Abstract
Material property variation in non-homogeneous internally pressurized thick-walled cylinders is investigated within the context of dynamic programming theory. The material is assumed to be linear, elastic, isotropic, and functionally graded in the radial direction. Based on the plane stress hypothesis, a state space formulation is given and the optimal control problem is stated and solved by means of Pontryagin’s Principle for different objective functionals. Optimal Young’s modulus distribution is found to be piecewise linear along the radial domain. A brief digression on the possible existence of switching points is addressed. Finally, a numerical example is performed within a special class of derived optimal solutions, showing promising results in terms of equivalent stress reduction with respect to the most used variations in literature.
Highlights
Graded (FG) materials are a kind of composite material whose physical and mechanical properties vary spatially along specific directions over the entire domain
Despite the fact that, when mechanical properties vary according to a power-law distribution, an analytic solution of the governing differential equations can be obtained, a more realistic optimization problem should consider all possible classes of property variations
We suppose that, in the formulation of Problem 1, the values of Young’s moduli are fixed at both radii, i.e., that x3 (1) is fixed and x3 ( Ro /Ri ). This assumption allows one to define the linear variation, between the two boundary values, whose importance is twofold: (i) it will be used in a geometrical interpretation of admissible solutions and (ii) it will serve a comparison for the effectiveness of the bang-bang optimal solution
Summary
Graded (FG) materials are a kind of composite material whose physical and mechanical properties vary spatially along specific directions over the entire domain. Despite the fact that, when mechanical properties vary according to a power-law distribution, an analytic solution of the governing differential equations can be obtained, a more realistic (and, in some sense, intrinsic) optimization problem should consider all possible classes of property variations. A designer may be interested in the search for mechanical or physical properties variation for a vessel of minimum compliance, a cylinder of minimum hoop stress at a specific radius or a disk of uniform strength or minimum mass, complying with other constraints which can be economic or related to the manufacturing process These problems can be formulated either in the context of optimal control theory (i.e., in a dynamic programming setting) or by means of calculus of variations.
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