UDC 539.371

THEORY OF ELASTICITY

V. Z. PARTON, B. A. KUDRYAVTSEV

DYNAMIC PROBLEM FOR A PLANE WITH A CRACK

(Presented by Academician L. I. Sedov, 15 VII 1968)

The problem of steady-state vibrations of an infinite plane made of an ideally brittle material with a crack of length \(2l\) along the real axis is considered. It is assumed that a normal load \(p(\theta)\cos \omega t\) is applied to the crack faces.

We introduce elliptic coordinates, in which the crack contour corresponds to the value \(\rho=0\) \((0 \leq \theta \leq 2\pi)\).

The equations of motion and the relations of elasticity theory for the amplitude values of stresses and displacements in the chosen coordinate system have the form

\[ \frac{\partial}{\partial \rho}(H\sigma_\rho) + \frac{\partial}{\partial \theta}(H\sigma_{\rho\theta}) + \frac{\partial H}{\partial \theta}\sigma_{\rho\theta} - \frac{\partial H}{\partial \rho}\sigma_\theta = -H^2\gamma\omega^2 u_\rho, \]

\[ \frac{\partial}{\partial \theta}(H\sigma_\theta) + \frac{\partial}{\partial \rho}(H\sigma_{\rho\theta}) + \frac{\partial H}{\partial \rho}\sigma_{\rho\theta} - \frac{\partial H}{\partial \theta}\sigma_\rho = -H^2\gamma\omega^2 u_\theta, \tag{1} \]

\[ \sigma_\rho = 2\mu\left( \frac{1}{H}\frac{\partial u_\rho}{\partial \rho} + \frac{1}{H^2}\frac{\partial H}{\partial \theta}u_\theta \right) + \lambda\frac{1}{H^2} \left[ \frac{\partial}{\partial \rho}(Hu_\rho) + \frac{\partial}{\partial \theta}(Hu_\theta) \right], \]

\[ \sigma_{\rho\theta} = \mu \left[ \frac{\partial}{\partial \rho}\left(\frac{u_\theta}{H}\right) + \frac{\partial}{\partial \theta}\left(\frac{u_\rho}{H}\right) \right], \]

\[ \sigma_\theta = 2\mu\left( \frac{1}{H}\frac{\partial u_\theta}{\partial \theta} + \frac{1}{H^2}\frac{\partial H}{\partial \rho}u_\rho \right) + \lambda\frac{1}{H^2} \left[ \frac{\partial}{\partial \rho}(Hu_\rho) + \frac{\partial}{\partial \theta}(Hu_\theta) \right]. \tag{2} \]

Here \(H^2=\tfrac{1}{2}l^2(\operatorname{ch}2\rho-\cos 2\theta)\); \(\gamma\) is the density of the material; \(\omega\) is the circular frequency; \(\lambda,\mu\) are Lamé constants.

Equations (1) are satisfied identically if the displacements and stresses are represented through two functions

\[ u_\rho=\frac{1}{H}\frac{\partial \varphi}{\partial \rho} + \frac{1}{H}\frac{\partial \psi}{\partial \theta}, \qquad u_\theta=\frac{1}{H}\frac{\partial \varphi}{\partial \theta} - \frac{1}{H}\frac{\partial \psi}{\partial \rho}, \]

where \(\varphi\) and \(\psi\) are solutions of the equations

\[ \nabla^2\varphi+2k_1(\operatorname{ch}2\rho-\cos 2\theta)\varphi=0, \qquad \nabla^2\psi+2k_2(\operatorname{ch}2\rho-\cos 2\theta)\psi=0, \]

\[ \nabla^2=\partial^2/\partial \rho^2+\partial^2/\partial \theta^2, \qquad k_1=\gamma\omega^2 l^2/4(\lambda+2\mu), \qquad k_2=\gamma\omega^2 l^2/4\mu. \tag{3} \]

With this representation, equations (2) are rewritten in the form

\[ \frac{1}{2\mu}\sigma_{\rho\theta} = \frac{1}{H}\frac{\partial}{\partial \theta} \left( \frac{1}{H}\frac{\partial \varphi}{\partial \rho} + \frac{1}{H}\frac{\partial \psi}{\partial \theta} \right) - \frac{1}{H^3}\frac{\partial H}{\partial \rho} \left( \frac{\partial \varphi}{\partial \theta} - \frac{\partial \psi}{\partial \rho} \right) + 2k_2\psi, \]

\[ \frac{1}{2\mu}\sigma_\rho = -\frac{1}{H}\frac{\partial}{\partial \theta} \left( \frac{1}{H}\frac{\partial \varphi}{\partial \theta} - \frac{1}{H}\frac{\partial \psi}{\partial \rho} \right) - \frac{1}{H^3}\frac{\partial H}{\partial \rho} \left( \frac{\partial \varphi}{\partial \rho} + \frac{\partial \psi}{\partial \theta} \right) - 2k_2\varphi, \]

\[ \frac{1}{2\mu}\sigma_\theta = \frac{1}{H}\frac{\partial}{\partial \theta} \left( \frac{1}{H}\frac{\partial \varphi}{\partial \theta} - \frac{1}{H}\frac{\partial \psi}{\partial \rho} \right) + \frac{1}{H^3}\frac{\partial H}{\partial \rho} \left( \frac{\partial \varphi}{\partial \rho} + \frac{\partial \psi}{\partial \theta} \right) - \frac{\lambda}{2\mu}\frac{\gamma\omega^2}{(\lambda+2\mu)}\varphi. \tag{4} \]

Separating variables in equations (3), we obtain ordinary differential equations whose solutions are Mathieu functions \((^1)\).

Using the symmetry of the stress state and the conditions at infinity, we represent the solutions of equations (3) in the form

\[ \varphi(\rho,\theta)= \sum_{m=0}^{\infty} C_m\,\mathrm{Fey}_{2m}(\rho,k_1)\,\mathrm{ce}_{2m}(\theta,k_1), \]

\[ \psi(\rho,\theta)= \sum_{m=0}^{\infty} D_m\,\mathrm{Gey}_{2m+2}(\rho,k_2)\,\mathrm{se}_{2m+2}(\theta,k_2). \tag{5} \]

Here \(C_m, D_m\) are constants; \(\mathrm{ce}_{2m}(\theta,k_1)\), \(\mathrm{se}_{2m+2}(\theta,k_2)\) are periodic Mathieu solutions; \(\mathrm{Fey}_{2m}(\rho,k_1)\), \(\mathrm{Gey}_{2m+2}(\rho,k_2)\) are the second solutions corresponding to the modified Mathieu equation (1), and in what follows representations of these functions for small values of \(k_1, k_2\) will be used.

The conditions on the contour of the cut \((\rho=0)\) have the form

\[ \sigma_{\rho\theta}=0,\qquad \sigma_\rho=q(\theta). \tag{6} \]

Taking (4) into account, from (6) we obtain

\[ \left(\frac{\partial \varphi}{\partial \rho}+\frac{\partial \psi}{\partial \theta}\right)_{\rho=0} = -2k_2\sin\theta\int_0^\theta \psi\big|_{\rho=0}\sin\theta\,d\theta +l\sin\theta\,\chi_1, \]

\[ \left(\frac{\partial \varphi}{\partial \theta}-\frac{\partial \psi}{\partial \rho}\right)_{\rho=0} = -2k_2\sin\theta\int_{\pi/2}^{\theta}\varphi\big|_{\rho=0}\sin\theta\,d\theta -\frac{l^2}{2\mu}\sin\theta\int_{\pi/2}^{\theta}\sigma_0\big|_{\rho=0}\sin\theta\,d\theta + \]

\[ \qquad\qquad\qquad\qquad +l\sin\theta+\chi_2. \tag{7} \]

where \(\chi_1=0,\ \chi_2=0\) by virtue of the obvious equalities

\[ u_\rho\big|_{\rho=0,\ \theta=0}=0,\qquad u_\theta\big|_{\rho=0,\ \theta=\pi/2}=0. \]

We expand the function \(q(\theta)\) specified on the contour of the cut in a series in Mathieu functions

\[ q(\theta)=\sum_{m=0}^{\infty}q_m\,\mathrm{ce}_{2m}(\theta,k_1), \qquad q_m=\frac{1}{\pi}\int_0^{2\pi}q(\theta)\mathrm{ce}_{2m}(\theta,k_1)\,d\theta. \tag{8} \]

Substituting the series (5), (8) into (7) and taking into account the expansions in Fourier series of the periodic solutions of Mathieu’s equation (1),

\[ \mathrm{ce}_{2m}(\theta,k_1)=\sum_{r=0}^{\infty}A_{2r}^{(2m)}\cos 2r\theta,\qquad \mathrm{se}_{2m+2}(\theta,k_2)=\sum_{r=0}^{\infty}B_{2r+2}^{(2m+2)}\sin(2r+2)\theta, \]

we obtain two infinite systems for determining the unknown constants \(C_m,D_m\)

\[ \sum_{m=0}^{\infty} \left\{ C_m\,\mathrm{Fey}'_{2m}(0,k_1)A_{2r}^{(2m)} + D_m\,\mathrm{Gey}_{2m+2}(0,k_2) \left[ 2rB_{2r}^{(2m+2)} -\frac{2r}{4r^2-1}k_2B_{2r}^{(2m+2)} +\right.\right. \]

\[ \left.\left. +\frac{k_2}{2(2r+1)}B_{2r+2}^{(2m+2)} +\frac{k_2}{2(2r-1)}B_{2r-2}^{(2m+2)} \right] \right\} =0,\qquad B_{-2}=B_0=0\quad (r=0,1,2\ldots), \tag{9} \]

\[ \sum_{m=0}^{\infty} \left\{ C_m\,\mathrm{Fey}_{2m}(0,k_1) \left[ 2rA_{2r}^{(2m)} -k_2\frac{2r}{(4r^2-1)}A_{2r}^{(2m)} +\right.\right. \]

\[ \left.\left. +\frac{1}{2}k_2\frac{(1+\delta_r^{(1)})}{(2r-1)}A_{2r-2}^{(2m)} +\frac{1}{2}k_2\frac{1}{(2r+1)}A_{2r+2}^{(2m)} \right] \right\} = \]

\[ =\frac{l^2}{4\mu}\sum_{m=0}^{\infty}q_m \left[ \frac{2r}{(4r^2-1)}A_{2r}^{(2m)} -\frac{1}{2}\frac{(1+\delta_r^{(1)})}{(2r-1)}A_{2r-2}^{(2m)} -\frac{1}{2}\frac{1}{(2r+1)}A_{2r+2}^{(2m)} \right] \]

\[ (r=0,1,2,\ldots),\qquad \delta_r^{(1)}=1\ \text{for } r=1;\qquad \delta_r^{(1)}=0\ \text{for } r=0. \tag{10} \]

We use the known relations (1)

\[ \mathrm{ce}_{2m}(0,k_1)=\sum_{r=0}^{\infty}A_{2r}^{(2m)},\qquad \mathrm{se}'_{2m+2}(0,k_2)=\sum_{r=0}^{\infty}(2r+2)B_{2r+2}^{(2m+2)}. \]

Adding all the equations of system (9), and analogously (10), we find, by virtue of the uniform continuity of the coefficients,

\[ C_m\,\mathrm{Fey}_{2m}(0,k_1)\mathrm{ce}_{2m}(0,k_1) + D_m\,\mathrm{Gey}_{2m+2}(0,k_2)\mathrm{se}'_{2m+2}(0,k_2) =0, \tag{11} \]

\[ C_m \operatorname{Fey}_{2m}(0,k_1)\left(\sum_{r=1}^{\infty}2rA_{2r}^{(2m)}+k_2A_0^{(2m)}\right)+ \]
\[ +\,D_m \operatorname{Gey}'_{2m+2}(0,k_2)\sum_{r=0}^{\infty}B_{2r+2}^{(2m+2)} =-\frac{l^2}{4\mu}\,q_m A_0^{(2m)} . \tag{11} \]

Equating to zero the determinant of the system (11), we obtain equations for determining the natural frequencies

\[ \Delta_m \equiv \operatorname{Fey}'_{2m}(0,k_1)\operatorname{Gey}'_{2m+2}(0,k_2)\operatorname{ce}_{2m}(0,k_1) \sum_{r=0}^{\infty}B_{2r+2}^{(2m+2)}- \]
\[ -\operatorname{Fey}_{2m}(0,k_1)\operatorname{Gey}_{2m+2}(0,k_2)\operatorname{se}'_{2m+2}(0,k_2) \left(\sum_{r=0}^{\infty}2rA_{2r}^{(2m)}+k_2A_0^{(2m)}\right)=0. \tag{12} \]

If the given frequency is such that \(\Delta_m\ne0\), then from (11) we determine the unknowns \(C_m,D_m\)

\[ D_m=-C_m\frac{\operatorname{Fey}'_{2m}(0,k_1)\operatorname{ce}_{2m}(0,k_1)} {\operatorname{Gey}_{2m+2}(0,k_2)\operatorname{se}'_{2m+2}(0,k_2)}, \]
\[ C_m=-\frac{l^2}{4\mu}\,q_m \frac{A_0^{(2m)}\operatorname{Gey}_{2m+2}(0,k_2)\operatorname{se}'_{2m+2}(0,k_2)} {\Delta_m}. \tag{13} \]

Let us use the known relations (1)

\[ \operatorname{Fey}'_{2m}(0,k_1)=\frac{2}{\pi}\frac{p_{2m}}{A_0^{(2m)}} \operatorname{ce}_{2m}\left(\frac{\pi}{2},k_1\right), \]

\[ \operatorname{Gey}_{2m+2}(0,k_2)=-\frac{2}{\pi} \frac{s_{2m+2}\operatorname{se}'_{2m+2}(\pi/2,k_2)} {k_2B_2^{(2m+2)}}, \]

\[ p_{2m}=\frac{\operatorname{ce}_{2m}(0,k_1)\operatorname{ce}_{2m}(\pi/2,k_1)} {A_0^{(2m)}},\qquad s_{2m+2}=\frac{\operatorname{se}'_{2m+2}(0,k_2)\operatorname{se}'_{2m+2}(\pi/2,k_2)} {k_2B_2^{(2m+2)}}. \]

Then, taking this into account, from (12), (13)

\[ \Delta_m=\frac{2}{\pi}\left[ p_{2m}^{\,2}\operatorname{Gey}'_{2m+2}(0,k_2)\sum_{r=0}^{\infty}B_{2r+2}^{(2m+2)}+ \right. \]
\[ \left. +\,s_{2m+2}^{\,2}\operatorname{Fey}_{2m}(0,k_1) \left(\sum_{r=0}^{\infty}2rA_{2r}^{(2m)}+k_2A_0^{(2m)}\right)\right], \tag{14} \]

\[ D_m=C_m\frac{p_{2m}^{\,2}}{s_{2m+2}^{\,2}},\qquad C_m=-\frac{l^2}{2\pi\mu}\,q_m \frac{A_0^{(2m)}s_{2m+2}^{\,2}}{\Delta_m}. \tag{15} \]

Taking (4) into account, the normal stresses at the points of the real axis on the continuation of the cut are

\[ \sigma_\theta\big|_{\theta=0}= \frac{2\mu}{l^2}\frac{1}{\operatorname{sh}^3\rho} \left[ \operatorname{sh}\rho\left(\frac{\partial^2\varphi}{\partial\theta^2} -\frac{\partial^2\psi}{\partial\rho\,\partial\theta}\right)_{\theta=0} + \operatorname{ch}\rho\left(\frac{\partial\varphi}{\partial\rho} +\frac{\partial\psi}{\partial\theta}\right)_{\theta=0} \right] -\frac{\lambda\gamma\omega^2}{\lambda+2\mu}\,\varphi\bigg|_{\theta=0} = \]
\[ =\frac{2\mu}{l^2}\frac{1}{\operatorname{sh}^3\rho} \sum_{m=0}^{\infty}\Omega_m(\rho) -\frac{\lambda\gamma\omega^2}{\lambda+2\mu}\,\varphi\bigg|_{\theta=0}. \tag{16} \]

Here it is denoted that

\[ \sum_{m=0}^{\infty}\Omega_m(\rho)= \operatorname{sh}\rho\left(\frac{\partial^2\varphi}{\partial\theta^2} -\frac{\partial^2\psi}{\partial\rho\,\partial\theta}\right)_{\theta=0} + \operatorname{ch}\rho\left(\frac{\partial\varphi}{\partial\rho} +\frac{\partial\psi}{\partial\theta}\right)_{\theta=0} = \]
\[ =\sum_{m=0}^{\infty}C_m\left\{ \operatorname{sh}\rho\left[ \operatorname{Fey}_{2m}(\rho,k_1)\operatorname{ce}''_{2m}(0,k_1) -\frac{p_{2m}^{\,2}}{s_{2m+2}^{\,2}} \operatorname{Gey}'_{2m+2}(\rho,k_2)\operatorname{se}'_{2m+2}(0,k_2) \right] +\right. \]
\[ \left. +\operatorname{ch}\rho\left[ \operatorname{Fey}'_{2m}(\rho,k_1)\operatorname{ce}_{2m}(0,k_1) +\frac{p_{2m}^{\,2}}{s_{2m+2}^{\,2}} \operatorname{Gey}_{2m+2}(\rho,k_2)\operatorname{se}'_{2m+2}(0,k_2) \right]\right\}. \tag{17} \]

To establish the dependence between the crack length and the applied load, we use Irwin’s relation [2]

\[ \lim_{s\to 0}\sqrt{2\pi s\,\sigma_\theta}\big|_{\theta=0}=K_c, \tag{18} \]

where

\[ s=x\big|_{\theta=0}-l=l(\operatorname{ch}\rho-1)=2l\,\operatorname{sh}^2{\rho\over 2} \]

is the distance from the end of the crack, and \(K_c\) is the fracture toughness [2].

Taking (16) and (17) into account, we obtain

\[ \lim_{s\to 0}\sqrt{2\pi s\,\sigma_\theta}\big|_{\theta=0} = \lim_{\rho\to 0}{\sqrt{2l}\,\mu\over l^2}\,{1\over 4} \sum_{m=0}^{\infty}{\Omega_m(\rho)\over \operatorname{sh}^2\rho/2} = {\mu\sqrt{2l}\over 2l^2} \sum_{m=0}^{\infty}\Omega_m''(0). \]

Fig. 1

We use the known relations

\[ \operatorname{Fey}_{2m}''(0,k_1) = (a_{2m}-2k_1)\operatorname{Fey}_{2m}(0,k_1), \]

\[ \operatorname{Gey}_{2m+2}''(0,k_2) = (b_{2m+2}-2k_2)\operatorname{Gey}_{2m+2}(0,k_2); \]

then we obtain the final result

\[ -{2\mu\over \sqrt{\pi\,l^{3/2}}} \sum_{m=0}^{\infty} C_m(2k_1-2k_2-a_{2m}+b_{2m+2})\,p_{2m}^{\,2} = K_c, \tag{19} \]

where \(a_{2m}\), \(b_{2m+2}\) are the eigenvalues of the Mathieu functions \(\operatorname{ce}_{2m}(\theta k_1)\) and \(\operatorname{se}_{2m+2}(\theta,k_2)\), respectively.

Fig. 2

Figures 1 and 2 show the curves of dependence (19) at different frequencies of oscillation of the external load for the case of concentrated forces and for the case of a uniform load applied to the crack faces (the dashed line corresponds to the static case \(\omega=0\)). The constructed solution shows that the inertial effect reduces the magnitude of the breaking load for a given crack length.

All-Union Correspondence
Civil Engineering Institute

Received
8 VII 1968

REFERENCES

1. N. V. Mak-Lakhlan, Theory and Applications of Mathieu Functions, Moscow, 1953.
2. G. R. Irwin, Fracture, Handb. Phys., Berlin, 6, 1958.