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Three-Dimensional Elastic Analysis of Transversely-Isotropic Composites

Published online by Cambridge University Press:  02 October 2017

Yu. V. Tokovyy  and
C. C. Ma
Yu. V. Tokovyy*
Affiliation:
Pidstryhach Institute for Applied Problems of Mechanics and MathematicsNational Academy of Sciences of UkraineLviv, Ukraine
C. C. Ma
Affiliation:
Mechanical Engineering DepartmentNational Taiwan UniversityTaipei, Taiwan
*
*Corresponding author ([email protected])
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Abstract

An exact analytical solution to the three-dimensional elasticity problem for a transversely-isotropic composite layer is constructed by making use of the direct integration method along with the Fourier double-integral transform. The original problem is reduced to a system of governing partial-differential equations for separate stress-tensor components. The governing equations are accompanied with corresponding local and integral boundary conditions, obtained on the basis of the original local boundary conditions imposing the normal and shearing forces on the limiting planes of the layer. The numerical analysis of the obtained solution is presented for certain transversely-isotropic composite materials.

Keywords

Transversely-isotropic materialComposite materialThree-dimensional problemElasticityExact analytical solution
Type
Research Article
Information
Journal of Mechanics , Volume 33 , Issue 6 , December 2017 , pp. 821 - 830
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2017 

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References

1. Mantič, V. (Ed.), Mathematical Methods and Models in Composites, Imperial College Press, London, p. 506 (2014).Google Scholar
2. Chernykh, K. F., An Introduction to Modern Anisotropic Elasticity, Begell House, New York, p. 248 (1998).Google Scholar
3. Rand, O. and Rovenski, V., Analytical Methods in Anisotropic Elasticity (with Symbolic Computational Tools), Birkhäuser, Boston-Basel-Berlin, p. 454 (2005).Google Scholar
4. Nemat-Nasser, S. and Yamada, M. “Harmonic Waves in Layered Transversely Isotropic Composites,” Journal of Sound and Vibration, 79, pp. 161–170 (1981).Google Scholar
5. Liu, G. R., Tani, J., Watanabe, K. and Ohyoshi, T., “Lamb Wave Propagation in Anisotropic Laminates,” Journal of Applied Mechanics, 57, pp. 923–929 (1990).Google Scholar
6. Christensen, R. M. and Zywicz, E., “A Three-Dimensional Constitutive Theory for Fiber Composite Laminated Media,” Journal of Applied Mechanics, 57, pp. 948–955 (1990).Google Scholar
7. Spencer, A. J. M. Deformations of Fibre Reinforced Materials, Oxford University Press, Oxford, p. 134 (1972)Google Scholar
8. Dong, X. N., Zhang, X., Huang, Y. and Guo, X. E., “A Generalized Self-Consistent Estimate for the Effective Elastic Moduli of Fiber-Reinforced Composite Materials with Multiple Transversely Isotropic Inclusions,” International Journal of Mechanical Sciences, 47, pp. 922–940 (2005).Google Scholar
9. Podil’chuk, Y. N., “Exact Analytic Solutions of Three-Dimensional Boundary-Value Problems of the Statics of a Transversely Isotropic Body of Canonical Form (Survey),” International Applied Mechanics, 33, pp. 763–787 (1997).Google Scholar
10. Nemish, Yu. N., “Development of Analytical Methods in Three-Dimensional Problems of the Statics of Anisotropic Bodies (Review),” International Applied Mechanics, 36, pp. 135–172 (2000).CrossRefGoogle Scholar
11. Wang, M. Z., Xu, B. X. and Gao, C. F., “Recent General Solutions in Linear Elasticity and Their Applications,” Applied Mechanics Reviews, 61, pp. 030803-1 – 030803-20 (2008).Google Scholar
12. Lekhnitskii, S. G., “On the Effect of Concentrated Forces on the Stress Distribution in an Aelotropic Elastic Solid,” Prikladnaya Matematika i Mekhanika, 3, pp. 66–69 (1936).Google Scholar
13. Lekhnitskii, S. G., “Symmetrical Deformation and Torsion of a Body of Revolution with a Special Kind of Anisotropy,” Prikladnaya Matematika i Mekhanika, 4, pp. 43–60 (1940).Google Scholar
14. Hu, H.-C., “On The Three-Dimensional Problems of the Theory of Elasticity of a Transversely Isotropic Body,” Acta Physica Sinica, 9, pp. 130–148 (1953).Google Scholar
15. Nowacki, W., “The Stress Function in Three- Dimensional Problems Concerning an Elastic Body Characterized by Transverse Isotropy,” Bulletin de l'Academie Polonaise des Sciences, 2, pp. 21–25 (1954).Google Scholar
16. Elliott, H. A., “Three-Dimensional Stress Distributions in Hexagonal Aeolotropic Crystals,” Mathematical Proceedings of the Cambridge Philosophical Society, 44, pp. 522–533 (1948).CrossRefGoogle Scholar
17. Lodge, A. S., “The Transformation to Isotropic Form of the Equilibrium Equations for a Class of Anisotropic Elastic Solids,” The Quarterly Journal of Mechanics and Applied Mathematics, 8, pp. 211–225 (1955).CrossRefGoogle Scholar
18. Podil'chuk, Y. N. and Tkachenko, V. F., “Stress State of a Transversely Isotropic Medium with a Parabolic Crack when Linearly Changing Pressure is Applied to its Surface,” International Applied Mechanics, 30, pp. 927–932 (1994).Google Scholar
19. Podil’chuk, Y. N. and Tkachenko, V. F., “Stress State of a Transversally Isotropic Medium with a Parabolic Crack under a Linearly Varying Shear Load,” International Applied Mechanics, 31, pp. 338–345 (1995).Google Scholar
20. Podil'chuk, Y. N. and Tkachenko, V. F., “Stressed State of a Transversally Isotropic Half-Space in the Case of a Flat Elliptical Die with Forces and Moments Acting on It,” International Applied Mechanics, 31, pp. 806–811 (1995).Google Scholar
21. Chen, T. W., “On Some Problems in Transversely Isotropic Elastic Materials,” Journal of Applied Mechanics, 33, pp. 347–355 (1966).Google Scholar
22. Liao, J. J. and Wang, C. D., “Elastic Solutions for a Transversely Isotropic Half-Space Subjected to a Point Load,” International Journal for Numerical and Analytical Methods in Geomechanics 22, pp. 425–447 (1998).Google Scholar
23. Wang, C.-D. and Liao, J.-J., “Elastic Solutions of Displacements for a Transversely Isotropic Half- Space Subjected to Three-Dimensional Buried Parabolic Rectangular Loads,” International Journal of Solids and Structures, 39, pp. 4805–4824 (2002).CrossRefGoogle Scholar
24. Shaldyrvan, V. A., “Three-Dimensional Thermoelasticity Problems for Transversally Isotropic Plates,” Soviet Applied Mechanics, 16, pp. 370–375 (1980).Google Scholar
25. Aleksandrov, A. Ya. and Vol'pert, V. S., “Solution of Three-Dimensional Problems of the Theory of Elasticity for a Transversally Isotropic Body Using Analytical Functions,” Izvestiya Academii Nauk SSSR: Mekhanika Tverdogo Tela, 5, pp. 82–91 (1967).Google Scholar
26. Goman, O. G., “Representation in Terms of p-Analytic Functions of the General Solution of Equations of the Theory of Elasticity of a Transversely Isotropic Body,” Journal of Applied Mathematics and Mechanics, 48, pp. 62–67 (1984).CrossRefGoogle Scholar
27. Pan, E. and Yuan, F. G., “Three-Dimensional Green's Functions in Anisotropic Bimaterials,” International Journal of Solids and Structures, 37, pp. 5329–5351 (2000).Google Scholar
28. Ding, H., Chen, W. and Zhang, L., Elasticity of Transversely Isotropic Materials, Springer, Dordrecht, 435 (2006).Google Scholar
29. Xie, L., Zhang, C., Sladek, J. and Sladek, V., “Unified Analytical Expressions of the Three-Dimensional Fundamental Solutions and Their Derivatives for Linear Elastic Anisotropic Materials,” Proceedings of the Royal Society A. Mathematical, Physical and Engineering Sciences, 472, 20150272 (2016).CrossRefGoogle ScholarPubMed
30. Távara, L., Ortiz, J. E., Mantič, V. and París, F., “Unique Real-Variable Expressions of Displacement and Traction Fundamental Solutions Covering All Transversely Isotropic Elastic Materials For 3D BEM,” International Journal for Numerical Methods in Engineering, 74, pp. 776–798 (2008).Google Scholar
31. Távara, L., Mantič, V., Ortiz, J. E. and París, F., “Unique Real-Variable Expressions of the Integral Kernels in the Somigliana Stress Identity Covering All Transversely Isotropic Elastic Materials for 3D BEM,” Computer Methods in Applied Mechanics and Engineering, 225, pp. 128–141 (2012).Google Scholar
32. Wang, M. Z. and Wang, W., “Completeness and Nonuniqueness of General Solutions of Transversely Isotropic Elasticity,” International Journal of Solids and Structures, 32, pp. 501–513 (1995).Google Scholar
33. Wang, W. and Shi, M. X., “On the General Solutions of Transversely Isotropic Elasticity,” International Journal of Solids and Structures, 35, pp. 3283–3297 (1998).Google Scholar
34. Eubanks, R. A. and Sternberg, E., “On the Axisymmetric Problems Elasticity Theory for a Medium with Transverse Isotropy,” Journal of Rational Mechanics and Analysis, 3, pp. 89–101 (1954).Google Scholar
35. Tokovyy, Y. V., “Direct Integration Method,” Encyclopedia of Thermal Stresses, Hetnarski, B. (ed.), Springer, Dordrecht, 2, pp. 951–960 (2014).Google Scholar
36. 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, pp. 545–555 (2015).Google Scholar
37. Tokovyy, Y. and Ma, C.-C., “Three-Dimensional Temperature and Thermal Stress Analysis of An Inhomogeneous Layer,” Journal of Thermal Stresses, 36, pp. 790–808 (2013).CrossRefGoogle Scholar
38. Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Body, Mir Publisher, Moscow, p. 340 (1981).Google Scholar
39. Lur'e, A. I., Three-Dimensional Problems of the Theory of Elasticity, Interscience Publisher, New York, p. 493 (1964).Google Scholar
40. Brigham, E. O., The Fast Fourier Transform and Its Applications, Prentice-Hall Inc., New Jersey, p. 448 (1988).Google Scholar