### Problem Statement**4.35** The equilibrium configuration of a deformed body is described by the mapping\[ \chi(X) = AX_1 \mathbf{e}_1 - BX_3 \mathbf{e}_2 + CX_2 \mathbf{e}_3, \]where \( A \), \( B \), and \( C \) are constants. If the Cauchy stress tensor for this body is\[[\sigma] = \begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & \sigma_0\end{bmatrix} \, \text{MPa},\]where \( \sigma_0 \) is a constant, determine:(a) The deformation tensor and its inverse in matrix form,(b) The matrices of the first and second Piola–Kirchhoff stress tensors, and(c) The pseudo stress vectors associated with the first and second Piola–Kirchhoff stress tensors on the \( \mathbf{e}_3 \)-plane in the deformed configuration. ### Explanation1. **Mapping Function**: The vector function \(\chi(X)\) describes how the coordinates of a point in a body deform. The components \(AX_1\), \(-BX_3\), and \(CX_2\) scale the original coordinates \(X_1\), \(X_3\), and \(X_2\) respectively.2. **Cauchy Stress Tensor**: Represented by a diagonal matrix where only the \((3,3)\)-component, \(\sigma_0\), is non-zero. This indicates that the stress is applied only in the \( \mathbf{e}_3 \) direction.3. **Tasks**: Determine the following: - **Deformation Tensor**: Describes the local deformation of the material. - **Inverse of Deformation Tensor**: Useful for relations in continuum mechanics. - **First and Second Piola–Kirchhoff Stress Tensors**: These tensors relate force and deformation in different configurations. - **Pseudo Stress Vectors**: Represent forces per unit area in the reference and current configurations on the specified plane.### Steps- **Deformation Tensor Calculation**: Derive the deformation tensor from the mapping function and solve for its inverse.- **Piola–Kirchhoff Stress

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The equilibrium configuration of a deformed body is described by the mapping X(X) = AX1 ê₁ - BX302 + CX203, where A, B, and C are constants. If the Cauchy stress tensor for this body is го о о [a] 0 0 0 MPa, 00 do where do is a constant, determine (a) the deformation tensor and its inverse in matrix form, (b) the matrices of the first and second Piola-Kirchhoff stress tensors, and (c) the pseudo stress vectors associated with the first and second Piola-Kirchhoff stress tensors on the ê3-plane in the deformed configuration.

The equilibrium configuration of a deformed body is described by the mapping X(X) = AX1 ê₁ - BX302 + CX203, where A, B, and C are constants. If the Cauchy stress tensor for this body is го о о [a] 0 0 0 MPa, 00 do where do is a constant, determine (a) the deformation tensor and its inverse in matrix form, (b) the matrices of the first and second Piola-Kirchhoff stress tensors, and (c) the pseudo stress vectors associated with the first and second Piola-Kirchhoff stress tensors on the ê3-plane in the deformed configuration.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

### Problem Statement

**4.35** The equilibrium configuration of a deformed body is described by the mapping

\[ \chi(X) = AX_1 \mathbf{e}_1 - BX_3 \mathbf{e}_2 + CX_2 \mathbf{e}_3, \]

where \( A \), \( B \), and \( C \) are constants. If the Cauchy stress tensor for this body is

\[
[\sigma] = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & \sigma_0
\end{bmatrix} \, \text{MPa},
\]

where \( \sigma_0 \) is a constant, determine:

(a) The deformation tensor and its inverse in matrix form,

(b) The matrices of the first and second Piola–Kirchhoff stress tensors, and

(c) The pseudo stress vectors associated with the first and second Piola–Kirchhoff stress tensors on the \( \mathbf{e}_3 \)-plane in the deformed configuration.

### Explanation

1. **Mapping Function**: The vector function \(\chi(X)\) describes how the coordinates of a point in a body deform. The components \(AX_1\), \(-BX_3\), and \(CX_2\) scale the original coordinates \(X_1\), \(X_3\), and \(X_2\) respectively.

2. **Cauchy Stress Tensor**: Represented by a diagonal matrix where only the \((3,3)\)-component, \(\sigma_0\), is non-zero. This indicates that the stress is applied only in the \( \mathbf{e}_3 \) direction.

3. **Tasks**: Determine the following:
- **Deformation Tensor**: Describes the local deformation of the material.
- **Inverse of Deformation Tensor**: Useful for relations in continuum mechanics.
- **First and Second Piola–Kirchhoff Stress Tensors**: These tensors relate force and deformation in different configurations.
- **Pseudo Stress Vectors**: Represent forces per unit area in the reference and current configurations on the specified plane.

### Steps

- **Deformation Tensor Calculation**: Derive the deformation tensor from the mapping function and solve for its inverse.

- **Piola–Kirchhoff Stress

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