均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.

ANALYTICAL SOLUTION FOR BENDING OF SIMPLY SUPPORTED FUNCTIONALLY GRADED BEAM SUBJECTED TO UNIFORM PRESSURE

Based on the semi-inverse stress function method,an analytical solution is obtained for bending of simply supported functionally graded beam subjected to uniform pressure with arbitrary property distribution across the thickness,including explicit analytical expressions of the stress,strain and displacement.Firstly, the system of partial differential equations for the stress function is established based on the fundamental equations for plane stress states,and the expressions of streses are obtained according to the boundary conditions for stresses.Then,the distributions of strains and displacements are obtained according to the constitutive relations of functionally graded materials and displacement boundary conditions.Finally,the proposed solution is validated by comparing the degenerated results for a homogeneous isotropic beam to the classic elastic solution.The distributions of stresses and displacements obtained in this paper are for the functionally graded beam whose material properties obey a power law distribution of the constituent volume fraction,and the effects of top-bottom surfaces’ Young’s modulus ratio A and volume fraction exponent n on the variation of the stresses and displacements of the functionally graded beam are discussed.