Nomenclature

$\pm{a_1}, \pm{b_1}, c_1, d_1$	crack tips
$\pm{a}, \pm{b}$	tips of the developed yield plastic zones
$D_i\ (i=0, 1, 2, 3)$	constants of the problem
$E$	Young's modulus
$F(\theta, k), E(\theta, k), \Pi(\theta, \alpha^2, k)$	incomplete elliptic integral of first, second, third kind, respectively
$L_i\ (i = 1, 2, 3)$	cracks
$P_n(z)$	polynomial of degree n
$p(t), q(t)$	applied stresses on the yield zones
$u, v$	components of displacement
$X_x, Y_y, X_y$	components of stress
$z=x+iy$	complex variable
$\gamma$	Poisson's ratio
$\mu$	shear modulus
$\delta(x)$	crack-tip-opening displacement at the crack tip x
$\Omega(z)=\omega^\prime(z), \Phi(z)=\phi^\prime(z)$	complex stress functions
$\Gamma^\prime$	$-\frac{1}{2}(N_1 - N_2)e^{-2i\alpha}, N_1$ and $N_2$ are the values of principal stresses at infinity, $\alpha$ be the angle between $N_1$ and the ox-axis
$\Gamma_i\ (i = 1, 2, \dots, 6)$	developed plastic/yield zones
$\kappa$	$=\frac{3-\gamma}{1+\gamma}$ for the plane-stress, $=3-4\gamma$ for the plane-strain
$\sigma_\infty$	remotely applied stress at infinite boundary of the plate
$\sigma_{ye}$	yield stress of the plate

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1. Introduction

Sometimes man-made structures are catastrophic due to decreasing the residual strength of the material used for the development and propagation of cracks or crack like defects ^[1]. Human life sometimes is in danger since these structures break unconditionally. A number of aircraft and ships were failed at initial stage due to the presence of multiple interacting cracks in the manufactured structures. Therefore, it becomes imperative to study the load bearing capacity of the plate containing defects. Dugdale ^[2] proposed a classical single crack model to evaluate load carrying capacity of a plate. This model was used and modified by Kanninen ^[3] for linear stress distribution. Effect of variable stress distribution on the rims of developed yield zones studied by Harrop ^[4].

Cracks in the structure increase initially as non-interacting and isolated defects. As the length of cracks increases (in other words distance between cracks decreases) a destructive interaction between cracks increases rapidly causes the failure of the structures. So the available results of single crack are not appropriate for the modelling of multiple site damage problem ^[5]. Hence, Collins et al. ^[6] and Chang et al. ^[7] applied Dugdale model to obtain the solution of two equal straight cracks in an infinite plate, they also discussed the conditions of coalescence of yield zones between cracks. Dugdale strip yield model for three collinear straight cracks was studied by Hasan et al. ^[8] and further modified the model for linear stress distribution ^[9].

Many efforts, in fact, have been made for the case of separated yield zones as far as multiple cracks are the concern. However, some efforts have been made by various researchers ^[10,11] to investigate the load bearing capacity of the cracked plate when yield zones were coalesced due to increase in applied stresses. As discussed by Gdoutos ^[12] that some of the structure fail at a stress, which is below the yield stress of the plate. Therefore, assuming the variable stress distribution acting on the rims of yield zones in order to arrest the cracks from further opening. Few cases of linear stress distribution have been considered by Tang et al. ^[13], Hasan ^[14] and Tada et al. ^[15]. Therefore, the purpose of this work is to study the interaction of multiple cracks when yield zones developed between two closely located cracks coalesced under the application of linearly varying stress distribution.

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2. Basic Mathematical Formulation

Components of stresses ($X_x, Y_y, X_y$) and displacements ($u, v$) are same as given by Muskhelishvili ^[16] in terms of two complex potential functions $\Phi(z), \Omega(z)$ as

$\label{e1} X_x + Y_y=2[\Phi(z)+\overline{\Phi(z)}],$

(1)

$\label{e2} Y_y-i X_y=\Phi(z)+\overline{\Omega(z)}-(z-\overline{z})\Phi^\prime(z),$

(2)

$\label{e3}2\mu(u+iv)=\chi\phi(z)-z\overline{\phi^{\prime}(z)}-\overline{\psi(z)}.$

(3)

where bar over the function or variable denotes its complex conjugate and prime its first order derivative.

If the stresses $X^{\pm}_y, Y^{\pm}_y$ are applied over the rims of the cracks, where superscript $(+)$ & $(-)$ denotes the values of stress components on the upper and lower rims of the cracks, the boundary value problem may be expressed as

$\label{e4}Y^+_y-iX^+_y=\Phi^+(t)+\Omega^-(t),$

(4)

$\label{e5}Y^-_y-iX^-_y=\Phi^-(t)+\Omega^+(t),$

(5)

under the assumption $\lim\limits_{y\rightarrow 0}y\Phi^\prime(t+iy) =0$.

General solution of the boundary value problems given in Eqs. (4) and (5) is obtained using Sokhotski Plemelj formula and written as

$\label{e8}\Phi(z)+\Omega(z)=\frac{1}{\pi iX(z)}\int\limits_L \frac{X(t)p(t)dt}{t-z}+\frac{2P_n(z)}{X(z)},$

(6)

$\label{e9}\Phi(z)-\Omega(z)=\frac{1}{\pi i}\int\limits_L \frac{q(t)dt}{t-z}-\overline{\Gamma'},$

(7)

where

$p(t)=\frac{1}{2}[Y^+_y +Y^-_y]-\frac{i}{2}[X^+_y +X^-_y], \quad q(t)=\frac{1}{2}[Y^+_y -Y^-_y]-\frac{i}{2}[X^+_y -X^-_y], \\ X(z)=\prod^n_{j=1}\sqrt{z-a_j}\sqrt{z-b_j}, \quad P_n(z)=D_0z^n + D_1z^{n-1} +D_2z^{n-2}+\dots+D_n,$

$a_j, b_j$ are the end points of $j^{th}$ crack.

Unknown constants $D_i\ (i = 0, 1, 2, \dots, n)$ of polynomial Pn(z) may be obtained using condition of single-valuedness of displacement around the rims of the cracks,

$2(\chi+1)\int\limits_{L_i}\frac{P_n(t)}{X(t)}dt+\chi\int\limits_{L_i}[\Phi_0^+(t)-\Phi_0^-(t)]dt+\int\limits_{L_i}[\Omega_0^+(t)-\Omega_0^-(t)]dt=0.$

(8)

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3. Problem Formulation

The present analysis is a modification of Dugdale model under the influence of linear stress distribution. Consider a plate is to be infinitely extended and contains three unequal cracks $L_1, L_2, L_3$ of real lengths $(-a_1, -b_1)$, $(b_1, c_1)$, $(d_1, a_1)$ along the x-axis. Boundary of the plate is subjected to uniform stress distribution, $Y_y=\sigma_\infty$ opens the rims of the cracks in mode-I type deformation. As a result, at each crack tip yield zone develop and grow on increasing stresses at the boundary of the plate. Therefore, yield zones developed at interior tips of two closely located cracks get coalesced. These zone are denoted by $\Gamma_i \ (i=1, 2, 3, \dots, 6)$ and occupy the intervals $(-a, -a_1)$, $(-b_1, -b)$, $(b, b_1)$, $(c_1, d_1)$, $(a_1, a)$ respectively, on ox-axis.

Further damage in the plate is seized by applying a linear stress distribution $Y_y=\frac{t}{a}\sigma_{ye}$ over the rims of developed yield zones. The entire configuration of this problem is given in Figure 1.

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4. Solution to the Problem

The solution of the problem stated in Section 3 is obtained by superposition solution of two states of the problem. The first state is the tensile case when the infinite boundary is subjected to uniform stress distribution. The second state of the problem is to apply a linear stress distribution on the rims of the yield zones.

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4.1. State-1: Tensile Case

Consider, two cracks exist on the real axis in an infinite plate which is subjected to uniform tension, $\sigma_\infty$. The entire configuration is shown in Figure 2.

The problem of tensile case is solved under the following boundary conditions,

$\label{e11} Y_y=\sigma_{\infty}, X_y=0, for\quad y\rightarrow \pm\infty, -\infty＜x＜\infty$

(9)

$\label{e12} Y_y=X_y=0, for\quad y=0, x\in \bigcup\limits_{i=1}^2 R_i$

(10)

The desired complex potential function $\Phi_{A}(z)$ for this case is calculated using mathematical formulation given in Section 2 and boundary conditions (9) and (10) as,

$\label{e14}\Phi_A(z)=\frac{\sigma_{\infty}}{2}\Big[\frac{1}{X(z)}(z^2-a^2\lambda^2)-\frac{1}{2}\Big]$

(11)

where $\lambda^2 =\frac{E(k)}{F(k)}, k^2=\frac{a^2-b^2}{a^2}$ and $F(k), E(k)$ are complete elliptical integrals of first and second kind, respectively, defined in Byrd ^[17].

Formula given here of calculating stress intensity factors for the configuration of Figure 2 at crack tip $z=z_1$ is taken from ^[6],

$\label{e19}K_I=2\sqrt{2\pi}\lim_{z\rightarrow z_1}\sqrt{z-z_1}\Phi_A(z).$

(12)

Hence, opening mode stress intensity factors at crack tips a and b are determine by substituting $\Phi_A(z)$ from Eq. (11) into Eq. (12) and after taking corresponding limits one can write

$\label{e20}(K_I^A)_a=\sigma_\infty\sqrt{\frac{\pi}{a}} \frac{a^2}{\sqrt{a^2-b^2}} (1-\lambda^2),$

(13)

$\label{e21}(K_I^A)_b=\sigma_\infty\sqrt{\frac{\pi}{b}} \frac{1}{\sqrt{a^2-b^2}} \big(b^2-a^2\lambda^2\big),$

(14)

where subscript I refers to mode-I type deformation, and superscript A refers to sub-problem-A.

Displacement components for the tensile case is obtained on putting the value of $\Phi_A(z)$ from Eq. (11) into Eq. (3) as,

$\label{ed1}(v^{\pm}_A)_a=\pm\frac{2a\sigma_{\infty}}{E}\Big[E(\theta_1, k)-\lambda^2F(\theta_1, k)\Big],$

(15)

$\label{ed2}(v^{\pm}_A)_b=\mp\frac{2a\sigma_{\infty}}{E}\Big[E(\theta_2, k)-\lambda^2F(\theta_2, k)-\frac{k^2\sin{\theta_2}\cos{\theta_2}}{\sqrt{1-k^2\sin^2{\theta_2}}}\Big],$

(16)

where $\theta_1=sin^{-1}\sqrt{\frac{a^2-a_1^2}{a^2-b^2}}, \quad \theta_2=sin^{-1}\sqrt{\frac{a^2-b_1^2}{a^2-b^2}}$.

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4.2. State-2: Yield Case

In this section, we aim to furnish a mathematical model for an isotropic infinite plate which is damaged by three collinear straight cracks with coalesced yield zones at the interior tip of two closely located cracks tips. These cracks are assumed to be located along ox-axis and denoted by $L_i \ (i=1, 2, 3)$. Cracks together with corresponding yield zones are treated as physical cracks of length $(\Gamma_{1}\bigcup L_1\bigcup\Gamma_{2}),$ and $(\Gamma_{3}\bigcup L_2\bigcup\Gamma_{4}\bigcup L_3\bigcup\Gamma_{5})$. The model may be illustrated by means of Figure 3. Rims of developed yield zones are subjected to a compressive stress distribution to detain the cracks from further opening. According to Godoutos ^[12], in history, a number of failures occurred at a stress which is well below the yield strength of the plate. Therefore a linearly varying stress $\frac{t}{a}\sigma_{ye}$ (stress which is below the yield strength of the plate) distribution is considered over the rims of developed yield zones.

The boundary conditions of the said problem are:

$\label{e22} Y_y=0, X_y=0, for\quad y\rightarrow \pm\infty, -\infty＜x＜\infty$

(17)

$\label{e23} Y_y=\frac{t}{a}\sigma_{ye}, \quad X_y=0, for\quad y\rightarrow 0, x\in\bigcup\limits_{n=1}^{6}\Gamma_n$

(18)

Using methodology given in Section 2 and boundary conditions (17) and (18), complex potential function for the sub-problem-B is obtained and written as:

$\label{e25}\Phi_B(z)=\frac{\sigma_{ye}}{2\pi iX(z)}\Big[\int\limits_{L'}\frac{tX(t)dt}{a(t-z)}+iD_2\Big]$

(19)

where $L'=[-a, -a_1] \bigcup [-b_1, -b] \bigcup [b, b_1] \bigcup [c_1, d_1] \bigcup [a_1, a]$, i.e., the loaded section in Figure 3. Using the fact that,

$\label{e26}X(t)=\sqrt{t-a}\sqrt{t+a}\sqrt{t-b}\sqrt{t+b}=i\sqrt{a^2-t^2}\sqrt{t^2-b^2},$

(20)

$\label{e27}X(-t)=\sqrt{-t-a}\sqrt{-t+a}\sqrt{-t-b}\sqrt{-t+b}= -i \sqrt{a^2-t^2}\sqrt{t^2-b^2}.$

(21)

The integral shown in Eq. (19) may be evaluated as,

$\int_{L'}\frac{tX(t)dt}{a(t-z)}=\frac{2}{a}\int\limits_{a_1}^a\frac{t^2X(t)}{t^2-z^2}dt+\frac{2}{a}\int\limits_{b}^{b_1}\frac{t^2X(t)}{t^2-z^2}dt +\frac{1}{a}\int\limits_{c_1}^{d_1}\frac{(t^2+tz)X(t)}{t^2-z^2}dt, \\ =\frac{2i}{a}\Bigg[\int\limits_{a_1}^a\frac{t^2\sqrt{a^2-t^2}\sqrt{t^2-b^2}}{t^2-z^2}dt+\int\limits_{b}^{b_1}\frac{t^2\sqrt{a^2-t^2}\sqrt{t^2-b^2}}{t^2-z^2}dt \\+\frac{1}{2}\int\limits_{c_1}^{d_1}\frac{t^2\sqrt{a^2-t^2}\sqrt{t^2-b^2}}{t^2-z^2}dt+\frac{z}{2}\int\limits_{c_1}^{d_1}\frac{t\sqrt{a^2-t^2}\sqrt{t^2-b^2}}{t^2-z^2}dt\Bigg],$

Hence,

$\label{e28}\int_{L^\prime}\frac{tX(t)dt}{a(t-z)}=2ia^2k^2\Bigg[I_1+\Big(1-\frac{1}{n^2}\Big)J_1+\frac{z}{a}\bigg\{I_2+\Big(1-\frac{1}{n^2}\Big)J_2\bigg\}\Bigg],$

(22)

where

$I_1=i_1(\theta_1)-i_1(\theta_2)+i_1\Big(\frac{\pi}{2}\Big)+\frac{1}{2}\Big(i_1(\theta_3)-i_1(\theta_4)\Big), \\ J_1=j_1(\theta_1)-j_1(\theta_2)+j_1\Big(\frac{\pi}{2}\Big)+\frac{1}{2}\Big(j_1(\theta_3)-j_1(\theta_4)\Big), \\ I_2=\frac{1}{2}\Big(i_2(\theta_3)-i_2(\theta_4)\Big), \quad J_2=\frac{1}{2}\Big(j_2(\theta_3)-j_2(\theta_4)\Big), \\ n^2=\frac{a^2-b^2}{a^2-z^2}, \quad \theta_3=sin^{-1}\sqrt{\frac{a^2-c_1^2}{a^2-b^2}}, \quad \theta_4=sin^{-1}\sqrt{\frac{a^2-d_1^2}{a^2-b^2}}, \\ i_1(\phi)=\frac{1}{3k^2}\Big[(1-k^2)F(\phi, k)+(2k^2-1)E(\phi, k)-k^2\sin\phi\cos\phi\sqrt{1-k^2\sin^2\phi}\Big], \\ j_1(\phi)=\Big[\frac{k^2}{n^2}F(\phi, k)-E(\phi, k)+\frac{n^2-k^2}{n^2}\Pi(\phi, n, k)\Big], \\ i_2(\phi)=\frac{1}{2}\big(\phi-\sin\phi\cos\phi\big), \quad j_2(\phi)=\Bigg[\frac{\tanh^{-1}\Big(\sqrt{n^2-1}\tan\phi\Big)}{\sqrt{n^2-1}}-\phi\Bigg].$

Constant $D_2$ is then obtained by using condition of single-valuedness of displacement around the rims of cracks given and written as ,

$\label{e29}D_2=-2a^2k^2(T_1-\lambda^2T_2),$

(23)

where

$T_1=t_1(\theta_1)-t_1(\theta_2)+t_1(\frac{\pi}{2})+\frac{1}{2}t_1(\theta_3)-\frac{1}{2}t_1(\theta_4), \\ T_2=t_2(\theta_1)-t_2(\theta_2)+t_2(\frac{\pi}{2})+\frac{1}{2}t_2(\theta_3)-\frac{1}{2}t_2(\theta_4), \\ t_1(\phi)=\frac{\sin\phi\cos\phi\sqrt{1-k^2\sin^2\phi}}{6}-\frac{1-k^2}{6k^2}F(\phi, k)+\Big(\frac{2-k^2}{3k^2}-\frac{1-k^2\sin^2\phi}{2k^2}\Big)E(\phi, k)\\ t_2(\phi)=\frac{E(\phi, k)}{2k^2}-\frac{1-k^2\sin^2\phi}{2k^2}F(\phi, k).$

Thus, after a long mathematical calculations, final expressions for complex potential function $\Phi_B(z)$ of closing case is obtained on substituting Eqs. (22) and (23) into Eq. (19),

$\label{e30}\Phi_B(z)=\frac{a^2k^2\sigma_{ye}}{\pi X(z)}\bigg[I_1+(1-\frac{1}{n^2})J_1+\frac{z}{a}\Big(I_2+(1-\frac{1}{n^2})J_2\Big)-(T_1-\lambda^2T_2)\bigg],$

(24)

Stress intensity factors at crack tips a, b for linear stress distribution is obtained by putting the value of $\Phi_B(z)$ from Eq. (24) into Eq. (12), therefore

$\label{e31}(K_I^B)_a=\frac{2a^2k^2\sigma_{ye}}{\sqrt{a\pi}\sqrt{a^2-b^2}}\Bigg(I_1-E_1+I_2-\phi_1-(T_1-\lambda^2T_2)\Bigg),$

(25)

$\label{e32}(K_I^B)_b=\frac{2a^2k^2\sigma_{ye}}{\sqrt{b\pi}\sqrt{a^2-b^2}}\Bigg(I_1+\frac{b}{a}I_2-(T_1-\lambda^2T_2)\Bigg).$

(26)

where

$E_1=E(\theta_1, k)-E(\theta_2, k)+E(\frac{\pi}{2}, k)+\frac{1}{2}E(\theta_3, k)-\frac{1}{2}E(\theta_4, k), \quad \phi_1=\frac{1}{2}\theta_3-\frac{1}{2}\theta_4$

Components of displacement are obtained on putting the value of $\Phi_B(z)$ in Eq. (3),

$\label{e35}(v^{\pm}_B)_{a_1}=\pm\frac{4a}{\pi E}\Big(A_1+A_2\Big),$

(27)

$\label{e36}(v^{\pm}_B)_{b_1}=\mp\frac{4a}{\pi E}\Big(B_1+B_2\Big),$

(28)

where

$A_1=\Big\{I_1-(T_1-\lambda^2T_2)\Big\}F(\theta_1, k)k^2-\frac{F_1}{2}\Big\{(1-k^2)F(\theta_1, k)+k^2S(\theta_1)\Big\}+\\ I_2k^2\theta_1-\frac{\phi_1k^2}{2}\Big\{\frac{\sin{2\theta_1}}{2}+\theta_1\Big\}-\frac{E_1}{2}\Big\{E(\theta_1, k)-2(1-k^2)F(\theta_1, k)\Big\}, \\ A_2=\frac{a_1^3\sqrt{a_1^2-b^2}}{2a^3\sqrt{a^2-a_1^2}}\Big\{\Pi(\theta_2, \alpha^2(a_1), k)-\Pi(\frac{\pi}{2}, \alpha^2(a_1), k)-\frac{1}{2}\Pi(\theta_3, \alpha^2(a_1), k)+\frac{1}{2}\Pi(\theta_4, \alpha^2(a_1), k)\Big\}\\ +\frac{1}{2}\Big\{\frac{1}{2}S(\theta_3)\Lambda_1(c_1)-S(\theta_2)\Lambda_1(b_1)-\frac{1}{2}S(\theta_4)\Lambda_1(d_1)\Big\}+\frac{1}{2a^2}\Big\{\Lambda_2(c_1)-\Lambda_2(d_1)\Big\}, \\ \nonumber % Remove numbering (before each equation) B_1=\Big\{I_1-(T_1-\lambda^2T_2)\Big\}F(\theta_5, k)k^2+\frac{F_1}{2}\Big\{N(\theta_5)-(1-k^2)F(\theta_5, k)\Big\}-\\ \frac{k^2\phi_1}{2}\Big\{\frac{\pi}{2}-\frac{\sin{2\theta_2}}{2}-\theta_2\Big\} -\frac{E_1}{2}\Big\{M(\theta_5)-2(1-k^2)F(\theta_5, k)\Big\}+k^2I_2\Big\{\frac{\pi}{2}-\theta_2\Big\}, \\ B_2=\frac{b_1^3\sqrt{b_1^2-b^2}}{2a^3\sqrt{a^2-b_1^2}}\Big\{\Pi(\theta_1, \alpha^2(b_1), k)+\Pi(\frac{\pi}{2}, \alpha^2(b_1), k)+\frac{1}{2}\Pi(\theta_3, \alpha^2(b_1), k)-\frac{1}{2}\Pi(\theta_4, \alpha^2(b_1), k)\Big\}\\ +\frac{b^2k^2}{2}\Big\{S(\theta_1)\Lambda_3(a_1)+\frac{1}{2}S(\theta_3)\Lambda_3(c_1)-\frac{1}{2}S(\theta_4)\Lambda_3(d_1)\Big\}+\frac{1}{2a^2}\Big\{\Lambda_4(c_1)-\Lambda_4(d_1)\Big\}, \\ F_1=F(\theta_1, k)-F(\theta_2, k)+F(\frac{\pi}{2}, k)+\frac{1}{2}F(\theta_3, k)-\frac{1}{2}F(\theta_4, k), \quad\alpha_1^2(t)=\frac{t^2(a^2-b^2)}{a^2(t^2-b^2)}, \\ \theta_5=\sin^{-1}\sqrt{\frac{a^2(b_1^2-b^2)}{b_1^2(a^2-b^2)}}, \quad S(\theta)=\sin{\theta}\cos{\theta}\sqrt{1-k^2\sin^2{\theta}}, \quad\alpha^2(t)=\frac{a^2-b^2}{a^2-t^2}, \\ M(\theta)=E(\theta, k)-\frac{k^2\sin{\theta}\cos{\theta}}{\sqrt{1-k^2\sin^2{\theta}}}, \quad N(\theta)=\frac{k^2(1-k^2)}{(1-k^2\sin^2{\theta})^{\frac{3}{2}}}, \\ \Lambda_1(t)=(\alpha^2(t)-k^2)\pi(\theta_1, \alpha^2(t), k)+k^2F(\theta_1, k), \quad \Lambda_2(t)=-\xi(t, a_1)-\frac{iX(t)}{2k^2}\Big(k^2\theta_1+\zeta(t, \theta_1)\Big), \\ \Lambda_3(t)=\frac{1}{t^2-b^2}\Pi(\theta_5, \alpha_1^2(t), k), \quad \Lambda_4(t)=\xi(t, b_1)-\frac{iX(t)}{2k^2}\bigg(k^2\Big(\frac{\Pi}{2}-\theta_2\Big)-\zeta(t, \theta_2)\bigg), \\ \xi(\beta, \gamma)=\frac{\gamma^2}{2}\tanh^{-1}\sqrt{\frac{(a^2-\beta^2)(\gamma^2-b^2)}{(a^2-\gamma^2)(\beta^2-b^2)}}, \quad \zeta(\beta, \gamma)=\frac{(\alpha^2(\beta)-k^2)\tanh^{-1}(\sqrt{\alpha^2(\beta)-1}\tan{\gamma})}{\sqrt{\alpha^2(\beta)-1}}.$

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5. Illustrative Study: Yield Zone Length and Crack-tip Opening Displacement

The techniques outlined have been used to evaluate yield zone lengths and crack tip opening displacements for three cracks with coalesced yield zones under linearly varying stress distribution. Yield zone length at each crack tip is evaluated using Dugdale ^[2] hypothesis that the stresses remain finite in the vicinity of crack, hence

$\label{e39}K=K^A_I(z_1)+K^B_I(z_1)=0.$

(29)

On substituting the corresponding values of $K_I^A$ from Eqs. (13) and (14) and the values of $K_I^B$ from Eqs. (25) and (26) into Eq. (29), one may get two non-linear equations as given below.

$\label{e40}\Big(\frac{\sigma_{\infty}}{\sigma_{ye}}\Big)_a(1-\lambda^2)+\frac{2k^2}{\pi}\Big[I_1-E_1+I_2-\phi_1-(T_1-\lambda^2T_2)\Big]=0,$

(30)

$\label{e41}\Big(\frac{\sigma_{\infty}}{\sigma_{ye}}\Big)_b(b^2-a^2\lambda^2)+\frac{2a^2k^2}{\pi}\Big[I_1+\frac{b}{a}I_2-(T_1-\lambda^2T_2)\Big]=0.$

(31)

It is almost impossible to obtain yield zone length for prescribed values of $\frac{\sigma_{\infty}}{\sigma_{ye}}$, therefore $\frac{\sigma_{\infty}}{\sigma_{ye}}$ is evaluated for given value of $|b_1-b|, |d_1-c_1|$ and $|a-a_1|$. Figure 4 shows the variation between normalized load ratio $\frac{\sigma_{\infty}}{\sigma_{ye}}$ and normalized yield zone length $\frac{|a-a_1|}{|a_1-b_1|}$ for different values of $\frac{a_1-b_1}{a_1+b_1}$ (say $\triangle$). It is clearly observed as the load applied at the boundary of the plate increases yield zone length $|a-a_1|$ increases as expected. The results of three collinear straight cracks with coalesced yield zones are compared with the results of two equal collinear cracks of the same length under same loading conditions, a significant difference is seen in yield zone length when cracks $L_1$ and $L_2$ are located far away from each other.

Figure 5 shows the increase in the length of yield zone at crack tip b when applied stresses increases. It is seen from the graph, yield zone length $|b_1-b|$ is highly affected by the presence of another crack. The plate can bear more load when cracks are located far away ($\triangle=0.1$) in comparison to closely located cracks ($\triangle=0.9$). Furthermore, on comparing the results with two equal cracks it is clear that no significant difference is seen in bearing capacity for closely located cracks $k=0.9$.

Crack tip opening displacement (CTOD) is obtained using the formula given by Feng et al. ^[18],

$\label{e44}\delta^{\pm}(x)=v_A^{\pm}(x)-v_B^{\pm}(x),$

(32)

Hence, on substituting corresponding values of $v_A^{\pm}$ and $v_B^{\pm}$ from Eqs. (15) and (16) and Eqs. (27) and (28) into Eq. (32). One may get two non-linear equations for CTOD at crack tip $a_1$ and $b_1$, as

$\label{e45}\delta^{\pm}(a_1)=\pm\frac{4a\sigma_{ye}}{E}\Big\{\Big(E(\theta_1, k)-\lambda^2F(\theta_1, k)\Big)\Big(\frac{\sigma_{\infty}}{\sigma_{ye}}\Big)_a-\frac{2}{\pi}\Big(A_1+A_2\Big)\Big\},$

(33)

$\label{e46}\delta^{\pm}(b_1)=\mp\frac{4a\sigma_{ye}}{E}\Big\{\Big(E(\theta_2, k)-\lambda^2F(\theta_2, k)-\frac{k^2\sin{\theta_2}\cos{\theta_2}}{\sqrt{1-k^2\sin^2{\theta_2}}}\Big)\Big(\frac{\sigma_{\infty}}{\sigma_{ye}}\Big)_b-\frac{2}{\pi}\Big(B_1+B_2\Big)\Big\}.$

(34)

In order to investigated CTODs shown in Eqs. (33) and (34) at crack tips $a_1$ and $b_1$ ratio of CTODs to crack length has been plotted against applied load ratio. Figure 6 shows the said variation at crack tip $a_1$ while Figure 7 at crack tip $b_1$.

As the load applied at the boundary of the plate increases cracks open in mode-I type deformation. In the case of $\triangle=0.9$ the opening of cracks are almost same in comparison to the opening of two equal cracks. But the opening at crack tip $a_1$ when $\triangle=0.1$ is significantly different in the said comparison.

Figure 7 shows the behaviour of normalized CTOD with respect to applied load ratio $\frac{\sigma_\infty}{\sigma_{ye}}$ at inner crack tip $b_1$. CTOD at inner crack tip $b_1$ is increasing gradually as the applied load at the boundary of the plate is increased. Moreover, a significant difference is seen in the CTOD of two configurations for different values of $\triangle$.

A comparison of the results obtained is shown in Figure 8 and 9 with the results of single cracks given by Harrop ^[4]. It has been observed that as $\triangle=0.9$, means cracks are located close to each other, behaviour of yield zone length and CTOD in case of configuration shown in Figure 1 is similar to single crack.

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6. Validation of Results

1) Results of opening mode stress intensity factor for sub problem A given in Eqs. (13) and (14) are the same as given by Tada ^[15];

2) Expression of yield zone length given in Eq. (30) and crack-tip opening displacement in Eq. (33) are same as the expressions given by Harrop ^[4] taking $b=b_1=c_1=d_1=0$ for a single crack.

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7. Conclusions

The following set of conclusions may be drawn from the work that has been reported. The problem of three asymmetrical collinear straight cracks with coalesced yield zones in an infinite sheet is analyzed.

a) Analytical expression for stress intensity factors, yield zone length and CTODs have been obtained under linearly varying stress distribution using the complex variable method.

b) Numerical results for applied stresses, the length of yield zones and crack-tip opening displacements are plotted and reported graphically. These results are compared with the results of an equivalent configuration of two equal collinear straight cracks.

c) It has been observed that the plate can bear more load when the coalesced yield zone $\Gamma_5$ bigger in size as compared to yield zone $\Gamma_3$.

d) Less opening of cracks is seen in the presence of coalesced yield zones when results are compared with the results of two equal cracks.

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Acknowledgement

First author is also grateful to University Grant Commission(UGC) for providing financial support under UGC-BSR scheme.

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Conflict of Interest

All authors declare no conflict of interest in this paper.

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