Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness G.J. Nie a , R.C. Batra b, * a School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China b Department of Engineering Science and Mechanics, M/C 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA article info Article history: Available online 13 September 2009 Keywords: Rotating disk Variable thickness Material tailoring Functionally graded rubberlike materials Thermal stress abstract We analyze axisymmetric deformations of a rotating disk with its thickness, mass density, thermal expansion coefficient and shear modulus varying in the radial direction. The disk is made of a rubberlike material that is modeled as isotropic, linear thermoelastic and incompressible. We note that the hydro- static pressure in the constitutive relation of the material is to be determined as a part of the solution of the problem since it cannot be determined from the strain field. The problem is analyzed by using an Airy stress function u. The non-homogeneous ordinary differential equation with variable coefficients for u is solved either analytically or numerically by the differential quadrature method. We have also analyzed the challenging problem of tailoring the variation of either the shear modulus or the thermal expansion coefficient in the radial direction so that a linear combination of the hoop stress and the radial stress is constant in the disk. For a rotating annular disk we present the explicit expression of the thermal expan- sion coefficient for the hoop stress to be uniform within the disk. For a rotating solid disk we give the exact expressions for the shear modulus and the thermal expansion coefficient as functions of the radial coordinate so as to achieve constant hoop stress. Numerical results for a few typical problems are pre- sented to illuminate effects of material inhomogeneities on deformations of a hollow and a solid rotating disk. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Functionally graded materials (FGMs) are composites in which the volume fraction, sizes, and shapes of material constituents can be varied to get desired smooth spatial variations of macro- scopic properties such as the elastic modulus, the mass density, the heat conductivity, etc. to optimize their performance. The FGMs abound in nature, e.g., human teeth, bamboo stick, sea shell. Engineered FGMs include ceramic–metal and fiber-reinforced polymeric composites, concrete, and rubberlike materials [1–5]. Vulcanized rubber components typically exhibit a spatial variation of mechanical properties caused either by thermal gradients dur- ing their fabrication or chemical changes induced due to interac- tion with the environment during their service [6–8]. For example, in a commercial butyl rubber sheet and a chlorosulfonat- ed polyethylene cable jacketing material the shear modulus was found to be a quadratic function of the radius [9]. Note that during the fabrication (vulcanization) of thick rubber parts the central portion is cured less than the material near the boundary surfaces unless the vulcanization time is sufficiently large [8,9]. Thus the shear modulus at the center is less than that at the surfaces. Rubberlike materials are widely used in aerospace, automotive, and biomedical fields. They are usually regarded as incompressible, can thus undergo only isochoric or volume preserving deforma- tions, and their constitutive relation involves hydrostatic pressure that cannot be determined from the deformation field but is to be found as a part of the solution of the boundary-value problem (BVP). The BVPs for functionally graded incompressible materials (FGIMs) are challenging since the governing differential equations have variable coefficients and it is difficult to find their exact solu- tions. In general, the solution of a BVP for a structure composed of an FGIM cannot be obtained from that of the corresponding prob- lem for a compressible material by setting Poisson’s ratio equal to 0.5. Furthermore, the solution for a plane stress problem cannot be obtained from that for a plane strain problem by modifying Young’s modulus E and Poisson’s ratio v. We briefly review below the literature on FG rotating disks and other works for FGIMs. Deformations of a rotating disk composed of a linear elastic, iso- tropic and homogeneous material have been studied thoroughly [10] and those of a FG rotating disk have been investigated by Hor- gan and Chan [11] by assuming that E is a power-law function of the radius r. Jahed et al. [12] presented a procedure for minimum 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.08.052 * Corresponding author. Tel.: +1 540 2316051; fax: +1 540 2314574. E-mail addresses: ngj@tongji.edu.cn (G.J. Nie), rbatra@vt.edu (R.C. Batra). Composite Structures 92 (2010) 720–729 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct