Material Tailoring of Linearly Elastic Functionally Graded Rubberlike Cylinders Under Combined Radial Expansion, Extension and Twisting Deformations

Abstract

The material tailoring problem for a hollow circular cylinder composed of an isotropic, incompressible, and linearly elastic functionally graded material has been analytically analyzed. The cylinder is deformed by torques and axial loads on the end faces, and pressures on its inner and outer surfaces. The cylinder material has one elastic parameter, the shear modulus \(\mu \left ( r \right ) \). For the direct problem it is a known positive and continuously varying function in the radial direction, \(r\). For the inverse problem \(\mu \left ( r \right )\) is a design variable and is found to provide the desired radial variation of either the strain energy density, \(W^{def} (r)\), or the von Mises stress, \(\sigma ^{VM} \left ( r \right )\), for the given loads and the cylinder geometry. If the three loads are simultaneously varied by a factor \(\gamma \) then \(W^{def} \left ( r \right ) \) and \(\sigma ^{VM} \left ( r \right ) \), respectively, change by \(\gamma ^{2} \) and \(\gamma \) for fixed \(\mu \left ( r \right ) \) in the direct problem and \(\mu (r)\) by \(\gamma ^{2} \) and \(\gamma \) in the inverse problem for preassigned \(W^{def} (r) = W_{cr} \left ( r \right ) \) and \(\sigma ^{VM} \left ( r \right ) = \sigma _{cr}^{VM} \left ( r \right )\). The \(W_{cr} \left ( r \right ) \) and \(\sigma _{cr}^{VM} \left ( r \right )\) are, respectively, values at failure of the strain energy density and the von Mises stress. For the cylinder material composed of two constituents having positive shear moduli as is often the case in experiments we use a homogenization technique to find the radial variations of their volume fractions and ensure \(\mu (r)\) is positive. We review three manufacturing techniques and propose an experimental program to find \(W_{cr} \left ( r \right )\) and \(\sigma _{cr}^{VM} \left ( r \right )\). The expression for \(\mu (r)\) is derived from the solution of the direct problem that has a unique solution. It provides reference solutions for similar nonlinear problems and verification of numerical algorithms. It supports the optimal design of cylinders.