Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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


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.


References [Enhancements On Off] (What's this?)

  • [1] M. S. Abid Mian and A. J. M. Spencer, Exact solutions for functionally graded laminated elastic materials, J. Mech. Phys. Solids 46, 2283-2295 (1998) MR 1658813
  • [2] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965
  • [3] B. A. Boley, Upper bounds and Saint-Venant's principle in transient heat conduction, Quart. Appl. Math. 18, 205-207 (1960) MR 0112591
  • [4] B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, John Wiley, New York, 1960 MR 0112414
  • [5] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, (2nd ed.), Oxford University Press, Oxford, 1959 MR 0022294
  • [6] 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, 227-242 (1998) MR 1668850
  • [7] F. Erdogan, Fracture mechanics of functionally graded materials, Composites Engineering 5, 753-770 (1995)
  • [8] 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)
  • [9] C. O. Horgan, Recent developments concerning Saint-Venant's principle: an update, Applied Mechanics Reviews 42, 295-303 (1989) MR 1021553
  • [10] C. O. Horgan, Recent developments concerning Saint-Venant's principle: a second update, Applied Mechanics Reviews 49, 101-111 (1996)
  • [11] C. O. Horgan and A. M. Chan, Torsion of functionally graded isotropic linearly elastic bars, J. of Elasticity 52, 181-199 (1999) MR 1685288
  • [12] C. O. Horgan and A.M. Chan, Vibration of inhomogeneous strings, rods and membranes, J. Sound and Vibration 225, 503-513 (1999)
  • [13] C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle, In : J. W. Hutchinson (ed.), Advances in Applied Mechanics, Vol. 23, Academic Press, New York, 1983, pp. 179-269 MR 889288
  • [14] C. O. Horgan and K. L. Miller, Anti-plane shear deformations for homogeneous and inhomogeneous anisotropic linearly elastic solids, J. Applied Mechanics 61, 23-29 (1994) MR 1266833
  • [15] 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, 277-303 (1993) MR 1237914
  • [16] C. O. Horgan, L. E. Payne, and L. T. Wheeler, Spatial decay estimates in transient heat conduction, Quart. Appl. Math. 42, 119-127 (1984) MR 736512
  • [17] C. O. Horgan and R. Quintanilla, Saint-Venant end effects in anti-plane shear for functionally graded linearly elastic materials, Math. and Mechanics of Solids 6, 115-132 (2001) MR 1817691
  • [18] C. O. Horgan and L. T. Wheeler, Spatial decay estimates for the heat equation via the maximum principle, J. of Appl. Math. and Phys. (ZAMP) 27, 371-376 (1976) MR 0441098
  • [19] J. Ignaczak, Saint-Venant type decay estimates for transient heat conduction in a composite rigid semispace, J. of Thermal Stresses 21, 185-204 (1998) MR 1810476
  • [20] Z. H. Jin and R. C. Batra, Some basic fracture mechanics concepts in functionally graded materials, J. Mech. Phys. Solids 44, 1221-1235 (1996)
  • [21] J. K. Knowles, On the spatial decay of solutions of the heat equation, J. of Appl. Math. and Phys. (ZAMP) 227, 1050-1056 (1971) MR 0298255
  • [22] C. T. Loy, K. Y. Lam, and J. N. Reddy, Vibration of functionally graded cylindrical shells, Internat. J. Mech. Sciences 41, 309-324 (1999)
  • [23] J. Aboudi, M. J. Pindera, and S. M. Arnold, Higher-order theory for functionally graded materials, Composites: Part B, 30, 777-832 (1999)
  • [24] R. Quintanilla, Spatial behavior for nonlinear heat equations, Math. Models Meth. Appl. Sci. 7, 633-647 (1997) MR 1460696
  • [25] F. T. Rooney and M. Ferrari, Torsion and flexure of inhomogeneous elements, Composites Engineering 5, 901-911 (1995)
  • [26] M. R. Scalpato and C. O. Horgan, Saint-Venant decay rates for an isotropic inhomogeneous linearly elastic solid in anti-plane shear, Journal of Elasticity 48, 145-166 (1997) MR 1607733

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 74G50, 35K05, 74E05, 74F05, 80A20

Retrieve articles in all journals with MSC: 74G50, 35K05, 74E05, 74F05, 80A20


Additional Information

DOI: https://doi.org/10.1090/qam/1848533
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society