zbMATH — the first resource for mathematics

Transient heat conduction in a functionally graded strip in contact with well stirred fluid with an outside heat source. (English) Zbl 1231.80036
Summary: A transient heat conduction model is established for a functionally graded strip which is in contact with a well stirred fluid. The fluid receives heat from an outside source. Exact solutions for the temperature distribution for both the functionally graded strip and the fluid are found using the Laplace transform and the inverse Laplace theorem. An iterative procedure is proposed to find the roots of the transcendental equation. Numerical results for the temperature are obtained and the computational accuracy in terms of the roots is proven. The effects of the graded parameter and the heat transfer parameter on the temperature distribution of both the functionally graded strip and the fluid are discussed, which is helpful for a design optimization of the graded material. In particular, the effect of the thickness of the functionally graded strip on the temperature distribution is analyzed, which provides practical references for a geometrical optimization of the structure.

80A20 Heat and mass transfer, heat flow (MSC2010)
44A10 Laplace transform
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
65H05 Numerical computation of solutions to single equations
Full Text: DOI
[1] Dai, K. Y.; Liu, G. R.; Han, X.; Lim, K. M.: Thermomechanical analysis of functionally graded material (FGM) plates using element-free Galerkin method, Comput. struct. 83, 1487-1502 (2005)
[2] Byrd, L.; Birman, V.: An investigation of numerical modeling of transient heat conduction in a one-dimensional functionally graded material, Heat transfer eng. 31, 212-221 (2010)
[3] Jin, Z. H.: An asymptotic solution of temperature field in a strip of functionally graded material, Inter. commun. Heat mass transfer 29, 887-895 (2002)
[4] Ayhan, A. O.: Three-dimensional mixed-mode stress intensity factors for cracks in functionally graded materials using enriched finite elements, Int. J. Solids struc. 46, 796-810 (2009) · Zbl 1215.74071 · doi:10.1016/j.ijsolstr.2008.09.026
[5] Ozturk, M.; Erdogan, F.: Axisymmetric crack problem in bonded materials with a graded interfacial region, Int. J. Solids struc. 33, 193-219 (1996) · Zbl 0929.74096 · doi:10.1016/0020-7683(95)00034-8
[6] Erdogan, F.: Fracture mechanics, Int. J. Solids struc. 37, 171-183 (2000) · Zbl 1075.74069 · doi:10.1016/S0020-7683(99)00086-4
[7] Delale, F.; Erdogan, F.: The crack problem for a nonhomogeneous plane, J. appl. Mech. 50, 609-614 (1983) · Zbl 0542.73118 · doi:10.1115/1.3167098
[8] Eischen, J. W.: Fracture of nonhomogeneous materials, Int. J. Frac. 34, 3-22 (1987)
[9] Erdogan, F.: Fracture mechanics of functionally graded materials, Compos. eng. 5, 753-770 (1995)
[10] Bao, G.; Wang, B. L.: Multiple cracking in functionally graded ceramic/metal coatings, Int. J. Solids struc. 32, 2853-2871 (1995) · Zbl 0919.73217 · doi:10.1016/0020-7683(94)00267-Z
[11] Erdogan, F.; Wu, B. H.: The surface crack problem for a plate with functionally graded properties, J. appl. Mech. 64, 449-456 (1997) · Zbl 0900.73615 · doi:10.1115/1.2788914
[12] Choi, H. J.; Lee, K. Y.; Jin, T. E.: Collinear cracks in a layered half-plane with a graded nonhomogeneous interracial zone – part A: mechanical response, Int. J. Frac. 94, 103-122 (1998)
[13] Cai, H.; Bao, G.: Crack bridging in functionally graded coatings, Int. J. Solids struc. 35, 701-717 (1998) · Zbl 0919.73245 · doi:10.1016/S0020-7683(97)00082-6
[14] Long, X.; Delale, F.: The mixed mode crack problem in an FGM layer bonded to a homogeneous half-plane, Int. J. Solids struc. 42, 3897-3917 (2005) · Zbl 1119.74559 · doi:10.1016/j.ijsolstr.2004.12.003
[15] Ding, S. H.; Li, X.; Zhou, Y. T.: Dynamic stress intensity factors of mode I crack problem for functionally graded layered structures, Comput. model. Eng. sci. 56, 43-84 (2010) · Zbl 1231.74377 · doi:10.3970/cmes.2010.056.043
[16] Santare, M. H.; Lambros, J.: Use of graded finite elements to model the behavior of nonhomogeneous materials, J. appl. Mech. 67, 819-822 (2000) · Zbl 1110.74660 · doi:10.1115/1.1328089
[17] Kim, J.; Paulino, G.: Mixed-mode fracture of orthotropic functionally graded materials using finite elements and the modified crack closure method, Eng. frac. Mech. 69, 1557-1586 (2002)
[18] Comi, C.; Mariani, S. Stefano: Extended finite element simulation of quasi-brittle fracture in functionally graded materials, Comput. methods appl. Mech. eng. 196, 4013-4026 (2007) · Zbl 1173.74406 · doi:10.1016/j.cma.2007.02.014
[19] Butcher, R. J.; Rousseau, C. E.; Tippur, H. V.: A functionally graded particulate composite: preparation, measurements and failure analysis, Acta mater. 47, 259-268 (1998)
[20] Abanto-Bueno, J.; Lambros, J.: An experimental study of mixed mode crack initiation and growth in functionally graded materials, Experiment. mech. 46, 179-196 (2006)
[21] Jin, Z. H.; Feng, Y. Z.: Thermal fracture resistance of a functionally graded coating with periodic edge cracks, Surface coatings tech. 202, 4189-4197 (2008)
[22] Sankar, B. V.; Tzeng, J. T.: Thermal stresses in functionally graded beams, Aiaa j. 40, 1228-1232 (2002)
[23] Vel, S. S.; Batra, R. C.: Exact solution for thermoelastic deformations of functionally graded thick rectangular plates, Aiaa j. 40, 1421-1433 (2002)
[24] Ohmich, M.; Noda, N.: Plane thermoelastic problem in a functionally graded plate with an oblique boundary to the functional graded direction, J. thermal stresses 30, 779-799 (2007)
[25] Tarn, J. Q.: Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads, Int. J. Solids struc. 38, 8189-8206 (2001) · Zbl 1016.74020 · doi:10.1016/S0020-7683(01)00182-2
[26] Liew, K. M.; Kitipornchai, S.; Zhang, X. Z.; Lim, C. W.: Analysis of the thermal stress behavior of functionally graded hollow circular cylinders, Int. J. Solids struc. 40, 2355-2380 (2003) · Zbl 1087.74529 · doi:10.1016/S0020-7683(03)00061-1
[27] Ootao, Y.; Tanigawa, Y.: Transient thermoelastic analysis for a laminated composite strip with an interlayer of functionally graded material, J. thermal stresses 32, 1181-1197 (2009)
[28] Noda, N.; Jin, Z.: A crack in a functionally gradient material under thermal shock, Arch. appl. Mech. 64, 99-110 (1994) · Zbl 0792.73060
[29] Jin, Z.; Noda, N.: Transient thermal stress intensity factors for a crack in a semi-infinite plane of a functionally gradient material, Int. J. Solids struc. 31, 203-218 (1994) · Zbl 0799.73055 · doi:10.1016/0020-7683(94)90050-7
[30] Jin, Z.; Batra, R.: Stress intensity relaxation at the tip of an edge crack in a functionally graded material subjected to a thermal shock, J. thermal stresses 19, 317-339 (1996)
[31] Ootao, Y.; Tanigawa, Y.: Transient thermoelastic problem of functionally graded thick strip due to nonuniform heat supply, Compos. struc. 63, 139-146 (2004) · Zbl 1119.74365
[32] Walters, M.; Jr., G. Paulino; Dodds, R.: Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading, Int. J. Solids struc. 41, 1081-1118 (2004) · Zbl 1075.74535 · doi:10.1016/j.ijsolstr.2003.09.050
[33] Ootao, Y.; Tanigawa, Y.: Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply, Int. J. Mech. sci. 47, 1769-1788 (2005) · Zbl 1192.74090 · doi:10.1016/j.ijmecsci.2005.06.003
[34] Zhou, Y. T.; Li, X.; Qin, J. Q.: Transient thermal stress analysis of orthotropic functionally graded materials with a crack, J. thermal stresses 30, 1211-1231 (2007)
[35] Zhou, Y. T.; Li, X.; Yu, D. H.: Transient thermal response of a partially insulated crack in an orthotropic functionally graded strip under convective heat supply, Comput. model. Eng. sci. 43, 191-221 (2009) · Zbl 1232.74091 · doi:10.3970/cmes.2009.043.191
[36] Schumann, T. E. W.: The diffusion problem for a solid in contact with a stirred fluid, Phys. rev. 37, 1508-1515 (1931) · Zbl 0002.22503
[37] Dag, S.; Erdogan, F.: A surface crack in a graded medium loaded by a sliding rigid stamp, Eng. frac. Mech. 69, 1729-1751 (2002)
[38] Carslaw, H. S.; Jaeger, J. C.: Conduction of heat in solids, (1956) · Zbl 0029.37801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.