Abstract
This paper applies a Hamiltonian method to study analytically the stress distributions of orthotropic two-dimensional elasticity in (x, z) plane for arbitrary boundary conditions without beam assumptions. It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns. Since coordinates (x, z) can not be easily separated, an alternative symplectic expansion is used. Similar to the Hamiltonian formulation in classical dynamics, we treat the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian matrix differential operator. The exponential of the Hamiltonian matrix is symplectic. There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions. The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues (zero eigen-solutions) and that of the well-behaved nonzero eigenvalues (nonzero eigen-solutions). The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions associated with averaged global behaviors such as rigid-body translation, rigid-body rotation or bending. On the other hand, the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant’s principle. Completed numerical examples are newly given to compare with established results.
Similar content being viewed by others
References
Williams M L. Stress singularities resulting from various boundary conditions in angular corners of plates in extension[J]. J Appl Mech-T ASME, 1952, 19(4):526–528.
Timoshenko S P, Goodier J N. Theory of elasticity[M]. New York: McGraw-Hill, 1970.
Gregory R D. The traction boundary-value problem for the elastostatic semi-infinite stripexistence of solution, and completeness of the Papkovich-Fadle eigenfunctions[J]. J Elasticity, 1980, 10(3):295–327.
Gregory R D, Gladwell I. The cantilever beam under tension, bending or flexure at infinity[J]. J Elasticity, 1982, 12(4):317–343.
Gregory R D, Wan F Y M. Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory[J]. J Elasticity, 1984, 14(1):27–64.
Horgan C O, Simmonds J G. Asymptotic analysis of an end-loaded, transversely isotropic, elastic, semi-infinite strip weak in shear[J]. Int J Solids Struct, 1991, 27(15):1895–1914.
Choi I, Horgan C O. Saint-Venants principle and end effects in anisotropic elasticity[J]. J Appl Mech-T ASME, 1977, 44(3):424–430.
Lin Y H, Wan F Y M. Bending and flexure of semi-infinite cantilevered orthotropic strips[J]. Comput Struct, 1990, 35(4):349–359.
Lin Y H, Wan F Y M. Semi-infinite orthotropic cantilevered strips and the foundations of plate theories[J]. Stud Appl Math, 1990, 82(3):217–244.
Savoia M, Tullini N. Beam theory for strongly orthotropic materials[J]. Int J Solids Struct, 1996, 33(17):2459–2484.
Tullini N, Savoia M. Logarithmic stress singularities at clamped-free corners of a cantilever orthotropic beam under flexure[J]. Compos Struct, 1995, 32(1/4):659–666.
Leung A Y T. An improved 3rd-order beam theory[J]. J Sound Vib, 1990, 142(3):527–528.
Leung A Y T, Chan J K W. Null space solution of Jordan chains for derogatory eigenproblems[J]. J Sound Vib, 1999, 222(4):679–690.
Leung A Y T, Su R K L. Mode-I crack problems by fractal 2-level finite-element methods[J]. Eng Fract Mech, 1994, 48(6):847–856.
Leung A Y T, Su R K L. Order of the singular stress fields of through-thickness cracks[J]. Int J Fracture, 1996, 75(1):85–93.
Leung A Y T, Xu X S, Gu Q, Leung C T O, Zheng J J. The boundary layer phenomena in two-dimensional transversely isotropic piezoelectric media by exact symplectic expansion[J]. Int J Numer Meth Eng, 2007, 69(11):2381–2408.
Levinson M. A new rectangular beam theory[J]. J Sound Vib, 1981, 74(1):81–87.
Heyliger P R, Reddy J N. A higher order beam finite element for bending and vibration problems[J]. J Sound Vib, 1988, 126(2):309–326.
Spence D A. A class of biharmonic end-strip problems arising in elasticity and stokes flow[J]. Ima J Appl Math, 1983, 30(2):107–139.
Mielke A. On Saint-Venant’s problem for an elastic strip[J]. P Roy Soc Edinb A, 1988, 110(1/2):161–181.
Zhong W X, Lin J H, Zhu J P. Computation of gyroscopic systems and symplectic eigensolutions of skew-symmetrical matrices[J]. Comput Struct, 1994, 52(5):999–1009.
Zhong W X, Williams F W. Physical interpretation of the symplectic orthogonality of the eigensolutions of a Hamiltonian or symplectic matrix[J]. Comput Struct, 1993, 49(4):749–750.
Zhong WX, Williams F W. On the direct solution of wave-propagation for repetitive structures[J]. J Sound Vib, 1995, 181(3):485–501.
Xu X S, Zhong W X, Zhang H W. The Saint-Venant problem and principle in elasticity[J]. Int J Solids Struct, 1997, 34(22):2815–2827.
Zhang H W, Zhong W X, Li Y P. Stress singularity analysis at crack tip on bi-material interfaces based on Hamiltonian principle[J]. Acta Mech Solida Sin, 1996, 9(2):124–138.
Yao W A, Zhong W X. Symplectic elasticity[M]. Beijing: Higher Education Press, 2002.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by GUO Xing-ming
Project supported by the Research Grant Council of Hong Kong (No. CERG1157/06)
Rights and permissions
About this article
Cite this article
Leung, A.Y.T., Zheng, Jj. Closed form stress distribution in 2D elasticity for all boundary conditions. Appl. Math. Mech.-Engl. Ed. 28, 1629–1642 (2007). https://doi.org/10.1007/s10483-007-1210-z
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10483-007-1210-z