﻿ 纳米尺度下半空间内浅埋圆形弹性夹杂对SH波的散射 Scattering of a Nanoscale Circular Elastic Inclusion on SH Wave in a Semi-Space

International Journal of Mechanics Research
Vol.07 No.02(2018), Article ID:25508,9 pages
10.12677/IJM.2018.72008

Scattering of a Nanoscale Circular Elastic Inclusion on SH Wave in a Semi-Space

Zhiying Ou, Yanbin Zheng

School of Science, Lanzhou University of Technology, Lanzhou Gansu

Received: Jun. 1st, 2018; accepted: Jun. 14th, 2018; published: Jun. 21st, 2018

ABSTRACT

Using complex function method, multi-polar coordinate method, image-source method, and Graf’s addition formula, the scattering of SH wave by a nano-circular inclusion in elastic half space was studied. Firstly, according to the boundary conditions, the reflected, scattered and refracted wave functions in medium were calculated. Then the wave functions of the corresponding stress field in complex coordinate system were derived. Finally, the calculation examples and results of the dynamic stress concentration around the interface of the circular inclusion were presented in the paper. The variation of circumferential dynamic stress around the boundary of circular inclusion with dimensionless wave number and the change of incidence angle were discussed in detail. The researcher also analyzed the influence of surface parameters on dynamic stress concentration and compared the dynamic stress concentration under macroscopic and microscopic conditions. The results show that the stress concentration at the boundary of the circular inclusion is stronger with the smaller the surface parameter and the higher the dimensionless wave number.

Keywords:Circular Inclusion, Complex Function Method, Dynamic Stress Concentration

1. 引言

2. 问题模型和分析

${\nabla }^{2}W+{K}^{2}W=0$ (1)

Figure 1. Scattering of a shallow buried circular inclusion in a half space for a steady incident plane SH wave

$\left\{\begin{array}{l}{\tau }_{rz}=\mu \left(\frac{\partial W}{\partial z}{\text{e}}^{i\theta }+\frac{\partial W}{\partial \stackrel{¯}{z}}{\text{e}}^{-i\theta }\right)\\ {\tau }_{\theta z}=i\mu \left(\frac{\partial W}{\partial z}{\text{e}}^{i\theta }-\frac{\partial W}{\partial \stackrel{¯}{z}}{\text{e}}^{-i\theta }\right)\end{array}$ (2)

${W}^{\left(\text{I}\right)}={W}_{0}{\text{e}}^{\frac{ik}{2}\left[\left(z{\text{e}}^{-i\alpha }+\stackrel{¯}{z}{\text{e}}^{i\alpha }\right)\right]{\text{e}}^{-iHk\mathrm{sin}\alpha }}$ (3)

${W}^{\left(\text{r}\right)}={W}_{0}{\text{e}}^{\frac{ik}{2}\left[\left(z{\text{e}}^{-i\alpha }+\stackrel{¯}{z}{\text{e}}^{i\alpha }\right)\right]{\text{e}}^{iHk\mathrm{sin}\alpha }}$ (4)

${W}^{\left(\text{s}\right)}=\underset{n=-\infty }{\overset{\infty }{\sum }}{A}_{n}\left\{{H}_{n}^{\left(1\right)}\left(k|z|\right){\left(\frac{z}{|z|}\right)}^{n}+{H}_{n}^{\left(1\right)}\left(k|z-2Hi|\right){\left(\frac{z-2Hi}{|z-2Hi|}\right)}^{-n}\right\}$ (5)

${W}^{\left(\text{f}\right)}=\underset{n=-\infty }{\overset{\infty }{\sum }}{B}_{n}{J}_{n}\left({k}_{i}|z|\right){\left(\frac{z}{|z|}\right)}^{n}$ (6)

${\tau }_{rz}^{\left(\text{I}\right)}=\frac{\text{i}k\mu {W}_{0}}{2}\left[{\text{e}}^{i\left(\theta -\alpha \right)}+{\text{e}}^{-i\left(\theta -\alpha \right)}\right]{\text{e}}^{-iHk\mathrm{sin}\left(\alpha \right)}{\text{e}}^{\frac{ik}{2}\left(z{\text{e}}^{-i\alpha }+\stackrel{¯}{z}{\text{e}}^{i\alpha }\right)}$ (7)

${\tau }_{\theta \text{z}}^{\left(\text{I}\right)}=\frac{-k\mu {W}_{0}}{2}\left[{\text{e}}^{i\left(\theta -\alpha \right)}+{\text{e}}^{-i\left(\theta -\alpha \right)}\right]{\text{e}}^{-iHk\mathrm{sin}\left(\alpha \right)}{\text{e}}^{\frac{ik}{2}\left(z{\text{e}}^{-i\alpha }+\stackrel{¯}{z}{\text{e}}^{i\alpha }\right)}$ (8)

${\tau }_{rz}^{\left(\text{r}\right)}=\frac{\text{i}k\mu {W}_{0}}{2}\left[{\text{e}}^{i\left(\theta \text{+}\alpha \right)}+{\text{e}}^{-i\left(\theta \text{+}\alpha \right)}\right]{\text{e}}^{iHk\mathrm{sin}\left(\alpha \right)}{\text{e}}^{\frac{ik}{2}\left(z{\text{e}}^{i\alpha }+\stackrel{¯}{z}{\text{e}}^{-i\alpha }\right)}$ (9)

${\tau }_{\theta z}^{\left(\text{r}\right)}=\frac{-k\mu {W}_{0}}{2}\left[{\text{e}}^{i\left(\theta \text{+}\alpha \right)}-{\text{e}}^{-i\left(\theta \text{+}\alpha \right)}\right]{\text{e}}^{iHk\mathrm{sin}\left(\alpha \right)}{\text{e}}^{\frac{ik}{2}\left(z{\text{e}}^{i\alpha }+\stackrel{¯}{z}{\text{e}}^{-i\alpha }\right)}$ (10)

$\begin{array}{c}{\tau }_{rz}^{\left(\text{s}\right)}=\frac{k\mu }{2}\underset{n=-\infty }{\overset{\infty }{\sum }}{A}_{n}\left[{H}_{n-1}^{\left(1\right)}\left(k|z|\right){\left(\frac{z}{|z|}\right)}^{n\text{-}1}{\text{e}}^{i\theta }-{H}_{n\text{+}1}^{\left(1\right)}\left(k|z|\right){\left(\frac{z}{|z|}\right)}^{n+1}{\text{e}}^{-i\theta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{H}_{n+1}^{\left(1\right)}\left(k|z-2Hi|\right){\left(\frac{z-2Hi}{|z-2Hi|}\right)}^{-n-1}{\text{e}}^{i\theta }+{H}_{n-1}^{\left(1\right)}\left(k|z-2Hi|\right){\left(\frac{z-2Hi}{|z-2Hi|}\right)}^{-n+1}{\text{e}}^{-i\theta }\right]\end{array}$ (11)

$\begin{array}{c}{\tau }_{\theta z}^{\left(\text{s}\right)}=\frac{\text{i}k\mu }{2}\underset{n=-\infty }{\overset{\infty }{\sum }}{A}_{n}\left[{H}_{n-1}^{\left(1\right)}\left(k|z|\right){\left(\frac{z}{|z|}\right)}^{n\text{-}1}{\text{e}}^{i\theta }+{H}_{n\text{+}1}^{\left(1\right)}\left(k|z|\right){\left(\frac{z}{|z|}\right)}^{n+1}{\text{e}}^{-i\theta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{H}_{n+1}^{\left(1\right)}\left(k|z-2Hi|\right){\left(\frac{z-2Hi}{|z-2Hi|}\right)}^{-n-1}{\text{e}}^{i\theta }-{H}_{n-1}^{\left(1\right)}\left(k|z-2Hi|\right){\left(\frac{z-2Hi}{|z-2Hi|}\right)}^{-n+1}{\text{e}}^{-i\theta }\right]\end{array}$ (12)

${\tau }_{rz}^{\left(\text{f}\right)}=\frac{{k}_{i}{\mu }_{i}}{2}\underset{n=-\infty }{\overset{\infty }{\sum }}{B}_{n}\left[{J}_{n-1}\left({k}_{i}|z|\right){\left(\frac{z}{|z|}\right)}^{n}+{J}_{n+1}\left({k}_{i}|z|\right){\left(\frac{z}{|z|}\right)}^{n}\right]$ (13)

${W}^{\left({\text{t}}_{1}\right)}={W}^{\left(\text{I}\right)}+{W}^{\left(\text{r}\right)}+{W}^{\left(\text{s}\right)}$ (14)

${W}^{\left({\text{t}}_{2}\right)}={W}^{\left(\text{f}\right)}$ (15)

${\tau }_{rz}^{\left({\text{t}}_{1}\right)}={\tau }_{rz}^{\left(\text{I}\right)}+{\tau }_{rz}^{\left(\text{r}\right)}+{\tau }_{rz}^{\left(\text{s}\right)}$ (16)

${\tau }_{\theta z}^{\left({\text{t}}_{1}\right)}={\tau }_{\theta z}^{\left(\text{I}\right)}+{\tau }_{\theta z}^{\left(\text{r}\right)}+{\tau }_{\theta z}^{\left(\text{s}\right)}$ (17)

${\tau }_{rz}^{\left({\text{t}}_{2}\right)}={\tau }_{rz}^{\left(\text{f}\right)}$ (18)

$\left\{\begin{array}{l}{\Gamma }_{H}:{\tau }_{yz}^{{\text{t}}_{1}}\left(x,H\right)=0\\ {\Gamma }_{R}:{\tau }_{rz}^{{\text{t}}_{1}}\left(R,\theta \right)-{\tau }_{rz}^{{\text{t}}_{2}}\left(R,\theta \right)=-S\frac{\partial {\tau }_{\theta z}^{{\text{t}}_{1}}}{\partial \theta }\\ {\Gamma }_{R}:{W}^{\left({\text{t}}_{1}\right)}\left(R,\theta \right)={W}^{\left({\text{t}}_{2}\right)}\left(R,\theta \right)\end{array}$ (19)

$\left\{\begin{array}{l}\underset{n=-\infty }{\overset{\infty }{\sum }}{a}_{11}{B}_{n}+{a}_{12}{A}_{n}={b}_{1}\\ \underset{n=-\infty }{\overset{\infty }{\sum }}{a}_{21}{B}_{n}+{a}_{22}{A}_{n}={b}_{2}\end{array}$ (20)

$\left\{\begin{array}{l}\underset{n=-\infty }{\overset{\infty }{\sum }}{c}_{11}{B}_{n}+{c}_{12}{A}_{n}={d}_{1}\\ \underset{n=-\infty }{\overset{\infty }{\sum }}{c}_{21}{B}_{n}+{c}_{22}{A}_{n}={d}_{2}\end{array}$ (21)

$\begin{array}{l}{c}_{jk}=\frac{1}{2\text{π}}{\int }_{-\text{π}}^{\text{π}}{a}_{jk}{\text{e}}^{-im\theta }\text{d}\theta \\ {d}_{j}=\frac{1}{2\text{π}}{\int }_{-\text{π}}^{\text{π}}{b}_{j}{\text{e}}^{-im\theta }\text{d}\theta \end{array}$ $\left(j,k=1,2\right)$ (22)

$\gamma =|\frac{{\tau }_{\theta \text{z}}^{{t}_{1}}}{ku{W}_{0}^{\left(1\right)}}|$ (23)

3. 算例及分析

Figure 2. The ring direction angle distribution of the cyclic dynamic stress concentration coefficient at the boundary of the circular inclusion (hole) the level of incident wave kR = 2

Figure 3. The ring direction angle distribution of the cyclic dynamic stress concentration coefficient at the boundary of the circular inclusion at the incident wave level

Figure 4. The ring direction angle distribution of the cyclic dynamic stress concentration coefficient at the incident wave at $\frac{\text{π}}{3}$ rad of incidence at the boundary of the circular inclusion

Figure 5. Circular angle distribution of the cyclic dynamic stress concentration coefficient at the boundary of a circular inclusion in a vertical incident wave incident

Figure 6. The ring direction angle distribution of the cyclic dynamic stress concentration coefficient at the boundary of the circular inclusion at the incident wave level when the incident wave is incident at kR = 0.2, S = 10−10, 0.1, 1

Figure 7. The ring direction angle distribution of the cyclic dynamic stress concentration coefficient at the incident wave level when the incident wave is incident at the level of kR = 0.2, S = 10−10, 0.1, 1

Figure 8. When the incident wave is $kR=0.2$ , $S={10}^{-10},0.1,1$ the ring direction of the dynamic stress concentration coefficient is distributed at the boundary of the circular inclusion at the $\frac{\text{π}}{3}$ rad incident

4. 结论

1) 在纳米尺度下圆形夹杂边界处的应力集中程度显著，表面参数 $S$ 越小，引起的圆形夹杂边界处的应力集中程度越强。

2) 在纳米尺度下，无量纲波数 $kR$ 越大，引起的圆形夹杂边界处的应力集中程度越强，和宏观条件下一致。

Scattering of a Nanoscale Circular Elastic Inclusion on SH Wave in a Semi-Space[J]. 力学研究, 2018, 07(02): 66-74. https://doi.org/10.12677/IJM.2018.72008

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