Abstract
For practical applications, singular solutions of classical problems in the theory of the elasticity associated with the infinite values of stresses in the vicinity of singular points can be viewed as an indication of specific zones with strong stress concentration at the points of disturbance of boundary smoothness, variation in the type of boundary conditions or contact points of dissimilar material. The existence or non-existence of singular solutions and their exponential dependence are defined by the geometry and mechanical characteristics of the material in the vicinity of singular points. The problem of optimization of geometry and material properties at singular points is considered. The objective of the optimization problem is minimization of the stress state in a solid body characterized by the stress intensity, a functional relation, which specifies the degree of nonhomogeneity of the stress state. A mathematical formulation of the examined optimization problem is considered and the key steps of the algorithm for its numerical implementation are discussed. For the sake of illustration, several optimization problems have been solved for different types of singular points. The analysis of solutions of optimization problems has shown that optimal solutions have the following common property: in the problems of geometry optimization in the vicinity of singular points, the parameters of optimal geometry and optimal characteristics of the material determine the boundary between the solutions with and without stress singularity, while in the problem of optimization of mechanical characteristics in the vicinity of singular points they determine absence of singular solutions.
Similar content being viewed by others
References
Baladi, A., Arezoodar, A.: Dissimilar materials joint and effect of angle junction on stress distribution at interface. Int. J. Mech. Aerosp. Ind. Mechatron. Manuf. Eng. 5(7), 1184–1187 (2011). http://waset.org/publications/4198
Borzenkov, S., Matveenko, V.: Optimization of elastic bodies in the vicinity of singular points. Izv. RAN. Mekhanika Tverdogo Tela 2, 93–100 (1996)
Chobanyan, K.: Stress State in Compound Elastic Bodies. Armenian Academy of Sciences Press, Yerevan (1987)
Comninou, M.: Stress singularity at a sharp edge in contact problems with friction. Z. Angew. Math. Phys. ZAMP 27(4), 493–499 (1976). https://doi.org/10.1007/BF01594906
Dempsey, J., Sinclair, G.: On the singular behavior at the vertex of a bi-material wedge. J. Elast. 11(3), 317–327 (1981). https://doi.org/10.1016/j.compstruct.2007.01.026
Dundurs, J.: Discussion: Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading (Bogy, D.B., 1968, ASME J. Appl. Mech., 35, pp. 460–466). ASME J. Appl. Mech. 36(3), 650–652 (1969). https://doi.org/10.1115/1.3564739
He, D., Sawa, T., Karami, A.: Stress analysis and strength evaluation of scarf adhesive joints with dissimilar adherends subjected to static tensile loadings. J. Solid Mech. Mater. Eng. 3(8), 1033–1044 (2009). https://doi.org/10.1299/jmmp.3.1033
Hu, N., Wang, B., Sekine, H., Yao, Z., Tan, G.: Shape-optimum design of a bi-material single-lap joint. Compos. Struct. 41(3–4), 315–330 (1998). https://doi.org/10.1016/S0263-8223(98)00054-3
Huang, C., Leissa, A.: Stress singularities in bimaterial bodies of revolution. Compos. Struct. 82(4), 488–498 (2008). https://doi.org/10.1016/j.compstruct.2007.01.026
Kondrat’ev, V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313 (1967)
Lang, T., Mallick, P.: Effect of spew geometry on stresses in single lap adhesive joints. Int. J. Adhes. Adhes. 18(1), 167–177 (1998). https://doi.org/10.1016/S0143-7496(97)00056-0
Lauke, B., Schller, T., Schneider, K.: Determination of interface strength between two polymer materials by a new curved interface tensile test. Compos. Interfaces 10(1), 1–15 (2003). https://doi.org/10.1163/156855403763586765
Matveenko, V.P., Fedorov, A.Y., Shardakov, I.N.: Analysis of stress singularities at singular points of elastic solids made of functionally graded materials. Dokl. Phys. 61(1), 24–28 (2016). https://doi.org/10.1134/S1028335816010031
Mihailov, S.: Stress singularity in the vicinity of an angle edge in an anisotropic composite and some applications to fibrous composites. Izv. Acad. Sci. USSR. Mech. Twerdogo Tela 5, 103–110 (1979)
Murakawa, H., Ueda, Y.: Shape optimization for reducing stress at ceramics/metal joints. Trans. JWRI 18(2), 295–302 (1989). http://ci.nii.ac.jp/naid/110006486876/en/
Murakawa, H., Ueda, Y.: Effect of singularity in stress field on optimum shape of ceramics/metal joint. Trans. JWRI 20(1), 109–116 (1991). http://ci.nii.ac.jp/naid/110006486941/en/
Paggi, M., Carpinteri, A.: On the stress singularities at multimaterial interfaces and related analogies with fluid dynamics and diffusion. Appl. Mech. Rev. 61(2), 020801–1–020801–22 (2008). https://doi.org/10.1115/1.2885134
Park, J., Anderson, W.: Geometric optimization of two bonded wedges involving stress singularities. Compos. Eng. 4(9), 901–912 (1994). https://doi.org/10.1016/0961-9526(94)90034-5
Parton, V., Perlin, P.: Methods of Mathematical Theory of Elasticity. Nauka, Moscow (1981)
Sinclair, G.: Stress singularities in classical elasticity—I: removal, interpretation, and analysis. Appl. Mech. Rev. 57(4), 251–298 (2004). https://doi.org/10.1115/1.1762503
Sinclair, G.: Stress singularities in classical elasticity—II: asymptotic identification. Appl. Mech. Rev. 57(5), 385–439 (2004). https://doi.org/10.1115/1.1767846
Suresh, S.: Fundamentals of Functionally Graded Materials, Processing and Thermomechanical Behavior of the Graded Metals and Metalceramic Composites. Cambridge University Press, Cambridge (1998)
Tsai, M., Morton, J.: The effect of a spew fillet on adhesive stress distributions in laminated composite single-lap joints. Compos. Struct. 32(1–4), 123–131 (1995). https://doi.org/10.1016/0263-8223(95)00059-3
Wang, P., Xu, L.: Convex interfacial joints with least stress singularities in dissimilar materials. Mech. Mater. 38(11), 1001–1011 (2006). https://doi.org/10.1016/j.mechmat.2005.10.002
Williams, M.: Stress singularities resulting from various boundary conditions in angular corners of plates in extension. ASME J. Appl. Mech. 19(4), 526–528 (1952)
Wu, Z.: Design free of stress singularities for bi-material components. Compos. Struct. 65(3–4), 339–345 (2004). https://doi.org/10.1016/j.compstruct.2003.11.009
Wu, Z.: Stress concentration analyses of bi-material bonded joints without in-plane stress singularities. Int. J. Mech. Sci. 50(4), 641–648 (2008). https://doi.org/10.1016/j.ijmecsci.2008.01.004
Xu, L., Kuai, H., Sengupta, S.: Dissimilar material joints with and without free-edge stress singularities: part I. A biologically inspired design. Exp. Mech. 44(6), 608–615 (2004). https://doi.org/10.1007/BF02428250
Xu, L.R., Sengupta, S.: Dissimilar material joints with and without free-edge stress singularities: part II. An integrated numerical analysis. Exp. Mech. 44(6), 616–621 (2004). https://doi.org/10.1007/BF02428251
Ziegler, F.: Selected Topics of Elastostatics, pp. 257–396. Springer US, New York, NY (1991). https://doi.org/10.1007/978-1-4684-0512-5_6
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to the memory of Franz Ziegler
Rights and permissions
About this article
Cite this article
Fedorov, A.Y., Matveenko, V.P. Optimization of geometry and mechanical characteristics of elastic bodies in the vicinity of singular points. Acta Mech 229, 645–658 (2018). https://doi.org/10.1007/s00707-017-1990-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-017-1990-5