Crack front perturbations revisited

Abstract

The problem of the in-plane dynamic perturbation of a crack propagating with a front that is nominally straight is solved, to second order in the perturbation. The method of approach is a streamlined and generalized version of that previously applied to first order by the author and co-workers. It emerges, however, that the analysis at second order requires for its consistency the introduction of a new singular term, of a type not present at first order. The analysis is restricted to the case of Mode I loading, for clarity of exposition. It is carried out at a level of generality that incorporates viscoelastic response as well as propagation in a “vertically stratified” medium including, as a special case, propagation in a slab of finite thickness. For illustration, the general solution is specialized to the case of a stationary crack in an infinite elastic medium and agreement with a solution recently developed by methodology that is specific to the static case is confirmed.

Appendix: Expansion of the Basic Identity to Second Order

The term that must be equated to the right side of (2.20) is composed from the limiting operations applied to $\frac{1}{2}(G - )^{ - 1} *[u]_{ - } + G_{ + } *\sigma_{ + }$. Completing the algebra gives

$$ \begin{aligned} \widehat{{G_{ + } *\sigma_{ + } }}\sim & - \frac{1}{2}i\xi_{1} \varepsilon^{2} \phi K - \varepsilon \phi K + \frac{1}{2}(\pi /2)^{1/2} \varepsilon^{2} \phi^{2} A & & \\ & + \frac{1}{2}\varepsilon^{2} Q_{1} *(\phi^{2} K) + \left\{ {K - \varepsilon (\pi /2)^{1/2} \phi A + \frac{3}{4}\varepsilon^{2} (\pi /2)^{1/2} \phi^{2} B} \right. \\ & \left. { - Q_{1} *[\varepsilon \phi K - \frac{1}{2}(\pi /2)^{1/2} \varepsilon^{2} \phi^{2} A] + \frac{1}{2}\varepsilon^{2} Q_{2} *(\phi^{2} K)} \right\}\frac{i}{{\xi_{1} + 0i}} \\ & - \left\{ {(\pi /2)^{1/2} (A - \frac{3}{2}\varepsilon \phi B) + Q_{1} *(K - \varepsilon (\pi /2)^{1/2} \phi A} \right. \\ & \left. { + \frac{3}{4}\varepsilon^{2} \phi^{2} B) - Q_{2} *[\varepsilon \phi K - \frac{1}{2}(\pi /2)^{1/2} \varepsilon^{2} \phi^{2} A]} \right\}\frac{1}{{(\xi + 0i)^{2} }} \\ & - \left\{ {\frac{3}{2}(\pi /2)^{1/2} B + (\pi /2)^{1/2} Q_{1} *(A - \frac{3}{2}\varepsilon \phi B)} \right. \\ & \left. { + Q_{2} *[K - \varepsilon (\pi /2)^{1/2} \phi A + \frac{3}{4}\varepsilon^{2} (\pi /2)^{1/2} \phi^{2} B]} \right\}\frac{i}{{(\xi 1 + 0i)^{3} }} \\ & + \widehat{{G_{ + } *\sigma_{ + } }}.| \\ \end{aligned} $$

(5.1)

Also,

$$ \begin{aligned} \frac{1}{2}\widehat{{(G - )^{ - 1} *[u]_{ - } }}\sim & - \frac{{(\pi /2)^{1/2} }}{{{\mathcal{A}}(V)}}\left[ { - \frac{1}{2}i\xi_{1} \varepsilon^{2} \phi^{2} K^{u} } \right. \\ & - \varepsilon \phi K^{u} - \frac{3}{4}\varepsilon^{2} \phi^{2} A^{u} + \frac{1}{2}\varepsilon^{2} R_{1} *(\phi^{2} K^{u} ) \\ & + \left\{ {K^{u} + \frac{3}{2}\varepsilon \phi A^{u} + \frac{15}{8}\varepsilon^{2} \phi^{2} B^{u} } \right. \\ & \left. { - R_{1} *(\varepsilon \phi K^{u} + \frac{3}{4}\varepsilon^{2} \phi^{2} A^{u} ) + \frac{1}{2}\varepsilon^{2} R_{2} *(\phi^{2} K^{u} )} \right\}\frac{i}{{\xi_{1} - 0i}} \\ & + \left\{ {\frac{3}{2}A^{u} + \frac{15}{4}\varepsilon \phi B^{u} - R_{1} *(K^{u} + \frac{3}{2}\varepsilon \phi A^{u} + \frac{15}{8}\varepsilon^{2} \phi^{2} B^{u} )} \right. \\ & \left. { + R_{2} *(\varepsilon \phi K^{u} + \frac{3}{4}\varepsilon^{2} \phi^{2} A^{u} )} \right\}\frac{1}{{\left( {\xi_{1} - 0i} \right)^{2} }} \\ & - \left\{ {\frac{15}{4}B^{u} - R_{1} *(\frac{3}{2}A^{u} + \frac{15}{8}\varepsilon \phi B^{u} )} \right. \\ & \left. {\left. { + R_{2} *(K^{u} + \frac{3}{2}\varepsilon \phi A^{u} + \frac{15}{8}\varepsilon^{2} \phi^{2} B^{u} )} \right\}\frac{i}{{\left( {\xi_{1} - 0i} \right)^{3} }}} \right] \\ & + \frac{1}{2}\widehat{{(G_{ - } )^{ - 1} *[u]_{ - }^{*} }}. \\ \end{aligned} $$

(5.2)

The exact forms taken for $\widehat{{G_{ + } *\sigma_{ + }^{*} }}$ and $\widehat{{G_{ - }^{ - 1} *u_{ - }^{*} }}$ are discussed in the main text.