Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T14:09:36.053Z Has data issue: false hasContentIssue false
  • English
  • Français

An Analytical Solution to the Three-Dimensional Problem on Elastic Equilibrium of an Exponentially-Inhomogeneous Layer

Published online by Cambridge University Press:  14 April 2015

Y. Tokovyy  and
C.-C. Ma
Y. Tokovyy*
Affiliation:
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine Lviv, Ukraine
C.-C. Ma
Affiliation:
Mechanical Engineering Department National Taiwan University Taipei, Taiwan
*
* Corresponding author (tokovyy@gmail.com)
Get access

Abstract

In this paper, we present an analytical solution to the general three-dimensional elasticity problem in a layer, whose Young’s modulus varies exponentially within the thickness coordinate and the Poisson’s ratio is constant. By making use of the direct integration method, the complete set of the governing equations in terms of stresses has been formed. The latter equations were reduced to separate equations for each stress-tensor component and then solved by means of the Fourier double-integral transformation with respect to the planar coordinates.

Keywords

Three-dimensional problemExponentially-inhomogeneous layerExact solutionDirect integration method
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 5 , October 2015 , pp. 545 - 555
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aichi, K., “On the Transversal Seismic Waves Travelling Upon the Surface of Heterogeneous Material,” Proceedings of the Physico-Mathematical Society of Japan, 3rd Series, 4, pp. 137–142 (1922).Google Scholar
2.Lock, M. H., “Axially Symmetric Elastic Waves in an Unbounded Inhomogeneous Medium with Exponentially Varying Properties,” Bulletin of the Seismological Society of America, 53, pp. 527–538 (1963).Google Scholar
3.Holl, D. L., “Stress Transmission in Earths,” Proceedings of the Twentieth Annual Meeting, Highway Research Board, Washington D.C., 20, pp. 709–721 (1940).Google Scholar
4.Gibson, R. E., “Some Results Concerning Displacements and Stresses in a Non-Homogeneous Elastic Half-Space,” Géotechnique, 17, pp. 58–67 (1967).Google Scholar
5.Popov, G. I., “On a Method of Solution of the Axisymmetric Contact Problem of the Theory of Elasticity,” Journal of Applied Mathematics and Mechanics, 25, pp. 105–118 (1961).Google Scholar
6.Teodorescu, P. P. and Predeleanu, M., “Quelques Considérations Sure le Problème des Corps Élastiques Hétérogènes,” Olszak, W. Ed., Non- Homogeneity in Elasticity and Plasticity, Olszak, W. Ed., Pergamon Press, pp. 31–38 (1959).Google Scholar
7.Rostovtsev, N. A., “On the Theory of Elasticity of a Nonhomoqeneous Medium,” Journal of Applied Mathematics and Mechanics, 28, pp. 745–757 (1964).CrossRefGoogle Scholar
8.Mossakovskii, V. I., “Pressure of a Circular Die [Punch] on an Elastic Half-Space, Whose Modulus of Elasticity is an Exponential Function of Depth,” Journal of Applied Mathematics and Mechanics, 22, pp. 168–171 (1958).Google Scholar
9.Giannakopoulos, A. E. and Suresh, S., “Indentation of Solids with Gradients in Elastic Properties. I. Point Force,” International Journal of Solids and Structures, 34, pp. 2357–2392 (1997).CrossRefGoogle Scholar
10.Selvadurai, A. P. S. and Katebi, A., “Mindlin’s Problem for an Incompressible Elastic Half-Space with an Exponential Variation in the Linear Elastic Shear Modulus,” International Journal of Engineering Science, 65, pp. 9–21 (2013).CrossRefGoogle Scholar
11.Katebi, A. and Selvadurai, A. P. S., “Undrained Behaviour of a Non-Homogeneous Elastic Medium: The Influence of Variations in the Elastic Shear Modulus with Depth,” Géotechnique, 63, pp. 1159–1169 (2013).Google Scholar
12.Plevako, V. P., “Equilibrium of a Nonhomogeneous Half-Plane under the Action of Forces Applied to the Boundary,” Journal of Applied Mathematics and Mechanics, 37, pp. 858–866 (1973).Google Scholar
13.Tokovyy, Y. and Ma, C.-C., “Analytical Solutions to the 2D Elasticity and Thermoelasticity Problems for Inhomogeneous Planes and Half-Planes,” Archive of Applied Mechanics, 79, pp. 441–456 (2009).Google Scholar
14.Ohmichi, M. and Noda, N., “Plane Thermoelastic Problem in a Functionally Graded Plate with an Oblique Boundary to the Functional Graded Direction,” Journal of Thermal Stresses, 30, pp. 779–799 (2007).Google Scholar
15.Teodorescu, P. P. and Predeleanu, M., “Über das Ebene Problem Nichthomogener Elastischer Körper,” Acta Technica Academiae Scientiarum Hungaricae, 27, pp. 349–369 (1959).Google Scholar
16.Chan, Y.-S., Gray, L. J., Kaplan, T. and Paulino, G H., “Green’s Function for a Two-Dimensional Exponentially Graded Elastic Medium,” Proceedings of the Royal Society of London A, 460, pp. 1689–1706 (2004).Google Scholar
17.Gray, L. J., Kaplan, T., Richardson, J. D. and Paulino, G. H., “Green’s Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction,” Journal of Applied Mechanics, Transactions of ASME, 70, pp. 543–549 (2003).Google Scholar
18.Ma, C. C. and Lee, J. M., “Theoretical Analysis of In-Plane Problem in Functionally Graded Nonhomogeneous Magnetoelectroelastic Bimaterials,” International Journal of Solids and Structures, 46, pp. 4208–4220 (2009).Google Scholar
19.Ma, C. C. and Chen, Y. T., “Theoretical Analysis of Heat Conduction Problems of Nonhomogeneous Functionally Graded Materials for a Layer Sandwiched Between Two Half-Planes,” Acta Mechanica, 221, pp. 223–237 (2011).CrossRefGoogle Scholar
20.Lee, J. M. and Ma, C. C., “Analytical Solutions for an Antiplane Problem of Two Dissimilar Functionally Graded Magnetoelectroelastic Half-Planes,” Acta Mechanica, 212, pp. 21–38 (2010).Google Scholar
21.Wang, C. D., Tzeng, C. S., Pan, E. and Liao, J. J., “Displacements and Stresses due to a Vertical Point Load in an Inhomogeneous Transversely Isotropic Half-Space,” International Journal of Rock Mechanics & Mining Sciences, 40, pp. 667–685 (2003).CrossRefGoogle Scholar
22.Ter-Mkrtichian, L. N., “Some Problems in the Theory of Elasticity of Nonhomogeneous Elastic Media,” Journal of Applied Mathematics and Mechanics, 25, pp. 1667–1675 (1961).CrossRefGoogle Scholar
23.George, O. D., “Torsion of an Elastic Solid Cylinder With a Radial Variation in the Shear Modulus,” Journal of Elasticity, 6, pp. 229–244 (1976).Google Scholar
24.Dağ, S., Dadıoğlu, S. and Selçuk Yakşi, O., “Circumferential Crack Problem for an FGM Cylinder under Thermal Stresses,” Journal of Thermal Stresses, 22, pp. 659–687 (1999).Google Scholar
25.Tutuncu, N., “Stresses in Thick-Walled FGM Cylinders with Exponentially-Varying Properties,” Engineering Structures, 29, pp. 2032–2035 (2007).Google Scholar
26.Nie, G. J. and Batra, R. C., “Material Tailoring and Analysis of Functionally Graded Isotropic and Incompressible Linear Elastic Hollow Cylinders,” Composite Structures, 92, pp. 265–274 (2010).Google Scholar
27.Keles, I. and Tutuncu, N., “Exact Analysis of Axisymmetric Dynamic Response of Functionally Graded Cylinders (or Disks) and Spheres,” Journal of Applied Mechanics, Transactions of ASME, 78, pp. 061014–1-7 (2011).Google Scholar
28.Shao, Z. S., Wang, T. J. and Ang, K. K., “Transient Thermo-Mechanical Analysis of Functionally Graded Hollow Circular Cylinders,” Journal of Thermal Stresses, 30, pp. 81–104 (2007).Google Scholar
29.Chakravorty, J. G., “Twisting of a Non-Homogeneous Sphere,” Pure and Applied Geophysics, 98, pp. 26–28 (1972).Google Scholar
30.Suresh, S., “Graded Materials for Resistance to Contact Deformation and Damage,” Science, 292, pp. 2447–2451 (2001).CrossRefGoogle ScholarPubMed
31.Jin, Z.-H. and Batra, R. C., “Some Basic Fracture Mechanics Concepts in Functionally Graded Materials,” Journal of Mechanics and Physics of Solids, 44, pp. 1221–1235 (1996).Google Scholar
32.Erdogan, F., “Fracture Mechanics of Functionally Graded Materials,” Composites Engineering, 5, pp. 753–770 (1995).CrossRefGoogle Scholar
33.Erdogan, F., “The Crack Problem for Bonded Non-homogeneous Materials under Antiplane Shear Loading,” Journal of Applied Mechanics, Transactions of ASME, 52, pp. 823–828 (1985).Google Scholar
34.Chen, Y. F. and Erdogan, F., “The Interface Crack Problem for a Nonhomogeneous Coating Bonded to a Homogeneous Substrate,” Journal Mechanical Physics and Solids, 44, pp. 771–87 (1996).Google Scholar
35.Jin, Z.-H. and Batra, R. C., “Interface Cracking Between Functionally Graded Coatings and a Substrate under Antiplane Shear,” International Journal of Engineering Science, 34, pp. 1705–1716 (1996).Google Scholar
36.Delale, F. and Erdogan, F., “On the Mechanical Modeling of the Interfacial Region in Bonded Half Planes,” Journal of Applied Mechanics, Transactions of ASME, 55, pp. 317–324 (1988).CrossRefGoogle Scholar
37.Ozturk, M. and Erdogan, F., “Axisymmetric Crack Problem in Bonded Materials with a Graded Interfacial Region,” International Journal of Solids and Structures, 33, pp. 193–219 (1996).Google Scholar
38.Jin, Z.-H. and Noda, N., “Transient Thermal Stress Intensity Factors for a Crack in a Semi-Infinite Plate of a Functionally Gradient Material,” International Journal of Solids and Structures, 31, pp. 203–218 (1994).Google Scholar
39.Jin, Z. H. and Noda, N., “Edge Crack in a Nonhomogeneous Half Plane under Thermal Loading,” Journal of Thermal Stresses, 17, pp. 591–599 (1994).Google Scholar
40.Noda, N. and Jin, Z. H., “Thermal Stress Intensity Factors for a Crack in a Strip of a Functionally Gradient Material,” International Journal of Solids and Structures, 30, pp. 1039–1056 (1993).Google Scholar
41.Gu, P. and Asaro, J., “Crack in Functionally Graded Materials,” International Journal of Solids and Structures, 34, pp. 1–17 (1997).Google Scholar
42.Erdogan, F., “The Crack Problem for Bonded Non-homogeneous Materials Under Antiplane Shear Loading,” Journal of Applied Mechanics, Transactions of ASME, 52, pp. 823–828 (1985).Google Scholar
43.Noda, N. and Jin, Z. H., “Thermal Stress Intensity Factors for a Crack in a Strip of a Functionally Gradient Material,” International Journal of Solids and Structures, 30, pp. 1039–1056 (1993).Google Scholar
44.Delale, F. and Erdogan, F., “On the Mechanical Modeling of the Interfacial Region in Bonded Half-Planes,” Journal of Applied Mechanics, Transactions of ASME, 55, pp. 317–324 (1988).CrossRefGoogle Scholar
45.Gu, P. and Asaro, R. J., “Cracks in Functionally Graded Materials,” International Journal of Solids and Structures, 34, pp. 1–17 (1997).CrossRefGoogle Scholar
46.Zhang, X. and Hasebe, N., “Elasticity Solution for a Radially Nonhomogeneous Hollow Circular cylinder,” Journal of Applied Mechanics, Transactions of ASME, 66, pp. 598–606 (1999).CrossRefGoogle Scholar
47.Liew, K. M., Kitipornchai, S., Zhang, X. Z. and Lim, C. W., “Analysis of the Thermal Stress Behaviour of Functionally Graded Hollow Circular Cylinders,” International Journal of Solids and Structures, 40, pp. 2355–2380 (2003).CrossRefGoogle Scholar
48.Ramirez, R., Heyliger, P. R. and Pan, E., “Static Analysis of Functionally Graded Elastic Anisotropic Plates Using a Discrete Layer Approach,” Composites Part B: Engineering, 37, pp. 10–20 (2006).Google Scholar
49.Guo, L.-C. and Noda, N., “Modeling Method for a Crack Problem of Functionally Graded Materials with Arbitrary Properties — Piecewise-Exponential Model,” International Journal of Solids and Structures, 44, pp. 6768–6790 (2007).Google Scholar
50.Sankar, B. V., “An Elasticity Solution for Functionally Graded Beams,” Composites Science and Technology, 61, pp. 689–696 (2001).Google Scholar
51.Abade, S., “Functionally Graded Plates Behave Like Homogeneous Plates,” Composites Part B: Engineering, 39, pp. 151–158 (2008).Google Scholar
52.Kashtalyan, M., “Three-Dimensional Elasticity Solution for Bending of Functionally Graded Rectangular Plates,” European Journal of Mechanics A/Solids, 23, pp. 853–864 (2004).Google Scholar
53.Pan, E., “Exact Solution for Functionally Graded Anisotropic Elastic Composite Laminates,” Journal of Composite Materials, 37, pp. 1903–1920 (2003).Google Scholar
54.Zhong, Z. and Shang, E. T., “Three-Dimensional Exact Analysis of a Simply Supported Functionally Gradient Piezoelectric Plate,” International Journal of Solids and Structures, 40, pp. 5335–5352 (2003).Google Scholar
55.Lu, P., Lee, H. P. and Lu, C., “Exact Solutions for Simply Supported Functionally Graded Piezoelectric Laminates by Stroh-like Formalism,” Composite Sructures, 72, pp. 352–363 (2006).Google Scholar
56.Martin, P. A., Richardson, J. D., Gray, L. J. and Berger, J. R., “On Green’s Function for a Three-Dimensional Exponentially Graded Elastic Solid,” Proceedings of the Royal Society of London A, 458, pp. 1931–1947 (2002).Google Scholar
57.Lur’e, A. I., Three-Dimensional Problems of the Theory of Elasticity, Interscience Publisher, New York, p. 493 (1964).Google Scholar
58.Borodachev, N. M., “Three-Dimensional Elasticity-Theory Problem in Terms of the Stress,” International Applied Mechanics, 31, pp. 991–996 (1995).Google Scholar
59.Teodorescu, P. P., Treatise on Classical Elasticity Theory and Related Problems, Springer, Dordrecht, p. 802 (2013).Google Scholar
60.Tokovyy, Y. V., “Direct Integration Method,” in Hetnarski, R. B., Encyclopedia of Thermal Stresses, Springer, New Jersey, 2, pp. 951–960 (2014).Google Scholar
61.Vigak, V. M. and Rychagivskii, A. V., “Solution of a Three-Dimensional Elastic Problem for a Layer,” International Applied Mechanics, 38, pp. 1094–1102 (2002).Google Scholar
62.Sadd, M. H., Elasticity: Theory Applications, and Numerics, Elsevier Academic Press, Oxford, p. 533 (2009).Google Scholar
63.Brigham, E. O., The Fast Fourier Transform and Its Applications, Prentice-Hall Inc., New Jersey, p. 448 (1988).Google Scholar
64.Brychkov, Y. A. and Prudnikov, A. P., Integral Transforms of Generalized Functions, Gordon and Breach Science Publishers, New York, p. 343 (1989).Google Scholar