Harmonic wave scattered by an inclusion in an elastic plane: The complete Gurtin-Murdoch model

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2024-03-13 DOI:10.1098/rspa.2023.0755
Ming Dai, Peter Schiavone
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Abstract

We consider the propagation of a harmonic elastic wave in a composite inclusion–matrix structure subjected to plane deformation. The interface between the inclusion and matrix is described by the complete Gurtin-Murdoch model with non-vanishing interface tension and interface stretching rigidity. We consider an inclusion of general shape and formulate the corresponding boundary value problem for the wave functions in the inclusion and matrix when a harmonic compressional or shear wave is incident on the edge of the matrix. The problem is then solved by series expansion methods for the case of a circular inclusion embedded in an infinite matrix. The series solution obtained is validated by checking its convergence and via comparisons with existing static and dynamic solutions for certain reduced cases. Numerical examples are presented for the case of a small-sized circular hole embedded in a soft matrix demonstrating the influence of surface tension on the incident wave-induced dynamic stress concentration in the matrix. We find that the presence of surface tension relieves the peak stress around the circular hole when the frequency of the incident wave is below a certain critical value, while it tends to intensify the peak stress for a high-frequency incident wave.

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由弹性平面中的包含体散射的谐波:完整的古尔廷-默多克模型
我们考虑了谐波弹性波在承受平面变形的复合包含体-基体结构中的传播问题。包含体和基体之间的界面由完整的 Gurtin-Murdoch 模型描述,该模型具有非消失的界面张力和界面拉伸刚度。我们考虑了一般形状的包含体,并提出了当谐波压缩波或剪切波入射到矩阵边缘时,包含体和矩阵中波函数的相应边界值问题。然后,针对嵌入无限矩阵中的圆形包含体,通过序列展开方法求解该问题。通过检查其收敛性,并与某些简化情况下的现有静态和动态解决方案进行比较,验证了所获得的序列解决方案。以嵌入软矩阵中的小尺寸圆孔为例,给出了数值示例,演示了表面张力对入射波在矩阵中引起的动态应力集中的影响。我们发现,当入射波频率低于某个临界值时,表面张力的存在会缓解圆孔周围的峰值应力,而当入射波频率较高时,表面张力则会加剧峰值应力。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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