Steady-state propagation of a mode II crack in couple stress elasticity

Abstract

The present work deals with the problem of a semi-infinite crack steadily propagating in an elastic body subject to plane-strain shear loading. It is assumed that the mechanical response of the body is governed by the theory of couple-stress elasticity including also micro-rotational inertial effects. This theory introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying microstructure and has proved to be very effective for modeling complex microstructured materials. It is assumed that the crack propagates at a constant sub-Rayleigh speed. An exact full field solution is then obtained based on integral transforms and the Wiener–Hopf technique. Numerical results are presented illustrating the dependence of the stress intensity factor and the energy release rate upon the propagation velocity and the characteristic material lengths in couple-stress elasticity. The present analysis confirms and extends previous results within the context of couple-stress elasticity concerning stationary cracks by including inertial and micro-inertial effects.

Appendices

Appendix 1: Branch cuts for the functions $\chi (z)$, $\beta (z)$ and $\gamma (z)$

The complex function $\chi (z)$ has four branch points (BPs) at $\pm ib_1$ and $\pm ib_2$, where $(b_1,b_2)$ are given in Eq. (72). In the case $h_0\le 1$, $(b_1,b_2)$ are always real, consequently, the BPs are located along the imaginary axis (Fig. 15a). On the other hand, when $h_0>1$, $(b_1,b_2)$ are complex and the BPS are symmetrically located with respect to the real axis at the four quadrants of the complex $z$-plane (Fig. 15b).

The complex functions $\beta (z)$ and $\gamma (z)$ are four-valued functions (having four Riemann sheets). Therefore, the BPs $\pm ib_1$ and $\pm ib_2$ are double valued BPs for $\beta (z)$ and $\gamma (z)$. Also, $z=0$ and $z= \pm ib_0$ are additional BPs. However, $z=\infty$ is not a BP as it can be readily shown by utilizing the transformation $z=1/t$ with $t \rightarrow 0$. Now, depending on the parameters $h_0$ and $m$, the branch cuts for the functions $\beta (z)$ and $\gamma (z)$ are illustrated in Figs. 16 and 17, respectively, with $\varepsilon$ being a real number such that $\varepsilon \rightarrow +0$. In fact, introducing $\varepsilon$ facilitates the introduction of the branch cuts corresponding to the branch point (BP) $z=0$ of the function $\gamma (z)$. It is also noted that in all cases the BPs of the function $\chi (z)$ are also BPs of $\beta (z)$ and $\gamma (z)$. We remark that the specific introduction of the branch cuts secures that the functions $(\chi ,\beta ,\gamma )$ are always single-valued and positive along the real axis. Finally, we note that $m<m_R$ with $m_R=\min \{1/h_0,1/h_0^*\}$ in order for the crack to propagate with a sub-Rayleigh speed.

Fig. 16

Branch cuts for the function $\beta (z)$

Fig. 17

Branch cuts for the function $\gamma (z)$

Appendix 2: Factorization of the kernel function $N(z)$

The kernel $N(z)$ is given by the expression

$$\begin{aligned} N(z)=\frac{1}{1-d}\left[ \frac{\theta ^2+\left[ z^2\right] ^{1/2} \theta +m^2}{\ell \theta (\beta +\gamma )}-d\right] , \end{aligned}$$

(133)

where the complex function $\theta \equiv \theta (z)$ is defined in (85). It is noted that the kernel function $N(z)$ does not have any poles or zeros in the finite complex domain in the sub-Rayleigh regime, only BPs which depend upon the values of $(h_0,m)$. In particular, we distinguish between the following three cases (see Fig. 18):

• Case I. $h_0\le 1$ and $m\le \dfrac{\sqrt{1-\sqrt{1-h_0^2}}}{h_0}$; then the BPs are: $0$ and $\pm ib_0$.

• Case II. $h_0\le 1$ and $m>\dfrac{\sqrt{1-\sqrt{1-h_0^2}}}{h_0}$; then the BPs are: $0$, $\pm ib_0$ and $\pm ib_1$.

• Case III. $h_0>1$; then the BPs are: $0$ and $\pm ib_0$.

Fig. 18

Branch cuts and integration paths for the factorization of the kernel function $N(z)$. The positive imaginary parts of $N(z)$ are given by (136) and (137), respectively

Note, that $m \le m_R$, always. It should be remarked that for the Cases I and III, the points: $\pm ib_1$ and $\pm ib_2$ are not BPs of the kernel function $N(z)$. Indeed, although these points are BPs for the functions $\beta (z)$, $\gamma (z)$ and $\chi (z)$ as we have shown, they are removable BPs for $N(z)$. This can be readily shown by expanding $N(z)$ in series around these points

$$\begin{aligned} N(z)&= a_0(m,h_0) \pm a_1(m,h_0) (z \pm ib_1)+O(z \pm ib_1)^2, \nonumber \\&\quad \text {for} \quad z \rightarrow \pm ib_1, \end{aligned}$$

(134)

$$\begin{aligned} N(z)&= a_2(m,h_0) \pm a_3(m,h_0) (z \pm ib_2)+O(z \pm ib_2)^2, \nonumber \\&\quad \text {for} \quad z \rightarrow \pm ib_2, \end{aligned}$$

(135)

where $a_n$ with $n=(0,1,2,3)$ are constants that depend on the parameters $(m,h_0)$. Thus, we see that around these points the function is single-valued. A similar situation applies for Case II where $\pm ib_2$ are removable BPs. Furthermore, for all Cases (I)–(III), when $\varepsilon <\text {Im}(z)<b_0$ and $\text {Re}(z)=+0$ (i.e. approaching the branch cut from the right), the real and imaginary parts of $N(z)$ take the following form

$$\begin{aligned}&\text {Re}\left( N(z)\right) =\frac{|z|(\theta ^2+m^2-\ell ^2\gamma ^2)}{\ell ^3(1-d)|\gamma |(\beta ^2-\gamma ^2)}-\frac{d}{1-d},\nonumber \\&\text {Im}\left( N(z)\right) =-\frac{|z|(\theta ^2+m^2-\ell ^2\beta ^2)}{\ell ^3(1-d)|\beta |(\beta ^2-\gamma ^2)}, \end{aligned}$$

(136)

whereas for Case II, when $b_0\!<\!\text {Im}(z)\!<\!b_1$ and $\text {Re}(z)\!=\!+0$, the real and imaginary parts of $N(z)$ become

$$\begin{aligned}&\text {Re}\left( N(z)\right) =-\frac{d}{1-d}, \nonumber \\&\text {Im}\left( N(z)\right) =\frac{|z|(\theta ^2+m^2+\ell ^2\beta \gamma )}{\ell ^3(1-d)\beta \gamma (\beta +\gamma )}. \end{aligned}$$

(137)

Next, taking into account that $N(z)\rightarrow 1$ as $|z| \rightarrow \infty$ and employing Jordan’s Lemma, we evaluate the functions $N^+(z)$ and $N^-(z)$ defined in Eqs. (91) and (92), by closing the original integration paths $C_u$ and $C_d$, which extend parallel to the real axis in the complex $z$-plane, with large semi-circles at infinity on the upper and lower half-planes respectively, as it is shown in Fig. 4. Then, a deformation of the integration contour along with the use of Cauchy’s theorem, allows taking as equivalent integration paths the contours $C'_u$ and $C'_d$, around the pertinent branch cuts of $N(z)$ (Fig. 18). In particular, we obtain for the function $N^-(z)$:

Cases I and III

$$\begin{aligned}&N^-(z)=\exp \left\{ -\frac{1}{2\pi i}\int \limits _{C_u}\frac{\log \left[ N(\zeta )\right] }{\zeta -z}\,d\zeta \right\} \nonumber \\&\quad =\exp \left\{ -\frac{1}{2\pi i}\int \limits _{C'_u}\frac{\log \left[ N(\zeta )\right] }{\zeta -z}\,d\zeta \right\} \nonumber \\&\quad =\exp \biggl \{-\frac{1}{2\pi i}\biggl (\int \limits _{C_{\varepsilon ^+}}+\int \limits _{+i\varepsilon }^{+ib_0}+\int \limits _{C_{b_0^+}}+\int \limits _{+ib_0}^{+i\varepsilon }\biggr )\frac{\log \left[ N(\zeta )\right] }{\zeta -z}\,d\zeta \biggr \}\nonumber \\&\quad =\exp \biggl \{-\frac{1}{2\pi }\biggl (\int \limits _{+i\varepsilon }^{+ib_0}\tan ^{-1}\biggl [\frac{\text {Im}\left( N(\zeta )\right) }{\text {Re}\left( N(\zeta )\right) }\biggr ]\frac{d\zeta }{\zeta -z}\\&\qquad +\int \limits _{+ib_0}^{+i\varepsilon } \tan ^{-1}\biggl [\frac{-\text {Im}\left( N(\zeta )\right) }{\text {Re}\left( N(\zeta )\right) }\biggr ]\frac{d\zeta }{\zeta -z}\biggr )\biggr \}\nonumber \\&\quad =\exp \biggl \{-\frac{1}{\pi }\int \limits _{i\varepsilon }^{ib_0}\tan ^{-1}\biggl [\frac{\text {Im} \left( N(\zeta )\right) }{\text {Re}\left( N(\zeta )\right) }\biggr ]\frac{d\zeta }{\zeta -z} \biggr \}.\nonumber \end{aligned}$$

(138)

Case II

$$\begin{aligned} N^-(z)&= \exp \left\{ -\frac{1}{2\pi i}\left( \int \limits _{C_{\varepsilon ^+}}+\int \limits _{+i\varepsilon }^{+ib_1}+\int \limits _{C_{b_1^+}}+\int \limits _{+ib_1}^{+i\varepsilon }\right) \frac{\log \left[ N(\zeta )\right] }{\zeta -z}\,d\zeta \right\} \nonumber \\&= \exp \left\{ -\frac{1}{\pi }\int \limits _{i\varepsilon }^{ib_1}\tan ^{-1}\left[ \frac{\text {Im}\left( N(\zeta )\right) }{\text {Re}\left( N(\zeta )\right) }\right] \frac{d\zeta }{\zeta -z}\right\} .\nonumber \\ \end{aligned}$$

(139)

where the function $\tan ^{-1}(\cdot )$ is defined as

$$\begin{aligned} \tan ^{-1}\frac{y}{x}\!=\!{{\mathrm{Tan}}}^{-1}\frac{y}{x} +{\left\{ \begin{array}{ll} 0, &{}\text {for} \quad x>0\\ \pi , &{}\text {for} \quad x\!<\!0 \quad \text {and} \quad y>0\\ -\pi , &{}\text {for} \quad x\!<\!0 \quad \text {and} \quad y \le 0\\ \end{array}\right. } \end{aligned}$$

(140)

in which ${{\mathrm{Tan}}}^{-1}(\cdot )$ is the principal value of the inverse tangent with branch cuts $(-i \infty ,-i]$ and $[i,i \infty )$. Also, it is noted that the contour integrals around the branch points of $N(\zeta )$ in Eqs. (138) and (139) are all zero. Indeed, by writing $\zeta =\zeta _k+r_0e^{i\theta }$, where $\zeta _k=\pm ik$ are the branch points of $N(\zeta )$, with $k=(\varepsilon , b_0, b_1)$, and by taking into account that $(\zeta -\zeta _k)\cdot \log N(\zeta ) \rightarrow 0$ uniformly as $r_0 \rightarrow 0$, it can readily shown that

$$\begin{aligned} \lim _{r_0 \rightarrow 0} \int \limits _{C_{k^{\pm }}} \frac{\log \left[ N(\zeta )\right] }{\zeta -z}\,d\zeta =0 \quad \text {with} \qquad k=(\varepsilon , b_0, b_1). \end{aligned}$$

(141)

Finally, by letting $\varepsilon \rightarrow 0$ and combining (138) and (139), yields a single formula for $N^{-}(z)$

$$\begin{aligned}&N^{-}(z)=\exp \left\{ -\frac{1}{\pi }\left[ \int \limits _{0}^{ib_0}\!\!\tan ^{-1}\!\left[ \frac{\text {Im}\left( N(\zeta )\right) }{\text {Re}\left( N(\zeta )\right) }\right] \frac{d\zeta }{\zeta -z}\right. \right. \nonumber \\&\qquad \quad \quad +\left. \left. a\int \limits _{ib_0}^{ib_1}\!\!\tan ^{-1}\!\left[ \frac{\text {Im}\left( N(\zeta )\right) }{\text {Re}\left( N(\zeta )\right) }\right] \frac{d\zeta }{\zeta -z}\right] \right\} , \end{aligned}$$

(142)

where the constant $a$ depends on the branches of $N(z)$ and it is defined as

$$\begin{aligned} a={\left\{ \begin{array}{ll} 1-\mathcal {H}\left( \sqrt{1-\sqrt{1-h_0^2}}-mh_0\right) , &{}\text {for}\, h_0 \le 1\\ 0, &{}\text {for}\, h_0>1 \end{array}\right. } \end{aligned}$$

(143)

with $\mathcal {H}()$ being the Heaviside step function. Similarly, integrating along $C'_d$, we evaluate the function $N^+(z)$ given in (91).

Appendix 3: The SIF in the equilibrium case

In the limit $m\rightarrow 0$, we have, according to (73) and (143), that $b_0=1$ and $a=0$. In addition, the real and imaginary parts of $N(z)$ become, respectively

$$\begin{aligned}&\text {Re}\left( N(z)\right) =\frac{1+4(1-\nu )z^2}{3-2\nu }, \nonumber \\&\qquad \text {Im}\left( N(z)\right) =\frac{4(1-\nu )}{3-2\nu }\,\frac{\left( -z^2\right) ^{\frac{3}{2}}}{(1+z^2)^{\frac{1}{2}}}, \end{aligned}$$

(144)

for $0 \le \text {Im}(z) \le 1$ and $\text {Re}(z)=+0$. In view of the above, the function $N^+(z)$ takes the following form

$$\begin{aligned}&N^+(z)\nonumber \\&\quad =\exp \biggl \{-\frac{1}{\pi }\biggl (\int \limits _{0}^{i}\tan ^{-1}\biggl [\frac{4(1-\nu ) \left( -\zeta ^2\right) ^{\frac{3}{2}}}{\left( 1+\zeta ^2\right) ^{\frac{1}{2}} \left( 1+4(1-\nu )\zeta ^2\right) }\biggr ]\frac{d\zeta }{\zeta +z}\biggr \},\nonumber \\ \end{aligned}$$

(145)

which after changing the variable of integration from $\zeta$ to $iq$ and by taking into account the properties of the inverse tangent function, yields

$$\begin{aligned}&N^+(z)\nonumber \\&=\exp \biggl \{\frac{1}{\pi }\biggl (\int \limits _{0}^{1}\left( -\frac{\pi }{2}+\tan ^{-1}\biggl [\frac{\left( 1-q^2\right) ^{\frac{1}{2}}\left( 1-4(1-\nu )q^2\right) }{4(1-\nu )q^3}\biggr ]\right) \frac{dq}{q+z/i}\biggr \}\nonumber \\&=\frac{1}{\left( 1+i/z\right) ^{\frac{1}{2}}}\exp \biggl \{\frac{1}{\pi }\biggl (\int \limits _{0}^{1}\tan ^{-1}\biggl [\frac{\left( 1-q^2\right) ^{\frac{1}{2}}\left( 1-4(1-\nu )q^2\right) }{4(1-\nu )q^3}\biggr ]\frac{dq}{q+z/i}\biggr \}.\nonumber \\ \end{aligned}$$

(146)

Finally, upon substituting $z=i\ell /L$ into the above expression, we evaluate the equilibrium SIF through the relation (117), which agrees exactly with the SIF given previously by Gourgiotis et al. (2012) for the stationary mode-II crack (see Eq. (126) in the cited work, with $\tau _0 \equiv T_0/L$).