Research Papers

Axisymmetric Stresses in an Elastic Radially Inhomogeneous Cylinder Under Length-Varying Loadings

[+] Author and Article Information
Yuriy Tokovyy

Pidstryhach Institute for Applied Problems of
Mechanics and Mathematics,
National Academy of Sciences of Ukraine,
3-B Naukova Street,
Lviv 79060, Ukraine
e-mail: tokovyy@gmail.com

Chien-Ching Ma

Fellow ASME
Department of Mechanical Engineering,
National Taiwan University,
No. 1 Roosevelt Road, Sec. 4
Taipei 10617, Taiwan
e-mail: ccma@ntu.edu.tw

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 1, 2016; final manuscript received August 13, 2016; published online September 8, 2016. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 83(11), 111007 (Sep 08, 2016) (7 pages) Paper No: JAM-16-1277; doi: 10.1115/1.4034459 History: Received June 01, 2016; Revised August 13, 2016

In this paper, we present an analytical solution to the axisymmetric elasticity problem for an inhomogeneous solid cylinder subjected to external force loadings, which vary within the axial coordinate. The material properties of the cylinder are assumed to be arbitrary functions of the radial coordinate. By making use of the direct integration method, the problem is reduced to coupled integral equations for the shearing stress and the total stress (given by the superposition of the normal ones). By making use of the resolvent-kernel solution, the latter equations were uncoupled and then solved in a closed analytical form. On this basis, the effect of variable material moduli in the stress distribution has been examined with special attention given to the negative Poisson's ratio.

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Obata, Y. , and Noda, N. , 1994, “ Steady Thermal Stresses in a Hollow Circular Cylinder and a Hollow Sphere of a Functionally Gradient Material,” J. Therm. Stresses, 17(3), pp. 471–487. [CrossRef]
Kushnir, R. M. , and Popovych, V. S. , 2006, “ Stressed State of a Thermosensitive Plate in a Central-Symmetric Temperature Field,” Mater. Sci., 42(2), pp. 145–154. [CrossRef]
Guo, L. , and Noda, N. , 2014, “ Investigation Methods for Thermal Shock Crack Problems of Functionally-Graded Materials—Part I: Analytical Method,” J. Therm. Stresses, 37(3), pp. 292–324. [CrossRef]
Kushnir, R. M. , Protsyuk, B. V. , and Synyuta, V. M. , 2004, “ Quasistatic Temperature Stresses in a Multilayer Thermally Sensitive Cylinder,” Mater. Sci., 40(4), pp. 433–445. [CrossRef]
Zimmerman, R. W. , and Lutz, M. P. , 1999, “ Thermal Stresses and Thermal Expansion in a Uniformly Heated Functionally Graded Cylinder,” J. Therm. Stresses, 22, pp. 177–188. [CrossRef]
Jabbari, M. , Meshkini, M. , and Eslami, M. R. , 2016, “ Mechanical and Thermal Stresses in FGPPM Hollow Cylinder Due to Radially Symmetric Loads,” ASME J. Pressure Vessel Technol., 138(1), p. 011207. [CrossRef]
Zhang, X. , and Hasebe, N. , 1999, “ Elasticity Solution for a Radially Nonhomogeneous Hollow Circular Cylinder,” ASME J. Appl. Mech., 66(3), pp. 598–606. [CrossRef]
Woodward, B. , and Kashtalyan, M. , 2015, “ A Piecewise Exponential Model for Three-Dimensional Analysis of Sandwich Panels With Arbitrarily Graded Core,” Int. J. Solids Struct., 75–76(1), pp. 188–198. [CrossRef]
Tokovyy, Y. V. , Kalynyak, B. M. , and Ma, C.-C. , 2014, “ Nonhomogeneous Solids: Integral Equations Approach,” Encyclopedia of Thermal Stresses, Vol. 7, R. B. Hetnarski , ed., Springer, Dordrecht, The Netherlands, pp. 3350–3356.
Krenev, L. I. , Tokovyy, Y. V. , Aizikovich, S. M. , Seleznev, N. M. , and Gorokhov, S. V. , 2016, “ A Numerical-Analytical Solution to the Mixed Boundary-Value Problem of the Heat-Conduction Theory for Arbitrarily Inhomogeneous Coatings,” Int. J. Therm. Sci., 107, pp. 56–65. [CrossRef]
Jha, D. K. , Kant, T. , and Singh, R. K. , 2013, “ A Critical Review of Recent Research on Functionally Graded Plates,” Compos. Struct., 96, pp. 883–849.
Soldatos, K. P. , 1994, “ Review of Three Dimensional Dynamic Analyses of Circular Cylinders and Cylindrical Shells,” ASME Appl. Mech. Rev., 47(10), pp. 501–516. [CrossRef]
Tokovyy, Y. V. , and Ma, C.-C. , 2008, “ Thermal Stresses in Anisotropic and Radially Inhomogeneous Annular Domains,” J. Therm. Stresses, 31(9), pp. 892–913. [CrossRef]
Jabbari, M. , Bahtui, A. , and Eslami, M. R. , 2006, “ Axisymmetric Mechanical and Thermal Stresses in Thick Long FGM Cylinders,” J. Therm. Stresses, 29(7), pp. 643–663. [CrossRef]
Jabbari, M. , Mohazzab, A. H. , Bahtui, A. , and Eslami, M. R. , 2007, “ Analytical Solution for Three-Dimensional Stresses in a Short Length FGM Hollow Cylinder,” Z. Angew. Math. Mech., 87(6), pp. 413–429. [CrossRef]
Hosseini Kordkheili, S. A. , and Naghdabadi, R. , 2008, “ Thermoelastic Analysis of Functionally Graded Cylinders Under Axial Loading,” J. Therm. Stresses, 31(1), pp. 1–17. [CrossRef]
Asgari, M. , 2015, “ Two Dimensional Functionally Graded Material Finite Thick Hollow Cylinder Axisymmetric Vibration Mode Shapes Analysis Based on Exact Elasticity Theory,” J. Theor. Appl. Mech., 45(2), pp. 3–20. [CrossRef]
Dai, H.-L. , Luo, W.-F. , and Dai, T. , 2016, “ Multi-Field Coupling Static Bending of a Finite Length Inhomogeneous Double-Layered Structure With Inner Hollow Cylinder and Outer Shell,” Appl. Math. Modell., 40(11–12), pp. 6006–6025. [CrossRef]
Chen, W. Q. , Bian, Z. G. , Lv, C. F. , and Ding, H. J. , 2004, “ 3D Free Vibration Analysis of a Functionally Graded Piezoelectric Hollow Cylinder Filled With Compressible Fluid,” Int. J. Solids Struct., 41, pp. 947–964. [CrossRef]
Alshits, V. I. , and Kirchner, H. O. K. , 2001, “ Cylindrically Anisotropic, Radially Inhomogeneous Elastic Materials,” Proc. R. Soc. London, Ser. A, 457(2007), pp. 671–693. [CrossRef]
Kim, K.-S. , and Noda, N. , 2002, “ Green's Function Approach to Unsteady Thermal Stresses in an Infinite Hollow Cylinder of Functionally Graded Material,” Acta Mech., 156(3–4), pp. 145–161. [CrossRef]
Dong, S. B. , Kosmatka, J. B. , and Lin, H. C. , 2001, “ On Saint-Venant's Problem for an Inhomogeneous, Anisotropic Cylinder—Part I: Methodology for Saint-Venant Solutions,” ASME J. Appl. Mech., 68(3), pp. 376–381. [CrossRef]
Ter-Mkrtich'ian, L. N. , 1961, “ Some Problems in the Theory of Elasticity of Nonhomogeneous Elastic Media,” J. Appl. Math. Mech., 25(6), pp. 1667–1675. [CrossRef]
Tokovyy, Y. V. , and Ma, C.-C. , 2015, “ Analytical Solutions to the Axisymmetric Elasticity and Thermoelasticity Problems for an Arbitrarily Inhomogeneous Layer,” Int. J. Eng. Sci., 92, pp. 1–17. [CrossRef]
Tokovyy, Y. V. , 2014, “ Direct Integration Method,” Encyclopedia of Thermal Stresses, Vol. 2, R. B. Hetnarski , ed., Springer, Dordrecht, The Netherlands, pp. 951–960.
Vihak, V. M. , Yasinskyy, A. V. , Tokovyy, Y. V. , and Rychahivskyy, A. V. , 2007, “ Exact Solution of the Axisymmetric Thermoelasticity Problem for a Long Cylinder Subjected to Varying With-Respect-to-Length Loads,” J. Mech. Behav. Mater., 18(2), pp. 141–148. [CrossRef]
Tokovyy, Y. V. , and Ma, C.-C. , 2011, “ Analysis of Residual Stresses in a Long Hollow Cylinder,” Int. J. Pressure Vessels Piping, 88(5–7), pp. 248–255. [CrossRef]
Hetnarski, R. B. , and Eslami, M. R. , 2009, Thermal Stresses—Advanced Theory and Applications, Springer, Dordrecht, The Netherlands.
Brychkov, Y. A. , and Prudnikov, A. P. , 1989, Integral Transforms of Generalized Functions, Gordon & Breach, New York.
Sneddon, I. N. , 1956, Special Functions of Mathematical Physics and Chemistry, Interscience Publication, New York.
Khan, K. A. , and Hilton, H. H. , 2010, “ On Inconstant Poisson's Ratios in Non-Homogeneous Elastic Media,” J. Therm. Stresses, 33(1), pp. 29–36. [CrossRef]
Lakes, R. , 1993, “ Advances in Negative Poisson's Ratio Materials,” Adv. Mater., 5(4), pp. 293–296. [CrossRef]
Hoang, T. M. , and Drugan, W. J. , 2016, “ Tailored Heterogeneity Increases Overall Stability Regime of Composites Having a Negative-Stiffness Inclusion,” J. Mech. Phys. Solids, 88, pp. 123–149. [CrossRef]


Grahic Jump Location
Fig. 2

Full-field distributions of the radial, circumferential, axial, and shearing stresses for the homogeneous material, normalized by p0, with n=0 and ν0=0.3 in Eq. (31) due to the loading profile (Eq. (30)) with λ=1

Grahic Jump Location
Fig. 3

Distribution of the radial stress in an elastic cylinder with material properties (Eq. (31)), where n=1 and m=0 (solid line) and m=2 (dashed line), in cross section z=0 due to force loading (Eq. (30)) with λ=1

Grahic Jump Location
Fig. 4

The axial stress in cross section z=0 of an elastic cylinder with properties (Eq. (31)), where n=1 and m=0 (solid line) and m=2 (dashed line), due to force loading (Eq. (30)) with λ=1

Grahic Jump Location
Fig. 5

The shearing stress in cross sections z=2/2 and z=2 of an elastic cylinder with properties (Eq. (31)), where n=1 and m=0 (solid line) and m=2 (dashed line), due to force loading (Eq. (30)) with λ=1

Grahic Jump Location
Fig. 6

The radial stress in cross section z=0 for: ν=0.5, n=1, and m=0 (curve 1); ν=0, n=1, and m=0 (curve 2); ν=−0.3, n=1, and m=0 (curve 3); ν=−0.5, n=1, and m=0 (curve 4); ν=−0.3, n=1 and m=2 (curve 5); and ν0=2,ν1=−1/2,κ=1,τ=2, n=1, and m=2 (curve 6)




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