One of the main requirements in the design of structures made of functionally graded materials is their best response when used in an actual environment. This optimum behaviour may be achieved by searching for the optimal variation of the mechanical and physical properties along which the material compositionally grades. In the works available in the literature, the solution of such an optimization problem usually is obtained by searching for the values of the so called heterogeneity factors (characterizing the expression of the property variations) such that an objective function is minimized. Results, however, do not necessarily guarantee realistic structures and may give rise to unfeasible volume fractions if mapped into a micromechanical model. This paper is motivated by the confidence that a more intrinsic optimization problem should a priori consist in the search for the constituents’ volume fractions rather than tuning parameters for prefixed classes of property variations. Obtaining a solution for such a class of problem requires tools borrowed from dynamic optimization theory. More precisely, herein the so-called Pontryagin Minimum Principle is used, which leads to unexpected results in terms of the derivative of constituents’ volume fractions, regardless of the involved micromechanical model. In particular, along this line of investigation, the optimization problem for axisymmetric bodies subject to internal pressure and for which plane elasticity holds is formulated and analytically solved. The material is assumed to be functionally graded in the radial direction and the goal is to find the gradation that minimizes the maximum equivalent stress. A numerical example on internally pressurized functionally graded cylinders is also performed. The corresponding solution is found to perform better than volume fraction profiles commonly employed in the literature.

设计功能梯度材料制成的结构的主要要求之一是在实际环境中使用时的最佳响应。可以通过搜索材料组成沿其分级的机械和物理性能的最佳变化来实现此最佳行为。在现有文献中，通常通过搜索所谓的异质性因子的值来获得这种优化问题的解决方案。（表征特性变化的表达），从而使目标函数最小化。但是，结果不一定能保证现实的结构，如果映射到微机械模型中，可能会导致体积分数不可行。本文的动机是相信，一个更本质的优化问题应该优先包含在搜索成分的体积分数中，而不是针对属性变化的前缀类来调整参数。要获得此类问题的解决方案，需要从动态优化理论中借用工具。更准确地说，这里是所谓的庞特里亚金最小原理不管涉及到的微机械模型如何，都使用了这种方法，导致在成分体积分数的导数方面产生了出乎意料的结果。尤其是，沿着这一研究方向，制定并分析解决了轴对称物体承受内部压力并保持平面弹性的优化问题。假定该材料在径向上是功能渐变的，目标是找到使最大等效应力最小的渐变。还执行了内部加压功能梯度气缸的数值示例。发现相应的解决方案的性能优于文献中常用的体积分数曲线。