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# 物理代写|弹性力学代写Elasticity代考|CEE2321 Traction vector and stress tensor

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## 物理代写|弹性力学代写Elasticity代考|Traction vector and stress tensor

In order to quantify the nature of the internal distribution of forces within a continuum solid, consider a general body subject to arbitrary (concentrated and distributed) external loadings, as shown in Fig. 3.2. To investigate the internal forces, a section is made through the body as shown. On this section consider a small area $\Delta A$ with unit outward normal vector $\boldsymbol{n}$. The resultant surface force acting on $\Delta A$ is defined by $\Delta \boldsymbol{F}$. Consistent with our earlier discussion, no resultant surface couple is included. The stress or traction vector is defined by
$$\boldsymbol{T}^n(\boldsymbol{x}, \boldsymbol{n})=\lim _{\Delta A \rightarrow 0} \frac{\Delta \boldsymbol{F}}{\Delta A}$$
Notice that the traction vector depends on both the spatial location and the unit normal vector to the surface under study. Thus, even though we may be investigating the same point, the traction vector still varies as a function of the orientation of the surface normal. Because the traction is defined as force per unit area, the total surface force is determined through integration as per relation (3.1.2). Note, also, the simple action-reaction principle (Newton’s third law)
$$\boldsymbol{T}^n(\boldsymbol{x}, \boldsymbol{n})=-\boldsymbol{T}^n(\boldsymbol{x},-\boldsymbol{n})$$

## 物理代写|弹性力学代写Elasticity代考|Stress transformation

Analogous to our previous discussion with the strain tensor, the stress components must also follow the standard transformation rules for second-order tensors established in Section 1.5. Applying transformation relation $(1.5 .1)3$ for the stress gives $$\sigma{i j}^{\prime}=Q_{i p} Q_{j q} \sigma_{p q}$$
where the rotation matrix $Q_{i j}=\cos \left(x_i^{\prime}, x_j\right)$. Therefore, given the stress in one coordinate system, we can determine the new components in any other rotated system. For the general three-dimensional case, the rotation matrix may be chosen in the form
$$Q_{i j}=\left[\begin{array}{lll} l_1 & m_1 & n_1 \ l_2 & m_2 & n_2 \ l_3 & m_3 & n_3 \end{array}\right]$$
Using this notational scheme, the specific transformation relations for the stress then become
\begin{aligned} &\sigma_x^{\prime}=\sigma_x l_1^2+\sigma_y m_1^2+\sigma_z n_1^2+2\left(\tau_{x y} l_1 m_1+\tau_{y z} m_1 n_1+\tau_{z x} n_1 l_1\right) \ &\sigma_y^{\prime}=\sigma_x l_2^2+\sigma_y m_2^2+\sigma_z n_2^2+2\left(\tau_{x y} l_2 m_2+\tau_{y z} m_2 n_2+\tau_{z x} n_2 l_2\right) \ &\sigma_z^{\prime}=\sigma_x l_3^2+\sigma_y m_3^2+\sigma_z n_3^2+2\left(\tau_{x y} l_3 m_3+\tau_{y z} m_3 n_3+\tau_{z x} n_3 l_3\right) \ &\tau_{x y}^{\prime}=\sigma_x l_1 l_2+\sigma_y m_1 m_2+\sigma_z n_1 n_2+\tau_{x y}\left(l_1 m_2+m_1 l_2\right)+\tau_{y z}\left(m_1 n_2+n_1 m_2\right)+\tau_{z x}\left(n_1 l_2+l_1 n_2\right) \ &\tau_{y z}^{\prime}=\sigma_x l_2 l_3+\sigma_y m_2 m_3+\sigma_z n_2 n_3+\tau_{x y}\left(l_2 m_3+m_2 l_3\right)+\tau_{y z}\left(m_2 n_3+n_2 m_3\right)+\tau_{z x}\left(n_2 l_3+l_2 n_3\right) \ &\tau^{\prime}{ }{z x}^{\prime}=\sigma_x l_3 l_1+\sigma_y m_3 m_1+\sigma_z n_3 n_1+\tau{x y}\left(l_3 m_1+m_3 l_1\right)+\tau_{y z}\left(m_3 n_1+n_3 m_1\right)+\tau_{z x}\left(n_3 l_1+l_3 n_1\right) \end{aligned}

## 物理代写|弹性力学代写Elasticity代考|牵引力矢量和应力张量

$$\boldsymbol{T}^n(\boldsymbol{x}, \boldsymbol{n})=-\boldsymbol{T}^n(\boldsymbol{x},-\boldsymbol{n})$$

## 物理代写|弹性力学代写Elasticity代考|应力转换

.

，其中旋转矩阵$Q_{i j}=\cos \left(x_i^{\prime}, x_j\right)$。因此，给定一个坐标系中的应力，我们可以确定任何其他旋转系统中的新分量。对于一般的三维情况，旋转矩阵可以选择形式为
$$Q_{i j}=\left[\begin{array}{lll} l_1 & m_1 & n_1 \ l_2 & m_2 & n_2 \ l_3 & m_3 & n_3 \end{array}\right]$$

\begin{aligned} &\sigma_x^{\prime}=\sigma_x l_1^2+\sigma_y m_1^2+\sigma_z n_1^2+2\left(\tau_{x y} l_1 m_1+\tau_{y z} m_1 n_1+\tau_{z x} n_1 l_1\right) \ &\sigma_y^{\prime}=\sigma_x l_2^2+\sigma_y m_2^2+\sigma_z n_2^2+2\left(\tau_{x y} l_2 m_2+\tau_{y z} m_2 n_2+\tau_{z x} n_2 l_2\right) \ &\sigma_z^{\prime}=\sigma_x l_3^2+\sigma_y m_3^2+\sigma_z n_3^2+2\left(\tau_{x y} l_3 m_3+\tau_{y z} m_3 n_3+\tau_{z x} n_3 l_3\right) \ &\tau_{x y}^{\prime}=\sigma_x l_1 l_2+\sigma_y m_1 m_2+\sigma_z n_1 n_2+\tau_{x y}\left(l_1 m_2+m_1 l_2\right)+\tau_{y z}\left(m_1 n_2+n_1 m_2\right)+\tau_{z x}\left(n_1 l_2+l_1 n_2\right) \ &\tau_{y z}^{\prime}=\sigma_x l_2 l_3+\sigma_y m_2 m_3+\sigma_z n_2 n_3+\tau_{x y}\left(l_2 m_3+m_2 l_3\right)+\tau_{y z}\left(m_2 n_3+n_2 m_3\right)+\tau_{z x}\left(n_2 l_3+l_2 n_3\right) \ &\tau^{\prime}{ }{z x}^{\prime}=\sigma_x l_3 l_1+\sigma_y m_3 m_1+\sigma_z n_3 n_1+\tau{x y}\left(l_3 m_1+m_3 l_1\right)+\tau_{y z}\left(m_3 n_1+n_3 m_1\right)+\tau_{z x}\left(n_3 l_1+l_3 n_1\right) \end{aligned}

## Matlab代写

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