### 数学代写|有限元方法代写Finite Element Method代考|MECH3300

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## 数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

An elastic deformation should not cause holes in a deformable body that does not have any holes before deformation. Moreover, no material overlap should be predicted by the displacement field. The strain compatibility conditions ensure that these constraints are satisfied [7].

In a planar deformation, where $u_x=u_x(x, y), u_y=u_y(x, y)$ and $u_z=0$, consider the following combination of the strains,
$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

Using the definitions given in Eq. (2.47) we will find,
$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$
Thus we see that the relationship (a) must be equal to zero. This is the strain compatibility equation for a two-dimensional deformation in the $x, y$ plane, which imposes a specific relationship between the strains and the strain-displacement relationships.

For three-dimensional deformations where $u_x=u_x(x, y, z), u_y=u_y(x, y, z)$ and $u_z=u_z(x, y, z)$ there are a total of six strain compatibility conditions. These can be found as follows:
\begin{aligned} & \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \ & \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \ & \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \ & 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \ & 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \ & 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right) \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

As stated in the introduction to Section 2.2, when a deformable body is subjected to external effects such as external forces and/or imposed displacements on its boundary, its shape will change and internal forces will develop throughout its volume. The level of deformation for given external effects depends on the material of the deformable body. Constitutive relations are empirically obtained, material specific relationships between the stress and the strain in the body. Here we are primarily interested in linear elastic relationships.

The deformation behavior of a specific material is determined experimentally. These experiments are designed such that only one of the stress components and the corresponding strain dominates the problem. This state is known as a simpleloading state.

For linear, isotropic materials tensile loading of a slender test specimen, i.e., the simple-tension test, reveals two fundamental material properties. The relationship between the normal stress and the normal strain is found by conducting a simple-tension test, as follows:
$$\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z$$
where $E$ is the elastic modulus of the material, also referred to as the Young’s modulus. The relationship between the longitudinal strain $\varepsilon_l$ and the transverse strain $\varepsilon_t$ represents the Poisson’s ratio, the second material property,
$$v=-\frac{\varepsilon_t}{\varepsilon_l}$$
The simple-shear test reveals the relationship between the shear strain and the shear stress,
$$\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j$$
where $G$ is the shear modulus, or modulus of rigidity. For a linear, elastic, isotropic material the following relationship holds:
$$G=\frac{E}{2(1+v)}$$

## 数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$

\begin{aligned} & \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \ & \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \ & \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \ & 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \ & 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \ & 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right) \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

$$\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z$$

$$v=-\frac{\varepsilon_t}{\varepsilon_l}$$

$$\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j$$

$$G=\frac{E}{2(1+v)}$$

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