Exact Solutions and Material Tailoring for Functionally Graded Hollow Circular Cylinders |
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Authors: | G J Nie R C Batra |
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Institution: | (1) State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao, 066004, P.R. China;(2) Department of Materials, University of Oxford, Oxford, OX1 3PH, UK |
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Abstract: | We employ the Airy stress function to derive analytical solutions for plane strain static deformations of a functionally graded
(FG) hollow circular cylinder with Young’s modulus E and Poisson’s ratio v taken to be functions of the radius r. For E
1 and v
1 power law functions of r, and for E
1 an exponential but v
1 an affine function of r, we derive explicit expressions for stresses and displacements. Here E
1 and v
1 are effective Young’s modulus and Poisson’s ratio appearing in the stress-strain relations. It is found that when exponents
of the power law variations of E
1 and v
1 are equal then stresses in the cylinder are independent of v
1; however, displacements depend upon v
1. We have investigated deformations of a FG hollow cylinder with the outer surface loaded by pressure that varies with the
angular position of a point, of a thin cylinder with pressure on the inner surface varying with the angular position, and
of a cut circular cylinder with equal and opposite tangential tractions applied at the cut surfaces. When v
1 varies logarithmically through-the-thickness of a hollow cylinder, then the maximum radial stress, the maximum hoop stress
and the maximum radial displacements are noticeably affected by values of v
1. Conversely, we find how E
1 and v
1 ought to vary with r in order to achieve desired distributions of a linear combination of the radial and the hoop stresses. It is found that for
the hoop stress to be constant in the cylinder, E
1 and v
1 must be affine functions of r. For the in-plane shear stress to be uniform through the cylinder thickness, E
1 and v
1 must be functions of r
2. Exact solutions and optimal design parameters presented herein should serve as benchmarks for comparing approximate solutions
derived through numerical algorithms. |
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Keywords: | |
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