A mode-III crack embedded in a homogeneous isotropic elastic layer of nanoscale finite thickness is studied in this article. The classical elasticity incorporating surface elasticity is employed to reduce a nonclassical mixed boundary value problem, where the layer interior obeys the traditional constitutive relation and the surfaces of the layer and the crack are dominated by the surface constitutive relation. Using the Fourier transform, we convert the problem to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces. By expanding the out-of-plane displacement as series of Chebyshev polynomials, the Galerkin method is invoked to reduce the singular integro-differential equation with Cauchy kernel to a set of algebraic linear equations for the unknown coefficients. An approximate solution is determined, and the influences of surface elasticity on the elastic field and stress intensity factor are examined and displayed graphically. It is shown that surface elasticity decreases the bulk stress and its intensity factor near the crack tips for positive surface shear modulus and gives rise to an opposite trend for a negative surface shear modulus.

中文翻译：

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反平面剪切载荷下具有表面弹性的裂纹弹性层

本文研究了嵌入在纳米级有限厚度的均匀各向同性弹性层中的 III 型裂纹。结合表面弹性的经典弹性用于减少非经典混合边界值问题，其中层内部服从传统本构关系，层表面和裂缝由表面本构关系支配。使用傅立叶变换，我们将问题转换为裂纹面上平面外位移的超奇异积分微分方程。通过将平面外位移扩展为一系列切比雪夫多项式，调用伽辽金方法将具有柯西核的奇异积分微分方程化简为一组未知系数的代数线性方程。确定了一个近似解，并以图形方式检查和显示表面弹性对弹性场和应力强度因子的影响。结果表明，对于正表面剪切模量，表面弹性降低了裂纹尖端附近的体积应力及其强度因子，而对于负表面剪切模量，则产生相反的趋势。