Development of consistent theories of functionally graded plates

功能梯度材料（FGM）是一种非均相复合材料，其组分的体积分数从材料的一个表面到另一个表面逐渐变化，从而导致材料的性能连续变化。女性生殖器切割有其应用，如牙科植入物、热交换管和发动机部件。通常，它们被用作中等厚度、各向同性和各向异性的板。因此，应采用一致板理论，考虑剪切变形、截面翘曲和厚度方向应力分布。这些理论是最近针对均质、各向同性和各向异性材料发展起来的。在研究项目中，它们将扩展到处理女性生殖器切割材料。首先，将考虑厚度方向上性能的幂律和指数律，然后，还将考虑在实际应用中最常用的具有和平常数材料性能的层合板。将特别强调这样一个事实，即一致性的概念是随着材料性质在厚度方向上的变化而改变的，必须进行调整，以满足不同阶近似的一致性要求。在特定荷载和边界条件下，存在功能梯度矩形板弯曲的三维弹性理论的封闭解析解，可作为相应板理论的基准。这些是用来访问板理论的质量和估计适用性的限制（厚度与长度比，刚度随厚度的变化）。下一步是开发功能梯度一致板的有限元（FE），并将该单元作为用户提供的子程序实现到商业有限元程序包（ABAQUS）中。测试实例的结果将与三维有限元计算结果进行比较，以检查单元的准确性及其适用范围。此外，结果将与文献中其他板块理论的结果进行比较，以检查其一致性。只有当材料性质和载荷分布满足一定的对称条件时，面内问题（圆盘）和面外问题（板）才解耦。在项目的最后一步，将考虑厚度方向上的非对称刚度分布，这在航空航天领域具有广泛的应用潜力。在这种程度上，一个耦合的，一致的盘-板理论将被开发，并与现有的三维弹性理论的解决方案进行比较。

Functionally graded materials (FGM) are a type of heterogeneous composite materials exhibiting gradual variations in volume fraction of their constituents from one surface of the material to the other, resulting in properties which vary continuously across the material. FGM find their applications, e.g., as dental implants, heat exchange tubes and engine components. Often, they are employed as moderately thick, isotropic and anisotropic plates. Therefore, consistent plate theories should be adopted, which take shear deformations, warping of the cross-section and stress distributions in thickness direction into account. These theories have been developed recently for homogeneous, isotropic and anisotropic materials. Within the research project, they will be extended to treat FGM materials. First of all, a power law and an exponential law of the properties in thickness directions will be considered, later on, also laminated plates with peace-wise constant material properties, which are used most frequently in practical applications. Special emphasis will be placed on the fact that the notion of consistency is changed by the variation of the material properties in thickness direction and must be adjusted in order to meet the requirements of consistency of different orders of approximations.For specific load and boundary conditions, closed-form analytical solutions of the three-dimensional theory of elasticity exist for the bending of functionally graded rectangular plates, which may serve as benchmark for the corresponding plate theories. These are employed to access the quality of the plate theories and to estimate limits of applicability (thickness-to-length ratio, stiffness variations over the thickness).The next step is to develop a finite element (FE) for functionally graded consistent plates and to implement the element into a commercial FE-program package (ABAQUS) as user supplied subroutine. The results of test examples will be compared with three-dimensional FE calculations in order to check the accuracy of the element and its limits of applicability. Further, the results will be compared with those of other plate theories available in the literature in order to check their consistency.The in-plane problem (disc) and the out-of-plane problem (plate) are decoupled only, if the material properties and the loading distribution satisfy certain symmetry conditions. In the last step of the project, unsymmetrical stiffness distributions in thickness direction will be considered, which enjoy a broad potential of application in aero- and astronautics. To this extent, a coupled, consistent disc-plate theory will be developed and compared with existing solutions of the three-dimensional theory of elasticity.