Spatial decay of transient end effects in functionally graded heat conducting materials

Horgan, C. O.; Quintanilla, R.

doi:10.1090/qam/1848533

Authors: C. O. Horgan and R. Quintanilla
Journal: Quart. Appl. Math. 59 (2001), 529-542
MSC: Primary 74G50; Secondary 35K05, 74E05, 74F05, 80A20
DOI: https://doi.org/10.1090/qam/1848533
MathSciNet review: MR1848533
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this research is to investigate the influence of material inhomogeneity on the spatial decay of end effects in transient heat conduction for isotropic inhomogeneous heat conducting solids. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The spatial decay of solutions to an initial-boundary value problem with variable coefficients on a semi-infinite strip is investigated. It is shown that the spatial decay of end effects in the transient problem is faster that that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. Explicit decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation with constant coefficients. The results may be interpreted in terms of a Saint-Venant principle for transient heat conduction in inhomogeneous solids.

M. Abid Mian and A. J. M. Spencer, Exact solutions for functionally graded and laminated elastic materials, J. Mech. Phys. Solids 46 (1998), no. 12, 2283–2295. MR 1658813, DOI https://doi.org/10.1016/S0022-5096%2898%2900048-9

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965

Bruno A. Boley, Upper bounds and Saint-Venant’s principle in transient heat conduction, Quart. Appl. Math. 18 (1960/61), 205–207. MR 112591, DOI https://doi.org/10.1090/S0033-569X-1960-0112591-8
Bruno A. Boley and Jerome H. Weiner, Theory of thermal stresses, John Wiley & Sons, Inc., New York London, 1960. MR 0112414
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
A. M. Chan and C. O. Horgan, End effects in anti-plane shear for an inhomogeneous isotropic linearly elastic semi-infinite strip, J. Elasticity 51 (1998), no. 3, 227–242. MR 1668850, DOI https://doi.org/10.1023/A%3A1007553422431

F. Erdogan, Fracture mechanics of functionally graded materials, Composites Engineering 5, 753-770 (1995)

F. Franchi and B. Straughan, Spatial decay estimates and continuous dependence on modeling for an equation from dynamo theory, Proc. Roy. Soc. London A 445, 437-451 (1994)

Cornelius O. Horgan, Recent developments concerning Saint-Venant’s principle: an update, AMR 42 (1989), no. 11, 295–303. MR 1021553, DOI https://doi.org/10.1115/1.3152414

C. O. Horgan, Recent developments concerning Saint-Venant’s principle: a second update, Applied Mechanics Reviews 49, 101-111 (1996)

C. O. Horgan and A. M. Chan, Torsion of functionally graded isotropic linearly elastic bars, J. Elasticity 52 (1998/99), no. 2, 181–199. MR 1685288, DOI https://doi.org/10.1023/A%3A1007544011803

C. O. Horgan and A.M. Chan, Vibration of inhomogeneous strings, rods and membranes, J. Sound and Vibration 225, 503-513 (1999)

Cornelius O. Horgan and James K. Knowles, Recent developments concerning Saint-Venant’s principle, Adv. in Appl. Mech. 23 (1983), 179–269. MR 889288
C. O. Horgan and K. L. Miller, Antiplane shear deformations for homogeneous and inhomogeneous anisotropic linearly elastic solids, Trans. ASME J. Appl. Mech. 61 (1994), no. 1, 23–29. MR 1266833, DOI https://doi.org/10.1115/1.2901416
C. O. Horgan and L. E. Payne, On the asymptotic behavior of solutions of linear second-order boundary value problems on a semi-infinite strip, Arch. Rational Mech. Anal. 124 (1993), no. 3, 277–303. MR 1237914, DOI https://doi.org/10.1007/BF00953070
C. O. Horgan, L. E. Payne, and L. T. Wheeler, Spatial decay estimates in transient heat conduction, Quart. Appl. Math. 42 (1984), no. 1, 119–127. MR 736512, DOI https://doi.org/10.1090/S0033-569X-1984-0736512-8
C. O. Horgan and R. Quintanilla, Saint-Venant end effects in antiplane shear for functionally graded linearly elastic materials, Math. Mech. Solids 6 (2001), no. 2, 115–132. MR 1817691, DOI https://doi.org/10.1177/108128650100600201
Cornelius O. Horgan and Lewis T. Wheeler, Spatial decay estimates for the heat equation via the maximum principle, Z. Angew. Math. Phys. 27 (1976), no. 3, 371–376 (English, with French summary). MR 441098, DOI https://doi.org/10.1007/BF01590509
Józef Ignaczak, Saint-Venant type decay estimates for transient heat conduction in a composite rigid semispace, J. Thermal Stresses 21 (1998), no. 3-4, 185–204. Special issue in honor of Richard B. Hetnarski on his seventieth birthday. MR 1810476, DOI https://doi.org/10.1080/01495739808956144

Z. H. Jin and R. C. Batra, Some basic fracture mechanics concepts in functionally graded materials, J. Mech. Phys. Solids 44, 1221-1235 (1996)

James K. Knowles, On the spatial decay of solutions of the heat equation, Z. Angew. Math. Phys. 22 (1971), 1050–1056 (English, with German summary). MR 298255, DOI https://doi.org/10.1007/BF01590873

C. T. Loy, K. Y. Lam, and J. N. Reddy, Vibration of functionally graded cylindrical shells, Internat. J. Mech. Sciences 41, 309-324 (1999)

J. Aboudi, M. J. Pindera, and S. M. Arnold, Higher-order theory for functionally graded materials, Composites: Part B, 30, 777-832 (1999)

R. Quintanilla, Spatial behavior for nonlinear heat equations, Math. Models Methods Appl. Sci. 7 (1997), no. 5, 633–647. MR 1460696, DOI https://doi.org/10.1142/S0218202597000335

F. T. Rooney and M. Ferrari, Torsion and flexure of inhomogeneous elements, Composites Engineering 5, 901-911 (1995)

M. R. Scalpato and C. O. Horgan, Saint-Venant decay rates for an isotropic inhomogeneous linearly elastic solid in anti-plane shear, J. Elasticity 48 (1997), no. 2, 145–166. MR 1607733, DOI https://doi.org/10.1023/A%3A1007408204283