Abstract The quasi-static frictional contact problem of a rigid rounded punch sliding over a bidirectionally graded half-plane is treated within the framework of plane-strain elasticity. The material heterogeneity in the half-plane is assumed to entail independent shear modulus variations along the depth and longitudinal directions in terms of exponential functions. In-plane field variables are converted through the Fourier transform method and subsequently reduced to a Cauchy singular integral equation (SIE) as a function of the unknown contact stress. The SIE is solved using a collocation approach on the roots of the Chebyshev polynomial of the first kind. A finite element approach (FEA) is also implemented for the same contact problem employing the homogeneous finite element method and the augmented Lagrange algorithm in the environment of a structural analysis software. Besides the verification results generated by the foregoing techniques, the effects of such factors as the friction coefficient and the degree of material heterogeneities are also addressed on the contact stress, the in-plane tensile stress, the normal contact force and on the centerline position of the punch. The bidirectional gradation is evidently shown superior to the conventional one-directional gradation, as far as their potentials are considered for the alleviation of the contact-induced failure risks.

中文翻译：

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滑动圆形冲头压入的双向渐变半平面的赫兹型摩擦接触问题

摘要 在平面应变弹性的框架内处理了刚性圆形冲头在双向渐变半平面上滑动的准静态摩擦接触问题。假设半平面中的材料不均匀性导致沿深度和纵向方向的独立剪切模量变化（根据指数函数）。面内场变量通过傅立叶变换方法进行转换，随后简化为柯西奇异积分方程 (SIE)，作为未知接触应力的函数。SIE 是在第一类 Chebyshev 多项式的根上使用搭配方法求解的。在结构分析软件的环境中，还使用齐次有限元方法和增强拉格朗日算法对相同的接触问题实施了有限元方法 (FEA)。除了上述技术产生的验证结果外，还讨论了摩擦系数和材料异质程度等因素对接触应力、面内拉应力、法向接触力和中心线位置的影响。冲床。双向渐变明显优于传统的单向渐变，就其在减轻接触引起的故障风险方面的潜力而言。摩擦系数和材料异质程度等因素对接触应力、面内拉伸应力、法向接触力和冲头中心线位置的影响也得到了解决。双向渐变明显优于传统的单向渐变，就其在减轻接触引起的故障风险方面的潜力而言。摩擦系数和材料异质程度等因素对接触应力、面内拉伸应力、法向接触力和冲头中心线位置的影响也得到了解决。双向渐变明显优于传统的单向渐变，就其在减轻接触引起的故障风险方面的潜力而言。