摘要
功能梯度材料是一种特殊的非均匀材料,具有材料组分沿某个方向连续分布的特点,结构不存在内部界面。与传统层合复合材料相比,具有无可比拟的优势。功能梯度梁在结构工程和智能结构中将有广泛的应用。本文系统地研究了各向异性功能梯度材料的有限长弹性梁、压电梁和磁电梁在多项式分布载荷和任意分布载荷作用时,各种端部条件下的弹性力学解。
首先从功能梯度各向异性弹性材料、压电材料或磁电材料的基本方程出发,导出平面问题的应力函数,电位移函数或磁感应强度函数所满足的偏微分方程。方程中的弹性柔度系数、压电电压系数、自由介电隔离率、磁电系数或磁通系数是梁厚度坐标的函数。当梁的上下表面受多项式载荷作用时,在假定梁的材料参数沿梁厚度方向是任意连续函数的情况下,都可以得到问题的解析解。当梁的上下表面受任意载荷作用,在求取解析解时,梁的材料参数假定为沿梁的厚度方向是指数函数或某种幂函数形式;在求取半解析解时,则可指定任意连续函数形式。其次,当梁的上下表面受多项式分布载荷作用时,设定应力函数、电位移函数或磁感应强度函数分别为长度方向的多项式形式,其中包含厚度方向的待定函数;当梁的上下表面受任意形式的分布载荷作用时,则设定应力函数、电位移函数或磁感应强度函数分别由两部分叠加而成。一是长度方向的三角函数与高度方向为待定函数的乘积,另一为长度方向的一次多项式函数形式。接着利用应变协调方程、电场协调方程和磁场协调方程等,求解出含待定常数的应力函数、电位移函数和磁感应强度函数表达式。然后得到应力、电位移、磁感应强度、轴力、剪力、弯矩、平均电位移、平均磁感应强度、位移和电势的表达式。再利用边界条件完全确定应力函数、电位移函数和磁感应强度函数。最后利用应力函数、电位移函数和磁感应强度函数得到应力、电位移、磁感应强度和位移、电势、磁势等物理量,从而得到了各向异性功能梯度梁的各种弹性力学解。
本文还给出若干解析解的数值算例,并与其他研究工作的数值算例和有限元算例进行了比较,发现并讨论了其中相同和不同之处。
Functionally graded materials(FGMs) are a kind of special non-homogenous materials,whose properties varying continuously along certain directions.There are no internal interfaces in FGMs.Compared with traditional composites,FGMs have a great deal of superiorities.FGMs will be widely applied in structure engineering and intelligent structures.Elasticity solutions are studied in this work for anisotropic functionally graded elastic,piezoelectric and magneto-electro-elastic finite-length beams with different kinds of conditions at their two ends,and subjected to polynomial and arbitrary loads on their upper and lower surfaces.
Firstly,based on elementary equations of anisotropic elastic materials,piezoelectric materials and magneto-electro-elastic materials,the partial differential equations are derided for the stress functions,electric displacement function and magnetic induction function of the plane problems.The coefficients of elastic compliance,piezoelectricity, dielectric impermeability,piezo-magnetism,magneto-electricity,and magnetic permeability are functions of the thickness coordinate.The material coefficients are assumed to be arbitrary continuous functions of the thickness coordinate so as to obtain the analytical solutions of the beams subjected to polynomial loads.When the beams are subjected to arbitrary loads,these coefficients are assumed to be exponential or power functions for obtaining analytical solutions,and to be any arbitrary functions for seeking semi-analytical solutions.Secondly,for beams subjected to polynomial loads,the stress function,electric displacement function and magnetic induction function are assumed in forms of polynomials in the longitudinal coordinate,with undetermined functions of the thickness coordinate.For beams subjected to arbitrary loads,the stress function,electric displacement function and magnetic induction function are assumed to consist of two parts,respectively.One is a product of a trigonometric function of the longitudinal coordinate and an undetermined function of the thickness coordinate,and the other a linear polynomial of longitudinal coordinate with unknown coefficients depending on thickness coordinate.Thirdly,the expressions of stress function,electric displacement function and magnetic induction function with undetermined constants are acquired by virtue of the compatibility equations of strain,electric field and magnetic field.The analytical expressions of stresses,electric displacements,magnetic inductions,axial force, bending moment,shear force,average electric displacement,average magnetic induction, displacements,electric potential and magnetic potential are then deduced,with integral constants determinable from the boundary conditions.Finally,the expressions are educed of the stresses,electric displacements,magnetic inductions,displacements,electric potential and magnetic potential.Thus the elasticity solutions are obtained for the anisotropic functionally graded beams.
Numerical examples of these elasticity solutions are presented.Comparisons are made among numerical examples of other literatures,by finite element method and by this work.Similarities and differences are found and discussed.
引文
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