### 物理代写|理论力学代写theoretical mechanics代考|PHYSICS 3544

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• 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|Solution of the System of Governing Equations

Solution of the system of Eq. (9) under conditions (10) can be constructed using the method of Chebyshev orthogonal polynomials, by reducing it to a quasi-completely regular system of algebraic equations [9]. However, more effective, in our opinion, is the method of mechanical quadratures [10], which we will use. Without loss of generality, we will assume that there is one crack and one inclusion in the base cell, which occupy intervals $(a, b)$ and $(c, d)$.
Turning to dimensionless quantities and introducing the notation

\begin{aligned} &a_{}=(b-a) / 2 h ; \quad b_{}=(b+a) / 2 h ; \quad c_{}=(d-c) / 2 h ; \quad d_{}=(d+c) / 2 h \ &\varphi_{1}(t)=V^{\prime}\left(h\left(a_{} t+b_{}\right)\right) ; \quad \varphi_{2}(t)=\frac{c_{} \tau\left(h\left(c_{} t+d_{}\right)\right)}{\mu_{1}} ; \ &R_{11}^{}(t, \xi)=\frac{a_{}}{\lambda_{1}} \int_{0}^{\infty} K_{11}(\zeta) \sin \left(\zeta a_{}(t-\xi)\right) d \zeta_{0} \ &R_{12}^{}(t, \xi)=\left(1-v_{1}\right) \int_{0}^{\infty} K_{12}(\zeta) \sin \left(\zeta\left(a_{} t+b_{}-c_{} \xi-d_{}\right)\right) d \zeta \ &R_{21}^{}(t, \xi)=-\frac{4 a_{} c_{}\left(1-v_{2}\right)}{\mu_{} K_{2}} \int_{0}^{\infty} K_{21}(\zeta) \sin \left(\zeta\left(c_{} t+d_{}-a_{} \xi-b_{}\right)\right) d \zeta \ &R_{22}^{}(t, \xi)=-\frac{c_{}}{\lambda_{2}} \int_{0}^{\infty} K_{22}(\zeta) \sin \left(\zeta c_{}(t-\xi)\right) d \zeta \ &f_{1}(t)=-\pi p_{1}\left[h\left(a_{} t+b_{}\right)\right] / \lambda_{1} ; \quad f_{2}(t)=\frac{2 \pi c_{} q_{2}\left(1-v_{2}^{2}\right)}{\mathbb{X}{2} \mu{1}} ; \ &P_{0}^{}=\frac{P_{0}^{(1)}}{h \mu_{1}} ; \quad \vartheta_{}=\frac{2 \pi h\left(1-v_{2}\right) c_{} E_{2}}{h_{1} E_{l}^{(1)}\left(1+v_{2}\right) \mathbb{T}{2}}, \end{aligned} we obtain the following system of defining equations: under conditions $$\int{-1}^{1} \varphi_{1}(s) d s=0 ; \quad \int_{-1}^{1} \varphi_{2}(s) d s=\frac{P_{0}^{*}}{2}$$

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

The numerical analysis is conducted based on the formulas of the preceding paragraph. It is assumed that the crack has a constant length equal to a quarter of the half-thickness of the layer $h$, and is located symmetrically about the axis $O y$, i.e. $a_{}=0.25, b_{}=0$. The location of the inclusion, whose length is equal to the length of the crack, can vary and is determined by the parameter $l$, which is the coordinate of the left end of the inclusion, i.e. $c_{}=a_{}, d_{}=l+a_{}$. In order to determine the effect of inclusion on the crack opening and on stress intensity factors (SIF) at its ends, we take the forces acting on the crack faces and the forces at infinity equal to zero $\left(p_{1}=0, q_{2}=0\right)$. The force applied to the left end of the inclusion $\left(t_{0}=-1\right)$, the ratio of the thickness of the inclusion to the half-thickness of the layer and the ratio of the Young’s modulus of the stringer to $E_{2}$ will be considered constants with values: $P_{0}^{*}=0.25, h_{1} / h=0.01, E_{I}^{(1)} / E_{2}=5$.

The calculations show that crack opens only when inclusion is located to the right of certain point, in other cases part of the crack is closed and the formulation of the problem is not valid. Note that the crack begins to close from the right end. The location of the above mentioned point can be found by equating the SIF at the right end of the crack to zero and it essentially depends only on the length of the inclusion. So, for example, if the inclusion length is equal to $a_{}$, this point is in the vicinity of the point $-0.8 a_{}$. If the inclusion length is equal to $2 a_{}$ the point is around $-2.4 a_{}$, and if the inclusion length is $0.5 a_{}$ the point is around $-0.1 a_{}$. Figure 2 shows the graphs of SIF at the right end of the crack depending on parameter $l$ for different values of the elastic constants of layers.

In Fig. 2, curve 1 corresponds to a homogeneous layer with $v_{1}=v_{2}=0.25$, curves $2,3,4$ correspond to inhomogeneous layers with parameters $E_{1} / E_{2}=1$,$v_{1}=0.25, v_{2}=0.35 ; E_{1} / E_{2}=3, v_{1}=v_{2}=0.25$ and $E_{1} / E_{2}=1 / 3, v_{1}=$ $0.25, v_{2}=0.35$ respectively.

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

We use two industrial US flaw detectors USD60-N and UD9812, shown in Fig. $2 .$ The low-frequency flaw detector USD60-N permits measurements in the frequency range $0.02-2.5 \mathrm{MHz}$ in the two regimes-the through-transmission method and echomethod. There is a possibility to display the full signal, the detected signal, as well as its spectrum. The second flaw detector UD9812 has the working frequency range $0.6-12 \mathrm{M \Gamma} ц$, and we use it to perform measurements at frequencies higher than $2.5 \mathrm{MHz}$. The both flaw detectors permit the transmission of the recorded data to a PC with the help of a special software. In the case of USD60-N for this aim one can use the network interface Ethernet, while the UD9812 can be attached to the PC with a USB 2.0.

As the generator and the receiver of US signals we use available US transducers of various frequencies and diameters.

Let us note that the values reflected in Table 1 are related to the maximum working frequency of the US transducer, while the spectrum generated by the probe contains a set of frequencies around the indicated carrier frequency. The measurements are

carried out by the through-transmitted method, when the radiating probe is placed on the top of the sample and another probe – on its bottom. To provide a good contact, we used a lubricating layer which permits the transition of the mechanical oscillations of the piezo-element inside the specimen at hand (Fig. 3).

A laboratory setup has been equipped to provide the experiments, see Fig. 4 , which is a device to fix the US probes and the sample. The device is a rack with three clamps. The first two clamps fix the receiving and radiated US transducers, between them there is a fixed sample for measurements, the third clamp fixes a spring which provides reliable contact between the transducers and the sample.

All experiments were performed without any additional amplifier with a fixed amplitude of $50 \mathrm{~V}$. The following filtration bands was applied to the received signal: at the frequency up to $0.2 \mathrm{MHz}$ we used a filtration over the interval $20-300 \mathrm{kHz}$; for the frequencies $0.4$ and $0.6 \mathrm{MHz}$ we put the filtration for the receiver $200-1250 \mathrm{kHz}$; the frequencies $1.25,1.8,2.5 \mathrm{MHz}$ were measured in the pass band $400-2500 \mathrm{kHz}$ for the frequencies 5 and $10 \mathrm{MHz}$-the frequency band $0.8-12 \mathrm{MHz}$.

## 物理代写|理论力学代写theoretical mechanics代考|Solution of the System of Governing Equations

∫−11披1(s)ds=0;∫−11披2(s)ds=磷0∗2

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