1 Introduction

The stress distribution in an infinite plate with circular hole subjected to uni-axial load was first obtained by Kirsch [1]. The details of the analytical solution can be found in Timoshenko [2], Sadd [3], Ukadgaonker [4]. Inglis [5] obtained stress concentration factor for an infinite plate with elliptical hole subjected to uni-axial load (Fig. 1). Complex variables approach was first introduced to solve plane elastic problems by Kolosov [6, 7] which was further utilised in solving various problems in elastostatics by Muskhelishvili [8] and others. With improvements in computer performance, numerical techniques like Finite Element Method (FEM) and Boundary Element Method (BEM) became popular methods of solving various engineering problems. Body force method, originally proposed by Nisitani [9,10,11,12], is a boundary type technique for stress analysis. Wang [13] first introduced Boundary Force Method (BFM), which is also a boundary type technique for stress analysis; using this method Wang determined stress intensity correction factors for collinear radial cracks emanating from circular hole under biaxial loading. Manjunath [14,15,16] has solved Flamant and Melan problems with circular holes using Body force method. Gao [17] obtained stresses and displacements around elliptical hole under biaxial loading using complex variable method in curvilinear coordinates. Kanezaki [18] found out closed-form solution for stress distribution around elliptical hole in cartesian coordinates. Batista [19] employed modified Muskhelishvili complex variable method and obtained stress distribution around a variety of hole shapes in a infinite plate. Sharma [20] obtained stress distribution around triangular, square, pentagonal, hexagonal, heptagonal and octagonal cut-out in an infinite plate and compared results with finite element models prepared in ANSYS. Sharma [21] obtained stresses around hypotrochoidal cutouts in infinite isotropic plate subjected to inplane loading using Muskhelishvili’s complex variable approach and some results were compared with values published in the literature. Jafari [22] investigated hoop stress distribution around triangular, square and pentagonal hole in an infinite composite plate under uniform heat flux using two-dimensional thermoelastic theory and Lekhnitskii’ complex variable technique in steady-state condition. Badiger [23] applied Modified Body Force Method to infinite plate with circular hole and compared computed stresses with those obtained from AE. Badiger [24] applied Boundary Force Method to infinite plate with circular hole, computed stresses were compared with results obtained from AE. Stress distribution along semi-major axis of an elliptical hole in infinite plate was obtained and compared with results from AE.

In the present work, BFM is applied to an elliptical hole in an infinite plate under uni-axial loading. This is extended to the case of a circular hole. For the purpose of demonstrating applicability of BFM, results from BFM are compared with known analytical solution. In the next section description of Boundary force method is provided. In Sect. 3 application of Boundary force method to elliptical hole and circular hole is discussed. In Sect. 4, AE related to elliptical hole are discussed. In Sect. 5, numerical results along with plots and discussion are presented followed by concluding remarks in Sect. 6.

Fig. 1
figure 1

An infinite plate with elliptical hole under uni-axial tension

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Fig. 2
figure 2

An infinite plate under uni-axial tension with boundary forces (\(\rho _{xi}\), \(\rho _{yi}\)) applied at the mid-points of 4 segments

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2 Boundary force method

Boundary force method is a hybrid technique (analytical and numerical) uses principle of superposition. In boundary force method, presence of a discontinuity like elliptic/circular hole in an infinite plate is transformed into a problem without discontinuity along with the application of boundary forces. The problem of infinite plate with discontinuity is solved by computing numerical values of boundary forces. In this paper equilibrium condition in terms of forces acting on the segments is used. This results in a set of linear equations, which are used in the determination of numerical values of boundary forces.

The problem of infinite plate with elliptical hole under uni-axial loading is considered. The elliptical hole is treated as imaginary, i.e. the infinite plate with elliptical hole is treated as a plate without a hole; the actual hole is regarded as imaginary, which is divided into \({{\textbf {M}}}\) number of segments, at the midpoints of the segments on its periphery, boundary forces (\(\rho _{\textbf{xi}}\), \(\rho _{\textbf{yi}}\)) are applied. This arrangement is shown in Fig. 2. In the beginning, magnitudes of boundary forces are unknown. The elasticity solution for a point load in an infinite plate is used as the fundamental solution (Refer Fig. 3). Balancing the forces acting on each segment (in x and y directions) due to the boundary forces and forces on the segments due to applied uni-axial tension, results in a set of linear equations which can be written in matrix form \({{\textbf {A}}}{{\textbf {x}}}={{\textbf {b}}}\), where \({{\textbf {A}}}\) is a square matrix (size \({{\textbf {2}}}{{\textbf {M}}} \times {{\textbf {2}}}{{\textbf {M}}}\)) known as Influence Coefficient Matrix (ICM), \({{\textbf {x}}}\) is boundary force column vector and column vector \({{\textbf {b}}}\) is the force on each segment in x and y directions due to the applied uni-axial tension. The stress at an arbitrary point \({\textbf {Q(x,y)}}\) is computed by the superposition of stresses due to the boundary forces and externally applied uni-axial tension. Here, boundary conditions are satisfied in terms of the resultant forces only. Poisson’s ratio is taken to be 0.33.

Fig. 3
figure 3

Point force \(X + i \, Y\) acting at point \(z_0\) in an infinite plate

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Fig. 4
figure 4

Variation of condition number of ICM with number of divisions of the ellipse/circle

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A concentrated point force \(X+i\,Y\) acting at point \(z_o=x_o+i\,y_o\) in an infinite plate is shown in Fig. 3. The two complex potentials describing this condition from England [7], Muskhelishvili [8] are

$$\begin{aligned} \phi (z)= & {} -\frac{( X + iY )}{2 \pi ( 1 + \kappa )}\ln (z-z_{o}) \end{aligned}$$
(1)
$$\begin{aligned} \psi (z)= & {} \frac{\kappa ( X - iY )}{2 \pi ( 1 + \kappa )}\ln (z-z_{o}) + \frac{( X + iY )}{2 \pi ( 1 + \kappa )}\frac{\overline{z_{o}}}{(z-z_{o})} \end{aligned}$$
(2)

The stresses at an arbitrary point \(z=x+i\,y\) are related to the above complex potentials [4, 7, 8] by the following relations

$$\begin{aligned} \sigma _{x} + \sigma _{y}= & {} 2 \left[ \phi '(z)+\overline{\phi '(z)}\right] \end{aligned}$$
(3)
$$\begin{aligned} \sigma _{y} - \sigma _{x} + 2i\tau _{xy}= & {} 2 \left[ {\overline{z}}\phi ''(z)+\psi '(z)\right] \end{aligned}$$
(4)

Substituting complex potentials from Eqs. (1) and (2) into Eqs. (3) and (4), stress components [25] obtained are as follows

$$\begin{aligned} \sigma _{x}= & {} \frac{-1}{2 \pi ( 1 + \kappa )}\, \left[ \frac{( X + iY )}{(z-z_0)} + \frac{( X - iY )}{({\overline{z}} - \overline{z_0})} \right. \nonumber \\{} & {} \left. + \Re \left\{ \frac{( X + iY )({\overline{z}} - \overline{z_0})}{(z-z_0)^2} + \kappa \,\frac{( X - iY )}{(z-z_0)} \right\} \right] \end{aligned}$$
(5)
$$\begin{aligned} \sigma _{y}= & {} \frac{-1}{2 \pi ( 1 + \kappa )}\, \left[ \frac{( X + iY )}{(z-z_0)} + \frac{( X - iY )}{({\overline{z}} - \overline{z_0})} \right. \nonumber \\{} & {} \left. - \Re \left\{ \frac{( X + iY )({\overline{z}} - \overline{z_0})}{(z-z_0)^2} + \kappa \,\frac{( X - iY )}{(z-z_0)} \right\} \right] \end{aligned}$$
(6)
$$\begin{aligned} \tau _{xy}= & {} \frac{1}{2 \pi ( 1 + \kappa )} \, \Im \left\{ \left( X+iY\right) \frac{({\overline{z}} - \overline{z_0})}{(z-z_0)^2} + \kappa \frac{( X - iY )}{(z-z_0)} \right\} \end{aligned}$$
(7)

In Eqs. (5)–(7), \(\Re\) represents real part and \(\Im\) represents imaginary part.

The resultant force acting on a segment connecting points A and B due to point load \(X+i\,Y\) is expressed in terms of complex potentials given by Eqs. (1) and (2) as presented in [4, 7, 8] is

$$\begin{aligned} P_{x} + iP_{y} = -i \left[ \phi (z) + z\overline{\phi '(z)}+\overline{\psi (z)}\right] ^B_A \end{aligned}$$
(8)

The resultant force acting on segment connecting points A and B in the x– and y– directions due to point load \(X+i\,Y\) acting at point \(z_o=x_o+i\,y_o\) is obtained by substituting Eqs. (1) and (2) in Eq. (8). When a x–direction unit force \(X=1\) acts at point \(z_o\) in an infinite plate, expression for force components obtained are

$$\begin{aligned} P_{x}= & {} \frac{-1}{2 \pi ( 1 + \kappa )} \left[ ( \kappa + 1 ) \tan ^{-1} \left( \frac{y-y_{o}}{x-x_{o}}\right) \right. \nonumber \\{} & {} \left. +\frac{2(x-x_{o})(y-y_{o})}{(x-x_{o})^2+(y-y_{o})^2} \right] ^B_A \end{aligned}$$
(9)
$$\begin{aligned} P_{y}= & {} \frac{-1}{2 \pi ( 1 + \kappa )} \left[ \frac{(\kappa - 1)}{2}\ln \left[ (x-x_{o})^2+(y-y_{o})^2\right] \right. \nonumber \\{} & {} \left. - \frac{(x-x_{o})^2-(y-y_{o})^2}{(x-x_{o})^2+(y-y_{o})^2} \right] ^B_A \end{aligned}$$
(10)

When a y-direction unit force \(Y = 1\) acts at point \(z_{o}\) in an infinite plate, expression for force components obtained are

$$\begin{aligned} P_{x}= & {} \frac{1}{2 \pi ( 1 + \kappa )} \left[ \frac{(\kappa - 1)}{2}\ln \left[ (x-x_{o})^2+(y-y_{o})^2\right] \right. \nonumber \\{} & {} \left. + \frac{(x-x_{o})^2-(y-y_{o})^2}{(x-x_{o})^2+(y-y_{o})^2} \right] ^B_A \end{aligned}$$
(11)
$$\begin{aligned} P_{y}= & {} \frac{-1}{2 \pi ( 1 + \kappa )} \left[ (\kappa + 1 ) \tan ^{-1} \left( \frac{y-y_{o}}{x-x_{o}}\right) \right. \nonumber \\{} & {} \left. - \frac{2(x-x_{o})(y-y_{o})}{(x-x_{o})^2+(y-y_{o})^2} \right] ^B_A \end{aligned}$$
(12)

where parameter \(\kappa\) depends on Poisson’s ratio \(\nu\)

$$\begin{aligned} \kappa = {\left\{ \begin{array}{ll} \dfrac{\left( 3 - \nu \right) }{\left( 1+ \nu \right) } &{} \quad \text {plane stress}\\ \left( 3 - 4 \nu \right) &{} \quad \text {plane strain} \end{array}\right. } \end{aligned}$$
(13)

When Poisson’s ratio \(\nu =0\), we get \(\kappa =3\), Eqs. (9) to (12) match with the Eqs. (6) to (9) from Wang [13, p. 121].

3 Boundary force method applied to infinite plate with elliptical hole subjected to uni-axial tension

An ellipse with \(a = 5\) mm, \(b = 4\) mm is considered for computation. The elliptical boundary is divided into four segments as shown in Fig. 2. All segments of the ellipse will experience forces in x and y directions due to boundary forces (\(\rho _{xi}\), \(\rho _{yi}\)) and the applied uni-axial tension. Equilibrium equation (due to force only) of each segment in x and y directions can be written as

$$\begin{aligned} P_{i}^{j} \rho _{j} = P_{i}^{EL} \end{aligned}$$
(14)

\(P_{i}^{j}\) is the force experienced by segment i (in both x and y directions) due to unit force on segment j (in both x and y directions). This is Influence Coefficient Matrix (ICM). \(\rho _{j}\) is the boundary force vector whose values needs to be determined. \(P_{i}^{EL}\) is the force on the segment i (in both x and y directions) due to external load (uni-axial tension in the current case). Equation (14) when put in matrix takes the form \({{\textbf {A}}}{{\textbf {x}}}={{\textbf {b}}}\). ICM is computed using Eqs. (9) to (12) for the elliptical hole shown in Fig. 2, with unit forces acting at the mid-point of each segment in x and y directions. Equation (15) shows ICM for imaginary ellipse (\(a = 5\) and \(b = 4\)) which is divided into 4 segments.

$$\begin{aligned} \begin{bmatrix} 0.365 &{} 0.081 &{} -\,0.165 &{} -\,0.047 &{} -\,0.135 &{} -\,0.081 &{} -\,0.066 &{} 0.047 \\ 0.065 &{} 0.389 &{} 0.047 &{} -\,0.039 &{} -\,0.065 &{} -\,0.111 &{} -\,0.047 &{} -\,0.239 \\ -\,0.165 &{} 0.047 &{} 0.365 &{} -\,0.081 &{} -\,0.066 &{} -\,0.047 &{} -\,0.135 &{} 0.081 \\ -\,0.047 &{} -\,0.039 &{} -\,0.065 &{} 0.389 &{} 0.047 &{} -\,0.239 &{} 0.065 &{} -\,0.111 \\ -\,0.135 &{} -\,0.081 &{} -\,0.066 &{} 0.047 &{} 0.365 &{} 0.081 &{} -\,0.165 &{} -\,0.047 \\ -\,0.065 &{} -\,0.111 &{} -\,0.047 &{} -\,0.239 &{} 0.065 &{} 0.389 &{} 0.047 &{} -\,0.039 \\ -\,0.066 &{} -\,0.047 &{} -\,0.135 &{} 0.081 &{} -\,0.165 &{} 0.047 &{} 0.365 &{} -\,0.081 \\ 0.047 &{} -\,0.239 &{} 0.065 &{} -\,0.111 &{} -\,0.047 &{} -\,0.039 &{} -\,0.065 &{} 0.389 \end{bmatrix} \end{aligned}$$
(15)

The ICM shown in Eq. (15) has the following properties. ICM is not a symmetric matrix. The condition number of ICM is \(4.1874\times 10^{16}\) which is very high hence \({{\textbf {A}}}{{\textbf {x}}}={{\textbf {b}}}\) cannot be inverted reliably. ICM for circle is symmetric. Elliptical geometry (Refer Fig. 2) is divided into 4 to 64 segments and ICMs are computed and the corresponding condition numbers of these ICMs are determined. Figure 4 shows variation of condition number of ICM with number of divisions of the ellipse/circle, it is noted that condition number remains very high irrespective of the number of divisions of the ellipse.

To overcome this problem, portion of ICM corresponding to quarter ellipse is used in obtaining boundary forces. It can be noted, this is similar to approach employed by Wang [13].

Force on each segments due to uni-axial tension (\(P_{i}^{EL}={{\textbf {b}}}\)) is determined by using complex potentials describing the uniform stress in an infinite plate under uni-axial tension without a hole. These complex potentials from [26] are given below.

$$\begin{aligned} \phi (z)= & {} \frac{z}{4} \cdot \sigma _{o} \end{aligned}$$
(16)
$$\begin{aligned} \psi (z)= & {} \frac{z}{2} \cdot \sigma _{o} \end{aligned}$$
(17)

Substituting Eqs. (16) and (17) in Eqs. (3) and (4) result in the following stress distribution.

$$\begin{aligned} \sigma _{x} = 0; \quad \sigma _{y} = \sigma _{o}; \quad \tau _{xy} = 0 \end{aligned}$$
(18)

This is the stress distribution of infinite plate under uni-axial tension (in y-direction) without elliptical hole (Refer Fig. 1).

Substitution of Eqs. (16) and (17) in Eq. (8) result in the following forces on a segment connecting points A to B.

$$\begin{aligned} P_{x}= & {} 0 \end{aligned}$$
(19)
$$\begin{aligned} P_{y}= & {} -\sigma _{o} \left[ x_{B} - x_{A}\right] \end{aligned}$$
(20)

Equations (19) and (20) applied to elliptical hole (\(a = 5\) mm, \(b = 4\) mm) in an infinite plate with uniform far field stress, \(\sigma _{o} = 1\) MPa results in the following column vector in N

$$\begin{aligned} {{\textbf {b}}}^T = \left[ 0 \; 5 \; 0 \; 5 \; 0 \; -5 \; 0 \; -5 \right] ^T \end{aligned}$$
(21)

For the quarter ellipse considered, \({{\textbf {b}}}^T = \left[ 0 \; 5 \right] ^T\) is used. Table 1 shows the set of linear equations for first quarter of the imaginary ellipse. Boundary forces \(\rho _x,\rho _y\) are obtained inverting matrix \({{\textbf {A}}}\), which is a sub-matrix of ICM (Eq. 15).

Table 1 A set of linear equations for first quarter of the imaginary ellipse
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Solving the above set of equations result in the following boundary forces: \(\rho _x = -2.9747 \, N,\, \rho _y = 13.3623 \,N\). The corresponding boundary force vector for the entire ellipse in newton is \(\left[ \,-2.9747\;13.3623\;2.9747\;13.3623\;2.9747 \;-\,13.3623 \;-\,2.9747 \;-\,13.3623 \right] ^T\)

The stress at arbitrary point Q(x, y) is obtained using principle of superposition as follows:

$$\begin{aligned} \sigma _{x}\left( Q(x,y) \right)= & {} \sum \sigma _{x}\left( Q(x,y):(\rho _{xi},\rho _{yi}) \right) \end{aligned}$$
(22)
$$\begin{aligned} \sigma _{y}\left( Q(x,y) \right)= & {} \sum \sigma _{y}\left( Q(x,y):(\rho _{xi},\rho _{yi}) \right) + \sigma _{o}\end{aligned}$$
(23)
$$\begin{aligned} \tau _{xy}\left( Q(x,y) \right)= & {} \sum \tau _{xy}\left( Q(x,y):(\rho _{xi},\rho _{yi}) \right) \end{aligned}$$
(24)

Stress, \(\sigma _{x}\) at point Q(x, y) is \(\sigma _{x}\left( Q(x,y) \right)\) obtained by the summation of stress \(\sigma _{x}\) due to boundary forces \((\rho _{xi},\rho _{yi})\) applied at the mid-point of each segment and obtained from Eq. (5).

Stress, \(\sigma _{y}\) at point Q(x, y) is \(\sigma _{y}\left( Q(x,y) \right)\) obtained by the summation of stress \(\sigma _{y}\) due to boundary forces \((\rho _{xi},\rho _{yi})\) applied at the mid-point of each segment and obtained from Eq. (6) and the uni-axial stress \(\sigma _{o}\).

Stress, \(\tau _{xy}\) at point Q(x, y) is \(\tau _{xy}\left( Q(x,y) \right)\) obtained by the summation of stress \(\tau _{xy}\) due to boundary forces \((\rho _{xi},\rho _{yi})\) applied at the mid-point of each segment and obtained from Eq. (7).

Radial, hoop and shear stresses are obtained from the following stress transformations.

$$\begin{aligned} \sigma _{\xi }= & {} \sigma _{x} \, \cos ^{2}\varphi + \sigma _{y} \, \sin ^{2}\varphi + 2 \, \tau _{xy} \, \sin \varphi \, \cos \varphi \end{aligned}$$
(25)
$$\begin{aligned} \sigma _{\eta }= & {} \sigma _{x} \, \sin ^{2}\varphi + \sigma _{y} \, \cos ^{2}\varphi - 2 \, \tau _{xy} \, \sin \varphi \, \cos \varphi \end{aligned}$$
(26)
$$\begin{aligned} \tau _{\xi \eta }= & {} (\sigma _{y} - \sigma _{x}) \, \sin \varphi \, \cos \varphi + \tau _{xy} \, (\cos ^{2}\varphi - \sin ^{2}\varphi ) \end{aligned}$$
(27)

where \(\varphi = \arctan (\frac{a^2}{b^2}\tan \theta )\).

4 Analytical results

Infinite plate with elliptical hole under bi-axial tension is shown in Fig. 5. Gao [17] obtained expressions for stresses in curvi-linear coordinates using complex variable approach. The expressions for stresses are reproduced here.

$$\begin{aligned} \sigma _{\xi }= & {} \frac{1}{2}\,(\lambda -1)\,\sigma \,e^{2\xi _{o}}\cos 2\beta \nonumber \\{} & {} + \frac{1}{2}\,\frac{(1-\lambda )\,\sigma \,e^{2\xi _{o}}}{(\cosh 2\xi - \cos 2\eta )} \nonumber \\{} & {} \times \left[ \sin 2\beta \sin 2\eta + \cos 2\beta \,(\sinh 2\xi + \cosh 2\xi _{o}) \right. \nonumber \\{} & {} \left. - \cos 2(\eta -\beta )\cosh 2(\xi -\xi _{o}) \right] \nonumber \\{} & {} + \frac{1}{2}\frac{(1-\lambda )\sigma e^{2\xi _{o}}}{(\cosh 2\xi - \cos 2\eta )^2}\nonumber \\{} & {} \times \left[ \sin 2\beta \sin 2\eta (\cosh 2\xi - \cosh 2\xi _{o} ) \right. \nonumber \\{} & {} \left. + \cos 2\beta \sinh 2\xi \left( \cos 2\eta \, - \cosh 2\xi _{o}\right) \right] \nonumber \\{} & {} + \frac{1}{2}\,\frac{(1+\lambda )\,\sigma }{\left( \cosh 2\xi - \cos 2\eta \right) ^2} \nonumber \\{} & {} \times \sinh 2\xi ( \cosh 2\xi \, - \cosh 2\xi _{o} ) \end{aligned}$$
(28)
$$\begin{aligned} \sigma _{\eta }= & {} \frac{1}{2}\,(\lambda -1)\,\sigma \,e^{2\xi _{o}}\cos 2\beta \nonumber \\{} & {} + \frac{1}{2}\,\frac{(1-\lambda )\,\sigma \,e^{2\xi _{o}}}{(\cosh 2\xi - \cos 2\eta )}\nonumber \\{} & {} \times \left[ \sin 2\beta \sin 2\eta + \cos 2\beta \, (\sinh 2\xi - \cosh 2\xi _{o} ) \right. \nonumber \\{} & {} \left. + \cos 2(\eta -\beta ) \cosh 2(\xi -\xi _{o}) \right] \nonumber \\{} & {} - \frac{1}{2}\frac{(1-\lambda )\sigma \,e^{2\xi _{o}}}{(\cosh 2\xi - \cos 2\eta )^2}\nonumber \\{} & {} \times \left[ \sin 2\beta \sin 2\eta (\cosh 2\xi - \cosh 2\xi _{o}) \right. \nonumber \\{} & {} \left. + \cos 2\beta \sinh 2\xi ( \cos 2\eta - \cosh 2\xi _{o} ) \right] \nonumber \\{} & {} + \frac{1}{2}\,\frac{(1+\lambda )\,\sigma }{(\cosh 2\xi - \cos 2\eta )^2} \nonumber \\{} & {} \times \sinh 2\xi ( \cosh 2\xi + \cosh 2\xi _{o} - 2\cos 2\eta ) \end{aligned}$$
(29)
$$\begin{aligned} \tau _{\xi \eta }= & {} \frac{1}{2}\frac{(1-\lambda )\sigma \,e^{2\xi _{o}}}{(\cosh 2\xi - \cos 2\eta )^2}\nonumber \\{} & {} \Big \lbrace (\cosh 2\xi - \cos 2\eta ) \bigl [ \sinh 2(\xi -\xi _{o}) \nonumber \\{} & {} \times \sin 2(\eta -\beta ) - \sinh 2\xi _{o}\sin 2\beta \bigr ]\nonumber \\{} & {} - \bigl [\sin 2\beta \sinh 2\xi (\cos 2\eta - \cosh 2\xi _{o}) \nonumber \\{} & {} + \cos 2\beta \sin 2\eta (\cosh 2\xi _{o} - \cosh 2\xi )\bigr ] \Big \rbrace \nonumber \\{} & {} -\frac{1}{2}\frac{(1+\lambda )\sigma }{(\cosh 2\xi - \cos 2\eta )^2} \sin 2\eta (\cosh 2\xi _{o}-\cosh 2\xi ) \end{aligned}$$
(30)
Fig. 5
figure 5

Infinite plate with elliptical hole under bi-axial loading

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In the equations for stresses (28), (29) and (30), \(\lambda\) the biaxial loading factor, is set to zero for the uni-axial loading case, \(\beta\) the orientation angle, is also set to zero and the far field uniform stress along y-direction, \(\sigma\) is set to unity.

The transformation relation employed is \(z = c\, \cosh (\zeta )\) where \(z = (x + iy)\) and \(\zeta = \xi + i \eta\). Ellipse boundary in curvilinear coordinates is related by the following relations.

$$\begin{aligned} a= & {} c\,\cosh \xi _{o} \end{aligned}$$
(31a)
$$\begin{aligned} b= & {} c\,\sinh \xi _{o} \end{aligned}$$
(31b)
$$\begin{aligned} c= & {} \sqrt{a^2 - b^2} \end{aligned}$$
(31c)
$$\begin{aligned} \xi _{o}= & {} \cosh ^{-1}{\left( \frac{a}{\sqrt{a^2 - b^2}}\right) } \end{aligned}$$
(31d)

Curvilinear coordinates of arbitrary point \(Q(\xi , \eta )\) corresponding to Q(x, y) are obtained using above relations and then stresses are computed.

The expression for critical value of hoop stress on the boundary of elliptical hole in infinite plate from Timoshenko [2] is given by

$$\begin{aligned} \sigma _{\theta } = \left( 1 + \frac{2a}{b}\right) \; \sigma _{0} \end{aligned}$$
(32)

Critical value of hoop stress occurs on the semi-major axis.

5 Numerical results and discussion

5.1 Numerical results from AE

Radial, hoop and shear stresses are computed for circular hole (\(a = 5\) mm and \(b = 4.999999\) mm), ellipical holes (\(a = 5\) mm, \(b = 4\) mm and \(b = 3\) mm) using Eqs. (28), (29) and (30). The following substitutions are used to obtain uni-axial load in y-direction: \(\beta = 0^{\circ }\), \(\lambda = 0\) and \(\sigma = 1\) MPa.

5.2 Numerical results from boundary force method

Stresses \(\sigma _{x}\), \(\sigma _{y}\) and \(\tau _{xy}\) are computed for circular hole (\(a = 5\) mm and \(b = 4.999999\) mm), elliptical holes (\(a = 5\) mm, \(b = 4\) mm and \(b = 3\) mm) using Eqs. (5), (6) and (7) and following superposition principle (Eqs. (22), (23) and (24)). Radial, hoop and shear stresses are obtained by using stress transformation (Eqs. (25), (26) and (27)). These stresses are computed when the ellipse/circle is divided into 4, 8 and 12 segments (nod).

Non-dimensional radial, hoop and shear stress variations with non-dimensional radial distance along \(0^{\circ }\) and \(90^{\circ }\) lines are shown in Fig. 6, 7, 8, 9, 10 and 11 for circular hole (\(a = 5, b = 4.999999\)).

Fig. 6
figure 6

Radial stress variation along \(0^{\circ }\) line for \(a=5\), \(b=5\)

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Fig. 7
figure 7

Radial stress variation along \(90^{\circ }\) line for \(a=5\), \(b=5\)

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Fig. 8
figure 8

Hoop stress variation along \(0^{\circ }\) line for \(a=5\), \(b=5\)

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Fig. 9
figure 9

Hoop stress variation along \(90^{\circ }\) line for \(a=5\), \(b=5\)

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Fig. 10
figure 10

Shear stress variation along \(0^{\circ }\) line for \(a=5\), \(b=5\)

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Fig. 11
figure 11

Shear stress variation along \(90^{\circ }\) line for \(a=5\), \(b=5\)

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Non-dimensional radial, hoop and shear stress variations with non-dimensional radial distance along \(0^{\circ }\) and \(90^{\circ }\) lines are shown in Figs. 12, 13, 14, 15, 16 and 17 for elliptical hole (\(a = 5, b = 4\)).

Non-dimensional radial, hoop and shear stress variations with non-dimensional radial distance along \(0^{\circ }\) and \(90^{\circ }\) lines are shown in Figs. 18, 19, 20, 21, 22 and 23 for elliptical hole (\(a = 5, b = 3\)).

For elliptical/circular hole, the stresses obtained from BFM along \(0^{\circ }\) and \(90^{\circ }\) lines show trends in line with AE.

Non-dimensional radial, shear and hoop stress variations with angle \(\theta\) on the boundary of the ellipse/circle are shown in Figs. 24, 25 and 26 respectively. Because of point forces applied at the mid-point of each segment, stresses computed near mid-point showed spikes which are removed and intermediate values are obtained using shape-preserving piecewise cubic interpolation.

Fig. 12
figure 12

Radial stress variation along \(0^{\circ }\) line for \(a=5\), \(b=4\)

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Fig. 13
figure 13

Radial stress variation along \(90^{\circ }\) line for \(a=5\), \(b=4\)

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Fig. 14
figure 14

Hoop stress variation along \(0^{\circ }\) line for \(a=5\), \(b=4\)

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Fig. 15
figure 15

Hoop stress variation along \(90^{\circ }\) line for \(a=5\), \(b=4\)

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Fig. 16
figure 16

Shear stress variation along \(0^{\circ }\) line for \(a=5\), \(b=4\)

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Fig. 17
figure 17

Shear stress variation along \(90^{\circ }\) line for \(a=5\), \(b=4\)

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Fig. 18
figure 18

Radial stress variation along \(0^{\circ }\) line for \(a=5\), \(b=3\)

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Fig. 19
figure 19

Radial stress variation along \(90^{\circ }\) line for \(a=5\), \(b=3\)

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Fig. 20
figure 20

Hoop stress variation along \(0^{\circ }\) line for \(a=5\), \(b=3\)

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Fig. 21
figure 21

Hoop stress variation along \(90^{\circ }\) line for \(a=5\), \(b=3\)

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Fig. 22
figure 22

Shear stress variation along \(0^{\circ }\) line for \(a=5\), \(b=3\)

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Fig. 23
figure 23

Shear stress variation along \(90^{\circ }\) line for \(a=5\), \(b=3\)

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Fig. 24
figure 24

Radial stress variation on the boundary of the ellipse/circle

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Fig. 25
figure 25

Shear stress variation on the boundary of the ellipse/circle

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Fig. 26
figure 26

Hoop stress variation on the boundary of the ellipse/circle

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Traction free boundary condition is fully satisfied with analytical equations for the case of circle but weakly satisfied for ellipse. Traction free boundary condition is weakly satisfied with BFM for ellipse/circle.

Tables 2, 3 and 4 show Stress Concentration Factor (SCF) obtained for circular hole (\(a = 5, b = 4.999999\)), elliptical holes (\(a = 5, b = 4\)) and (\(a = 5, b = 3\)) respectively using AE and Boundary Force Method (BFM) for various values of nod. It can be noted, with increasing in value of nod, results of BFM improves up to a certain value.

Table 2 Stress concentration factor obtained from AE and BFM for circular hole (\(a = 5\) and \(b = 4.999999\))
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Table 3 Stress concentration factor obtained from AE and BFM for elliptic hole (\(a = 5\) and \(b = 4\))
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Table 4 Stress concentration factor obtained from AE and BFM for elliptic hole (\(a = 5\) and \(b = 3\))
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6 Conclusion

Boundary Force Method is applied to an isotropic infinite plate with elliptic/circular hole subjected to far field uni-axial tension. The boundary condition satisfied in terms of the resultant forces, gets translated into weakly satisfied traction free boundary condition.

Accurate SCF values can be estimated by BFM based on the value of ratio \(\frac{a}{b}\). As the ratio \(\frac{a}{b}\) increases, number of divisions (nod) required to obtain accurate value of SCF also increases.

The Boundary Force Method can be useful as a first estimate in determining SCF in plates having various shapes of discontinuities (for e.g. irregular shaped hole), interaction of multiple discontinuities etc. for which theoretical solutions are not available.