**3. Methodology**

The series of analytical expressions used to produce the solutions in Section 4 are outlined in Section 3. We start out with the generic elastic displacement equations for a single hole in an infinite plate subjected to far-field stress (Section 3.1), which can be solved for plane strain (Section 3.2) or plane stress (Section 3.3) boundary conditions. Although these expressions are basic, some confusing errors occur in both primary (journal papers) and secondary (textbooks) literature, which need to be pointed out (see respective sections below). What is new in our approach is that we do not solely focus on the stress concentrations at the hole but solve the stress magnitudes and principal stress orientations throughout the plates for a finite domain near the hole (Section 4) based on the equations given in Section 3.

The difference or delta between the stress magnitudes due to a plane strain or plane stress assumption is quantified in an explicit expression (Section 3.4). Because boreholes are commonly pressured from the inside by a net mud pressure, we also evaluate the displacements due to the internal pressure on the wellbore (Section 3.5). The equations of Sections 3.1–3.5 are all given in polar coordinates, but the far-field stresses in nature are typically uniform in Cartesian directions, which is why we switch to Cartesian coordinates in Sections 3.6. The use of Cartesian coordinates is essential for our analysis of both single-hole problems (Section 4.1) and multi-hole solutions (Section 4.2). Ultimately, when all the vector displacements have been computed and converted to strains, constitutive equations are needed (Section 3.7) to convert certain strain tensor fields, for any given set of elastic moduli to the corresponding stress field. The systematic series of equations in Sections 3.1 to 3.7 was used to produce the results in Section 4.

#### **3.1 Hole displacement equations**

In the theory of linear elasticity, stress quantities are linear functions of the displacement gradients expressed as strain quantities. Let us analyze the elastic displacements around a circular cylindrical hole of radius *a*, in an infinite plate subjected to far-field stress, *σxx*�<sup>∞</sup>, acting along the *x*-axis. Analytical solutions for the displacement equations in polar coordinates ð Þ *r*, *θ* are (see ref. [17]):

$$u\_r = \frac{\sigma\_{\text{xx}-\text{os}}}{8\text{ G}} a \left\{ \frac{r}{a} (\kappa - 1 + 2\cos 2\theta) + \frac{2a}{r} \left[ 1 + \left( \kappa + 1 - \frac{a^2}{r^2} \right) \cos 2\theta \right] \right\} \tag{1}$$

$$u\_{\theta} = \frac{\sigma\_{\text{xx} \to \infty}}{8 \text{ G}} a \left[ -\frac{2r}{a} + \frac{2a}{r} \left( 1 - \kappa - \frac{a^2}{r^2} \right) \right] \sin 2\theta \tag{2}$$

Above expressions capture the displacements for either plane strain or plane stress, depending upon the value inserted for *κ*, to be readily able to convert solutions for plane stress to plane strain, and vice-versa. In the above example, the solution for plane strain is given by substituting *κ* ¼ 3 � 4 *ν*; for plane stress, one should use *κ* ¼ ð Þ 3 � *ν =*ð Þ 1 þ *v* . Physically, the plane stress boundary condition applies to thin plates, while the plane strain condition applies to thick plates. The delta between the displacements and associated stress concentrations outcomes of the two approaches has not been made explicit, either for single or multiple holes, in any prior study.

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*

#### **3.2 Plane strain solution for the hole with uniaxial far-field stress**

Using Eqs. (1) and (2) and substituting *κ* ¼ 3 � 4 *ν* for plane strain, the displacement solutions are:

$$u\_r = \frac{\sigma\_{\text{xx}-\text{os}}}{4\,\text{Gr}} \left\{ r^2 (\mathbf{1} - 2\,\nu + \cos 2\theta) + a^2 \left[ \mathbf{1} + \left( 4(\mathbf{1} - \nu) - \frac{a^2}{r^2} \right) \cos 2\theta \right] \right\} \tag{3}$$

$$u\_{\theta} = -\frac{\sigma\_{\text{xx}-\text{os}}}{4\text{ Gyr}} \left[ r^2 + a^2 \left( 2(\mathbf{1} - 2\,\nu) + \frac{a^2}{r^2} \right) \right] \sin 2\theta \tag{4}$$

The above expressions for plane strain may be formulated using Young's modulus, *E*, instead of the shear modulus, *G*, substituting *G* ¼ *E=*2 1ð Þ þ *ν* into Eqs. (3) and (4):

$$u\_r = \sigma\_{\text{xx}-\text{os}} \left( \frac{\mathbf{1} + \boldsymbol{\nu}}{2Er} \right) \left\{ r^2 (\mathbf{1} - 2\boldsymbol{\nu} + \cos 2\theta) + a^2 \left[ \mathbf{1} + \left( 4(\mathbf{1} - \boldsymbol{\nu}) - \frac{a^2}{r^2} \right) \cos 2\theta \right] \right\} \tag{5}$$

$$u\_{\theta} = -\sigma\_{\text{xx}-\text{os}} \left( \frac{\mathbf{1} + \nu}{2Er} \right) \left[ r^2 + a^2 \left( 2(\mathbf{1} - 2\nu) + \frac{a^2}{r^2} \right) \right] \sin 2\theta \tag{6}$$

It is worth noting that a textbook by Goodman [18] has wrongly truncated terms in his Eq. (7.2a) and a sign error occurs in his Eq. (7.2b). Moreover, Kirsch's Equations [1] are quoted in his Eq. (7.1a–c) with a wrong statement that these would be valid for plane strain; the quoted equations are for plane stress boundary conditions. Several other sources [19–20] have promulgated the use of wrong equations similar to Goodman's (without mentioning the source). The original Kirsch equations are widely used, but also widely misused or marred by misprinted equations in the literature. For example, Eq. (3.15) in [21] has a typo, and dropped a plus sign between two terms, for the radial stress around a single borehole.

#### **3.3 Plane stress solution for the hole with uniaxial far-field stress**

With the expressions of Sections 2.1 and 2.3 in place, we now solve Eqs. (1) and (2) for plane stress by substituting *κ* ¼ ð Þ 3 � *ν =*ð Þ 1 þ *v* ; the corresponding displacement solutions are:

$$u\_r = \frac{\sigma\_{\text{xx}-\text{os}}}{4\operatorname{Gr}} \left\{ \frac{1-\nu}{1+\nu}r^2 + a^2 + \left(\frac{4a^2}{1+\nu} + r^2 - \frac{a^4}{r^2}\right) \cos 2\theta \right\} \tag{7}$$

$$u\_{\theta} = -\frac{\sigma\_{\text{xx}-\text{os}}}{4\operatorname{Gr}} \left[ \frac{1-v}{1+v} \left. 2a^2 + r^2 + \frac{a^4}{r^2} \right| \sin 2\theta \tag{8}$$

The above expressions for plane stress may be formulated using Young's modulus, *E*, instead of the shear modulus, *G*, substituting *G* ¼ *E=*2 1ð Þ þ *ν* into Eqs. (7) and (8):

$$u\_r = \sigma\_{\text{xx}-\text{os}} \left( \frac{\mathbf{1} + \boldsymbol{\nu}}{2Er} \right) \left\{ \frac{\mathbf{1} - \boldsymbol{\nu}}{\mathbf{1} + \boldsymbol{\nu}} r^2 + a^2 + \left( \frac{4a^2}{\mathbf{1} + \boldsymbol{\nu}} + r^2 - \frac{a^4}{r^2} \right) \cos 2\theta \right\} \tag{9}$$

$$u\_{\theta} = -\sigma\_{\text{xx}-\text{os}} \left( \frac{\mathbf{1} + v}{2Er} \right) \left[ \frac{\mathbf{1} - v}{\mathbf{1} + v} \ 2a^2 + r^2 + \frac{a^4}{r^2} \right] \sin 2\theta \tag{10}$$

Eqs. (9) and (10) are identical to those given in [22] (p. 740–742) and were used in a prior study focused on multi-hole problems under plane stress [3].

Separately, we checked for the computational integrity of the plane strain displacement solutions of Section 3.2 by applying a standard conversion substitution, as explained in Appendix A.
