### 物理代写|理论力学代写theoretical mechanics代考|PHYC20014

statistics-lab™ 为您的留学生涯保驾护航 在代写理论力学theoretical mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写理论力学theoretical mechanics代写方面经验极为丰富，各种代写理论力学theoretical mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|理论力学代写theoretical mechanics代考|Constitutive Relations

Let us consider an elastic solid occupying volume $V$ with the boundary $S=\partial V$. In what follows we consider infinitesimal deformations, so the kinematics is based on the displacement field
$$\mathbf{u}=\mathbf{u}(\mathbf{x}, t)$$
where $\mathbf{x}$ is the position vector and $t$ is time. In Cartesian coordinates $x_{k}, k=1,2,3$, (1) takes the form
$$u_{k}=u\left(x_{1}, x_{2}, x_{3}, t\right)$$
with $\mathbf{u}=\mathbf{u}{k} \mathbf{i}{k}$. Here $\mathbf{i}_{k}$ are Cartesian base vectors and the Einstein summation rule is utilized. In what follows we use the direct (coordinate-free) tensor analysis as described in Lebedev et al. [26], Eremeyev et al. [15].

For simplicity we consider an isotropic material in the bulk. So we have the following constitutive equations
\begin{aligned} \mathscr{W} &=\mu \mathbf{e}: \mathbf{e}+\frac{1}{2} \lambda(\operatorname{tr} \mathbf{e})^{2} \ \mathscr{K} &=\frac{1}{2} \rho \dot{\mathbf{u}} \cdot \dot{\mathbf{u}} \ \boldsymbol{\sigma} & \equiv \frac{\partial \mathscr{W}}{\partial \mathbf{e}}=2 \mu \mathbf{e}+\lambda \mathbf{I} \operatorname{tr} \mathbf{e} \end{aligned}

where $\mathscr{W}$ and $\mathscr{K}$ are the strain energy and kinetic energy densities, $\lambda$ and $\mu$ are Lamé elastic moduli, $\boldsymbol{\sigma}$ is the stress tensor, $\mathrm{e}$ is the linear strain tensor,
$$\mathbf{e}=\frac{1}{2}\left(\nabla \mathbf{u}+(\nabla \mathbf{u})^{T}\right), \quad \nabla \mathbf{u}=\frac{\partial u_{j}}{\partial x_{i}} \mathbf{i}{i} \otimes \mathbf{i}{j}$$
“tr” is the trace operator, and $\rho$ is the mass density. The overdot stands for the derivative with respect to $t$, the superscript ” $T$ ” means the transpose operation,” “” denotes the scalar product of second-order tensors, $\nabla$ is the $3 \mathrm{D}$ nabla operator, and ” $\otimes$ ” stands for dyadic product. In what follows for brevity we use the notation $\frac{\partial}{\partial x_{j}}=\partial_{j}$, so, for example, $\nabla \mathbf{u}=\partial_{j} u_{i} \mathbf{i}{j} \otimes \mathbf{i}{i}$.

Within the surface elasticity in addition to the constitutive equations in the bulk, we introduce the surface strain energy and the surface kinetic energy. For example, within the Gurtin-Murdoch linear isotropic model the strain energy is given by
$$\begin{gathered} \mathscr{W}{s}=\mu{s} \boldsymbol{\varepsilon}: \boldsymbol{\varepsilon}+\frac{1}{2} \lambda_{s}(\operatorname{tr} \boldsymbol{\varepsilon})^{2}, \ \mathbf{s} \equiv \frac{\partial \mathscr{W}{s}}{\partial \boldsymbol{\varepsilon}}=\mu{s} \boldsymbol{\varepsilon}+\lambda_{s}(\operatorname{tr} \boldsymbol{\varepsilon}) \mathbf{P}, \ \boldsymbol{\varepsilon}=\frac{1}{2}\left(\mathbf{P} \cdot\left(\nabla_{s} \mathbf{u}\right)+\left(\nabla_{s} \mathbf{u}\right)^{T} \cdot \mathbf{P}\right) \end{gathered}$$

## 物理代写|理论力学代写theoretical mechanics代考|Anti-plane Motions of an Elastic Half-Space

In order to demonstrate some peculiarities of the model let us consider the propagation of the surface anti-plane waves. Earlier such analysis was performed within the Gurtin-Murdoch model by Eremeyev et al. [14] and it was compared with the

Toupin-Mindlin strain gradient elasticity by Eremeyev et al. [16]. Following these works, let us consider an elastic half-space $x_{3} \leq 0$. The anti-plane motions have one of the forms, see Achenbach [2],
$$\mathbf{u}=u_{1}\left(x_{2}, x_{3}, t\right) \mathbf{i}{1}, \quad \text { or } \quad \mathbf{u}=u{2}\left(x_{1}, x_{3}, t\right) \mathbf{i}{2},$$ which correspond two different direction of wave propagation. With (15) the general motion equations reduce into two wave equations with respect to $u{1}$ and $u_{2}$, respectively,
\begin{aligned} &\mu\left(\partial_{2}^{2}+\partial_{3}^{2}\right) u_{1}=\rho \partial_{t}^{2} u_{1} \ &\mu\left(\partial_{1}^{2}+\partial_{3}^{2}\right) u_{2}=\rho \partial_{t}^{2} u_{2} \end{aligned}
Here $\partial_{t}$ stands for the derivative with respect to $t$.
Making standard assumption on steady-state behaviour, we are looking for solution of $(16)$ and $(17)$ in the form
$$u_{\alpha}=U_{\alpha}\left(x_{\beta}, x_{3}\right) \exp (i \omega t), \quad \alpha=1,2, \beta=2,1$$
where $\omega$ is a circular frequency, $i$ is the imaginary unit, and $U_{\alpha}$ is a amplitude. As a result, (16) and (17) transform into
\begin{aligned} &\mu\left(\partial_{2}^{2}+\partial_{3}^{2}\right) U_{1}=-\rho \omega_{t}^{2} U_{1} \ &\mu\left(\partial_{1}^{2}+\partial_{3}^{2}\right) U_{2}=-\rho \omega^{2} U_{2} \end{aligned}

## 物理代写|理论力学代写theoretical mechanics代考|Problem Statement and Derivation of the Governing

Suppose we have a piecewise-uniform elastic plane, made by alternately connecting layers of thickness $2 h$ from two dissimilar materials. The abscissa axis of the Cartesian coordinate system $O x y$ is directed along the dividing line of materials. On median lines of dissimilar layers $y=(4 n+1) h$ and $y=(4 n-1) h(n \in Z)$ on systems of intervals $L_{1}=\bigcup_{j=1}^{N}\left(a_{j}, b_{j}\right)$ and $L_{2}=\bigcup_{j=1}^{M}\left(c_{j}, d_{j}\right)$ are located cracks and elastic thin inclusions of thickness $h_{j}$ and reduced elastic moduli $E_{I}^{(j)}=E_{j} I\left(1-v_{j}^{2}\right)(j=1, M)$ respectively. We assume that the plane is deformed under the influence of distributed loads $p_{j}(x)$, applied to the cracks $\left(a_{j}, b_{j}\right)(j=1, N)$, concentrated loads $P_{0}^{(j)}(j=1, M)$ applied to inclusions at points $x_{0}^{(j)} \in\left[c_{j}, d_{j}\right](j=1, M)$ and uniformly distributed loads $q_{1}$ and $q_{2}$, applied to the layers at infinity (Fig. 1 ).

Obviously, with this formulation of the problem, the lines $y=(2 n+1) h(n \in Z)$ are lines of symmetry. As a result, the stated problem can be formulated as a problem for a piecewise homogeneous layer (base cell) occupying the region $\Omega{-\infty<x<\infty ;|y| \leq h}$, on the boundaries $y=\pm h$ of which outside cracks and inclusions, symmetry conditions are specified, on $L_{1}$ normal stresses are specified, and on $L_{2}$ contact conditions of inclusion with a base are specified. Here, the inclusions are interpreted as one-dimensional continua, which under the influence of concentrated loads applied to them and tangential contact stresses are in a uniaxial stress state [9]. Also, we assume that due to the smallness of the thickness of inclu-sions and the symmetry of the problem with respect to the axes of the inclusions, the vertical displacements of the points of the inclusions are zero.

The task is to determine the patterns of change in the tangential contact stresses acting on the long sides of the inclusions, crack opening and intensity factors of the fracture stresses at the end points of the cracks depending on the mechanical and geometric parameters.

Based on this assumptions, we will have the following conditions on $L_{1}$ and $L_{2}$ :
$$\begin{gathered} \tau_{x y}^{(1)}(x, h)=0 ; \quad \sigma_{y}^{(1)}(x, h)=-p_{j}(x) \quad\left(a_{j}<x<b_{j}, \quad j=1, N\right) \ V_{2}(x,-h)=0 ; \quad \frac{d U_{2}(x,-h)}{d x}=\varepsilon_{j}(x) \quad\left(c_{j}<x<d_{j}, \quad j=1, M\right) \end{gathered}$$

## 物理代写|理论力学代写theoretical mechanics代考|Constitutive Relations

“tr”是跟踪运算符，并且ρ是质量密度。过点代表关于的导数吨, 上标”吨” 表示转置操作，” “” 表示二阶张量的标量积，∇是个3Dnabla 运营商，和”⊗”代表二元乘积。下面为简洁起见，我们使用符号∂∂Xj=∂j，所以，例如，∇在=∂j在一世一世j⊗一世一世.

## 物理代写|理论力学代写theoretical mechanics代考|Anti-plane Motions of an Elastic Half-Space

Eremeyev 等人的 Toupin-Mindlin 应变梯度弹性。[16]。在这些工作之后，让我们考虑一个弹性半空间X3≤0. 反平面运动具有其中一种形式，参见 Achenbach [2]，

μ(∂22+∂32)在1=ρ∂吨2在1 μ(∂12+∂32)在2=ρ∂吨2在2

μ(∂22+∂32)在1=−ρω吨2在1 μ(∂12+∂32)在2=−ρω2在2

## 物理代写|理论力学代写theoretical mechanics代考|Problem Statement and Derivation of the Governing

τX是(1)(X,H)=0;σ是(1)(X,H)=−pj(X)(一个j<X<bj,j=1,ñ) 在2(X,−H)=0;d在2(X,−H)dX=ej(X)(Cj<X<dj,j=1,米)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。