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Question

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VIPSolutions · Accepted Answer

Let's break down and address each part of the question based on what's visible in the image.

---

## 2(a)
**(i) How do the components of an objective 3rd order tensor, given by $ Q = G_{ijk} e_i \otimes e_j \otimes e_k $, transform under the coordinate change $ Q_{ijk} = (\bar{e}_i, e_p) (\bar{e}_j, e_q) (\bar{e}_k, e_r) G_{pqr} $? ($ \bar{e}_i $ is the "new" basis vector)**

**Answer:**
The components transform as:
$$
Q_{ijk} = a_{ip} a_{jq} a_{kr} G_{pqr}
$$
where $ a_{ip} = (\bar{e}_i, e_p) $ is the direction cosine matrix.

---

**(ii) How many unique elements are in a tensor whose components are $ C_{ijkl} $?**

**Answer:**
For a 3D space (indices 1,2,3), there are $ 3^4 = 81 $ components **if there are no symmetry constraints**.

---

## 2(c)
Given assertions:
- $ T_{ij} = C_{ijkl} B_{kl} $
- $ T_{ij} = \frac{\partial U}{\partial B_{ij}} $

**How many unique elements can $ C_{ijkl} $ possess?**

**Answer:**
If the above assertions are true and **no symmetry is assumed**, $ C_{ijkl} $ has 81 components.
However, usually, from the symmetry of mixed partial derivatives and the symmetry of $ B_{kl} $, $ C_{ijkl} $ is symmetric with respect to $ (i, j) $ and $ (k, l) $:
$$
C_{ijkl} = C_{jikl} = C_{ijlk}
$$
For a symmetric 2nd-order tensor $ B_{kl} $, the number of independent components for $ C_{ijkl} $ in 3D is **21** (for each index pair), so $ 21 \times 21 = 441 $, but considering major and minor symmetry reduces it to **21 independent components** (this is the case for isotropic elasticity).

---

## 2(d)
Given assertions:
- $ T_{ij} = C_{ijkl} B_{kl} $
- $ T_{ij} = \frac{\partial U}{\partial B_{ij}} $
- $ T_{ij} = T_{ji} $
- $ B_{ij} = B_{ji} $

**How many unique elements can $ C_{ijkl} $ possess?**

**Answer:**
Because both $ T_{ij} $ and $ B_{kl} $ are symmetric, and with the properties of elasticity tensors, $ C_{ijkl} $ has the following symmetries:
$$
C_{ijkl} = C_{jikl} = C_{ijlk} = C_{klij}
$$
Therefore, in 3D, **21 independent components** (for the most general anisotropic case).

---

## 2(e) Deformation gradient, strains, and geometry

Given:
$$
\begin{align*}
x_1 &= X_1 + u - X_3 \frac{\partial w}{\partial X_1} - X_2 \frac{\partial v}{\partial X_1} \
x_2 &= X_2 + v - X_3 \frac{\partial w}{\partial X_2} - X_1 \frac{\partial u}{\partial X_2} \
x_3 &= X_3 + w
\end{align*}
$$
Assume $ u, v, w $ do not depend on $ X_3 $.

---

**i. Express the components of the deformation gradient as a matrix.**

The deformation gradient $ F_{ij} = \frac{\partial x_i}{\partial X_j} $:

$$
F = \begin{pmatrix}
1 + \frac{\partial u}{\partial X_1} - X_3 \frac{\partial^2 w}{\partial X_1^2} - X_2 \frac{\partial^2 v}{\partial X_1^2} & \frac{\partial u}{\partial X_2} - X_3 \frac{\partial^2 w}{\partial X_1 \partial X_2} - X_2 \frac{\partial^2 v}{\partial X_1 \partial X_2} & -\frac{\partial w}{\partial X_1} \
\frac{\partial v}{\partial X_1} - X_3 \frac{\partial^2 w}{\partial X_1 \partial X_2} - X_1 \frac{\partial^2 u}{\partial X_1 \partial X_2} & 1 + \frac{\partial v}{\partial X_2} - X_3 \frac{\partial^2 w}{\partial X_2^2} - X_1 \frac{\partial^2 u}{\partial X_2^2} & -\frac{\partial w}{\partial X_2} \
\frac{\partial w}{\partial X_1} & \frac{\partial w}{\partial X_2} & 1
\end{pmatrix}
$$

---

**ii. For the infinitesimal case, write down the strains in Voigt notation.**

Infinitesimal strain tensor:
$$
E_{ij} = \frac{1}{2} \left(\frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i} \right)
$$

Voigt notation:
$$
\begin{bmatrix}
E_{11} \
E_{22} \
E_{33} \
E_{12} \
E_{13} \
E_{23}
\end{bmatrix}
=
\begin{bmatrix}
\frac{\partial u}{\partial X_1} \
\frac{\partial v}{\partial X_2} \
\frac{\partial w}{\partial X_3} \
\frac{1}{2} \left( \frac{\partial u}{\partial X_2} + \frac{\partial v}{\partial X_1} \right) \
\frac{1}{2} \left( \frac{\partial u}{\partial X_3} + \frac{\partial w}{\partial X_1} \right) \
\frac{1}{2} \left( \frac{\partial v}{\partial X_3} + \frac{\partial w}{\partial X_2} \right)
\end{bmatrix}
$$

Since $ u, v, w $ are independent of $ X_3 $, $ \frac{\partial u}{\partial X_3} = 0 $, etc.

---

**iii. Total length of the edge from $(a,-a,t)$ to $(a,a,t)$:**

Edge: $ X_2 $ varies from $-a$ to $a$, $ X_1 = a $, $ X_3 = t $.

The deformed edge:
$$
\ell = \int_{-a}^{a} \sqrt{ \left( \frac{\partial x_1}{\partial X_2} \right)^2 + \left( \frac{\partial x_2}{\partial X_2} \right)^2 + \left( \frac{\partial x_3}{\partial X_2} \right)^2 }\, dX_2
$$
Plug in the expressions for derivatives from the deformation mapping.

---

**iv. Angle at the corner initially at $( -a, -a, 0 )$:**

Find the vectors along the two edges meeting at this corner after deformation, then use the dot product to find the angle.

---

**v. Relative volume change of an infinitesimal element:**

Relative volume change is given by the determinant of the deformation gradient:
$$
\frac{dV}{dV_0} = \det(F)
$$

---

**vi. If $ u = v = 0 $, give mathema

VIPSolutions · Answer

Let's systematically address each part of the question based on the given image and the text:

---

### 2(a) Transformation of a third-order objective tensor $Q_{ijk}$

**Question:** How do the components $Q_{ijk}$ transform under a change of basis $ \bar{e}_i $?

**Answer:**

The components of a third-order tensor transform via the rotation matrix $a_{ip} = (\bar{e}_i, e_p)$ as:

$$
Q_{ijk}' = a_{ip} \, a_{jq} \, a_{kr} \, Q_{pqr}
$$

**In words:** Each index of the tensor transforms with the rotation matrix $a_{ip}$ associated with the basis change.

---

### 2(b) Number of unique elements in $ C_{ijkl} $

**Question:** How many unique elements are in a fourth-order tensor $ C_{ijkl} $?

**Answer:**

- Total components without symmetry: $3^4 = 81$.

- **Assuming no symmetries:** all 81 are potentially unique.

**So the answer is:**

$$
\boxed{81}
$$

---

### 2(c) Structure of $ C_{ijkl} $ with $ T_{ij} = C_{ijkl} B_{kl} $ and $ T_{ij} = \frac{\partial U}{\partial B_{ij}} $

**Question:** How many independent elements can $ C_{ijkl} $ have?

**Analysis:**

- From the relation $ T_{ij} = \frac{\partial U}{\partial B_{ij}} $, second derivatives of $U$ with respect to $B_{ij}$ imply symmetry:

$$
C_{ijkl} = C_{jikl} = C_{ijlk}
$$

- These are **minor symmetries**.

- **Major symmetry** $ C_{ijkl} = C_{klij} $ also holds because mixed partial derivatives commute.

- **Result:** The tensor is symmetric with respect to interchange of $ (i,j) $ and $ (k,l) $, and also symmetric within these pairs.

- **Number of independent components:**

Map $ C_{ijkl} $ to a 6x6 matrix (Voigt notation):

$$
\boxed{21}
$$

**Answer:**

- **Number of independent elements:** $ \boxed{21} $.

---

### 2(d) Additional symmetry: $ T_{ij} = T_{ji} $, $ B_{ij} = B_{ji} $

**Question:** How many unique elements can $ C_{ijkl} $ have?

**Analysis:**

- Both $ T_{ij} $ and $ B_{ij} $ are symmetric.

- This implies $ C_{ijkl} $ must satisfy:

$$
C_{ijkl} = C_{jikl} = C_{ijlk} = C_{klij}
$$

- These symmetries are **the same as in classical elasticity tensors**, leading to **21** independent components.

**Answer:**

$$
\boxed{21}
$$

---

### 2(e) Deformation of a square plate

Given the deformation:

$$
x_1 = X_1 + u - X_3 \frac{\partial w}{\partial X_1} - X_2 \frac{\partial v}{\partial X_1}
$$
$$
x_2 = X_2 + v - X_3 \frac{\partial w}{\partial X_2} - X_1 \frac{\partial u}{\partial X_2}
$$
$$
x_3 = X_3 + w
$$

with $ u, v, w $ independent of $ X_3 $.

---

### (i) **Deformation gradient $ F_{ij} $:**

Calculate:

$$
F_{ij} = \frac{\partial x_i}{\partial X_j}
$$

- $ F_{11} $:

$$
\frac{\partial}{\partial X_1} \left( X_1 + u - X_3 w_{,1} - X_2 v_{,1} \right) = 1 + u_{,1} - X_3 w_{,11} - X_2 v_{,11}
$$

- $ F_{12} $:

$$
\frac{\partial}{\partial X_2} \left( X_1 + u - X_3 w_{,1} - X_2 v_{,1} \right) = u_{,2} - X_3 w_{,12} - v_{,1} - X_2 v_{,12}
$$

- $ F_{13} $:

$$
\frac{\partial}{\partial X_3} \left( X_1 + u - X_3 w_{,1} - X_2 v_{,1} \right) = -w_{,1}
$$

Similarly for the others:

- $ F_{21} = -X_3 w_{,12} - u_{,2} - X_1 u_{,12} $
- $ F_{22} = 1 + v_{,2} - X_3 w_{,22} - X_1 u_{,22} $
- $ F_{23} = -w_{,2} $
- $ F_{31} = w_{,1} $
- $ F_{32} = w_{,2} $
- $ F_{33} = 1 $

---

### (ii) **Infinitesimal strains in Voigt notation:**

$$
E_{ij} = \frac{1}{2} (u_{i,j} + u_{j,i})
$$

Given $ u, v, w $ independent of $ X_3 $:

$$
\boxed{
\begin{bmatrix}
E_{11} \ E_{22} \ E_{33} \ E_{12} \ E_{13} \ E_{23}
\end{bmatrix}
=
\begin{bmatrix}
u_{,1} - X_3 w_{,11} - X_2 v_{,11} \
v_{,2} - X_3 w_{,22} - X_1 u_{,22} \
0 \
\frac{1}{2}(u_{,2} + v_{,1} - 2X_3 w_{,12} - X_2 v_{,12} - X_1 u_{,12}) \
0 \
0
\end{bmatrix}
}
$$

---

### (iii) **Total length of the edge from $(a, -a, t)$ to $(a, a, t)$:**

The initial length:

$$
L_0 = 2a
$$

The deformed length:

$$
L = \int_{-a}^{a} \left| \frac{\partial \mathbf{x}}{\partial X_2} \right| dX_2
$$

where

$$
\frac{\partial \mathbf{x}}{\partial X_2} = \left( u_{,2} - X_3 w_{,12} - v_{,1} - X_2 v_{,12}, \quad 1 + v_{,2} - X_3 w_{,22} - X_1 u_{,22}, \quad w_{,2} \right)
$$

---

### (iv) **Angle at the corner initially at $(-a, -a, 0)$:**

Calculate the deformation of the two edges meeting at this point:

- Along $ X_1 $-direction:

$$
\mathbf{v}_1 = \frac{\partial \mathbf{x}}{\partial X_1}
$$

- Along $ X_2 $-direction:

$$
\mathbf{v}_2 = \frac{\partial \mathbf{x}}{\partial X_2}
$$

The angle $ \theta $ is:

$$
\cos \theta = \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{|\mathbf{v}_1| |\mathbf{v}_2|}
$$

---

### (v) **Relative volume change:**

$$
\boxed{
\frac{dV}{dV_0} = \det \mathbf{F}
}
$$

The volume change is given by the determinant of the deformation gradient $ \mathbf{F} $.

---

### (vi) **Condition for volume conservation when $ u = v = 0 $:**

$$
\det \mathbf{F} = 1
$$

which implies the Jacobian $ J = 1 $.

---

**Summary:**

| Part | Result |
|---|---|
| (a) | $ Q_{ijk}' = a_{ip} a_{jq} a_{kr} Q_{pqr} $

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