An isotropic elastic strip, with a continuous inhomogeneity profile for Young’s modulus is considered, subject to a self-equilibrated load on one of its axial ends and free of traction on the remainder of its surface boundaries. By taking advantage of the analytical flexibility of an exponential inhomogeneity profile, the full equations of linear theory of elasticity are employed to find the two-dimensional Saint-Venant decay rate in terms of an inhomogeneity parameter that measures gradation steepness in a Functionally Graded Material (FGM). The results show that softening axial inhomogeneity may be introduced to engineered FGM strips and beams to accelerate the decay of stresses. For a hardening axial inhomogeneity on the other hand, the end effects extend beyond a one-width-size distance from the loaded end. The plane problem end effects are shown to decay faster compared to the anti-plane shear counterparts, consistent with what is observed in homogeneous media. For inhomogeneity in the lateral direction from the core toward the outer edges, the qualitative behaviour changes with the degree of inhomogeneity. Whether the core is softer or harder relative to the outer edges, a steep lateral gradation of elastic modulus can significantly increase the decay length of end effects.

中文翻译：

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线弹性功能梯度材料平面问题中的圣维南端效应

考虑了具有连续不均匀杨氏模量分布的各向同性弹性条，在其轴向端之一上承受自平衡载荷，而在其表面边界的其余部分上不受牵引。通过利用指数不均匀分布的分析灵活性，使用线性弹性理论的完整方程根据测量功能梯度材料中梯度陡度的不均匀参数来找到二维圣维南衰减率（女性生殖器切割）。结果表明，软化的轴向不均匀性可能会被引入到工程 FGM 条带和梁中，以加速应力衰减。另一方面，对于硬化轴向不均匀性，端部效应会延伸到距负载端的一个宽度尺寸的距离之外。与反平面剪切对应物相比，平面问题的端部效应衰减得更快，这与在均质介质中观察到的结果一致。对于从核心到外边缘的横向不均匀性，定性行为随着不均匀性的程度而变化。无论核心相对于外边缘更软或更硬，弹性模量的陡峭横向梯度都可以显着增加末端效应的衰减长度。