Numerical modeling of the nucleation of facets ahead of a primary crack under mode I + III loading

Abstract

A numerical study of crack front segmentation under mode I + III loading is proposed. Facets initiation ahead of a parent crack is predicted through a tridimensional application of the coupled criterion. Crack initiation shape, orientation and spacing are determined for any mode mixity ratio by coupling a stress and an energy criterion using matched asymptotic expansions. The stress and the energy conditions are computed through a 3D finite element modeling of a periodic network of facets ahead of the parent crack. The initiation shape, loading and spacing of facets depend on the blunted parent crack tip radius. A good estimate of facet orientations is obtained based on the direction of maximum tensile stress. The facet shapes, determined using the stress isocontours, are qualitatively similar to those observed experimentally. The order of magnitude of numerical predictions of facets spacing is very close to experimental measurements.

Appendix

The matched asymptotic expansions

The actual domain embeds the parent blunted crack (root radius R) and a group of small slanted crack regularly spaced. The outer domain is obtained for $R \rightarrow 0$, and as a consequence $l \rightarrow 0$ since it is assumed that l is smaller or of the same order of magnitude than R (to be checked afterwards). In the outer domain, the blunted crack becomes a slit with no thickness. Both actual and outer domains are spanned by the Cartesian space variables $x_i$ ($i=1,3$) ($r,\theta ,x_3$, in cylindrical coordinates).The outer expansion with respect to the small parameter R can be written ($\nabla _x$ is the gradient operator with respect to the $x_i$’s)

$$\begin{aligned} \left\{ \begin{array}{ll} \underline{U}^R(x_1,x_2,x_3,l) = \underline{U}^0(x_1,x_2,x_3)+\cdots \\ \underline{\underline{\sigma }}(\underline{U}^R) = \mathbf {C}:\nabla _x\underline{U}^R=\mathbf {C}:\nabla _x\underline{U}^0+\cdots \end{array}\right. \end{aligned}$$

(13)

The dots denote a remainder that is small and decreases with R. The behavior of the leading term of Eq. (13) near the crack tip is described by the Williams series (polar and Cartesian coordinates are mixed without risk of confusion)

$$\begin{aligned} \begin{aligned} \underline{U}^0(x_1,x_2,x_3) =\,&\underline{C}+K_{\mathrm {I}}\sqrt{r}\underline{u}_{\mathrm {I}}(\theta )+K_{\mathrm {III}}\sqrt{r}\underline{u}_{\mathrm {III}}(\theta )+\cdots \\ = \,&\underline{C}+K_{\mathrm {I}}\sqrt{r}[\underline{u}_{\mathrm {I}}(\theta )+m\underline{u}_{\mathrm {III}}(\theta )]+\cdots \\&\mathrm {with}\, m=\frac{K_{\mathrm {III}}}{K_{\mathrm {I}}}=(\frac{1}{\beta }-1)^{-1} \end{aligned} \end{aligned}$$

(14)

The leading term is an irrelevant constant (rigid translation), $K_{\mathrm {I}}$ and $K_{\mathrm {III}}$ are the modes I and III stress intensity factors and $\underline{u}_{\mathrm {I}}$ and $\underline{u}_{\mathrm {III}}$ are the associated opening and shear modes. The intensity factors $K_{\mathrm {I}}$ and $K_{\mathrm {III}}$ are independent of $x_3$ as a consequence of the first assumption stated at the end of Sect. 2.

The inner domain is obtained by zooming in the actual domain by 1 / R and considering again the limit as $R\rightarrow 0$. It is an unbounded domain spanned by the dimensionless space variables $y_i=x_i/R$ ($i=1,3$); it embeds the semi-infinite parent crack and an infinite number of regularly spaced slanted cracks with length $\lambda =l/R$ along $y_1$ axis. In the $y_3$ direction, a periodicity assumption is done reducing the domain to a single period of length $\epsilon =e/R$ in this direction (see Figs. 1–3).

After fulfilling the matching conditions (Leguillon and Sanchez-Palencia 1987), the inner expansion can be written

$$\begin{aligned} \begin{aligned} \underline{U}^R(x_1,x_2,x_3,l)&= \underline{U}^R(Ry_1,Ry_2,Ry_3,R\lambda )\\&= \underline{C}+K_{\mathrm {I}}\sqrt{R}\underline{V}_{\mathrm {I}}(y_1,y_2,y_3,\lambda ) \\&\quad +K_{\mathrm {III}}\sqrt{R}\underline{V}_{\mathrm {III}}(y_1,y_2,y_3,\lambda )+\cdots \\&= \underline{C}+K_{\mathrm {I}}\sqrt{R}[\underline{V}_{\mathrm {I}}(y_1,y_2,y_3,\lambda ) \\&\quad +m\underline{V}_{\mathrm {III}}(y_1,y_2,y_3,\lambda )]+\cdots \end{aligned} \end{aligned}$$

(15)

where $\underline{V}_{\mathrm {I}}$ (respectively $\underline{V}_{\mathrm {III}}$) behaves like $\sqrt{\rho } \underline{u}_{\mathrm {I}}(\theta )$ (respectively $\sqrt{\rho } \underline{u}_{\mathrm {III}}(\theta )$) at infinity ($\rho =r/R \rightarrow \infty$). Relationship (15) can be rewritten for the purpose of full FE calculations

$$\begin{aligned} \left\{ \begin{array}{ll} \underline{U}^R(x_1,x_2,x_3,l) = \underline{C}+ K_{\mathrm {I}}\sqrt{R}\underline{W}\left( y_1,y_2,y_3,\lambda \right) +\cdots \\ \underline{\underline{\sigma }}(\underline{U}^R) = \frac{1}{R}\mathbf {C}:\nabla _y\underline{U}^R=\frac{K_{\mathrm {I}}}{\sqrt{R}}\mathbf {C}:\nabla _y\underline{W}=\frac{K_{\mathrm {I}}}{\sqrt{R}}\underline{\underline{\tilde{\sigma }}}(\underline{W}) \end{array}\right. \end{aligned}$$

(16)

where $\underline{W}=\underline{V}_{\mathrm {I}}+m\underline{V}_{\mathrm {III}}$ behaves at infinity like $\sqrt{\rho } \underline{u}_{\mathrm {I}}(\theta )+m\sqrt{\rho } \underline{u}_{\mathrm {III}}(\theta )$. According to Leguillon et al. (2007), the change in potential energy takes the form

$$\begin{aligned} \begin{aligned} -\delta W_{\mathrm {p}} =\,&\varPsi _x\left( \underline{U}^R\left( x_1,x_2,x_3,l\right) , \underline{U}^R(x_1,x_2,x_3,0) \right) \\ =\,&\varPsi _x\left( \underline{U}^R(x_1,x_2,x_3,l), \underline{U}^0(x_1,x_2,x_3)\right) \\&- \varPsi _x\left( \underline{U}^R(x_1,x_2,x_3,0), \underline{U}^0(x_1,x_2,x_3)\right) \end{aligned} \end{aligned}$$

(17)

where

$$\begin{aligned} \varPsi _x(\underline{F},\underline{G})=\frac{1}{2}\int _{\Gamma _x} \left[ \underline{\underline{\sigma }}(\underline{F}){\cdot }\underline{n}{\cdot }\underline{G}-\underline{\underline{\sigma }}(\underline{G}){\cdot }\underline{n}{\cdot }\underline{F}\right] \mathrm {d}s \end{aligned}$$

(18)

The contour $\Gamma _x$ starts from the stress free edges of the crack and embeds the blunted notch and the crack extension. The integral (18) is path indepedent for any two functions $\underline{F}$ and $\underline{G}$ fulfilling the equilibrium equations. The index x in Eqs. (17) and (18) recalls that the calculations are a priori carried out in the actual (using the actual solution) or the outer domain [using the outer expansion but this requires calculating the remainder in (13) (Leguillon 2011)].

Taking into account the change of variables defined by the dilatation in the inner domain, (18) rewrites

$$\begin{aligned} \varPsi _x(\underline{F},\underline{G})&=\frac{R}{2}\int _{\Gamma _y}\left[ \underline{\underline{\tilde{\sigma }}}(\underline{F}){\cdot }\underline{n}{\cdot }\underline{G}-\underline{\underline{\tilde{\sigma }}}(\underline{G}){\cdot }\underline{n}{\cdot }\underline{F}\right] \mathrm {d}S\nonumber \\&=R\varPsi _y(\underline{F},\underline{G}) \end{aligned}$$

(19)

Because $\mathrm {d}S=R^2 \times \mathrm {d}s$ (while $\mathrm {d}S=R \times \mathrm {d}s$ in 2D leading to two similar relationships expressed either with respect to the $x_i$’s or the $y_i$’s). Then

$$\begin{aligned} -\delta W_{\mathrm {p}}&= K_{\mathrm {I}}^2R^2\left[ \varPsi _y\left( \underline{W}(y_1,y_2,y_3,\lambda ), \sqrt{\rho }[\underline{u}_{\mathrm {I}}(\theta )+m\underline{u}_{\mathrm {III}}(\theta )\right] \right) \nonumber \\&\quad - \varPsi _y\left( \underline{W}(y_1,y_2,y_3,0), \sqrt{\rho }\left[ \underline{u}_{\mathrm {I}}(\theta )+m\underline{u}_{\mathrm {III}}(\theta )\right] \right) \nonumber \\&= K_{\mathrm {I}}^2R^2\left( B(\lambda )-B(0)\right) +\cdots \end{aligned}$$

(20)

Note that $B(\lambda )-B(0)$ is nothing but the change in potential energy computed in the $y_i$ variables when moving from $\underline{W}(y_1,y_2,y_3,0)$ to $\underline{W}(y_1,y_2,y_3,\lambda )$.

With a newly created crack surface $S=\alpha l^2$ ($\alpha$ is a scaling coefficient) it comes the energy condition

$$\begin{aligned} G_{\mathrm {inc}}=-\frac{\delta W_{\mathrm {P}}}{S} =K_{\mathrm {I}}^2 \frac{B(\lambda )-B(0)}{\alpha \lambda ^2}\ge G_{\mathrm {c}} \end{aligned}$$

(21)

Denoting $\sigma$ and $\tilde{\sigma }$ the tensile component of the corresponding stress tensors, the stress condition can be written

$$\begin{aligned} \sigma (\underline{U}^R)=\frac{K_{\mathrm {I}}}{\sqrt{R}}\tilde{\sigma }(\underline{W})\ge \sigma _{\mathrm {c}} \end{aligned}$$

(22)

here $\tilde{\sigma }(\underline{W})$ is nothing but the tensile stress computed in the $y_i$ variables.

Combining (21) and (22) gives the equation for the dimensionless crack extension length $\lambda _{\mathrm {c}}$

$$\begin{aligned} \frac{1}{\tilde{\sigma }(\lambda _{\mathrm {c}})}\frac{B(\lambda )-B(0)}{\alpha \lambda _{\mathrm {c}}^2}=\frac{1}{R} \frac{G_{\mathrm {c}}}{\sigma _{\mathrm {c}}^2} \end{aligned}$$

(23)

In this equation $\tilde{\sigma }(\lambda _{\mathrm {c}})$ denotes the tensile stress associated with $\underline{W}$, expressed in the $y_i$ variables and computed at a dimensionless distance $\lambda _{\mathrm {c}}$ of the notch root along the $y_1$ axis. Then the load at failure can be derived from (21)

$$\begin{aligned} K_{\mathrm {Ic}}^{\mathrm {app}}=\sqrt{\frac{ \alpha \lambda _{\mathrm {c}}^2G_{\mathrm {c}}}{B(\lambda _{\mathrm {c}})-B(0)}} \ ;\ K_{\mathrm {IIIc}}^{\mathrm {app}}=mK_{\mathrm {Ic}}^{\mathrm {app}} \end{aligned}$$

(24)