Mathematical model for three equal collinear straight cracks: A modified Dugdale approach

Abstract

The present paper discuss a multiple site damage (MSD) problem of three collinear straight cracks exist in an infinite isotropic elastic perfectly plastic plate. Uniform stress distribution is applied at the infinite boundary of the plate in a direction perpendicular to the faces of the cracks. As a result, yield zones are developed ahead each crack tip. These yield zones are sensitive about the load applied at the boundary of the plate. Since the regions near the crack tips are very week as compared to other parts of the plate, therefore, it is assumed that the behaviour of yield stress of the plate near the yield zones is in linear fashion. Hence, rims of the yield zones are subjected to linearly varying stress distribution to stop further opening of cracks along the real axis. The problem is solved using widely used complex variable technique and closed form expressions are derived for stress intensity factor (SIF), crack tip opening displacement (CTOD) and length of developed yield zones at each crack tip. Under small-scale yielding, numerical results for yield zone length and crack-tip opening displacement as a function of remotely applied stress are obtained and presented graphically.

Keywords

Crack opening displacement Dugdale model multiple cracks stress intensity factor yield zone

Nomenclature

$D_{i}$ ( $i = 0, 1, 2, 3$ )

constants of the problem

Young’s modulus

F (θ, k)

E (θ, k)

Π (θ, α^{2}, k)

incomplete elliptic integral of first, second, third kind, respectively

L_{i}

(

i = 1, 2, 3

)

cracks

δ (x)

crack-tip opening displacement at the crack tip x

\pm a_{1}

\pm b_{1}

\pm c_{1}

crack tips

\pm a

\pm b

\pm c

tips of the developed yield plastic zones

p (t)

q (t)

applied stresses on the yield zones

u, v

components of displacement

z = x + i y

complex variable

Γ^{'}

$- \frac{1}{2} (N_{1} - N_{2}) e^{- 2 i α}$ , $N_{1}$ and $N_{2}$ are the values of principal stresses at infinity, α be the angle between $N_{1}$ and the $o x$ -axis

Γ_{i}

(

i = 1, 2, \dots, 6

)

developed plastic/yield zones

Ω (z) = ω^{'} (z)

Φ (z) = ϕ^{'} (z)

complex stress functions

Poisson’s ratio

shear modulus

= $\frac{3 - γ}{1 + γ}$ for the plane-stress, = $3 - 4 γ$ for the plane-strain

X_{x}

Y_{y}

X_{y}

components of stress

σ_{\infty}

remotely applied stress at infinite boundary of the plate

σ_{ye}

yield stress of the plate

1. Introduction

Multiple site damage (MSD) problems are characterised by the presence of multiple interacting cracks which may occur due to the application of repeated load or by environmental attacks [1]. Strength of the structures beset by the presence of cracks or crack like defects. Therefore, its becomes imperative to estimate the residual strength of the structures. In order to evaluate the residual strength of the plate in the presence of cracks, a model was developed by Dugdale [2] for an infinite plate weakened by a crack. This model was used to estimate yield zones length at the crack tips. Similar work was done by Barenblatt [3] for the brittle materials. After that some more fracture parameters like stress intensity factor, plastic zone size and crack tip opening displacement are estimated approximately at the crack tip by Rice [4] using Dugdale–Barenblatt model. Dugdale–Barenblatt model was further modified by Theocaris et al. [5] in case of strain hardening material and show that the model is not limited to constant yield stress but also useful for variable pressure. In addition to that the case of linearly varying tensile loading studied by Kanninen [6] and quadratically yield stress distribution was used by Harrop [7]. A massive collection of SIFs for various cracks geometries under different types of linear/non-linear stress distributions are given in [8].

Further, Dugdale model was widely used to solve multiple crack problems. For example, Theocaris [9] solved a problem of two equal/unequal cracks problem under general yielding conditions, Collins et al. [10] derived closed form solution for two equal cracks and also discussed the criterion of coalescence of yield zones. A numerical algorithm was presented to obtained crack tip opening displacement by Kaminsky et al. [11] for three collinear cracks. Recently, Hasan et al. [12] studied load carrying capacity of an infinite plate weakened by three equal/unequal collinear straight cracks under general yielding conditions.

Moreover, Nishimura [13] obtained plastic zone sizes and crack tip opening displacements for two collinear cracks subjected to remotely applied stress. Bhargava et al. [14] also evaluate yield zone sizes and crack tip opening displacements for two unequal straight cracks under quadratically yield stress distribution. A comparative study has been carried to study the nonlinear interaction between two equal-length collinear cracks; analytically and numerically by Chang et al. [15] using Dugdale model.

On the basis of cracks geometries and mechanical loading conditions, many methods have been presented to solve crack problems. Few examples, Chen [16] used integral equation method to convert multiple crack problems into a system of Fredholm integral equations and solved them numerically. Bhargava et al. [17] used complex variable method to obtain closed form solution of SIFs and CTODs for the multiple crack problem. Weight function approach was applied by Wu [18] to approximate plastic zone sizes and crack tip opening displacements for multiple collinear cracks.

The complex variable method due its mathematical simplicity provides a flexible way to solve the multiple crack problems [19]. Therefore, complex variable method has been used in this paper to solve the problem of three collinear straight cracks exist in an infinite elastic perfectly plastic plate. Developed yield zones, due to uniform tensile stresses acting at the infinite boundary of the plate, are subjected to linearly varying stress distribution to stop further opening of cracks. It is mentioned by Gdoutos [20] that some of the structure fails at stress which is well below the yield stress of the material, therefore, linearly varying stress distribution is considered in this work such a stress distribution. Analytical expressions for stress intensity factor and crack-tip opening displacement are obtained in the form of elliptical integrals defined in [21]. Numerical results of yield zone length and crack-tip opening displacements as a function of applied load ratio is presented graphically. Finally, the results are compared with the results of two equal cracks and/or a single crack under similar mechanical loading conditions.

2. Complex variable formulation

Consider an isotropic elastic-perfectly plastic plate occupy entire $x y$ -plane containing n collinear straight cracks $L_{i}$ ( $i = 1, 2, 3, \dots$ ) lying along real axis. Let $X_{x}^{\pm}$ , $X_{y}^{\pm}$ , $Y_{y}^{\pm}$ be the components of stress acting on the rims of the cracks in the strip yield model, where superscript (+), (−) denotes the boundary values of stress components on the upper and lower rims of the cracks. Components of stress $X_{x}$ , $Y_{y}$ , $X_{y}$ and displacements u, v can be expressed in terms of two complex potential functions $Φ (z)$ , $Ω (z)$ as given in [22] $\begin{array}{l} (1) & X_{x} + Y_{y} = 2 [Φ (z) + \overline{Φ (z)}], \\ (2) & Y_{y} - i X_{y} = Φ (z) + Ω (\overline{z}) + (z - \overline{z}) \overline{Φ^{'} (z)}, \\ (3) & 2 μ (u + i v) = κ ϕ (z) - ω (\overline{z}) - (z - \overline{z}) \overline{Φ (z)}, \end{array}$ where the bar over a function or a variable denotes its complex conjugate.

It is assumed that functions $Φ (z)$ and $Ω (z)$ are sectionally holomorphic functions on entire complex plane except L. So, under the assumption ${lim}_{y \to 0} y Φ^{'} (t + i y) = 0$ Eqs (1) and (2) may be converted into two problems of linear relationship: $\begin{array}{l} (4) & Φ^{+} (t) + Ω^{-} (t) = Y_{y}^{+} - i X_{y}^{+}, \\ (5) & Φ^{-} (t) + Ω^{+} (t) = Y_{y}^{-} - i X_{y}^{-} \end{array}$ on $L = ⋃_{i = 1}^{n} L_{i}$ .

On adding and subtracting Eqs (4) and (5) we get $\begin{array}{l} (6) & {[Φ (t) + Ω (t)]}^{+} + {[Φ (t) + Ω (t)]}^{-} = 2 p (t), \\ (7) & {[Φ (t) - Ω (t)]}^{+} + {[Φ (t) - Ω (t)]}^{-} = 2 q (t), \end{array}$ where $p (t)$ , $q (t)$ are the following functions and satisfy H condition on L: $\begin{matrix} p (t) = \frac{1}{2} [Y_{y}^{+} + Y_{y}^{-}] - \frac{i}{2} [X_{y}^{+} + X_{y}^{-}], q (t) = \frac{1}{2} [Y_{y}^{+} - Y_{y}^{-}] - \frac{i}{2} [X_{y}^{+} - X_{y}^{-}] . \end{matrix}$

The general solution of the boundary problems given in Eqs (6) and (7) may be written using Sokhotski–Plemelj formula as $\begin{array}{l} (8) & Φ (z) + Ω (z) = \frac{1}{π i X (z)} \int_{L} \frac{X (t) p (t) d t}{t - z} + \frac{2 P_{n} (z)}{X (z)}, \\ (9) & Φ (z) - Ω (z) = \frac{1}{π i} \int_{L} \frac{q (t) d t}{t - z} - \overline{Γ^{'}}, \\ (10) & X (z) = \prod_{k = 1}^{n} {(z - a_{k})}^{1 / 2} {(z - b_{k})}^{1 / 2}, \\ (11) & P_{n} (z) = D_{0} z^{n} + D_{1} z^{n - 1} + D_{2} z^{n - 2} + \dots + D_{n}, \end{array}$ where $a_{k}$ , $b_{k}$ denotes the boundary points of kth crack.

Constants appear in the polynomial $P_{n} (z)$ may be determine using loading condition at the infinite boundary and condition of single-valuedness of the displacements, $\begin{matrix} (12) & 2 (κ + 1) \int_{L_{k}} \frac{P_{n} (t)}{X (t)} d t + κ \int_{L_{k}} [Φ_{0}^{+} (t) - Φ_{0}^{-} (t)] d t + \int_{L_{k}} [Ω_{0}^{+} (t) - Ω_{0}^{-} (t)] d t = 0, \end{matrix}$ where $\begin{array}{l} Φ_{0} (z) = \frac{1}{2 π i X (z)} \int_{L} \frac{X (t) p (t)}{t - z} d t + \frac{1}{2 π i} \int_{L} \frac{q (t)}{t - z} d t, \\ Ω_{0} (z) = \frac{1}{2 π i X (z)} \int_{L} \frac{X (t) p (t)}{t - z} d t - \frac{1}{2 π i} \int_{L} \frac{q (t)}{t - z} d t, \end{array}$ and $L_{k} = [a_{k}, b_{k}]$ , $k = 1, 2, 3, \dots$ .

The formulation discussed in this section is taken from Muskhelishvili [22] to make the paper self-sufficient.

3. Analysis of the problem

Consider an isotropic infinite plate in which three collinear straight cracks exist. Strength of the plate is decreases in presence of theses cracks. These cracks are open in mode-I type deformation due to stresses applied, in a direction perpendicular to the rims of the cracks, at the infinite boundary of the plate. As a result, yield zones are develop at each crack tip. These yield zones are subjected to linearly varying stress distribution $Y_{y} = \frac{| t |}{a} σ_{ye}$ (say $σ_{a}$ ) to stop further opening of cracks, where t is any point on the rims of yield zones and $σ_{ye}$ is the yield stress of the plate. Cracks are denoted by $L_{i}$ ( $i = 1, 2, 3$ ) and occupy intervals $[- a_{1}, - b_{1}]$ , $[- c_{1}, c_{1}]$ , $[b_{1}, a_{1}]$ respectively on real axis. The developed yield zones are denoted by $Γ_{i}$ ( $i = 1, 2, 3, \dots, 6$ ) and occupy intervals $(- a, - a_{1}]$ , $[- b_{1}, - b)$ , $(- c, - c_{1}]$ , $[c_{1}, c)$ , $(b, b_{1}]$ , $[a_{1}, a)$ respectively. The entire configuration of the problem is depicted in Fig. 1.

Fig. 1.

Modified Dugdale model for three straight cracks.

4. Solution of the problem

Problem posed in Section 3 is solved by dividing it into two subproblems, representing the case of opening and closing of cracks (linearly varying stress distribution applied over the rims of the yield zones). These two subproblems are denoted by subproblem-I (opening case, $Y_{y} = σ_{\infty}$ ) and subproblem-II (closing case, $Y_{y} = \frac{| t |}{a} σ_{ye}$ ). Solution of the main problem is then obtained by superposing the solutions of two component problems.

4.1. Subproblem-I and its solution

Opening of the cracks, due to stresses applied at the infinite boundary of the plate, is discussed in this section. Consider an infinite elastic perfectly plastic plate weakened by three collinear straight cracks. Boundary of the plate is subjected to a tensile stress, $Y_{y} = σ_{\infty}$ . Therefore, yield zones are developed ahead of each crack tip. These yield zones are assumed to be inline with the cracks. Cracks along with corresponding yield zones are denoted by $R_{1} (- a, - b)$ , $R_{2} (- c, c)$ and $R_{3} (b, a)$ and termed as extended/physical cracks as shown Fig. 2.

Fig. 2.

Configuration of the subproblem-I.

Mathematical form of the boundary conditions of the problem are $\begin{array}{l} (13) & Y_{y} = σ_{\infty}, X_{y} = 0 for y \to \pm \infty, - \infty < x < \infty, \\ (14) & Y_{y} = X_{y} = 0 for y \to 0, - \infty < x < \infty . \end{array}$

Complex potential function for this case under the boundary conditions given in Eqs (13) and (14) and formulation given in Section 2 may be written as: $\begin{matrix} (15) & Φ_{A} (z) = \frac{σ_{\infty}}{2} [\frac{z ((z^{2} - c^{2}) - (a^{2} - c^{2}) λ^{2})}{\sqrt{z^{2} - a^{2}} \sqrt{z^{2} - b^{2}} \sqrt{z^{2} - c^{2}}} - \frac{1}{2}], \end{matrix}$ where $λ^{2} = \frac{E (k)}{F (k)}$ , $k^{2} = \frac{a^{2} - b^{2}}{a^{2} - c^{2}}$ and $F (k)$ , $E (k)$ are complete elliptical integrals of first and second kind respectively, defined in Byrd [21].

Opening mode stress intensity factor at crack tip $z = α$ may be obtained by using formulae given in [10], $\begin{matrix} (16) & K_{I} = 2 \sqrt{2 π} lim_{z \to α} \sqrt{z - α} Φ (z) . \end{matrix}$

Therefore, analytical expression for stress intensity factor at each external crack tip of three cracks may be obtained by putting Eq. (15) into Eq. (16). Hence, $\begin{array}{l} (17) & {(K_{I}^{A})}_{a} = \frac{1 - λ^{2}}{k} σ_{\infty} \sqrt{π a}, \\ (18) & {(K_{I}^{A})}_{b} = \frac{λ^{2} + k^{2} - 1}{k \sqrt{1 - k^{2}}} σ_{\infty} \sqrt{π b}, \\ (19) & {(K_{I}^{A})}_{c} = \frac{λ^{2}}{\sqrt{1 - k^{2}}} σ_{\infty} \sqrt{π c} . \end{array}$

These stress intensity factors are validated with the results was given by Tada [8] and Hasan et al. [12].

Analytical expressions for the component of displacement, $v_{\infty}^{\pm}$ , at each crack tip $\pm a_{1}$ , $\pm b_{1}$ and $\pm c_{1}$ due to remotely applied stress $Y_{y} = σ_{\infty}$ may be written using Eqs (3) and (15) as: $\begin{array}{l} (20) & v_{\infty}^{\pm} (\pm a_{1}) = \pm \frac{2 σ_{\infty}}{E} \sqrt{a^{2} - c^{2}} [E (θ_{1}, k) - λ^{2} F (θ_{1}, k)], \\ (21) & v_{\infty}^{\pm} (\pm b_{1}) = \mp \frac{2 σ_{\infty}}{E} \sqrt{a^{2} - c^{2}} [E (θ_{2}, k) - λ^{2} F (θ_{2}, k) - \frac{k^{2} sin θ_{2} cos θ_{2}}{\sqrt{1 - k^{2} {sin}^{2} θ_{2}}}], \\ (22) & v_{\infty}^{\pm} (\pm c_{1}) = \mp \frac{2 σ_{\infty}}{E} \sqrt{a^{2} - c^{2}} [E (θ_{3}, k) - λ^{2} F (θ_{3}, k) - tan θ_{3} \sqrt{1 - k^{2} {sin}^{2} θ_{3}}], \end{array}$ where $\begin{matrix} sin θ_{1} = \sqrt{\frac{a^{2} - a_{1}^{2}}{a^{2} - b^{2}}}, sin θ_{2} = \frac{1}{k} \sqrt{\frac{b_{1}^{2} - b^{2}}{b_{1}^{2} - c^{2}}}, sin θ_{3} = \sqrt{\frac{c^{2} - c_{1}^{2}}{b^{2} - c_{1}^{2}}} . \end{matrix}$

Results written as Eqs (20) and (21) are validated with the results given in [17] by taking $c_{1} = c = 0$ .

4.2. Subproblem-II and its solution

Consider, an isotropic infinite elastic perfectly plastic plate weakened by three collinear straight cracks. Developed yield zones $Γ_{i}$ ( $i = 1, 2, \dots, 6$ ) are subjected to a linearly varying yield stress distribution, $\frac{| t |}{a} σ_{ye}$ , as shown in Fig. 3 in order to detain the cracks from further opening. Since, the infinite boundary of the plate is stress-free and a linearly varying stress distribution is acting over the rims of the yield zones, the boundary conditions of the problem may be written as $\begin{array}{l} (23) & Y_{y} = 0, X_{y} = 0 for y \to \pm \infty, - \infty < x < \infty, \\ (24) & Y_{y} = \frac{| t |}{a} σ_{ye}, X_{y} = 0 for y \to 0, x \in ⋃_{i = 1}^{6} Γ_{i} . \end{array}$

Fig. 3.

Configuration of the subproblem-II.

The complex potential function $Φ_{B} (z)$ for this case may be obtained by using boundary conditions given in Eqs (23), (24) and formulation given in Section 2. Finally, $\begin{matrix} (25) & Φ_{B} (z) = \frac{σ_{ye}}{2 π i X (z)} [\int_{⋃_{i = 1}^{6} Γ_{i}} \frac{t X (t) d t}{a (t - z)} + i (z^{2} D_{1} + z D_{2} + D_{3})] . \end{matrix}$

Due to symmetrical loading conditions the constants $D_{1}$ and $D_{3}$ are zero and constant $D_{2}$ may be obtained by Eqs (12) and (25), $\begin{array}{rcl} D_{2} & = & \frac{m}{k} [a_{1}^{2} (E (θ_{1}, k) - λ^{2} F (θ_{1}, k)) - b_{1}^{2} (E (θ_{6}, k) - λ^{2} F (θ_{6}, k)) \\ - c_{1}^{2} (E (θ_{3}, k) - λ^{2} F (θ_{3}, k)) + J_{1} (θ_{1}) + J_{2} (θ_{2}) + 2 J_{3} (θ_{3}) \\ - a^{2} (E (θ_{1}, k) + M (θ_{2})) \\ + b^{2} (E (θ_{3}, k) - (1 - k_{2}^{2}) (F (θ_{3}, k) - N (θ_{3}))) \\ + λ^{2} (c^{2} (F (θ_{1}, k) + F (θ_{2}, k) - F (θ_{3}, k)) \\ (26) & + \frac{a^{2} m^{2}}{k^{2}} (E (θ_{1}, k) + M (θ_{2}) - N (θ_{3})))], \end{array}$ where $\begin{array}{l} J_{1} (θ) = \frac{a^{2} m^{2}}{3 k^{2}} ((2 k^{2} - 1) E (θ, k) + (1 - k^{2}) F (θ, k) - k^{2} S (θ, k, 0)), \\ J_{2} (θ) = \frac{a^{2} m^{2}}{3 k^{2}} [(2 k^{2} - 1) M (θ) + (1 - k^{2}) F (θ, k) + (\frac{k^{2} (1 - k^{2})}{1 - k^{2} {sin}^{2} θ}) S (θ, 0, k)], \\ J_{3} (θ) = \frac{a^{2} m^{2}}{3 k^{2}} [(2 k^{2} - 1) N (θ) + (1 - k^{2}) F (θ, k) - \frac{(1 - k^{2})}{{cos}^{4} θ} S (θ, k, 0)], \\ N (θ) = E (θ, k) - tan θ \sqrt{1 - k^{2} {sin}^{2} θ}, S (θ, p, q) = \frac{sin θ cos θ \sqrt{1 - p^{2} {sin}^{2} θ}}{\sqrt{1 - q^{2} {sin}^{2} θ}}, \\ M (θ) = E (θ, k) - \frac{k^{2} sin θ cos θ}{\sqrt{1 - k^{2} {sin}^{2} θ}}, k_{2}^{2} = \frac{c^{2}}{b^{2}}, m^{2} = \frac{a^{2} - b^{2}}{a^{2}}, sin θ_{6} = \sqrt{\frac{a^{2} - b_{1}^{2}}{a^{2} - b^{2}}} . \end{array}$

The integral over yield zones $⋃_{i = 1}^{6} Γ_{i}$ given in Eq. (25) may be evaluate using the facts, $\begin{array}{l} X (t) = X (- t) = i \sqrt{a^{2} - t^{2}} \sqrt{t^{2} - b^{2}} \sqrt{t^{2} - c^{2}}, \\ (27) & when - a < - b < - c < c < b < t < a, \\ X (t) = X (- t) = - i \sqrt{a^{2} - t^{2}} \sqrt{b^{2} - t^{2}} \sqrt{c^{2} - t^{2}}, \\ (28) & when - a < - b < - c < t < c < b < a . \end{array}$

Hence, $\begin{array}{l} \int_{⋃_{i = 1}^{6} Γ_{i}} \frac{t X (t) d t}{a (t - z)} & = \frac{2 z}{a} [\int_{a_{1}}^{a} \frac{t X (t)}{t^{2} - z^{2}} d t + \int_{b}^{b_{1}} \frac{t X (t)}{t^{2} - z^{2}} d t + \int_{c_{1}}^{c} \frac{t X (t)}{t^{2} - z^{2}} d t] \\ (29) & = 2 i a^{2} z [\frac{m^{3}}{k} {ζ - (1 - \frac{1}{n^{2} (z)}) ρ_{1} (z)} + \frac{m_{2}^{2}}{k_{2}} {ξ + (1 - \frac{1}{n_{2}^{2} (z)}) ρ_{2} (z)}], \end{array}$ where $\begin{array}{l} ζ = I_{1} (θ_{1}) - I_{1} (θ_{6}) + I_{1} (\frac{π}{2}), ξ = I_{2} (\frac{π}{2}) - I_{2} (θ_{5}), \\ I_{1} (θ) = \frac{1}{3 k^{2}} ((1 - k^{2}) F (θ, k) + (2 k^{2} - 1) E (θ, k) - k^{2} S (θ, k, 0)), \\ I_{2} (θ) = \frac{\sqrt{1 - m_{2}^{2}}}{3 k_{2}} ((1 - 2 k_{2}^{2} + \frac{k_{2}^{2}}{m_{2}^{2}}) E (ϕ_{1} (θ), k) - 2 (1 - k_{2}^{2}) F (ϕ_{1} (θ), k)) \\ - \frac{S (θ, m_{2}, k_{2}) k_{2}^{2}}{3 sin θ} (\frac{2}{k_{2}^{2}} - 2 - {sin}^{2} θ + \frac{1}{m_{2}^{2}}), \\ ρ_{1} (z) = H_{1} (θ_{1}, z) - H_{1} (θ_{6}, z) + H_{1} (\frac{π}{2}, z), ρ_{2} (z) = H_{2} (\frac{π}{2}, z) - H_{2} (θ_{5}, z), \\ H_{1} (θ, z) = E (θ, k) - \frac{k^{2}}{n^{2} (z)} F (θ, k) - (1 - \frac{k^{2}}{n^{2} (z)}) Π (θ, n^{2} (z), k), \\ H_{2} (θ, z) = - \frac{m_{2}^{2} S (θ, k_{2}, m_{2})}{sin θ} - k_{2} \sqrt{1 - m_{2}^{2}} (E (ϕ_{2} (θ), k) - F (ϕ_{2} (θ), k)) \\ + \frac{m_{2}^{2} (n_{2}^{2} (z) - k_{2}^{2})}{k_{2} n_{2}^{2} (z) \sqrt{1 - m_{2}^{2}}} Π (ϕ_{2} (θ), α_{4}^{2} (z), k), \\ ϕ_{1} (θ) = {tan}^{- 1} (\frac{k_{2} cos θ}{\sqrt{1 - k_{2}^{2}}}), ϕ_{2} (θ) = {tan}^{- 1} (\frac{\sqrt{1 - m_{2}^{2}}}{m_{2} cos θ}), \\ m_{2}^{2} = \frac{c^{2}}{a^{2}}, n^{2} (z) = \frac{a^{2} - b^{2}}{a^{2} - z^{2}}, n_{2}^{2} (z) = \frac{c^{2}}{z^{2}}, sin θ_{5} = \frac{c_{1}}{c} . \end{array}$

Complex potential function $Φ_{B} (z)$ for the case of linearly varying stress distribution may be evaluated by putting Eqs (26) and (29) into Eq. (25). Hence, $\begin{matrix} (30) & Φ_{B} (z) = \frac{z a^{2} σ_{ye}}{π X (z)} [\frac{m^{3}}{k} {ζ - (1 - \frac{1}{n^{2} (z)}) ρ_{1} (z)} + \frac{m_{2}^{2}}{k_{2}} {ξ + (1 - \frac{1}{n_{2}^{2} (z)}) ρ_{2} (z)} + \frac{D_{2}}{2 a^{2}}] . \end{matrix}$

Closed form expressions for stress intensity factor at the crack tips a, b and c may be obtained by putting Eq. (30) into Eq. (16). Hence, $\begin{array}{l} {(K_{I}^{B})}_{a} = \frac{2 k σ_{ye} \sqrt{a π}}{m^{2} π} [\frac{m^{3}}{k} (ζ - (E (θ_{1}, k) - E (θ_{6}, k) + E (k))) \\ (31) & + \frac{m_{2}^{2}}{k_{2}} (ξ + (1 - \frac{1}{m_{2}^{2}}) ρ_{2} (a)) + \frac{D_{2}}{2 a^{2}}], \\ (32) & {(K_{I}^{B})}_{b} = \frac{- 2 k σ_{ye} \sqrt{b π}}{m^{2} π \sqrt{1 - k^{2}}} [\frac{m^{3}}{k} ζ + \frac{m_{2}^{2}}{k_{2}} (ξ + (1 - \frac{1}{k_{2}^{2}}) ρ_{2} (b)) + \frac{D_{2}}{2 a^{2}}], \\ (33) & {(K_{I}^{B})}_{c} = \frac{2 k^{2} σ_{ye} \sqrt{c π}}{m^{2} π \sqrt{1 - k^{2}}} [\frac{m^{3}}{k} (ζ - (1 - \frac{1}{k^{2}}) ρ_{1} (c)) + \frac{m_{2}^{2}}{k_{2}} ξ + \frac{D_{2}}{2 a^{2}}] . \end{array}$

Using Eqs (3) and (30), one can obtain components of displacement at cracks tips $\pm a_{1}$ , $\pm b_{1}$ , and $\pm c_{1}$ respectively, $\begin{array}{l} (34) & v_{ye}^{\pm} (\pm a_{1}) = \mp \frac{2 σ_{ye}}{π E} [A_{1} (θ_{1}) + A_{2} (θ_{1}) + A_{3} (θ_{1}) - \frac{k D_{2}}{a m} F (θ_{1}, k)], \\ (35) & v_{ye}^{\pm} (\pm b_{1}) = \pm \frac{2 σ_{ye}}{π E} [B_{1} (θ_{2}) + B_{2} (θ_{2}) + B_{3} (θ_{2}) - \frac{k D_{2}}{a m} F (θ_{2}, k)], \\ (36) & v_{ye}^{\pm} (\pm c_{1}) = \mp \frac{2 σ_{ye}}{π E} [C_{1} (θ_{3}) + C_{2} (θ_{3}) + C_{3} (θ_{3}) + \frac{k D_{2}}{a m} F (θ_{3}, k)], \end{array}$ where $\begin{array}{l} A_{1} (θ) = \frac{i k}{a^{2} m} [\frac{a_{1}^{2} X (a_{1})}{a^{2} - a_{1}^{2}} (F (θ, k) - Π (θ, α_{4}^{2} (a_{1}), k)) - \frac{b_{1}^{2} X (b_{1})}{a^{2} - b_{1}^{2}} Π (θ, α_{1}^{2} (b_{1}), k) \\ + \frac{c_{1}^{2} X (c_{1})}{a^{2} - c_{1}^{2}} Π (θ, α_{1}^{2} (c_{1}), k)], \\ A_{2} (θ) = \frac{E (θ, k)}{a} [c^{2} (F (θ_{1}, k) + F (θ_{2}, k) - F (θ_{3}, k)) \\ + (\frac{a^{2} m^{2}}{k^{2}} - \frac{a^{2} F (θ, k)}{E (θ, k)}) (E (θ_{1}, k) + M (θ_{2}) - N (θ_{3}))] \\ + \frac{F (θ, k)}{a} [J_{1} (θ_{1}) + J_{2} (θ_{2}) - J_{3} (θ_{3})], \\ A_{3} (θ) = a m^{2} S (θ, k, 0) [F (θ_{1}, k) + F (θ_{2}, k - F (θ_{3}, k) \\ + \frac{a_{1}^{2}}{b^{2} - a_{1}^{2}} {(1 - α_{1}^{2} (a_{1})) (F (θ_{1}, k) - Π (θ_{1}, α_{4}^{2} (a_{1}), k)) + α_{3}^{2} (a_{1}) F (θ_{2}, k) \\ + (1 - α_{3}^{2} (a_{1})) Π (θ_{2}, α_{2}^{2} (a_{1}), k) - (F (θ_{3}, k) - (1 - α_{3}^{2} (a_{1})) Π (θ_{3}, α_{3}^{2} (a_{1}), k))}], \\ B_{1} (θ) = \frac{a m^{2}}{k^{2}} [(E (θ_{1}, k) - E (θ_{6}, k)) (M (θ) - (\frac{b^{2} k^{2}}{a^{2} m^{2}} + (1 - k^{2})) F (θ, k)) \\ + E (θ_{4}, k) (M (θ) - \frac{m_{2}^{2} k^{2}}{m^{2}} F (θ, k)) - 2 (1 - k^{2}) E (k) F (θ, k)] \\ + \frac{i b_{1}^{2} k X (b_{1})}{a^{2} m (a^{2} - b_{1}^{2})} [Π (θ_{1}, α_{1}^{2} (b_{1}), k) - F (θ_{6}, k) + Π (θ_{6}, α_{4}^{2} (b_{1}), k) - Π (θ_{4}, α_{4}^{2} (b_{1}), k)], \\ B_{2} (θ) = \frac{1}{a} [(F (θ_{1}, k) - F (θ_{6}, k)) (a^{2} M (θ) - J_{2} (θ) - 2 J_{4} (θ)) \\ + \frac{a^{2} m^{2}}{k^{2}} F (θ_{4}, k) (\frac{k^{2}}{m^{2}} N (θ) - (1 - k^{2}) F (θ, k) - M (θ)) \\ - 2 F (k) (J_{4} (θ) + J_{5} (θ) - \frac{a^{2} m^{2}}{k^{2}} M (θ))] - 2 F (θ, k) (\frac{b k m_{2}}{m} ξ + a m^{2} ζ) \\ + \frac{2 b m m_{2}}{k sin θ_{5}} M (θ) S (θ_{5}, k_{2}, m_{2}), \\ B_{3} (θ) = - a m^{2} [\frac{S (θ_{1}, k, 0)}{c^{2} - a_{1}^{2}} (c^{2} F (θ, k) + \frac{a_{1}^{2} (1 - α_{3}^{2} (a_{1}))}{α_{3}^{2} (a_{1})} Π (θ, α_{2}^{2} (a_{1}), k)) - \frac{S (θ_{6}, k, 0)}{b^{2} - b_{1}^{2}} η (θ, b_{1})] \\ + \frac{α_{4}^{2} (c_{1})}{a} S (θ_{4}, k, 0) {\frac{a^{2} m^{2}}{k^{2}} (\frac{c^{2} F (θ, k)}{c^{2} - c_{1}^{2}} - M (θ)) + \frac{c_{1}^{2} (1 - α_{3}^{2} (c_{1}))}{1 - α_{1}^{2} (c_{1})} Π (θ, α_{2}^{2} (c_{1}), k)}, \\ C_{1} (θ) = \frac{1}{a} [(F (θ_{1}, k) - F (θ_{6}, k)) (J_{3} (θ) - a^{2} N (θ) - 2 J_{6} (θ)) \\ + \frac{a^{2} m^{2}}{k^{2}} F (θ_{4}, k) (N (θ) + (1 - k^{2}) F (θ, k) - \frac{k^{2}}{m^{2}} M (θ)) \\ - 2 F (k) (J_{6} (θ) - J_{7} (θ) + \frac{a^{2} m^{2}}{k^{2}} N (θ))] \\ + a m^{2} [\frac{S (θ_{1}, k, 0)}{b^{2} - a_{1}^{2}} η (θ, a_{1}) - \frac{S (θ_{6}, k, 0)}{b^{2} - b_{1}^{2}} η (θ, b_{1})], \\ C_{2} (θ) = \frac{a m^{2}}{k^{2}} [(E (θ_{1}, k) - E (θ_{6}, k)) ((\frac{b^{2} k^{2}}{a^{2} m^{2}} + (1 - k^{2})) F (θ, k) - N (θ)) \\ + E (θ_{4}, k) (\frac{k^{2} m_{2}^{2}}{m^{2}} F (θ, k) - N (θ)) + 2 (1 - k^{2}) E (k) F (θ, k)] \\ - \frac{i k c_{1}^{2} X (c_{1})}{a^{2} m (a^{2} - c_{1}^{2})} (Π (θ_{1}, α_{1}^{2} (c_{1}), k) - F (θ_{4}, k) + Π (θ_{4}, α_{1}^{2} (c_{1}), k) - Π (θ_{6}, α_{1}^{2} (c_{1}), k)), \\ C_{3} (θ) = 2 F (θ, k) (\frac{b k m_{2}}{m} ξ + a m^{2} ζ) - \frac{2 b m m_{2}}{k sin θ_{5}} N (θ) S (θ_{5}, k_{2}, m_{2}) \\ - \frac{α_{4}^{2} (c_{1})}{a} S (θ_{4}, k, 0) {\frac{a^{2} m^{2}}{k^{2}} (\frac{F (θ, k)}{{cos}^{2} θ_{5}} - N (θ)) + \frac{c_{1}^{2} (1 - α_{3}^{2} (c_{1}))}{1 - α_{1}^{2} (c_{1})} Π (θ, α_{2}^{2} (c_{1}), k)}, \\ α_{1}^{2} (x) = \frac{a^{2} - b^{2}}{a^{2} - x^{2}}, α_{3}^{2} (x) = \frac{x^{2} - b^{2}}{x^{2} - c^{2}}, α_{2}^{2} (x) = \frac{k^{2}}{α_{3}^{2} (x)}, \\ α_{4}^{2} (x) = \frac{a^{2} - x^{2}}{a^{2} - c^{2}}, sin θ_{4} = \sqrt{\frac{a^{2} - c^{2}}{a^{2} - c_{1}^{2}}}, \\ J_{4} (θ) = \frac{a^{2} m^{2}}{3 k^{2}} [(2 - k^{2}) M (θ) - 2 (1 - k^{2}) F (θ, k) + S (θ, 0, k) (\frac{k^{2} (1 - k^{2})}{1 - k^{2} {sin}^{2} θ})], \\ J_{5} (θ) = \frac{a^{2} m^{2}}{3 k^{2}} [(1 + k^{2}) M (θ) - (1 - k^{2}) F (θ, k) - S (θ, 0, k) (\frac{k^{2} (1 - k^{2})}{1 - k^{2} {sin}^{2} θ})], \\ J_{6} (θ) = \frac{a^{2} m^{2}}{3 k^{2}} [(k^{2} - 2) N (θ) + 2 (1 - k^{2}) F (θ, k) + S (θ, k, 0) \frac{(1 - k^{2})}{{cos}^{4} θ}], \\ J_{7} (θ) = \frac{a^{2} m^{2}}{3 k^{2}} [(1 + k^{2}) N (θ) - (1 - k^{2}) F (θ, k) + S (θ, k, 0) \frac{(1 - k^{2})}{{cos}^{4} θ}], \\ η (θ, t) = b^{2} F (θ, k) + t^{2} (α_{3}^{2} (t) - 1) Π (θ, α_{3}^{2} (t), k) . \end{array}$

4.3. Analytical expression for remotely applied stress

σ_{\infty}

According to Dugdale’s hypothesis [2] that the stresses remains finite at each crack tip. Hence, stress intensity factors for both the cases (applied and yield) must vanishes together. Therefore, after superposing the solution of two component problems final stress intensity factor must be equal to zero [13] at the extended crack tip $z = α$ , $\begin{matrix} (37) & K \equiv {(K_{I}^{A})}_{α} + {(K_{I}^{B})}_{α} = 0 . \end{matrix}$

So, after putting the corresponding values of ${(K_{I}^{A})}_{α}$ from Eqs (17)–(19) and ${(K_{I}^{B})}_{α}$ from Eqs (31)–(33) in the formula given in Eq. (37). One gets three non-linear equations to determine yield zone lengths at each crack tip. These equations are $\begin{array}{l} \frac{m^{2}}{k^{2}} (1 - λ^{2}) {(\frac{σ_{\infty}}{σ_{ye}})}_{a} + \frac{2}{π} [\frac{m^{3}}{k} (ζ - (E (θ_{1}, k) - E (θ_{6}, k) + E (k))) \\ (38) & + \frac{m_{2}^{2}}{k_{2}} (ξ + (1 - \frac{1}{m_{2}^{2}}) ρ_{2} (a)) + \frac{D_{2}}{2 a^{2}}] = 0, \\ (39) & \frac{m^{2}}{k^{2}} (1 - k^{2} - λ^{2}) {(\frac{σ_{\infty}}{σ_{ye}})}_{b} + \frac{2}{π} [\frac{m^{3}}{k} ζ + \frac{m_{2}^{2}}{k_{2}} (ξ + (1 - \frac{1}{k_{2}^{2}}) ρ_{2} (b)) + \frac{D_{2}}{2 a^{2}}] = 0, \\ (40) & \frac{m^{2}}{k^{2}} λ^{2} {(\frac{σ_{\infty}}{σ_{ye}})}_{c} - \frac{2}{π} [\frac{m^{3}}{k} (ζ - (1 - \frac{1}{k^{2}}) ρ_{1} (c)) + \frac{m_{2}^{2}}{k_{2}} ξ + \frac{D_{2}}{2 a^{2}}] = 0 . \end{array}$

4.4. Analytical expression for crack-tip opening displacement $δ (x)$

Closed form expressions for the CTODs $δ (x)$ at each original crack tip ( $α = a_{1}, b_{1}, c_{1}$ ) may be obtained using formula given by Feng et al. [23]. Here, $\begin{matrix} (41) & δ (x = \pm α) = (v_{\infty}^{+} (x) + v_{ye}^{+} (x)) - (v_{\infty}^{-} (x) + v_{ye}^{-} (x)) . \end{matrix}$

Therefore, the expressions for CTOD at each crack tip $a_{1}$ , $b_{1}$ and $c_{1}$ may be written by putting the corresponding values of $v_{\infty}^{\pm} (x)$ and $v_{ye}^{\pm} (x)$ from Eqs (20)–(22), (34), (35) and (36) into Eq. (41). One may found CTODs at each crack tip, as: $\begin{array}{l} δ (a_{1}) = \frac{4 σ_{ye}}{E} [\frac{a m}{k} (E (θ_{1}, k) - λ^{2} F (θ_{1}, k)) {(\frac{σ_{\infty}}{σ_{ye}})}_{a} \\ (42) & - \frac{1}{π} (A_{1} (θ_{1}) + A_{2} (θ_{1}) + A_{3} (θ_{1}) - \frac{k D_{2}}{a m} F (θ_{1}, k))], \\ δ (b_{1}) = \frac{4 σ_{ye}}{E} [\frac{a m}{k} (M (θ_{2}) - λ^{2} F (θ_{2}, k)) {(\frac{σ_{\infty}}{σ_{ye}})}_{b} \\ (43) & - \frac{1}{π} (B_{1} (θ_{2}) + B_{2} (θ_{2}) + B_{3} (θ_{2}) - \frac{k D_{2}}{a m} F (θ_{2}, k))], \\ δ (c_{1}) = \frac{4 σ_{ye}}{E} [\frac{a m}{k} (- N (θ_{3}) + λ^{2} F (θ_{3}, k)) {(\frac{σ_{\infty}}{σ_{ye}})}_{c} \\ (44) & - \frac{1}{π} (C_{1} (θ_{3}) + C_{2} (θ_{3}) + C_{3} (θ_{3}) + \frac{k D_{2}}{a m} F (θ_{3}, k))] . \end{array}$

The results for remotely applied load stresses given in Eq. (38) and CTODs given in Eq. (42) are validated with the results of a single crack subjected to linearly varying stress distribution, as given by Harrop [7].

5. Illustrative examples

5.1. Applied stresses and yield zone size

Numerical results are obtained for applied load ratio, $\frac{σ_{\infty}}{σ_{ye}}$ , as a function of yield zone length at each crack tip. The results are compared with the results of two equal collinear straight cracks similarly positioned as the two outer cracks of the configuration shown in Fig. 1. Moreover, results of the inner crack shown in Fig. 1 are compared with the results of a single crack under same mechanical loading conditions.

5.1.1. Case of three equal cracks

In this section, growth of normalized yield zone size, $\frac{a - a_{1}}{a_{1} - b_{1}}$ , is measured with respect to increasing ratio of applied stress to the yield stress, $\frac{σ_{\infty}}{σ_{ye}}$ . It may be noted from Fig. 4 that the yield zones length increases as the stresses applied at the infinite boundary of the plate increases.

Fig. 4.

A relation between yield zone ratio $\frac{Γ_{6}}{L_{3}}$ and applied stress ratio $\frac{σ_{\infty}}{σ_{ye}}$ for $L_{1} = L_{2} = L_{3}$ .

Fig. 5.

Normalized yield zone ratio $\frac{Γ_{5}}{L_{3}}$ vs applied stress ratio $\frac{σ_{\infty}}{σ_{ye}}$ for $L_{1} = L_{2} = L_{3}$ .

A parameter $ρ = \frac{2 c_{1}}{b_{1} + c_{1}}$ shows the separation between cracks, a higher value of ρ (say 0.9) means cracks are very close to each other and a smaller value of ρ (say 0.1) means cracks are situated far away from each other. When $ρ = 0.1$ bearing capacity of an infinite plate is more than $ρ = 0.9$ because two outer cracks are less affected by inner crack due to the big separation between them. To confirm this finding, the results are compared with the results of two equal cracks situated at the same position as outer cracks in Fig. 1 (means inner crack absent). It is seen from the said comparison that the results are almost same for both the configurations (three cracks and two cracks) are same for $ρ = 0.1$ , but significantly different for $ρ = 0.9$ . Therefore, if outer cracks are situated far away from each other as shown in Fig. 1 then they behave like two equal collinear cracks.

Same variation has been plotted in Fig. 5 at the crack tip $t = b_{1}$ . A slightly different view of the load bearing capacity is seen at crack tip $t = b_{1}$ , its follows the results of two equal cracks unless ρ becomes very high. Figure 6 shows the variation between normalized yield zone length $\frac{c - c_{1}}{2 c_{1}}$ and load ratio $\frac{σ_{\infty}}{σ_{ye}}$ at the crack tip $t = c_{1}$ . It may be noted that the length of yield zone $Γ_{4}$ is much bigger than in case of $ρ = 0.1$ in comparison to $ρ = 0.9$ . This is because internal crack is behave like a single crack of length, $2 c_{1}$ , when outer cracks are situated far away from it.

Fig. 6.

A relation between yield zone ratio $\frac{Γ_{4}}{L_{3}}$ and applied stress ratio $\frac{σ_{\infty}}{σ_{ye}}$ for $L_{1} = L_{2} = L_{3}$ .

5.1.2. Case of three unequal cracks

To investigate the importance of inner crack, it is taken five times longer than the outer cracks. At the outermost tip, $t = a_{1}$ yield zone length increases as the value of ρ and applied stresses increase as shown in Fig. 7. It should also be noted that all (for different ρ) case of two equal cracks are almost same due to the absence of a big crack among them.

Fig. 7.

A relation between yield zone ratio $\frac{Γ_{6}}{L_{2}}$ and applied stress ratio $\frac{σ_{\infty}}{σ_{ye}}$ for $5 L_{1} = L_{2} = 5 L_{3}$ .

Fig. 8.

Normalized yield zone ratio $\frac{Γ_{5}}{L_{3}}$ vs applied stress ratio $\frac{σ_{\infty}}{σ_{ye}}$ for $5 L_{1} = L_{2} = 5 L_{3}$ .

Fig. 9.

Normalized yield zone ratio $\frac{Γ_{4}}{L_{2}}$ vs applied stress ratio $\frac{σ_{\infty}}{σ_{ye}}$ for $5 L_{1} = L_{2} = 5 L_{3}$ .

Fig. 10.

Normalized CTOD $\frac{δ (a_{1}) E}{σ_{ye} L_{3}}$ vs applied load ratio $\frac{σ_{\infty}}{σ_{ye}}$ when $L_{1} = L_{2} = L_{3}$ .

Figure 8 shows the same variation at crack tip $t = b_{1}$ as in Fig. 7. Since, crack tip $t = b_{1}$ is under the influence of a large dominating crack, therefore, load bearing capacity is almost similar to the case of two equal crack when $ρ = 0.1$ and significantly different is case of $ρ = 0.9$ . Yield zone $| c_{1} - c |$ is under the influence of a very fragile stress due to linearly varying stress distribution, $\frac{| t |}{a} σ_{ye}$ as shown in Fig. 9. Therefore, length of yield zone $Γ_{4}$ is bigger at $ρ = 0.1$ rather than $ρ = 0.9$ . It may also be noted that the results are almost similar to the results of a single crack of length $2 c_{1}$ due to its dominating nature.

5.2. Applied stresses and crack-tip opening displacement

It this section, the effect of increasing tensile stress is examined on the opening of cracks when all cracks are equal/unequal in nature. Graphs are plotted between applied load ratio $\frac{σ_{\infty}}{σ_{ye}}$ and normalized CTODs $\frac{δ (x) E}{σ_{ye} L_{i}}$ , where $x = a_{1}, b_{1}, c_{1}$ and $i = 2, 3$ . Opening of three cracks is compared with the opening of two equal cracks and a single crack under same mechanical loading conditions just like Section 5.1.

5.2.1. Case of three equal cracks

Figure 10 shows opening of cracks at the outermost tip $t = a_{1}$ in the configuration of three cracks, when all cracks are equal in size. CTOD at $ρ = 0.9$ is bigger in comparison to CTOD at $ρ = 0.1$ . It may be noted from Fig. 10 that increasing inter crack distance marginally affect the opening of cracks at tip $t = a_{1}$ , while at the crack tip $t = b_{1}$ opening of cracks are highly affected by the size of inter crack distance as shown in Fig. 11. Moreover, the inner crack opens more in case of $ρ = 0.1$ (outer cracks are situated far away) due less load applied on the rims of yield zone as shown in Fig. 11. Although, opening of the crack at tip $c_{1}$ is almost similar to the opening of a single crack when $ρ = 0.1$ but significantly different in the case of closely located cracks, $ρ = 0.9$ as shown in Fig. 12.

Fig. 11.

Normalized CTOD $\frac{δ (b_{1}) E}{σ_{ye} L_{3}}$ vs applied load ratio $\frac{σ_{\infty}}{σ_{ye}}$ when $L_{1} = L_{2} = L_{3}$ .

Fig. 12.

Normalized CTOD $\frac{δ (c_{1}) E}{σ_{ye} L_{3}}$ vs applied load ratio $\frac{σ_{\infty}}{σ_{ye}}$ when $L_{1} = L_{2} = L_{3}$ .

5.2.2. Case of three unequal cracks

Further studies are conducted to investigate the influences of sizes of cracks on CTOD. Consider, the internal crack $L_{2}$ be five times larger than outer cracks $L_{1}$ , $L_{3}$ . Cracks are open in self-similar manner and it is observed from the Fig. 13 that CTOD at the crack tip $t = a_{1}$ is more in comparison to the opening for three equal cracks. CTOD $δ (a_{1})$ is further compared with CTOD for two equal cracks at the same tip, it may be observed that the absence of a big inner crack creates a large space between two cracks. Hence, whatever be the value of ρ; CTOD is almost same.

Figure 14 shows the variation between normalized CTOD and applied load ratio at the crack tip $t = b_{1}$ . It may be noted that CTOD is less in case of $ρ = 0.1$ in comparison to $ρ = 0.9$ , as expected. Furthermore, bearing capacity of the plate is more in case of two equal cracks rather than three unequal cracks. At crack tip $t = c_{1}$ , a graph has been plotted between normalized CTOD and applied load ratio as depicted in Fig. 15. Opening of the inner crack is same as the opening of an equivalent single crack. This is because, a big inner crack in the configuration shown in Fig. 1 dominate over entire configuration.

Fig. 13.

$\frac{δ (a_{1}) E}{σ_{ye} L_{3}}$ vs $\frac{σ_{\infty}}{σ_{ye}}$ when $5 L_{1} = L_{2} = 5 L_{3}$ .

Fig. 14.

$\frac{δ (b_{1}) E}{σ_{ye} L_{3}}$ vs $\frac{σ_{\infty}}{σ_{ye}}$ when $5 L_{1} = L_{2} = 5 L_{3}$ .

Fig. 15.

$\frac{δ (c) E}{σ_{ye} L_{2}}$ vs $\frac{σ_{\infty}}{σ_{ye}}$ when $5 L_{1} = L_{2} = 5 L_{3}$ .

6. Comparison to yield stress distribution at crack tip a

At extended crack tip $t = a$ linearly varying yield stress distribution is behaved like a constant yield stress distribution. Therefore, a comparative study is carried out between linearly varying and constant yield stress distribution for yield zone length and CTOD, where $r_{0}$ , $δ_{0}$ are yield zone length, CTOD for two equal collinear straight cracks and $r_{A}$ , $δ_{A}$ are yield zone length, CTOD for three collinear straight cracks under same mechanical loading conditions, respectively.

6.1. Three equal cracks

Ratio of yield zones lengths $\frac{r_{A}}{r_{0}}$ is plotted against the increasing value of applied load ratio $\frac{σ_{\infty}}{σ_{ye}}$ at the outer crack tip of $L_{3}$ in Fig. 16. From the plotted results, we see that the length of yield zone $r_{A}$ depends on the inter crack distance and stresses applied at the infinite boundary of the plate. Difference between yield zone lengths $r_{A}$ and $r_{0}$ is very small when $ρ = 0.1$ and $\frac{σ_{\infty}}{σ_{ye}} < 0.6$ . It may also be noted that yield zone lengths $r_{A}$ and $r_{0}$ are significantly different when $ρ = 0.9$ (for closely located cracks). As far as CTOD at crack tip $a_{1}$ is concern it shows the same behaviour as yield zone length as depicted in Fig. 17.

Fig. 16.

$\frac{σ_{\infty}}{σ_{ye}}$ vs $\frac{r_{A}}{r_{0}}$ when $L_{1} = L_{2} = L_{3}$ .

Fig. 17.

$\frac{σ_{\infty}}{σ_{ye}}$ vs $\frac{δ_{A}}{δ_{0}}$ when $L_{1} = L_{2} = L_{3}$ .

6.2. Three unequal cracks

In this section inner crack $L_{2}$ is considered five times larger than the outer cracks $L_{1}$ and $L_{3}$ . Figures 18 and 19 are plotted for load bearing capacity and CTODs respectively. Yield zones $r_{A}$ and $r_{0}$ behave similarly for $ρ = 0.1$ and $\frac{σ_{\infty}}{σ_{ye}} < 0.8$ due a big separation between inner and outer cracks. But, when ρ becomes higher (cracks located closer) significant difference is seen in yield zone lengths $r_{A}$ and $r_{0}$ as shown in Fig. 18. Opening of cracks at tip $t = a_{1}$ is plotted in Fig. 19, it may be noted that the ratio of $\frac{δ_{A}}{δ_{0}}$ behave almost similar to $\frac{r_{A}}{r_{0}}$ under similar mechanical conditions.

Fig. 18.

$\frac{σ_{\infty}}{σ_{ye}}$ vs $\frac{r_{A}}{r_{0}}$ when $5 L_{1} = L_{2} = 5 L_{3}$ .

Fig. 19.

$\frac{σ_{\infty}}{σ_{ye}}$ vs $\frac{δ_{A}}{δ_{0}}$ when $5 L_{1} = L_{2} = 5 L_{3}$ .

7. Conclusion

The plane problem of three collinear straight cracks is considered in the paper when the cracks were subjected to a mode-I loading and yield zones are subjected to a linearly varying stress distribution. Closed form solution is obtained for SIFs and CTODs in terms of elliptic integrals using Muskhelishvili’s complex variable method. Numerical results are obtained for applied stresses and CTODs. These results are compared with the results of two equal collinear straight cracks and/or a single crack under similar mechanical loading conditions. It may be noted from the results that the yield zones at the outermost tip approximately behave like the yield zones of two equal cracks when all cracks are equal in sizes and far away located. The middle crack will dominate over entire configuration and behave like a single crack when taken five times bigger than outer cracks.

Footnotes

Acknowledgements

The second author is grateful to University Grant Commission, Government of India for providing financial support under UGC-BSR fellowship scheme. Authors are grateful to the referees and the editor for their suggestions, which improved the understandability of the paper.

Conflict of interest

The authors have no conflict of interest to report.

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