### 物理代写|理论力学代写theoretical mechanics代考|PHYS4103

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## 物理代写|理论力学代写theoretical mechanics代考|The Propagation of High-Frequency Shear Elastic Waves

The features of the formation and propagation of forms of an elastic shear wave, concatenated with a canonical (rectangular, periodic in section) protrusions of surfaces each with the other one in elastic isotropic half-spaces (Fig. 7) is investigated [17]. The connection of two half-spaces with surface canonical protrusions is modeled as a composite waveguide consisting of periodically, longitudinally inhomogeneous embedded inner layer in two homogeneous half-spaces.

It is shown from the formation of half-spaces with protrusions, that for the convenience of the mathematical boundary value problem, the coordinate plane yoz

(coordinate plane $x=0$ ) is allocated on one of lateral surfaces of the protrusion contact of the half-spaces $\Omega_{1}{x ; y}$ and $\Omega_{2}{x ; y}$, and the coordinate axis $o z$ is parallel to the forming of these projections. The canonicity of projections (the forms of pins and their linear dimensions) allows us to provide the full mechanical contact along the entire line of contact of half-spaces.

By input of virtual cross-sections, in fact a three-layer waveguide is formed from two homogeneous half-spaces and virtually separated longitudinally inhomogeneous (piecewise-homogeneous) layer of periodically distributed cells of protrusions pairs $\Omega_{1 n}{x ; y}$ and $\Omega_{2 n}{x ; y}$. The mathematical boundary problem on the propagation of normal wave signal (SH) of elastic shear is formulated from the equations of the corresponding homogeneous half-spaces and their respective protrusions:

• in $\Omega_{1}{x ; y}$ and $\Omega_{1 n}{x ; y}$
$$\partial^{2} \mathrm{w}{1}(x ; y) / \partial x^{2}+\partial^{2} \mathrm{w}{1}(x ; y) / \partial y^{2}=-\omega^{2} / c_{1 t}^{2} \cdot \mathrm{w}_{1}(x ; y)$$
• in $\Omega_{2}{x ; y}$ and $\Omega_{2 n}{x ; y}$
$$\partial^{2} \mathrm{w}{2}(x ; y) / \partial x^{2}+\partial^{2} \mathrm{w}{2}(x ; y) / \partial y^{2}=-\omega^{2} / c_{2 t}^{2} \cdot \mathrm{w}{2}(x ; y)$$ One group of boundary conditions of full mechanical contact is satisfied on the virtual cross-sections $y=h{0}$ and $y=-h_{0}$ along the widths of surface protrusions, respectively. Along the width of each protrusion $\Omega_{1 n}{x ; y}$, the continuity surface conditions of mechanical fields will be
\begin{aligned} &\mathrm{w}{1}\left(x ;-h{0} ; t\right) \equiv \mathrm{w}{1}\left(x ;-h{0} ; t\right), \ &G_{1} \cdot \partial \mathrm{w}{1}(x ; y ; t) /\left.\partial y\right|{\mathrm{y}=-h_{0}} \equiv G_{1} \cdot \partial \mathrm{w}{1}(x ; y ; t) /\left.\partial y\right|{y=-h_{0}} \ &\mathrm{w}{1}\left(x ; h{0} ; t\right)=\mathrm{w}{2}\left(x ; h{0} ; t\right), \ &G_{1} \cdot \partial \mathrm{w}{1}(x ; y ; t) /\left.\partial y\right|{y=h_{0}}=G_{2} \cdot \partial \mathrm{w}{2}(x ; y ; t) /\left.\partial y\right|{y=h_{0}} \end{aligned}

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

To study the filtration properties of the metamaterials, let us consider the normal incidence of a plane longitudinal wave, propagating in an unbounded medium $p^{i n c}=$ $\mathrm{e}^{i k x_{1}}$, on a doubly-periodic system of finite number $M(>2)$ of identical vertical arrays, which are finite or infinite along $x_{2}$ and infinite in the direction $x_{3}$. Each of them is an ordinary periodic system of coplanar linear cracks located at $x=0, d, 2 d, \ldots,(M-$ 1)d. In the infinite case, under the natural symmetry, the problem is reduced to the consideration of a plane waveguide of the width $2 a$, which includes $M$ cracks (Fig. 1). For the finite case it is necessary to solve the corresponding boundary integral equation over all available contours of the crack system.

It is assumed that with the normal wave incidence $\mathrm{e}^{i\left(k_{1} x_{1}-\omega t\right)}$ there is a regime of one-mode propagation at $k_{1} a<\pi$, where $k_{1}$-the wave number of the longitudinal wave, $2 a$-the period of the system in the vertical direction, $d$-in the horizontal one. The semi-analytical method is used when the distance between the adjacent parallel arrays $d$ and the incident wave length $\lambda=2 \pi / k_{1}$ are such that the condition $\lambda / d \gg$ 1 is satisfied. A comparative analysis of the properties of the scattering parameters is carried out for the three diffraction problems for a finite and infinite periodic system in a scalar formulation, as well as for an infinite periodic system under the conditions of the plane problem of the elasticity theory.

## 物理代写|理论力学代写theoretical mechanics代考|Infinite Periodic System. Anti-plane Problem

The solution for elastic problems with infinite periodic arrays of cracks, in the antiplane formulation is presented in $[5,7,15]$. Omitting some routine transformations, the problem can be reduced to the following system of $M$ integral equations regarding the unknown functions $g^{s}(y) ;|y|<b ; s=1, \ldots, M,[8]$ :
$\frac{1}{2 a} \int_{-b}^{b} g^{\prime}(t)\left{\frac{1}{2}-\frac{K(y-t)}{i k_{2}}\right} d t+\frac{e^{k_{1} d d}}{4 a} \int_{-b}^{b} g^{2}(t) d t+\frac{e^{2 k k_{2} d}}{4 a} \int_{-b}^{b} g^{3}(t) d t+\ldots+\frac{e^{u k_{2}(M-1) d}}{4 a} \int_{-b}^{b} g^{M}(t) d t=1$
$\frac{e^{a_{1} d}}{4 a} \int_{-6}^{b} g^{1}(t) d t+\frac{1}{2 a} \int_{-1}^{b} g^{2}(t)\left{\frac{1}{2}-\frac{K(y-t)}{i k_{2}}\right) d t+\frac{e^{k_{2} d}}{4 a} \int_{-+}^{h} g^{3}(t) d t+\ldots+\frac{e^{a_{2}(M-2) d}}{4 a} \int_{-h}^{h} g^{M}(t) d t=e^{u_{2} d}$;
$\frac{e^{u_{2} 2 d}}{4 a} \int_{-t}^{b} g^{1}(t) d t+\frac{e^{u_{2} d}}{4 a} \int_{-b}^{b} g^{2}(t) d t+\frac{1}{2 a} \int_{-b}^{b} g^{3}(t)\left{\frac{1}{2}-\frac{K(y-t)}{i k_{2}} \int d t+\ldots+\frac{e^{k_{2}(M-5) d}}{4 a} \int_{-b}^{b} g^{M}(t) d t=e^{k_{2} 2 d} ;\right.$
where the kernel has the following form: $K(y)=\sum_{n=1}^{\infty} r_{n} \cos \left(a_{n} y\right), r_{n}=$ $\sqrt{(\pi n / a)^{2}-k_{2}^{2}}, a_{n}=\pi n / a, k_{2}$-the wave number of the incident transverse wave. As mentioned for some aspects of the proposed semi-analytical method [16, 17], it is necessary to consider the auxiliary integral equation, whose kernel $K(y)$ requires a special treatment:$\frac{1}{2 a} \int_{-b}^{b} h(\eta) K(y-\eta) d \eta=1, K(y)=\sum_{n=1}^{\infty} r_{n} \cos \left(a_{n} y\right), \quad|y|<b .$

## 物理代写|理论力学代写theoretical mechanics代考|The Propagation of High-Frequency Shear Elastic Waves

（坐标平面X=0) 分配在半空间的突起接触的一个侧面上Ω1X;是和Ω2X;是, 和坐标轴○和平行于这些突起的形成。投影的规范性（销的形式及其线性尺寸）使我们能够沿半空间的整个接触线提供完全的机械接触。

• 在Ω1X;是和Ω1nX;是
∂2在1(X;是)/∂X2+∂2在1(X;是)/∂是2=−ω2/C1吨2⋅在1(X;是)
• 在Ω2X;是和Ω2nX;是
∂2在2(X;是)/∂X2+∂2在2(X;是)/∂是2=−ω2/C2吨2⋅在2(X;是)在虚拟截面上满足一组全机械接触边界条件是=H0和是=−H0分别沿着表面突起的宽度。沿每个突起的宽度Ω1nX;是, 力学场的连续面条件为
在1(X;−H0;吨)≡在1(X;−H0;吨), G1⋅∂在1(X;是;吨)/∂是|是=−H0≡G1⋅∂在1(X;是;吨)/∂是|是=−H0 在1(X;H0;吨)=在2(X;H0;吨), G1⋅∂在1(X;是;吨)/∂是|是=H0=G2⋅∂在2(X;是;吨)/∂是|是=H0

## 物理代写|理论力学代写theoretical mechanics代考|Infinite Periodic System. Anti-plane Problem

\frac{1}{2 a} \int_{-b}^{b} g^{\prime}(t)\left{\frac{1}{2}-\frac{K(yt)}{i k_{2}}\right} d t+\frac{e^{k_{1} d d}}{4 a} \int_{-b}^{b} g^{2}(t) d t+\frac{ e^{2 k k_{2} d}}{4 a} \int_{-b}^{b} g^{3}(t) d t+\ldots+\frac{e^{uk_{2}( M-1) d}}{4 a} \int_{-b}^{b} g^{M}(t) d t=1\frac{1}{2 a} \int_{-b}^{b} g^{\prime}(t)\left{\frac{1}{2}-\frac{K(yt)}{i k_{2}}\right} d t+\frac{e^{k_{1} d d}}{4 a} \int_{-b}^{b} g^{2}(t) d t+\frac{ e^{2 k k_{2} d}}{4 a} \int_{-b}^{b} g^{3}(t) d t+\ldots+\frac{e^{uk_{2}( M-1) d}}{4 a} \int_{-b}^{b} g^{M}(t) d t=1
$\frac{e^{a_{1} d}}{4 a} \int_{-6}^{b} g^{1}(t) d t+\frac{1}{2 a} \int_{ -1}^{b} g^{2}(t)\left{\frac{1}{2}-\frac{K(yt)}{i k_{2}}\right) d t+\frac{ e^{k_{2} d}}{4 a} \int_{-+}^{h} g^{3}(t) d t+\ldots+\frac{e^{a_{2}(M-2 ) d}}{4 a} \int_{-h}^{h} g^{M}(t) dt=e^{u_{2} d};\frac{e^{u_{2} 2 d}}{4 a} \int_{-t}^{b} g^{1}(t) d t+\frac{e^{u_{2} d} }{4 a} \int_{-b}^{b} g^{2}(t) d t+\frac{1}{2 a} \int_{-b}^{b} g^{3}( t)\left{\frac{1}{2}-\frac{K(yt)}{i k_{2}} \int d t+\ldots+\frac{e^{k_{2}(M-5) d}}{4 a} \int_{-b}^{b} g^{M}(t) dt=e^{k_{2} 2 d} ;\right.在H和r和吨H和ķ和rn和lH一个s吨H和F○ll○在一世nGF○r米:K(y)=\sum_{n=1}^{\infty} r_{n} \cos \left(a_{n} y\right), r_{n}=\sqrt{(\pi n / a)^{2}-k_{2}^{2}}, a_{n}=\pi n / a, k_{2}−吨H和在一个在和n在米b和r○F吨H和一世nC一世d和n吨吨r一个ns在和rs和在一个在和.一个s米和n吨一世○n和dF○rs○米和一个sp和C吨s○F吨H和pr○p○s和ds和米一世−一个n一个l是吨一世C一个l米和吨H○d[16,17],一世吨一世sn和C和ss一个r是吨○C○ns一世d和r吨H和一个在X一世l一世一个r是一世n吨和Gr一个l和q在一个吨一世○n,在H○s和ķ和rn和lK(y)r和q在一世r和s一个sp和C一世一个l吨r和一个吨米和n吨:\frac{1}{2 a} \int_{-b}^{b} h(\eta) K(y-\eta) d \eta=1, K(y)=\sum_{n=1}^ {\infty} r_{n} \cos \left(a_{n} y\right), \quad|y|<b .$

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