**C.3 The superposed Cases 1–1 and 1–2 (Case 1–3)**

In this section, we consider Case 1–3 of the superposed of Cases 1–1 and 1–2. The plane strain displacement solutions in both directions are the sum of Eqs. (15), (16) and (27), (28). Similarly, the plane stress displacement solutions in both directions are the sum of Eqs. (17) and (18) and (27), (28). The strain tensor components, principal strains, stress tensor components, and principal stresses will be computed accordingly. As in the results of the previous two cases, the first column images are for plane strain; the second column images are for plane stress; and the third column quantifies the delta between plane strain and plane stress solutions.

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

## 1. Displacement fields.

#### **Figure C12.**

*The elastic displacement solutions around the single hole in an elastic plate subjected to both far-field stress and internal pressure. First column is for the plan strain solution, which is the sum of the results in the first column of Figures C1 and C7. Second column is the plane stress solution, which is the sum of the results in the second column of Figures C1 and C7. Third column quantifies the delta between plane stress and plane strain solutions. Since there is no displacement difference corresponding to the internal pressure effect, the deltas presented in the third column of this figure are identical to those in the third column of Figures C1.*

#### 2. Strain tensor components.

#### **Figure C13.**

*Strain tensor components around the single hole in an elastic plate subjected to both far-field stress and internal pressure. These solutions are obtained by analytical differentiation of the displacement components in Figure C12 in both x and y directions. εzz* ¼ 0 *for the plane strain solution and therefore there is no delta in z*�*direction.*

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

### 3. Principal strains.

#### **Figure C14.**

*Principles strains around the single hole in an elastic plate subjected to both far-field stress and internal pressure. These solutions are obtained by applying Eq. (36) for the principal strain magnitude solution using the strain tensors in Figure C13.*

#### 4. Stress tensor components.

#### **Figure C15.**

*Stress tensor components around the single hole in an elastic plate subjected to both far-field stress and internal pressure. These solutions are obtained from the relations between the strain tensor components and the stress tensor components shown in Eqs. (32)–(34). Note that σzz* ¼ 0 *for plane stress solution and therefore there is no delta in the z*�*direction.*

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