The present work proposes a semi-analytical technique for solution of mixed boundary value problem in functionally graded circular annulus wherein shear modulus varies radially in power-law form while Poisson’s ratio is constant. The technique relies on two main steps. In the first step, corresponding to terms in periodic Fourier series applied individually as traction along the annulus surface, stress and displacement field in the annulus is computed harnessing Airy stress functions approach. In the second step, leveraging the strain–displacement relations in polar co-ordinates, mixed boundary conditions are rendered in terms of displacement all along the annulus surface. Assuming the unknown traction along the annulus surface in terms of periodic Fourier series with finite terms, the modified displacement boundary condition, family of solution from the first step and orthogonality of sine and cosine functions is used to generate a system of simultaneous linear equations for the series coefficients. Knowing the coefficients of periodic Fourier series, stress and displacement field can be computed everywhere in the annulus. The first step is digitized in terms of MAPLE functions, exhaustively validated through traction distribution comprising of normal, shear traction on part of the boundary and pair of equal and diametrically opposite point load along the boundary. The second step is corroborated through a problem where the inner surface is subjected to a specified traction distribution and mixed boundary conditions exist along the outer surface in the form of complete constraint along a part and traction-free condition elsewhere.

本工作提出了一种半解析技术，用于求解功能梯度圆环中的混合边值问题，其中，剪切模量以幂律形式径向变化，而泊松比恒定。该技术依赖于两个主要步骤。第一步，与沿环面表面分别作为牵引力的周期性傅立叶级数项相对应，利用艾里应力函数方法计算环面中的应力和位移场。第二步，利用极坐标中的应变-位移关系，沿整个环面的位移给出混合边界条件。假设以带有限项的周期性傅立叶级数沿着环面的未知牵引力，修改后的位移边界条件，第一步的解族以及正弦和余弦函数的正交性用于生成一系列系数的同时线性方程组。知道了周期性傅立叶级数的系数，就可以在环空中的任何地方计算应力和位移场。第一步是根据MAPLE函数进行数字化，并通过牵引力分布全面验证，该分布包括部分边界上的法向，剪力牵引力以及沿边界的一对相等且直径相对的点载荷。第二步是通过以下问题得到证实的：内表面受到指定的牵引力分布，并且沿外表面存在混合边界条件，其形式为沿零件的完全约束和其他位置的无牵引力状态。