### 数学代写|计算方法代写computational method代考|Equations of linear elasticity – strong form

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## 数学代写|计算方法代写computational method代考|Strain-displacement relationships

Mathematical problems of linear elasticity belong in the category of vector elliptic boundary value problems. The unknown functions are the components of the displacement vector. In Cartesian coordinates the displacement vector is:
\begin{aligned} \mathbf{u} & \stackrel{\text { def }}{ }=u_{x}(x, y, z) \mathbf{e}{x}+u{y}(x, y, z) \mathbf{e}{y}+u{z}(x, y, z) \mathbf{e}{z} \ & \equiv\left{u{x}(x, y, z) u_{y}(x, y, z) u_{z}(x, y, z)\right}^{T} \ & \equiv u_{i}\left(x_{j}\right) \end{aligned}
where $\mathbf{e}{x}, \mathbf{e}{y}, \mathbf{e}_{z}$ are the Cartesian basis vectors.
The formulation of problems of linear elasticity is based on three fundamental relationships: the strain-displacement equations, the stress-strain relationships and the equilibrium equations.

1. Strain-displacement relationships. We introduce the infinitesimal strain-displacement relationships here. A detailed derivation of these relationships is presented in Section 9.2.1. By definition, the infinitesimal normal strain components are:
$$\epsilon_{x} \equiv \epsilon_{x x} \stackrel{\text { def }}{=} \frac{\partial u_{x}}{\partial x} \quad \epsilon_{y} \equiv \epsilon_{y y} \stackrel{\text { def }}{=} \frac{\partial u_{y}}{\partial y} \quad \epsilon_{z} \equiv \epsilon_{z z} \stackrel{\text { def }}{=} \frac{\partial u_{z}}{\partial z}$$
and the shear strain components are:
\begin{aligned} &\left.\epsilon_{x y}=\epsilon_{y x} \equiv \frac{\gamma_{x y}}{2} \stackrel{\operatorname{def} 1}{=} \frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{\partial x}\right) \ &\epsilon_{y z}=\epsilon_{z y} \equiv \frac{\gamma_{y z}}{2} \stackrel{\operatorname{det} 1}{=}\left(\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}\right) \ &\epsilon_{z x}=\epsilon_{x z} \equiv \frac{\gamma_{z x}}{2} \stackrel{\text { def }}{=} \frac{1}{2}\left(\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{\partial z}\right) \end{aligned}

where $\gamma_{x y}, \gamma_{y z}, \gamma_{z x}$ are called the engineering shear strain components. In index notation, the infinitesimal strain at a point is characterized by the strain tensor
$$\epsilon_{i j} \stackrel{\text { def }}{=} \frac{1}{2}\left(u_{i, j}+u_{j, i}\right) .$$

1. Stress-strain relationships Mechanical stress is defined as force per unit area $\left(\mathrm{N} / \mathrm{m}^{2} \equiv \mathrm{Pa}\right)$. Since one Pascal (Pa) is a very small stress, the usual unit of mechanical stress is the Megapascal (MPa) which can be understood to mean either $10^{6} \mathrm{~N} / \mathrm{m}^{2}$ or $1 \mathrm{~N} / \mathrm{mm}^{2}$.
The usual notation for stress components is illustrated on an infinitesimal volume element shown in Fig. 2.7. The indexing rules are as follows: Faces to which the positive $x, y, z$ axes are normal are called positive faces, the opposite faces are called negative faces. The normal stress components are denoted by $\sigma$, the shear stresses components by $\tau$. The normal stress components are assigned one subscript only, since the orientation of the face and the direction of the stress component are the same. For example, $\sigma_{x}$ is the stress component acting on the faces to which the $x$-axis is normal and the stress component is acting in the positive (resp. negative) coordinate direction on the positive (resp. negative) face. For the shear stresses, the first index refers to the coordinate direction of the normal to the face on which the shear stress is acting. The second index refers to the direction in which the shear stress component is acting.
On a positive (resp. negative) face the positive stress components are oriented in the positive (resp. negative) coordinate directions. The reason for this is that if we subdivide a solid body into infinitesimal hexahedral volume elements, similar to the element shown in Fig. 2.7, then each negative face will be coincident with a positive face. By the action-reaction principle, the forces acting on those faces must have equal absolute value and opposite sign. In index notation $\sigma_{11} \equiv \sigma_{x}, \sigma_{12} \equiv \sigma_{x y} \equiv \tau_{x y}$, etc.

## 数学代写|计算方法代写computational method代考|Boundary and initial conditions

As in the case of heat conduction, we will consider three kinds of boundary conditions: prescribed displacements, prescribed tractions and spring boundary conditions. Tractions are forces per unit area acting on the boundary. Prescribed displacements and tractions are often specified in a normal-tangent reference frame.

1. Prescribed displacement. One or more components of the displacement vector is prescribed on all or part of the boundary. This is called a kinematic boundary condition.
2. Prescribed traction. One or more components of the traction vector is prescribed on all or part of the boundary. The definition of traction vector is given in Appendix K.1.
3. Linear spring. A linear relationship is prescribed between the traction and displacement vector components. The general form of this relationship is:
$$T_{i}=c_{i j}\left(d_{j}-u_{j}\right)$$
where $T_{i}$ is the traction vector, $c_{i j}$ is a positive-definite matrix that represents the distributed spring coefficients; $d_{j}$ is a prescribed function that represents displacement imposed on the spring and $u_{j}$ is the (unknown) displacement vector function on the boundary. The spring coefficients $c_{i j}$ (in $\mathrm{N} / \mathrm{m}^{3}$ units) may be functions of the position $x_{k}$ but are independent of the displacement $u_{i}$. This is called a “Winkler spring ${ }^{12}$ “.
A schematic representation of this boundary condition on an infinitesimal boundary surface element is shown in Fig. $2.8$ under the assumption that $c_{i j}$ is a diagonal matrix and therefore three spring coefficients $c_{1} \stackrel{\text { def }}{=} c_{11}, c_{2} \stackrel{\text { def }}{=} c_{22}, c_{3} \stackrel{\text { def }}{=} c_{33}$ characterize the elastic properties of the boundary condition.
Fig. $2.8$ should be interpreted to mean that the imposed displacement $d_{i}$ will cause a differential force $\Delta F_{i}$ to act on the centroid of the surface element. Suspending the summation rule, the magnitude of $\Delta F_{i}$ is
$$\Delta F_{i}=c_{i} \Delta A\left(d_{i}-u_{i}\right), \quad i=1,2,3$$
where $u_{i}$ is the displacement of the surface element. The corresponding traction vector is:
$$T_{i}=\lim {\Delta A \rightarrow 0} \frac{\Delta F{i}}{\Delta A}=c_{i}\left(d_{i}-u_{i}\right), \quad i=1,2,3 .$$

## 数学代写|计算方法代写computational method代考|Symmetry, antisymmetry and periodicity

Symmetry and antisymmetry of vectors in two dimensions with respect to the $y$ axis is illustrated in Fig. $2.9$

The definition of symmetry and antisymmetry of vectors in three dimensions is analogous: the corresponding vector components parallel to a plane of symmetry (resp. antisymmetry) have the same absolute value and the same (resp. opposite) sign. The corresponding vector components normal to a plane of symmetry (resp. antisymmetry) have the same absolute value and opposite (resp. same) sign.

In a plane of symmetry the normal displacement and the shearing traction components are zero. In a plane of antisymmetry the normal traction is zero and the in-plane components of the displacement vector are zero.

When the solution is periodic on $\Omega$ then a periodic sector of $\Omega$ has at least one periodic boundary segment pair denoted by $\partial \Omega_{p}^{+}$and $\partial \Omega_{p}^{-}$. On corresponding points of a periodic boundary segment pair, $P^{+} \in \partial \Omega_{p}^{+}$and $P^{-} \in \partial \Omega_{p}^{-}$the normal component of the displacement vector and the periodic in-plane components of the displacement vector have the same value. The normal component of the traction vector and the periodic in-plane components of the traction vector have the same absolute value but opposite sign.

Owing to the complexity of three-dimensional problems in elasticity, dimensional reduction is widely used. Various kinds of dimensional reduction are possible in elasticity, such as planar, axisymmetric, shell, plate, beam and bar models. Each of these model types is sufficiently important to have generated a substantial technical literature. In the following models for planar and axially symmetric problems are discussed. Models for beams, plates and shells will be discussed separately.

## 数学代写|计算方法代写computational method代考|Strain-displacement relationships

\begin{aligned} \mathbf{u} & \stackrel{\text { def }}{ }=u_{x}(x, y, z) \mathbf{e}{x}+u{y}(x, y, z) \mathbf{e}{y}+u{z}(x, y, z) \mathbf{e}{z} \ & \equiv\left{u{x}(x, y, z) u_{y}(x, y, z) u_{z}(x, y, z)\right}^{T} \ & \equiv u_{i}\left(x_{j}\right) \end{对齐}\begin{aligned} \mathbf{u} & \stackrel{\text { def }}{ }=u_{x}(x, y, z) \mathbf{e}{x}+u{y}(x, y, z) \mathbf{e}{y}+u{z}(x, y, z) \mathbf{e}{z} \ & \equiv\left{u{x}(x, y, z) u_{y}(x, y, z) u_{z}(x, y, z)\right}^{T} \ & \equiv u_{i}\left(x_{j}\right) \end{对齐}

1. 应变-位移关系。我们在这里介绍了无穷小的应变-位移关系。这些关系的详细推导在第 9.2.1 节中介绍。根据定义，无穷小的法向应变分量为：
εX≡εXX= 定义 ∂在X∂Xε是≡ε是是= 定义 ∂在是∂是ε和≡ε和和= 定义 ∂在和∂和
剪应变分量为：
εX是=ε是X≡CX是2=定义⁡1∂在X∂是+∂在是∂X) ε是和=ε和是≡C是和2=这⁡1(∂在是∂和+∂在和∂是) ε和X=εX和≡C和X2= 定义 12(∂在和∂X+∂在X∂和)

ε一世j= 定义 12(在一世,j+在j,一世).

1. 应力-应变关系 机械应力定义为每单位面积的力(ñ/米2≡磷一种). 由于一个帕斯卡 (Pa) 是非常小的应力，因此机械应力的常用单位是兆帕 (MPa)，可以理解为106 ñ/米2或者1 ñ/米米2.
应力分量的常用符号在图 2.7 所示的一个无穷小体积单元上进行了说明。索引规则如下：X,是,和轴正常的称为正面，相反的面称为负面。法向应力分量表示为σ, 剪应力分量由τ. 法向应力分量仅分配一个下标，因为面的方向和应力分量的方向相同。例如，σX是作用在面的应力分量X- 轴是法线，应力分量在正（或负）面的正（或负）坐标方向上作用。对于剪切应力，第一个指标是指剪切应力作用面的法线坐标方向。第二个指标是指剪切应力分量作用的方向。
在正（或负）面上，正应力分量朝向正（或负）坐标方向。这样做的原因是，如果我们将一个实体细分为无穷小的六面体体积单元，类似于图 2.7 中所示的单元，那么每个负面将与一个正面重合。根据作用-反作用原理，作用在这些面上的力必须具有相等的绝对值和相反的符号。在索引符号中σ11≡σX,σ12≡σX是≡τX是， ETC。

## 数学代写|计算方法代写computational method代考|Boundary and initial conditions

1. 规定的位移。位移矢量的一个或多个分量被规定在全部或部分边界上。这称为运动学边界条件。
2. 规定的牵引力。牵引矢量的一个或多个分量被规定在全部或部分边界上。牵引矢量的定义见附录 K.1。
3. 线性弹簧。在牵引力和位移矢量分量之间规定了线性关系。这种关系的一般形式是：
吨一世=C一世j(dj−在j)
在哪里吨一世是牵引矢量，C一世j是一个正定矩阵，表示分布的弹簧系数；dj是一个规定的函数，表示施加在弹簧上的位移，并且在j是边界上的（未知）位移矢量函数。弹簧系数C一世j（在ñ/米3单位）可能是位置的函数Xķ但与位移无关在一世. 这被称为“温克勒弹簧”12“。
这种边界条件在一个无穷小的边界面单元上的示意图如图 1 所示。2.8在假设C一世j是对角矩阵，因此是三个弹簧系数C1= 定义 C11,C2= 定义 C22,C3= 定义 C33表征边界条件的弹性特性。
如图。2.8应该被解释为意味着施加的位移d一世会产生不同的力ΔF一世作用于面元的质心。暂停求和规则，大小ΔF一世是
ΔF一世=C一世Δ一种(d一世−在一世),一世=1,2,3
在哪里在一世是面元的位移。对应的牵引向量为：
吨一世=林Δ一种→0ΔF一世Δ一种=C一世(d一世−在一世),一世=1,2,3.

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## MATLAB代写

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