Multiscale cohesive zone model for propagation of segmented crack fronts in mode I+III fracture

Abstract

Quasistatic crack propagation in mixed-mode I+III fracture is widely observed to be unstable, the instability being characterized by the segmentation of the parent crack into a periodic array of daughter cracks shaped as flat facets rotated towards the principal stress axis. While there has been recent progress to characterize this instability, no global theory is presently available to describe all aspects of the propagation of the segmented front, including both “local” features like the angle of rotation of the facets and the ratio of their width to their spacing, and “global” ones like the effective energy-release-rate of the segmented crack front and the tendency of the facets to coarsen. This paper embarks on the development of such a theory, based on the assumption that the spacing of the facets is much smaller than their length, and asymptotic matching of outer and inner solutions for the mechanical fields on scales comparable to the facet length and spacing, respectively. The inner problem is shown to reduce to a 2D linear elastic fracture mechanics problem in the plane perpendicular to the crack propagation axis. The solution of this problem is used to develop an effective cohesive zone description of the crack front on a scale much larger than the facet spacing. Such a description leads to a system of 1D integral equations for the outer mechanical fields on the cohesive zone, which may be solved numerically. Numerical examples are given that notably illustrate the prediction of the effective energy-release-rate of the segmented crack front in terms of the various geometrical parameters; this energy-release-rate is predicted to be smaller for a segmented front than for the parent planar front, with the conclusion that segmentation acts as a toughening mechanism. Implications upon the phenomenon of facet coarsening are also briefly discussed.

Appendices

Appendix 1: Approximate expressions of the functions $F_{ij}^{p}$

The approximate expressions of the functions $F_{ij}^{p}(c/d\,,\alpha )$ proposed by Leblond and Frelat (2014) (in the sole case $c<d$) are as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle F^I_{11}\left( \frac{c}{d},\,\alpha \right) \simeq \frac{1-\cos (2\alpha )}{2\sqrt{\cos \alpha }};\\ \displaystyle F^I_{22}\left( \frac{c}{d},\,\alpha \right) \simeq \frac{1+\cos (2\alpha )}{2\sqrt{\cos \alpha }}; \\ \displaystyle F^I_{12}\left( \frac{c}{d},\,\alpha \right) \simeq - \left[ 3 + \frac{\pi c/d}{\sin (\pi c/d)} \right] \frac{\sin (2\alpha )}{4\sqrt{\cos \alpha }}; \\ \displaystyle F^{II}_{11}\left( \frac{c}{d},\,\alpha \right) \simeq - \frac{\sin (2\alpha )}{2\sqrt{\cos \alpha }};\\ \displaystyle F^{II}_{22}\left( \frac{c}{d},\,\alpha \right) \simeq \left[ 3 - \frac{\pi c/d}{\sin (\pi c/d)} \right] \frac{\sin (2\alpha )}{4\sqrt{\cos \alpha }}; \\ \displaystyle F^{II}_{12}\left( \frac{c}{d},\,\alpha \right) \simeq \frac{\cos (2\alpha )}{\sqrt{\cos \alpha }}\,. \\ \end{array} \right. \end{aligned}$$

(62)

These formulae present the following nice features:

• for small values of $\alpha$ but arbitrary values of $c/d\,(<1)$, they match the exact first-order solution in $\alpha$ (Melin 1983; Leblond and Frelat 2014);

• for infinitesimal values of $c/d$ but arbitrary values of $\alpha$, they again match the exact, trivial solution corresponding to isolated cracks;

• they yield acceptable results in all cases except when $\alpha$ and $c/d$ are simultaneously large (see Leblond and Frelat 2014’s comparisons with the results of some finite element calculations).

Appendix 2: Approximate expressions of the coefficients ${{\mathcal {A}}}_{\lambda \mu }$

Approximate expressions of the coefficients ${{\mathcal {A}}}_{\lambda \mu }$ may be derived using their definition (18) and the approximate expressions (62) of the functions $F_{ij}^{p}$. All integrals involved reduce to elementary integrals plus a single, non-elementary one defined by

$$\begin{aligned} J(u) \equiv \int _0^u \frac{v^2}{\sin v}\,{\mathrm{d}}v; \end{aligned}$$

(63)

but practical calculation of this integral does not raise any problem since it is given by the following very quickly converging series (Gradshteyn and Ryzhik 1980, formula 2.643.3):

$$\begin{aligned} J(u) = \frac{u^2}{2} + \sum _{k=1}^{+\infty } (-1)^{k+1}\frac{2^{2k-1}-1}{(k+1)(2k)!}\,B_{2k}u^{2(k+1)}\nonumber \\ \end{aligned}$$

(64)

where $B_i$ is the $i$th Bernoulli number.

Defining

$$\begin{aligned} {\bar{x}} \equiv \frac{\pi x}{2}, \end{aligned}$$

(65)

the expressions found are as follows:

$$\begin{aligned}&{{\mathcal {A}}}_{11}(x,\alpha ) \simeq - \frac{2}{\pi } \, \tan ^2\alpha \, \ln \left( \cos {\bar{x}} \right) ; \end{aligned}$$

(66)

$$\begin{aligned}&\begin{array}{lll} {{\mathcal {A}}}_{22}(x,\alpha ) &{} \simeq &{} \displaystyle \frac{1}{2\pi \cos ^2\alpha } \bigg \{ - \ln (\cos {\bar{x}})\left[ -3\cos ^2(2\alpha ) + 2\cos (2\alpha ) + 5 \right] \\ {} &{} {} &{} \displaystyle + \left[ -\frac{7}{4}\,{\bar{x}}\tan {\bar{x}} + \frac{{\bar{x}}^2}{8\cos ^2{\bar{x}}} +\, \frac{J(2{\bar{x}})}{16} \right] \sin ^2(2\alpha ) \bigg \}; \\ {} &{} {} &{} {} \end{array} \end{aligned}$$

(67)

$$\begin{aligned}&\begin{array}{lll} {{\mathcal {A}}}_{33}(x,\alpha ) &{} \simeq &{} \displaystyle \frac{1}{2\pi \cos ^2\alpha } \bigg \{ \!-\! \ln (\cos {\bar{x}})\left[ 3\cos ^2(2\alpha ) \!+\! 1 \right] \\ &{}&{}\displaystyle \quad {}+ \left[ \frac{5}{4}\,{\bar{x}}\tan {\bar{x}} +\, \frac{{\bar{x}}^2}{8\cos ^2{\bar{x}}} + \frac{J(2{\bar{x}})}{16} \right] \sin ^2(2\alpha ) \bigg \};\\ {} &{} {} &{} {} \end{array} \end{aligned}$$

(68)

$$\begin{aligned}&{{\mathcal {A}}}_{12}(x,\alpha ) = {{\mathcal {A}}}_{21}(x,\alpha ) \simeq \frac{2}{\pi } \, \sin ^2\alpha \left[ \ln \left( \cos {\bar{x}} \right) + \frac{{\bar{x}}}{2}\tan {\bar{x}}\right] ; \end{aligned}$$

(69)

$$\begin{aligned}&{{\mathcal {A}}}_{13}(x,\alpha ) = {{\mathcal {A}}}_{31}(x,\alpha ) \simeq \frac{\tan \alpha }{\pi } \left\{ \ln \left( \cos {\bar{x}} \right) \left[ \cos (2\alpha ) + 1 \right] + \frac{{\bar{x}}}{2}\tan {\bar{x}} \left[ \cos (2\alpha ) - 1 \right] \right\} ; \end{aligned}$$

(70)

$$\begin{aligned}&{{\mathcal {A}}}_{23}(x,\alpha ) = {{\mathcal {A}}}_{32}(x,\alpha ) \simeq \frac{\tan \alpha }{\pi } \left\{ \ln \left( \cos {\bar{x}} \right) \left[ - 3\cos (2\alpha ) + 1 \right] - \frac{{\bar{x}}}{2}\tan {\bar{x}} \left[ 3\cos (2\alpha ) + 1 \right] \right\} . \end{aligned}$$

(71)

Appendix 3: Expressions of the coefficients $M_{ij}$ and $N_{ij}$

The definitions (59) of the coefficients $M_{ij}$ and $N_{ij}$ may be rewritten in the form

$$\begin{aligned} \left\{ \begin{array}{l} M_{ij} \equiv \phi (w'_{j-1},w'_j\,;w_i), \\ \phi (u,v;w) \equiv \displaystyle PV \int _{u}^{v} \sqrt{\frac{1-w'}{w'}} \,\frac{{\mathrm{d}}w'}{w'-w} \\ N_{ij} \equiv \psi (w'_{j-1},w'_j\,;w_i) , \\ \psi (u,v;w) \equiv \displaystyle PV \int _{u}^{v} \sqrt{w'(1-w')} \ \frac{{\mathrm{d}}w'}{w'-w}. \end{array} \right. \end{aligned}$$

(72)

These integrals may be reduced to ordinary, easily calculable integrals plus a single one in principal value, by using the change of variable $w'=\sin ^2\theta$ and then expanding the numerator in powers of $\sin ^2\theta - w$; one thus gets

$$\begin{aligned} \left\{ \begin{array}{lll} \phi (u,v;w) &{} = &{} \displaystyle 2\left[ \arcsin (\sqrt{u})-\arcsin (\sqrt{v})\right] \\ &{}&{} +\, 2(1-w)\chi (u,v;w) \\ \psi (u,v;w) &{} = &{} \displaystyle (1\!-\!2w)\left[ \arcsin (\sqrt{v})\!-\!\arcsin (\sqrt{u})\right] \\ &{}&{} +\, \sqrt{v(1-v)} - \sqrt{u(1-u)} \\ {} &{} {} &{} \displaystyle +\, 2w(1-w)\chi (u,v;w) \end{array} \right. \nonumber \\ \end{aligned}$$

(73)

where

$$\begin{aligned} \chi (u,v;w) \equiv PV \int _{\arcsin (\sqrt{u})}^{\arcsin (\sqrt{v})} \frac{{\mathrm{d}}\theta }{\sin ^2\theta - w}. \end{aligned}$$

(74)

The problem is thus reduced to calculating the single integral in principal value $\chi (u,v;w)$, which is easily done by using Gradshteyn and Ryzhik (1980)’s formula (2.562.1); the three cases $w<\arcsin (\sqrt{u})$, $\arcsin (\sqrt{u})<w<\arcsin (\sqrt{v})$, $\arcsin (\sqrt{v})<w$ must be distinguished in the calculation but the results may be expressed in a single formula:

$$\begin{aligned} \chi (u,v;w)&= \frac{\Xi (u;w)-\Xi (v;w)}{\sqrt{w(1-w)}}, \nonumber \\ \Xi (t;w)&\equiv \left\{ \begin{array}{lll} \displaystyle \mathrm{argtanh}\left( \sqrt{\frac{t(1-w)}{w(1-t)}} \right) &{} \quad \text{ if } &{} t<w \\ \displaystyle \mathrm{argcoth}\left( \sqrt{\frac{t(1-w)}{w(1-t)}} \right) &{} \quad \text{ if } &{} t>w. \end{array} \right. \nonumber \\ \end{aligned}$$

(75)

Formulae (72), (73) and (75) provide the desired expressions of the coefficients $M_{ij}$ and $N_{ij}$.