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物理代写|弹性力学代写Elasticity代考|ME340 Curvilinear cylindrical and spherical coordinates

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物理代写|弹性力学代写Elasticity代考|Curvilinear cylindrical and spherical coordinates

The solution to many problems in elasticity requires the use of curvilinear cylindrical and spherical coordinates. It is therefore necessary to have the field equations expressed in terms of such coordinate systems. We now pursue the development of the strain-displacement relations in cylindrical and spherical coordinates. Starting with form (2.2.7)
$$\boldsymbol{e}=\frac{1}{2}=\left[\nabla \boldsymbol{u}+(\nabla \boldsymbol{u})^T\right]$$
the desired curvilinear relations can be determined using the appropriate forms for the displacement gradient term $\nabla u$.

The cylindrical coordinate system previously defined in Fig. $1.5$ establishes new components for the displacement vector and strain tensor
\begin{aligned} &\boldsymbol{u}=u_r \boldsymbol{e}r+u\theta \boldsymbol{e}\theta+u_z \boldsymbol{e}_z \ &\boldsymbol{e}=\left[\begin{array}{lll} e_r & e{r \theta} & e_{r z} \ e_{r \theta} & e_\theta & e_{\theta z} \ e_{r z} & e_{\theta z} & e_z \end{array}\right] \end{aligned}
Notice that the symmetry of the strain tensor is preserved in this orthogonal curvilinear system. Using results (1.9.17) and (1.9.10), the derivative operation in cylindrical coordinates can be expressed by
\begin{aligned} \nabla u=& \frac{\partial u_r}{\partial r} \boldsymbol{e}r \boldsymbol{e}_r+\frac{\partial u\theta}{\partial r} \boldsymbol{e}r \boldsymbol{e}\theta+\frac{\partial u_z}{\partial r} \boldsymbol{e}r \boldsymbol{e}_z \ &+\frac{1}{r}\left(\frac{\partial u_r}{\partial \theta}-u\theta\right) \boldsymbol{e}\theta \boldsymbol{e}_r+\frac{1}{r}\left(u_r+\frac{\partial u\theta}{\partial \theta}\right) \boldsymbol{e}\theta \boldsymbol{e}\theta+\frac{1}{r} \frac{\partial u_z}{\partial \theta} \boldsymbol{e}\theta \boldsymbol{e}_z \ &+\frac{\partial u_r}{\partial z} \boldsymbol{e}_z \boldsymbol{e}_r+\frac{\partial u\theta}{\partial z} \boldsymbol{e}z \boldsymbol{e}\theta+\frac{\partial u_z}{\partial z} \boldsymbol{e}z \boldsymbol{e}_z \end{aligned} Placing this result into the strain-displacement form (2.2.7) gives the desired relations in cylindrical coordinates. The individual scalar equations are given by \begin{aligned} &e_r=\frac{\partial u_r}{\partial r}, e\theta=\frac{1}{r}\left(u_r+\frac{\partial u_\theta}{\partial \theta}\right), e_z=\frac{\partial u_z}{\partial z} \ &e_{r \theta}=\frac{1}{2}\left(\frac{1}{r} \frac{\partial u_r}{\partial \theta}+\frac{\partial u_\theta}{\partial r}-\frac{u_\theta}{r}\right) \ &e_{\theta z}=\frac{1}{2}\left(\frac{\partial u_\theta}{\partial z}+\frac{1}{r} \frac{\partial u_z}{\partial \theta}\right) \ &e_{z r}=\frac{1}{2}\left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right) \end{aligned}

物理代写|弹性力学代写Elasticity代考|Body and surface forces

When a structure is subjected to applied external loadings, internal forces are induced inside the body. Following the philosophy of continuum mechanics, these internal forces are distributed continuously within the solid. In order to study such forces, it is convenient to categorize them into two major groups, commonly referred to as body forces and surface forces.

Body forces are proportional to the body’s mass and are reacted with an agent outside of the body. Examples of these include gravitational-weight forces, magnetic forces, and inertial forces. Fig. 3.1A shows an example body force of an object’s self-weight. By using continuum mechanics principles, a body force density (force per unit volume) $\boldsymbol{F}(\boldsymbol{x})$ can be defined such that the total resultant body force of an entire solid can be written as a volume integral over the body
$$\boldsymbol{F}_R=\iiint_V \boldsymbol{F}(\boldsymbol{x}) d V$$
Surface forces always act on the surface and generally result from physical contact with another body. Fig. 3.1B illustrates surface forces existing in a beam section that has been created by sectioning the body into two pieces. For this particular case, the surface $S$ is a virtual one in the sense that it was artificially created to investigate the nature of the internal forces at this location in the body. Again, the resultant surface force over the entire surface $S$ can be expressed as the integral of a surface force density function $T^n(x)$
$$\boldsymbol{F}_S=\iint_S \boldsymbol{T}^n(\boldsymbol{x}) d S$$

物理代写|弹性力学代写弹性代考|曲线柱坐标和球坐标

$$\boldsymbol{e}=\frac{1}{2}=\left[\nabla \boldsymbol{u}+(\nabla \boldsymbol{u})^T\right]$$

\begin{aligned} &\boldsymbol{u}=u_r \boldsymbol{e}r+u\theta \boldsymbol{e}\theta+u_z \boldsymbol{e}z \ &\boldsymbol{e}=\left[\begin{array}{lll} e_r & e{r \theta} & e{r z} \ e_{r \theta} & e_\theta & e_{\theta z} \ e_{r z} & e_{\theta z} & e_z \end{array}\right] \end{aligned}

\begin{aligned} \nabla u=& \frac{\partial u_r}{\partial r} \boldsymbol{e}r \boldsymbol{e}r+\frac{\partial u\theta}{\partial r} \boldsymbol{e}r \boldsymbol{e}\theta+\frac{\partial u_z}{\partial r} \boldsymbol{e}r \boldsymbol{e}_z \ &+\frac{1}{r}\left(\frac{\partial u_r}{\partial \theta}-u\theta\right) \boldsymbol{e}\theta \boldsymbol{e}_r+\frac{1}{r}\left(u_r+\frac{\partial u\theta}{\partial \theta}\right) \boldsymbol{e}\theta \boldsymbol{e}\theta+\frac{1}{r} \frac{\partial u_z}{\partial \theta} \boldsymbol{e}\theta \boldsymbol{e}_z \ &+\frac{\partial u_r}{\partial z} \boldsymbol{e}_z \boldsymbol{e}_r+\frac{\partial u\theta}{\partial z} \boldsymbol{e}z \boldsymbol{e}\theta+\frac{\partial u_z}{\partial z} \boldsymbol{e}z \boldsymbol{e}_z \end{aligned}将该结果代入应变-位移形式(2.2.7)，得到柱坐标下所需的关系。单个标量方程由\begin{aligned} &e_r=\frac{\partial u_r}{\partial r}, e\theta=\frac{1}{r}\left(u_r+\frac{\partial u\theta}{\partial \theta}\right), e_z=\frac{\partial u_z}{\partial z} \ &e_{r \theta}=\frac{1}{2}\left(\frac{1}{r} \frac{\partial u_r}{\partial \theta}+\frac{\partial u_\theta}{\partial r}-\frac{u_\theta}{r}\right) \ &e_{\theta z}=\frac{1}{2}\left(\frac{\partial u_\theta}{\partial z}+\frac{1}{r} \frac{\partial u_z}{\partial \theta}\right) \ &e_{z r}=\frac{1}{2}\left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right) \end{aligned} 给出

物理代写|弹性力学代写弹性代考|身体和表面力

$$\boldsymbol{F}_R=\iiint_V \boldsymbol{F}(\boldsymbol{x}) d V$$

$$\boldsymbol{F}_S=\iint_S \boldsymbol{T}^n(\boldsymbol{x}) d S$$ 的积分

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