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Elastic Analysis of Inhomogeneous Solids: History and Development in Brief

Published online by Cambridge University Press:  18 July 2019

Yuriy Tokovyy Open the ORCID record for Yuriy Tokovyy [Opens in a new window]  and
Chien-Ching Ma
Yuriy Tokovyy*
Affiliation:
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine 3-b Naukova St., 79060Lviv, Ukraine
Chien-Ching Ma
Affiliation:
Department of Mechanical Engineering National Taiwan University No 1 Roosevelt Rd., Sec. 4, 10617Taipei, Taiwan ROC
*
*Corresponding author (tokovyy@gmail.com)
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Abstract

Inhomogeneous materials (the ones exhibiting spatial variations in their, all or specific, material properties) present a great deal of interest for scientists and engineers in both academia and industry. For over a hundred years, the mechanical behavior of inhomogeneous materials excites numerous attempts in mathematical modeling and development of methods for proper analysis and verification. With this concern, the reach experience was gained by scientists of different scientific schools in many countries. Despite numerous significant achievements, some results, unfortunately, remain unnoticed by the wide scientific community. On the other hand, one can observe a growing number of publications which repeatedly publish the solutions reported years ago or deal with the similar problems with slight modifications. The main objective of this paper is to: i) present a brief survey of the development history of the elastic analysis of inhomogeneous solids and ii) characterize some dominant analytical and semi-analytical methods. It was not our intent here to provide a comprehensive list of references on the topic, which is nearly impossible to make in view of rapidly growing number of publications and other restrictions, but to emphasize some important, in our opinion, stages of the development, methods and results.

Keywords

inhomogeneous materialselastic analysisdevelopment historyanalytical methods
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 5 , October 2019 , pp. 613 - 626
Copyright
© The Society of Theoretical and Applied Mechanics 2019 

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References

REFERENCES

Timoshenko, S. P., History of Strength of Materials: With a Brief Account of the History of Theory of Elasticity and Theory of Structures, McGraw-Hill, New York, 452 p. (1953).Google Scholar
Todhunter, I. and Pearson, K., A History of the Theory of Elasticity and of the Strength of Materials: from Galilei to Lord Kelvin. V 1. Galilei to Saint-Venant, 1639 – 1850, Dover Pub., New York, 924 p. (1960).Google Scholar
Maugin, G. A., Material Inhomogeneities in Elasticity, Chapman and Hall, London, 280 p. (1993).CrossRefGoogle Scholar
Olszak, W., Rychlewski, J. and Urbanowski, W., “Plasticity under Non-Homogeneous Conditions,” Advances in Applied Mechanics, 7, pp. 131–214 (1962).CrossRefGoogle Scholar
Ilyushin, A. A., Basic Directions of the Development in Problems of Durability and Plasticity, Nauka, Moscow, pp. 5 – 18 (1971).Google Scholar
Suresh, S. and Mortensen, A., Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behaviour of Graded Metals and Metal-Ceramic Composites, Ashgate Publishing, London, 165 p. (1998).Google Scholar
Suresh, S., “Graded Materials for Resistance to Contact Deformation and Damage,” Science, 292, No. 5526, pp. 2447–2451 (2001).CrossRefGoogle ScholarPubMed
Maxwell, J.C., A Treatise on Electricity and Magnetism, Vol 1., Clarendon Press, Oxford, 425 p. (1873).Google Scholar
Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M. and Carrera, E., “Stress, Vibration and Buckling Analyses of FGM Plates – A State-of-the-Art Review,” Composite Structures 120, pp. 10–31 (2017).CrossRefGoogle Scholar
Swaminathan, K. and Sangeetha, D.M., “Thermal Analysis of FGM Plates – A Critical Review of Various Modeling Techniques and Solution Methods,” Composite Structures, 160, pp. 43–60 (2017).CrossRefGoogle Scholar
Dai, H.-L., Rao, Y.-N. and Dai, T., “A Review of Recent Researches on FGM Cylindrical Structures under Coupled Physical Interactions, 2000–2015,” Composite Structures, 152, pp. 199–225 (2016).CrossRefGoogle Scholar
Thai, H.-T. and Kim, S.-E., “A Review of Theories for the Modeling and Analysis of Functionally Graded Plates and Shells,” Composite Structures, 128, pp. 70–86 (2015).CrossRefGoogle Scholar
Jha, D. K., Kant, T. and Singh, R.K., “A Critical Review of Recent Research on Functionally Graded Plates,” Composite Structures, 96, pp. 883–849 (2013).CrossRefGoogle Scholar
Markworth, A. J., Ramesh, K. S. and Parks, W. P., “Modelling Studies Applied to Functionally Graded Materials,” Journal of Materials Science, 30 (9), pp. 2183–2193 (1995).CrossRefGoogle Scholar
Jasinsky, F. S. Collection of Papers, Vol. 3, Izd. Instituta Inzhenerov Putey Soobscheniy Imperatora Aleksandra I, St-Petersburg, 206 p. (1904).Google Scholar
Olszak, W. (ed.), Non-Homogeneity in Elasticity and Plasticity, Pergamon Press, New York, 529 p. (1959).Google Scholar
Kupradze, V.D. (ed.), Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Publ. Co., Amsterdam, 929 p. (1979).Google Scholar
Podstrigach, Ya. S., Lomakin, V. A. and Kolyano, Yu. M., Thermoelasticity of Bodies of Nonhomogeneous Structure, Nauka, Moscow, 378 p. (1984) [in Russian].Google Scholar
Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity, Groningen, Noordhoff, 704 p. (1953).Google Scholar
Lekhnitskii, S. G., Anisotropic Plates, Gordon and Breach, New York, 534 p. (1968).Google Scholar
Lekhnitskii, S. G., Theory of elasticity of an Anisotropic Body, Mir Publishers, Moscow, 340 p. (1981).Google Scholar
Murzewski, J., “Elastic-Plastic Stochastically Non-Homogeneous Bodies,” in. Olszak, W. (ed.), Non-Homogeneity in Elasticity and Plasticity, Pergamon Press, New York, pp. 479–489. (1959).Google Scholar
Hashin, Z., “Theory of Mechanical Behavior of Heterogeneous Media,” Applied Mechanics Reviews, 17 (1), pp. 1–9 (1964).Google Scholar
Noda, N., “Thermal Stresses in Materials with Temperature-Dependent Properties,” Applied Mechanics Reviews, 44 (9), pp. 383–397, (1991).CrossRefGoogle Scholar
Kushnir, R. M. and Popovych, V. S, “Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange,” in: Vikhrenko, V. S. (ed.) Heat Conduction – Basic Research, InTech, Rijeka, pp. 131–154 (2011).Google Scholar
Ewing, W. M., Jardetzki, W. S. and Press, F., Elastic Waves in Layered Media, McGraw Hill Book Company Inc., New York, Toronto, London, 405 p. (1957).Google Scholar
Meissner, E. “Elastische Oberflächenwellen mit Dispersion in einem inhomogenen Medium,” Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 66, pp. 181–195 (1921).Google Scholar
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(4-5), pp. 137–142 (1922).Google Scholar
Sezawa, K., “A Kind of Waves Transmitted over a Semi-Infinite Solid Body of Varying Elasticity,” Bulletin of the Earthquake Research Institute University of Tokyo, 9(3), pp. 310–315 (1931).Google Scholar
Stoneley, R., “The Transmission of Rayleigh Waves in a Heterogeneous Medium,” Geophysical Journal International, 3(6), pp. 222–232 (1934).CrossRefGoogle Scholar
Pekeris, C. L., “The Propagation of Rayleigh Waves in Heterogeneous Media,” Physics, 6, pp. 133–138 (1935).CrossRefGoogle Scholar
Muravskii, B. G., Mechanics of Non-Homogeneous and Anisotropic Foundations, Springer, Berlin, 364 p. (2001).CrossRefGoogle Scholar
Winkler, E., Die Lehre von der Elastizität und Festigkeit: mit besonderer Rücksicht auf ihre Anwendung in der Technik, für polytechnische Schulen Bauakademien, Ingenieure, Maschinenbauer, Architecten, etc., Dominicus, Prague, 388 p. (1867).Google Scholar
Fröhlich, O.K., Druckferteilung im Baugrunde, Springer, Wien, 185 p. (1934).CrossRefGoogle Scholar
Ohde, J., “Zur Theorie der Druckverteilung im Baugrund,” Der Bauingenieur, 20, pp. 451–459 (1939).Google Scholar
Carrier, W.D. and Christian, J.T., “Rigid Circular Plate Resting on a Non-Homogeneous Elastic Half-Space,” Géotechnique, 23 (1), pp. 67–84 (1973).CrossRefGoogle Scholar
Chuong, T.L., “Sur les Contraintes et les Déplacements d’un Semi-Espace non Homogène,” Comptes Rendus Hebdomadaires des Séances de L’Académies des Sciences, 273 (4), pp. 254–257 (1971).Google Scholar
Gibson, R.E., “Some Results Concerning Displacements and Stresses in a Non-Homogeneous Elastic Halfspace,” Géotechnique, 17, pp. 58–67 (1967).CrossRefGoogle Scholar
Kassir, M.K., “Boussinesq Problems for Nonhomogeneous Solid,” Journal of the Engineering Mechanics Division, 98 (2), pp. 457–470 (1972).Google Scholar
Rostovtsev, N. A., “An Integral Equation Encountered in the Problem of a Rigid Foundation Bearing on Nonhomogeneous Soil,” Applied Mathematics and Mechanics, 25, pp. 238–246 (1961).Google Scholar
Ward, W.H., Burland, J.B. and Gallois, R.W., “Geotechnical Assessment of a Site at Mundford, Norfolk, for a Large Proton Acceleration,” Géotechnique, 18 (4), pp. 399–431 (1968).CrossRefGoogle Scholar
Cristescu, N. D., Craciun, E-M. and Soós, E., Mechanics of Elastic Composites, Chapman & Hall / CRC, New York, 682 p. (2004).Google Scholar
Jones, R. M., Mechanics of Composite Materials, Taylor & Francisc, Philadelphia, 538 p. (1999).Google Scholar
Nielsen, L. F., Composite Materials. Properties as Influenced by Phase Geometry, Springer, Berlin Heidelberg, 259 p. (2005).CrossRefGoogle Scholar
Vasiliev, V. V. and Morozov, E.V., “Advanced Mechanics of Composite Materials,” Elsevier, Amsterdam, 491 p. (2007).Google Scholar
Tauchert, T. R., “A Review: Quasistatic Thermal Stresses in Anisotropic Bodies, with Applications to Composite Materials,” Acta Mechanica, 23, pp. 113– 135 (1975).Google Scholar
Hashin, Z., “Analyzis of Composite Materials – a Survey,” Journal of Applied Mechanics, 50, pp. 481–505 (1983).CrossRefGoogle Scholar
Soldatos, K. P., “Review of Three Dimensional Dynamic Analyses of Circular Cylinders and Cylindrical Shells,” Applied Mechanics Reviews, 47(10), pp. 501–516 (1994).Google Scholar
Rabin, B.H. and Shiota, I., “Functionally Gradient Materials,” Materials Research Society Bulletin, 20(1), pp. 14–18 (1995).CrossRefGoogle Scholar
Koizumi, M., “FGM Activities in Japan,” Composites Part B: Engineering, 28(1–2), pp. 1–4 (1997).Google Scholar
Miyamoto, Y., Niino, M. and Koizumi, M., “FGM Research Programs in Japan – from Structural to Functional Uses,” in Shiota, I., Miyamoto, M.Y. (eds.) Functionally Graded Materials 1996, Elsevier, Amsterdam, pp. 1–8 (1997).Google Scholar
Rabin, B.H. and Shiota, I. (eds.), Materials Research Society Bulletin, 20(1) (1995)CrossRefGoogle Scholar
Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A. and Ford, R.G. (eds.), Functionally Graded Materials: Design, Processing and Applications, Springer, 330 p. (1999).Google Scholar
Watanabe, Y. and Sato, H., “Review Fabrication of Functionally Graded Materials under a Centrifugal Force,” in Cuppoletti, J. (ed.) Nanocomposites with Unique Properties and Applications in Medicine and Industry, InTech, Rijeka, pp.133–150 (2011).Google Scholar
Birman, V. and Byrd, L.W., “Modeling and Analysis of Functionally Graded Materials and Structures,” Applied Mechanics Reviews, 60(5), pp. 195–216 (2007).CrossRefGoogle Scholar
Mortensen, A. and Suresh, S., “Functionally Graded Metals and Metal-Ceramic Composites: Part 1 Processing,” International Materials Reviews, 40(6), 239–265 (1995).Google Scholar
Noda, N., “Thermal Stresses in Functionally Graded Materials,” Journal of Thermal Stresses, 22 (4–5), pp. 477–512 (1999).CrossRefGoogle Scholar
Tanigawa, Y., “Some Basic Thermoelastic Problems for Nonhomogeneous Structural Materials,” Applied Mechanics Reviews, 48:6, pp. 287–300 (1995).CrossRefGoogle Scholar
Teodorescu, P. P., Treatise on Classical Elasticity. Theory and Related Problems, Dordrecht, Springer, 802 p. (2013)Google Scholar
Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, McGraw-Hill Book Company, Inc., New York, 506 p. (1951).Google Scholar
Tokovyy, Yu.V., “Direct Integration Method,” in Hetnarski, R. B. (ed.) Encyclopedia of Thermal Stresses, 1, Springer, Dordrecht, pp. 951–960 (2014).CrossRefGoogle Scholar
Gibson, R.E., “Fourteenth Rankine Lecture: The Analytical Method in Soil Mechanics,” Géotechnique, 24(2), pp. 115–140 (1974).CrossRefGoogle Scholar
Awojobi, A.O., “The Settlement of a Foundation on Gibson Soil of the Second Kind,” Géotechnique, 25(2), 221–228 (1975).CrossRefGoogle Scholar
Simons, N.E. and Rodrigues, J.S.N., “Finite Element Analysis of the Surface Deformation due to Uniform Loading on a Layer of Gibson Soil Resting on a Smooth Rigid Base,” Géotechnique, 25(2), pp. 375 – 379 (1975).CrossRefGoogle Scholar
Korenev, B.G., “A Die Resting on an Elastic Half-Space, the Modulus of Elasticity of Which is an Exponential Function of the Depth,” Doklady Akademii Nauk SSSR, 112(5), pp. 823 – 826 (1957) [in Russian].Google Scholar
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(1), pp. 168–171 (1958).CrossRefGoogle Scholar
Popov, G.Ia., “The Contact Problem of the Theory of Elasticity for the Case of a Circular Area of Contact,” Journal of Applied Mathematics and Mechanics, 26(1), pp. 207–225 (1962).CrossRefGoogle Scholar
Lekhnitskii, S.G., “Radial Distribution of Stresses in a Wedge and in a Half-Plane with Variable Modulus of Elasticity,” Journal of Applied Mathematics and Mechanics, 26(1), pp.199–206 (1962).CrossRefGoogle Scholar
Rostovtsev, N.A., “On the Theory of Elasticity of a Nonhomogeneous Medium,” Journal of Applied Mathematics and Mechanics, 28(4), pp. 745–757 (1964).CrossRefGoogle Scholar
Giannakopoulos, A.E. and Pallot, P., “Two-Dimensional Contact Analysis of Elastic Graded Materials,” Journal of Mechanics and Physics of Solids, 48, pp. 1597–1631 (2000).CrossRefGoogle Scholar
Belik, G. I. and Protsenko, V.S. “The Contact Problem of a Half-Plane for Which the Modulus of Elasticity of the Material is Expressed by a Power Function of the Depth,” International Applied Mechanics, 3(6), pp. 80 – 82 (1967).Google Scholar
Tokovyy, Y. and Ma, C.-C., “An Analytical Solution to the Three-Dimensional Problem on Elastic Equilibrium of an Exponentially-Inhomogeneous Layer,” Journal of Mechanics, 31(5), pp. 545–555 (2015).CrossRefGoogle Scholar
Zimmerman, R.W. and Lutz, M.P., “Thermal Stresses and Thermal Expansion in a Uniformly Heated Functionally Graded Cylinder,” Journal of Thermal Stresses, 22, pp. 177–188 (1999).Google Scholar
Horgan, C.O. and Chan, A.M., “The Stress Response of Functionally Graded Isotropic Linearly Elastic Rotating Disks,” Journal of Elasticity, 55, pp. 219–230 (1999).CrossRefGoogle Scholar
Oral, A. and Anlas, G., “Effect of Radially Varying Moduli on Stress Distribution of Nonhomogeneous Anisotropic Cylindrical Bodies,” International Journal of Solids and Structures, 42, pp. 5568–5588 (2005).CrossRefGoogle Scholar
Jabbari, M., Sohrabpour, S. and Eslami, M. R., “Mechanical and Thermal Stresses in a Functionally Graded Hollow Cylinder due to Radially Symmetric Loads,” International Journal of Pressure Vessels and Piping, 79, pp. 493–497 (2002).CrossRefGoogle Scholar
Jabbari, M., Sohrabpour, S. and Eslami, M. R., “General Solution for Mechanical and Thermal Stresses in a Functionally Graded Hollow Cylinder due to Nonaxisymmetric Steady-State Loads,” Journal of Applied Mechanics, 70, pp. 111–118 (2003).CrossRefGoogle Scholar
Tarn, J.Q., “Exact Solutions for Functionally Graded Anisotropic Cylinders Subjected to Thermal and Mechanical Loads,” International Journal of Solids and Structures, 38, pp. 8189–8206 (2001).CrossRefGoogle Scholar
Tarn, J.Q. and Chang, H.H., “Extension, Torsion, Bending, Pressuring, and Shearing of Piezoelectric Circular Cylinders with Radial Inhomogeneity,” Journal of Intelligent Material Systems and Structures, 16, pp. 631–641 (2005).CrossRefGoogle Scholar
Zhang, X. and Hasebe, N., “Elasticity Solution for a Radially Nonhomogeneous Hollow Circular Cylinder,” Journal of Applied Mechanics, 66, pp. 598–606 (1999).Google Scholar
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 and Mining Sciences, 40, pp. 667–685 (2003).CrossRefGoogle Scholar
Plevako, V. P., “The Deformation of a Nonhomogeneous Half-Space under the Action of a Surface Load,” International Applied Mechanics, 9(6), pp. 593–598 (1973).Google Scholar
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(5), pp. 858–866 (1973).CrossRefGoogle Scholar
Tanigawa, Y., Morishita, H. and Ogaki, S. “Derivation of Systems of Fundamental Equations for a Three-Dimensional Thermoelastic Field with Nonhomogeneous Material Properties and Its Application to a Semi-Infinite Body,” Journal of Thermal Stresses, 22(7), pp. 689–711 (1999).Google Scholar
Selvadurai, A.P.S., “The Settlement of a Rigid Circular Foundation Resting on a Half-Space Exhibiting a Near Surface Elastic Non-Homogeneity,” International Journal for Numerical and Analytical Methods in Geomechanics, 20, pp. 351–364 (1996).3.0.CO;2-L>CrossRefGoogle Scholar
Selvadurai, A.P.S. and Lan, Q., “Axisymmetric Mixed Boundary Value Problems for an Elastic Halfspace with a Periodic Nonhomogeneity,” International Journal of Solids and Structures, 35(15), pp. 1813–1826 (1998).Google Scholar
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(5), pp. 441–456 (2009).Google Scholar
Choi, H.J. and Paulino, G.H., “Thermoelastic Contact Mechanics for a Flat Punch Sliding over a Graded Coating/Substrate System with Frictional Heat Generation,” Journal of Mechanics and Physics of Solids, 56, pp. 1673–1692 (2008).CrossRefGoogle Scholar
Yang, J., Ke, L.L. and Kitipornchai, S., “Thermo-Mechanical Analysis of an Inhomogeneous Double-Layer Coating System under Hertz Pressure and Tangential Traction,” Mechanics of Advanced Materials and Structures, 16, pp. 308–318 (2009).CrossRefGoogle Scholar
Yevtushenko, A.A., Rozniakowska, M. and Kuciej, M., “Transient Temperature Processes in Composite Strip and Homogeneous Foundation,” International Communications in Heat and Mass Transfer, 34(9-10), pp. 1108–1118 (2007).CrossRefGoogle Scholar
Mao, J.J., Ke, L.L. and Wang, Y.S., “Thermoelastic Contact Instability of a Functionally Graded Layer and a Homogeneous Half-Plane,” International Journal of Solids and Structures, 51(23-24), pp. 3962–3972 (2014).Google Scholar
Chen, P. and Chen, S., “Thermo-Mechanical Contact Behavior of a Finite Graded Layer under a Sliding Punch with Heat Generation,” International Journal of Solids and Structures, 50(7–8), pp. 1108–1119 (2013).Google Scholar
Khan, K. A. and Hilton, H.H., “On Inconstant Poisson's Ratios in Non-Homogeneous Elastic Media,” Journal of Thermal Stresses, 33(1), pp. 29– 36 (2010).CrossRefGoogle Scholar
Golecki, J. and Knops, R.J., “Introduction to Linear Elastoplastics with Variable Poisson’s Ratio,” Zeszyty Naukowe Akademii Górniczo Hutniczej, 30, pp. 81 – 91 (1969).Google Scholar
Hilton, H.H., “Elastic and Viscoelastic Poisson’s Ratios: The Theoretical Mechanics Perspective,” Materials Sciences and Applications, 8, pp. 291–332 (2017).Google Scholar
Golecki, J.J., “On Non-Radial Stress Distribution in Non-Homogeneous Elastic Half-Plane under Concentrated Load,” Meccanica, 6(3), pp. 147–156 (1971).CrossRefGoogle Scholar
Golecki, J., “Elastic Half-Plane with Variable Poisson’s Ratio. Displacement Boundary Problems,” Bulletin L'Académie Polonaise des Science, Série des Sciences Techniques, 16(4), pp. 175–182 (1968).Google Scholar
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(1), pp. 10–20 (2006).CrossRefGoogle Scholar
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).Google Scholar
Liu, J., Ke, L.L. and Wang, Y.S., “Thermoelastic Contact Analysis of Functionally Graded Materials with Properties Varying Exponentially,” Advanced Materials Research, 189-193, pp. 988–992 (2011).Google Scholar
Liu, J., Ke, L.L., Wang, Y.S., Yang, J., and Alam, F., “Thermoelastic Frictional Contact of Functionally Graded Materials with Arbitrarily Varying Properties,” International Journal of Mechanical Sciences, 63, pp. 86–98 (2012).Google Scholar
Ke, L.-L. and Wang, Y.-S., “Two-Dimensional Contact Mechanics of Functionally Graded Materials with Arbitrary Spatial Variations of Material Properties,” International Journal of Solids and Structures, 43(18-19), pp. 5779–5798 (2006).CrossRefGoogle Scholar
Alshits, V.I. and Kirchner, H.O.K., „Cylindrically Anisotropic, Radially Inhomogeneous Elastic Materials,” Proceedings of the Royal Society of London A, 457, pp. 671–693 (2001).CrossRefGoogle Scholar
Kim, K.-S. and Noda, N., “Green’s Function Approach to Unsteady Thermal Stresses in an Infinite Hollow Cylinder of Functionally Graded Material,” Acta Mechanica, 156, pp. 145–161 (2002).Google Scholar
Shao, Z.S., “Mechanical and Thermal Stresses of a Functionally Graded Circular Hollow Cylinder with Finite Length,” International Journal of Pressure Vessels and Piping, 82, pp. 155–163 (2005).Google Scholar
Awaji, H. and Sivakumar, R., “Temperature and Stress Distributions in a Hollow Cylinder of Functionally Graded Material: the Case of Temperature-Independent Material Properties,” Journal of the American Ceramic Society, 84(5), pp. 1059–1065 (2001).Google Scholar
Shevlyakov, Y.A., Naumov, Y.A. and Chistyak, V.I., “On Calculating Inhomogeneous Foundations,” Soviet Applied Mechanics, 4(9), pp. 42 – 45 (1968).Google Scholar
Ootao, Y. and Tanigawa, Y., “Three-Dimensional Transient Piezothermoelasticity in Functionally Graded Rectangular Plate Bonded to a Piezoelectric Plate,” International Journal of Solids and Structures, 37, pp. 4377–4401 (2000).Google Scholar
Kim, K. S. and Noda, N., “Green’s Function Approach to Three-Dimensional Heat Conduction Equation of Functionally Graded Materials,” Journal of Thermal Stresses, 24(5), pp. 457–477 (2001).Google Scholar
Watremetz, B., Baietto-Dubourg, M.C. and Lubrecht, A.A., “2D Thermo-Mechanical Contact Simulations in a Functionally Graded Material: A Multigrid-Based Approach,” Tribology International, 40(5), pp. 754– 762 (2007).CrossRefGoogle Scholar
Tokova, L., Yasinskyy, A. and Ma, C.-C., “Effect of the Layer Inhomogeneity on the Distribution of Stresses and Displacements in an Elastic Multilayer Cylinder,” Acta Mechanica, 228(8), pp. 2865–2877 (2017).Google Scholar
Plevako, V., “Influence of the Change of Material Parameters on the Stressed State of an Inhomogeneous Cylinder,” In: Proceedings of the International Scientific-Technical Conference Problems of Mathematical Modeling of the Modern Technologies PMM-2002, Khmelnyts’kyy, Ukraine., p. 11 (2002) [in Ukrainian].Google Scholar
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).CrossRefGoogle Scholar
Vihak, V.M., “Solution of the Thermoelastic Problem for a Cylinder in the Case of a Two-Dimensional Nonaxisymmetric Temperature Field,” Zeitschrift für Angewandte Mathematik und Mechanik, 76(1), pp. 35–43 (1996).Google Scholar
Vihak, V., Tokovyi, Yu. and Rychahivskyy, A., “Exact Solution of the Plane Problem of Elasticity in a Rectangular Region,” Journal of Computational and Applied Mechanics, 3(2), pp. 193–206 (2002).Google Scholar
Vigak, V.M. and Rychagivskii, A.V., “Solution of a Three-Dimensional Elastic Problem for a Layer,” International Applied Mechanics, 38(9), pp. 1094 – 1102 (2002).CrossRefGoogle Scholar
Michell, J. H., “On the Direct Determination of Stress in an Elastic Solid, with Application to the Theory of Plates,” Proceedings of the London Mathematical Society, 31(687), pp. 100–124 (1899).CrossRefGoogle Scholar
Vigak, V.M., “Solution of One-Dimensional Problems of Elasticity and Thermoelasticity in Stresses for a Cylinder,” Journal of Mathematical Sciences, 96(1), pp. 2887–2891 (1999).CrossRefGoogle Scholar
Vigak, V.M., “Correct Solutions of Plane Elastic Problems for a Semi-Plane,” International Applied Mechanics, 40(3), pp. 283 – 289 (2004).Google Scholar
Rychahivskyy, A.V. and Tokovyy, Yu.V., “Correct Analytical Solutions to the Thermoelasticity Problems in a Semi-Plane,” Journal of Thermal Stresses, 31(11), pp. 1125–1145 (2008).CrossRefGoogle Scholar
Tokovyy, Yu. and Ma, C.-C., “Steady-State Heat Transfer and Thermo-Elastic Analysis of Inhomogeneous Semi-Infinite Solids,” in: Vikhrenko, V. S. (ed.) Heat Conduction – Basic Research, InTech, Rijeka, pp. 249–268 (2011).Google Scholar
Tokovyy, Y. V. and Ma, C.-C. “Analytical Solutions to the Axisymmetric Elasticity and Thermoelasticity Problems for an Arbitrarily Inhomogeneous Layer,” International Journal of Engineering Science, 92, pp. 1–17 (2015).CrossRefGoogle Scholar
Tokovyy, Y., Lozynskyy, Y. and Ma, C.-C., “Two-Dimensional Thermal Stresses and Displacements in an Arbitrarily Inhomogeneous Elastic Layer,” Applied Mechanics and Materials, 627, pp. 141–144 (2014).Google Scholar
Tokovyy, Yu.V., Kalynyak, B.M. and Ma, C.-C., “Nonhomogeneous solids: Integral equations approach,” in Hetnarski, R. B. (ed.) Encyclopedia of Thermal Stresses, 7, Springer, Dordrecht, pp. 3350–3356 (2014).Google Scholar
Lopatynsky, Ya.B., “On a Class of Reduction of Boundary-Value Problems for a System of Differential Equations of Elliptic Type to Regular Integral Equations,” Ukrainian Mathematical Journal, 5(2), pp. 123–151 (1953).Google Scholar
Fichera, G., “Linear Elliptic Equations of Higher Order in Two Independent Variables and Singular Integral Equations, With Applications to Anisotropic inhomogeneous Elasticity,” in: Partial Differential Equations and Continuum Mechanics, Univ. Wisconsin Press, Madison, pp. 55–80 (1961).Google Scholar
Panferov, V.M. and Leonova, É.A., “A Solution of the Problems of Thermal Elasticity with Variable Moduli,” Strength of Materials, 7(6), pp. 674–681 (1975).Google Scholar
Clements, D.L. and Rogers, C., “On the Bergman Operator Method and Anti-Plane Contact Problems Involving an Inhomogeneous Half-Space,” SIAM Journal of Applied Mathematics, 34(4), pp. 764–773 (1978).Google Scholar
Furuhashi, R., “On the Uniqueness and the Existence of Solution in Elastostatic for Inhomogeneous Materials,” The Japan Society of Mechanical Engineers: Bulletin of the JSME, 15(83), pp. 657–662 (1972).CrossRefGoogle Scholar
Furuhashi, R. “On Green's Function in Elastostatics for Inhomogeneous Materials,” Research Reports of the Faculty of Engineering, Meiji University, 26/27, pp. 11–19 (1972).Google Scholar
Furuhashi, R. and Kataoka, M., “On the Integral Equations of the Basic Boundary Value Problems of Elasticity of Inhomogeneous Media,” Transactions of the Japan Society of Mechanical Engineers, 33(253), pp. 1331–1343 (1967).Google Scholar
Furuhashi, R. and Kataoka, M., “Theory of Elastic Potential of Inhomogeneous Materials,” The Japan Society of Mechanical Engineers: Bulletin of the JSME, 11(48), pp. 972–982 (1968).Google Scholar
Li, X.-F., Peng, X.-L. and Lee, K.-Y. “Radially Polarized Functionally Graded Piezoelectric Hollow Cylinders as Sensors and Actuators,” European Journal of Mechanics A Solids, 29(4), pp. 413–437 (2010).Google Scholar
Peng, X.L. and Li, X.L., “Transient Response of Temperature and Thermal Stresses in a Functionally Graded Hollow Cylinder,” Journal of Thermal Stresses, 33(5), pp. 485–500 (2010).CrossRefGoogle Scholar
Peng, X.L. and Li, X.L., “Thermoelastic Analysis of a Cylindrical Vessel of Functionally Graded Materials,” International Journal of Pressure Vessels and Piping, 87(5), pp. 203–210 (2010).CrossRefGoogle Scholar
Tokovyy, Y. and Ma, C.-C., “An Explicit-Form Solution to the Plane Elasticity and Thermoelasticity Problems for Anisotropic and Inhomogeneous Solids,” International Journal of Solids and Structures, 46(21), pp. 3850 – 3859 (2009).Google Scholar
Tokovyy, Y., Chyzh, A. and Ma, C.-C., “An Analytical Solution to the Axisymmetric Thermoelasticity Problem for a Cylinder with Arbitrarily Varying Thermomechanical Properties,” Acta Mechanica, https://doi.org/10.1007/s00707-017-2012-3 (2017).Google Scholar
Kalynyak, B. M., “Integration of Equations of One-Dimensional Problems of Elasticity and Thermoelasticity for Inhomogeneous Cylindrical Bodies,” Journal of Mathematical Sciences, 99(5), pp. 1662–1670 (2000).CrossRefGoogle Scholar
Shevchuk, V.A. and Kalynyak, B.M., “Stressed State of Cylindrical Bodies with Multilayer Inhomogeneous Coatings,” Materials Science, 46, pp. 747–756 (2011).CrossRefGoogle Scholar
Vigak, V. M., “Solutions of One-Dimensional Problems of Elasticity and Thermoelasticity for Cylindrical Piecewise-Homogeneous Bodies,” Journal of Mathematical Sciences, 96(2), pp. 3057–3064 (1999).CrossRefGoogle Scholar
Tokova, L. and Yasinskyy, A., “Approximate Solution of the One-Dimensional Problem of the Theory of Elasticity for an Inhomogeneous Solid Cylinder,” Journal of Mathematical Sciences, 228(2), pp. 133–141 (2018).Google Scholar
Popovych, V. S. and Kalynyak, B. M., “Mathematical Modeling and Methods for the Determination of the Static Thermoelastic State of Multilayer Thermally Sensitive Cylinders,” Journal of Mathematical Sciences, 215(2), pp. 218 – 242 (2016).CrossRefGoogle Scholar
Artemyuk, V. Yu. and Kalynyak, B. M., “Integral Equation for the Radial Stresses in a Radially Inhomogeneous Heat-Sensitive Hollow Sphere,” Journal of Mathematical Sciences, 223(2), pp. 132–144 (2017).CrossRefGoogle Scholar
Tokovyy, Y. V. and Ma, C.-C. “Analysis of 2D Non-Axisymmetric Elasticity and Thermoelasticity Problems for Radially Inhomogeneous Hollow Cylinders,” Journal of Engineering Mathematics, 61(24), pp. 171–184 (2008).Google Scholar
Tokovyy, Y. V. and Ma, C.-C., “Analytical Solutions to the Planar Non-Axisymmetric Elasticity and Thermoelasticity Problems for Homogeneous and Inhomogeneous Annular Domains,” International Journal of Engineering Science, 47(3), pp. 413–437 (2009).CrossRefGoogle Scholar
Tokovyy, Y. V. and Ma, C.-C., “Thermal Stresses in Anisotropic and Radially Inhomogeneous Annular Domains,” Journal of Thermal Stresses, 31(9), pp. 892–913 (2008).CrossRefGoogle Scholar
Tokovyy, Y., and Ma, C.-C., “Axisymmetric Stresses in an Elastic Radially Inhomogeneous Cylinder Under Length-Varying Loadings,” Journal of Applied Mechanics, 83, pp. 111007–1–7 (2016).Google Scholar
Tokovyy, Y. and Ma, C.-C., “Three-Dimensional Temperature and Thermal Stress Analysis of an Inhomogeneous Layer,” Journal of Thermal Stresses, 36(8), pp. 790–808 (2013).Google Scholar