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TABLE 1. Four examples of coefficients P and Q for spherical and nonspherical scatterers. The superscripts h and i refer to host and inclusion phases, respectively. Special characters are defined by $\beta= \mu[(3K+\mu)/(3K+4\mu)]$,$\gamma= \mu[(3K+\mu)/(3K+7\mu)]$, and $\zeta= (\mu/6)[(9K+8\mu)/(K+2\mu)]$.The expressions for spheres, needles, and disks were derived by Wu (1966) and Walpole (1969). The expressions for penny-shaped cracks were derived by Walsh (1969) and assume K_i/K_h << 1 and $\mu_i/\mu_h << 1$.The aspect ratio of the cracks is $\alpha$.

$$0.1 in] \begin{tabular} {lll} \hline\hline Inclusion shape ... ...ver3}\mu_i+\pi\alpha\beta_h}}\right)$ \hline\hline\end{tabular}$$