Three-Dimensional Elasticity Solutions for Isotropic and Generally Anisotropic Bodies


Article Preview

Classical methods of two-dimensional elasticity can be extended to give an exact solution of the three-dimensional problem for the beam — i.e. a general solution for the pris- matic bar loaded on its lateral surfaces, subject only to the restriction that the tractions can be expanded as power series in the axial coordinate z. A series of sub-problems Pj is defined by successive partial differentiations with respect to z. For isotropic materials, a recursive al- gorithm can be used for generating the solution to Pj+1 from that for Pj in the context of the Papkovich-Neuber solution. For the generally anisotropic material, a similar strategy is proposed, based on partial integrations of Stroh’s formulation of the two-dimensional problem.



Edited by:

Patrick Sean Keogh






J.R. Barber "Three-Dimensional Elasticity Solutions for Isotropic and Generally Anisotropic Bodies", Applied Mechanics and Materials, Vols. 5-6, pp. 541-550, 2006

Online since:

October 2006





[1] Almansi, E., Sopra la Deformazione dei Cilinri Solecitati Lateralmente, Atti Real Accad. naz. Lincei Rend, Cl. sci fis., mat e natur. Ser. 5 10 (I), pp.333-338, 1901a.

DOI: 10.4171/rlm

[2] Almansi, E., Sopra la Deformazione dei Cilinri Solecitati Lateralmente, Atti Real Accad. naz. Lincei Rend, Cl. sci fis., mat e natur. Ser. 5 10 (II), pp.400-408, 1901b.

DOI: 10.4171/rlm

[3] Barber, J. R., Elasticity, 2nd edn, Kluwer, Dordrecht, (2002).

[4] Barber, J. R., Three-dimensional Elasticity Problems for the Prismatic Bar, Proc. Roy. Soc. (London), 462, pp.1877-1896, (2006).

[5] El Fatmi, R. & Zenzri, H., A Numerical Method for the Exact Elastic Beam Theory. Applications to Homogeneous and Composite Beams, Int.J. Solids Structures, 41, pp.2521-2537, (2004).

DOI: 10.1016/j.ijsolstr.2003.12.011

[6] Green, A. E. & Zerna, W., Theoretical Elasticity, Clarendon Press, Oxford, (1954).

[7] Huang, C.H. & Dong, S.B., Analysis of Laminated Circular Cylinders of Materials with the most General Form of Cylindrical Anisotropy: I Axially Symmetric Deformations, Int.J. Solids Structures, 68, pp.6163-6182, (2001).

DOI: 10.1016/s0020-7683(00)00374-7

[8] Ie¸san, D., On Saint-Venant's Problem, Arch. Rational Mech. Anal., 91, pp.363-373, (1986).

[9] Ladev`eze, P., Sanchez, Ph., & Simmonds, J.G., Beamlike (Saint-Venant) Solutions for Fully Anisotropic Elastic Tubes of Arbitrary Closed Cross Section, Int.J. Solids Structures, 41, pp.1925-1944, (2004).

DOI: 10.1016/j.ijsolstr.2003.11.006

[10] Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, (1963).

[11] Michell, J. H., The Theory of Uniformly Loaded Beams, Quart. J. Math. 32, pp.28-42, (1901).

[12] Milne-Thomson, L. M., Plane Elastic Systems, 2nd edn, Springer, Berlin, (1968).

[13] Milne-Thomson, L. M., Antiplane Elastic Systems, Springer, Berlin, (1962).

[14] Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, (1963).

[15] Rand, O. & Rovenski, V.Y., Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkh¨auser, Boston, (2005).

[16] Stevenson, A. C., Some Boundary Problems of Two-dimensional Elasticity, Phil. Mag., 34, pp.766-793, (1943).

[17] Stevenson, A. C., Complex Potentials in Two-dimensional Elasticity, Proc. Roy. Soc. (London)., 184A, pp.129-179, (1945).

[18] Stroh, A. N., Dislocations and Cracks in Anisotropic Elasticity, Phil. Mag., 3, pp.625-646, (1958).

[19] Stroh, A. N., Steady-state Problems in Anisotropic Elasticity, J. Math. Phys., 41, pp.77-103, (1962).

[20] Sveklo V.A., Boussinesq Type Problems for the Anisotropic Half-space, J. Appl. Math. Mech., 28, pp.1099-1105, (1964).

[21] Taciroglu, E. & Liu, C.W., Analysis and Design of Multimodal Piezoelectric Layered Tubular Sensors and Actuators, Smart Matls. Structures, 11, pp.605-614, (2005).

DOI: 10.1088/0964-1726/14/4/019

[22] Ting, T. C. T., Anisotropic Elasticity, Oxford University Press, New York, (1996).

In order to see related information, you need to Login.