Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T15:52:14.424Z Has data issue: false hasContentIssue false
  • English
  • Français

The Three-Dimensional Infinite Space and Half-Space Green's Functions for Orthotropic Materials

Published online by Cambridge University Press:  05 June 2014

V.-G. Lee
V.-G. Lee*
Affiliation:
Department of Civil Engineering, National Chi Nan University, Puli, Nantou, Taiwan 54561, R.O.C.
*
[email protected]
Get access

Abstract

Common materials, ranging from natural wood to modern composites, have been recognized as ortho-tropic materials. The elastic properties of such materials are governed by nine elastic constants. In this paper the complete set of Green's functions for an infinite medium and a half space is given, which were not reported completely before. Analytic expressions for the infinite Green's functions are derived through the explicit form of the sextic equation given explicitly. For an orthotopic half space, the Green's function is derived by a superposition method. The mathematical concept is based on the addition of a complementary term to the Green's function in an orthotropic infinite domain to fulfill the boundary condition on the free surface. Both solutions are illustrated in certain directions to demonstrate the nature of orthotropy.

Keywords

CompositesOrthotopic materialsGreen's functionsSuperposition method
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 1 , February 2015 , pp. 21 - 28
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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

REFERENCES

1.Taylor, W. R., Roland, E., Ploeg, H., Hertig, D., Klabunde, R., Warner, M. D., Hobatho, M. C., Rakotomanana, L. and Clift, S. E., “Determination of Orthotropic Bone Elastic Constants Using FEA and Modal Analysis,” Journal of Biomechanics, 35, pp. 767–773 (2002).Google Scholar
2.Reddy, J. N. and Phan, N. D., “Stability and Vibration of Isotropic, Orthotropic and Laminated Plates According to a Higher-Order Shear Deformation Theory,” Journal of Sound and Vibration, 98, pp. 157–170 (1985).Google Scholar
3.Hou, P. F., Wang, L. and Yi, T., “2D Green's Functions for Semi-Infinite Orthotropic Thermoelastic Plane,” Applied Mathematical Modelling, 33, pp. 1674–1682 (2009).Google Scholar
4.Hou, P. F., He, S. and Chen, C. P., “2D General Solution and Fundamental Solution for Orthotropic Thermoelastic Materials,” Engineering Analysis with Boundary Elements, 35, pp. 56–60 (2011)Google Scholar
5.Kumar, R. and Chawla, V., “Green's Functions in Orthotropic Thermoelastic Diffusion Media,” Engineering Analysis with Boundary Elements, 36, pp. 1272–1277 (2012).Google Scholar
6.Swanson, S. R., “Hertzian Contact of Orthotropic Materials,” International Journal of Solids and Structures, 41, pp. 1945–1959 (2004).Google Scholar
7.Swanson, S. R., “Contact Deformation and Stress in Orthotropic Plates,” Composites A, 36, pp. 1421–1429 (2005).Google Scholar
8.Bonnet, G., “Orthotropic Elastic Media Having a Closed Form Expression of the Green Tensor,” International Journal of Solids and Structures, 46, pp. 1240–1250 (2009).Google Scholar
9.Ting, T. C. T. and Lee, V. G., “The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Quarterly Journal of Mechanics and Applied Mathematics, 50, pp. 407–426 (1997).Google Scholar
10. Lee, N. G., “Elastic Green's Function for an Infinite Half-Space of a Hexagonal Continuum with its Basal Plane as Surface,” International Journal of Engineering Science, 17, pp. 681–689 (1979).Google Scholar
11. Couteau, B., Labey, L., Hobatho, M. C., Vander, Sloten J., Arlaud, J. Y. and Brignola, J. C., “Validation of a Three Dimensional Finite Element Model of a Femur with a Customized Hip Implant. In: Middleton,” Computer Methods in Biomechanics & Biomedical Engineering, 2, pp. 147–154 (1998).Google Scholar
12. Herakovich, C. T., Mechanics of Fibrous Composites, John Wiley & Sons, Inc., Hoboken, New Jersey, p. 57 (1998).Google Scholar
13. Ting, T. C. T., Anisotropic Elasticity: Theory and Application, Oxford University Press, United Kingdom, pp. 528–530 (1996)Google Scholar
14. Lee, V. G., “Superposing Scheme for the Three-Dimensional Green's Functions of an Anisotropic Half-Space,” International Journal of Solids and Structures, 50, pp. 2407–2415 (2013).Google Scholar