摘要

題 目：複合楔形板之應力奇異性
研 究 生：李文仁
指導教授：胡潛濱

主要以異向性熱彈性力學為基礎，來探討複合楔形板尖端之應力奇異性，即應力集中的現象，此現象常出現於結構體中幾何、或材料不連續的位置，因而造成結構體於受外在負載時，會在這些位置先行產生破壞，導致結構體的毀損。
文中以r1-d來描述應力奇異性的現象，其中r表示自尖端處到楔形板中任一位置之距離，d為應力奇異性階數，與楔形板角度、材料性質及邊界條件有關。文中探討的邊界條件包括：固定邊界、自由邊界、及固定-自由邊界。對於熱效應對應力奇異性階數的影響在本文中也有陳述，結果顯示有熱效應時與無熱效應時，兩者是相互獨立的，即可由不同方程式求得奇異性階數，其對應力奇異性的影響則視所求的的應力奇異性階數大小而定。
對於一般纖維加強的複合材料，我們可以藉由纖維方向的排列或是幾何形狀的修改，來使應力奇異性的現象減至最低，減少結構體在外在負荷時產生破壞。

Abstract

Subject：Stress Singularities of Composite Wedges
Student：Wen-jen Lee
Advisor：Chyanbin Hwu
Based on the two-dimensional linear thermo-anisotropic elasticity theory, the stress singularity near the apex of the multi-bonded wedges is studied. The stress singularities often occur near the locations where geometry and material properties are discontinuous. Damages will then occur near these locations when loaded with external forces.
The mathematical form r1-d is used to describe the stress singularity where r represents the distance ahead of the wedge apex. d is the order of stress singularity, which is related to the wedge angle, material properties, and boundary conditions. The boundary conditions considered in this paper include: fixed-fixed, fixed-free, and free-free boundary conditions. For thermal effect, the results show that the derived orders of stress singularity with and without thermal effect are independent. Their effect on stress singularity depends on their values.
If each wedge is composed of the unidirectional fiber-reinforced composites, the stress singularities may be reduced by the re-arrangement of fiber orientations or re-modification of wedge angles. Through proper arrangements, the multi-bonded wedge structure may have long life endurance and resistance to external forces.

參考文獻
[1] Bogy, D. B., “Edge-Bonded Dissimilar Orthogonal Elastic Wedges under Normal and Shear Loading”, Journal of Applied Mechanics, Vol. 35, pp. 460-466, 1968.
[2] Bogy, D. B., “On the Problem of Edge-Bonded Elastic Quarter- Planes Loaded at the Boundary”, International Journal of Solids and Structures, Vol. 6, pp. 1287-1313, 1970.
[3] Bogy, D. B. & Wang, K. C., “Stress Singularities at Interface Corners in Bonded Dissimilar Isotropic Elastic Materials”, Journal of Solids and Structures, Vol. 7, pp. 993-1005, 1971.
[4] Bogy, D. B., “Two Edge-Bonded Elastic Wedges of Different Materials and Wedge Angles under Surface Tractions”, Journal of Applied Mechanics, Vol. 38, 377-386, 1971.
[5] Barnett, D. M. & Lothe, J., Synthesis of the sextic and the integral formalism for dislocations, Greens function and surfaces waves in anisotropic elastic solids. Phys. Norv. 7: 13-19, 1973.
[6]Berger, J. R., Martin, P. A. and Lien, J. P., “Reduction of free-edges stress intensities in anisotropic bi-materials”, International Journal of Fracture, Vol. 91, pp. 165~177, 1998.
[7]Clements, D. L., ”Thermal Stress in an Anisotropic Elastic Half-Space”, SIAM J. Appl. Math., Vol. 24, pp. 332~337, 1973.
[8]Chen, H. P., “Stress Singularities in Anisotropic Multi-Material Wedges and Junctions”, International Journal of Solids and Structures, Vol. 35, pp. 1057-1073, 1998.
[9]Chue, C. H., Liu, C. I., “A General Solution on Stress Singularities in Anisotropic Wedge”, International Journal of Solids and Structures, Vol. 38, pp. 6889-6906, 2001.
[10]Delale, F., “Stress Singularities in Bonded Anisotropic Materials”, International Journal of Solids and Structures, Vol. 20, pp. 31-40, 1984.
[11]Hein, V. L., & Erdogan, F., “Stress Singularities in a Two-Material Wedge”, International Journal of Fracture Mechanics, Vol. 7, pp. 317-330, 1971.
[12] Hwu, C., “Thermal Stresses in an Anisotropic Plate Disturbed by an Insulated Elliptic Hole or Crack”, Journal of Applied Mechanics, Vol. 57, pp. 916~922, 1990.
[13]Hwu, C., Oomiya, M., Kishimoto, K., “A Key Matrix for the Stress Singularity of the Anisotropic Elastic Composite Wedges”, Submitted for publication.
[14]Leknitskii, S. G., Theory of elasticity of an anisotropic elastic body. Gostekhizdat, Moscow (in Russian), Holden-Day, San Francisco, 1950. (in English 1963) and Mir Publication, Moscow (in English, 1981)
[15]Stroh, A. N., “Steady State Problems in Anisotropic Elasticity”, Journal of Mathematical Physics, Vol. 41, pp. 77-103, 1962.
[16]Theocaris, P. S., “The Order of Singularity at a Multi-Wedge Corner of a Composite Plate”, International Journal of Engineering Science, Vol. 12, pp. 107-120, 1974.
[17]Ting, T. C. T. & Yan, Gongpu, “The Singularity at Interface Cracks in Anisotropic Bimaterials due to Heat Flow”, Journal of Thermal Stress, pp. 85-99, 1992.
[18]Ting, T. C. T., “Stress Singularities at the tip of interface in polycrystals”, Damage and Failure of Interfaces, Rossmanith (ed.), pp.75-82, 1997.
[19]Ting, T. C. T., Anisotropic Elasticity: Theory and Applications. Oxford University Press, 1996.
[20] Williams, M. L., “Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates in Extension”, Journal of Applied Mechanics, Vol. 74, pp. 526-528, 1952.
[21]Wu, C. H., ”Plane Anisotropic Thermoelasticity”, Journal of Applied Mechanics, Vol. 51, pp. 724-726, 1984.
[22]Yang, Y. Y. & Munz, D. “Stress Distribution in a Dissimilar Materials Joint for Complex Singular Eigenvalues under Thermal Loading”, Journal of Thermal Stress, pp. 407-419, 1995.
[23] 林靜娟, ”絕緣或導熱剛體介質之熱應力分析”, 國立成功大學航空太空工程研究所, 1991。
[24]褚晴暉, 劉全益, ”非等向性楔形結構之應力奇異性分析”, 第六屆破壞力學研討會, pp. (18-1)-(18-9), 2000。
[25] 劉全益, ”異種非等向性楔形結構之應力奇異性分析”, 國立成功大學機械工程研究所, 2001。