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

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## 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)

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