Mathematical modeling of the stress-strain state of composite wedge-shaped structural elements
DOI:
https://doi.org/10.31734/agroengineering2020.24.121Keywords:
composite wedge, multi-wedge system, generalized functions, radial defects, stress asymptotes, mode III stress intensity factorAbstract
The stress-strain state of structural elements such as a homogeneous wedge or a plate with a wedge-shaped cut under antiplane shear deformation has been thoroughly studiedю. However, it is not true for those elements that can be modeled as composites wedge. This paper describes a procedure that allows writing the singular integral equations to determine the stress-strain state in a composite, composed of any number of interconnected wedges that converge at one point, and there are finite defects on the lines of their connection.
The proposed method is based on the method of the generalized
conjugation problem and of the jump function method, according to which the multi-wedge composite is considered as a whole, and its physical-mechanical
characteristics are described by piecewise constant functions – , (, - the Heaviside function), and the radially located defects are modelled by the stress and displacements jumps – , , , ,( is the area occupied by the inclusion). That approach reduces the definition of the stress-strain state in a multi-wedge composite under longitudinal shear to the solution of the
boundary value problem for one partially degenerated differential equation.
The work supplies a detailed description of the proposed technique, and with its help the stress and displacement fields in a composite wedge are written in the form of the Mellin transformants. The problem of the stress-strain state of a two-wedge system with a radial crack of finite length under the action of a concentrated shear load is considered in the research. A singular integral equation for determining the displacement field in such system is constructed and an algorithm of reducing it to an equation with a Cauchy kernel is proposed. This equation is analytically solved in the case of a system consisting of two wedges with the same opening angles. The stress fields near the ends of the interfacial crack in such a system are investigated.
References
Bateman, H., & Erdeyi, A. (1954). Tables of Integral Transforms (Vol. 1, 391). McGraw-Hill Book Company.
Bodzhy, D. B. (1971). Deistvie poverkhnostnykh nahruzok na systemu iz dvukh soedinennykh po hraniam upruhykh klynev izhotovlennykh iz razlichnykh materialov i imeiushchikh proizvolnye uhly rastvora. Tr. AOYM. Ser. Prykladnaia mekhanika, 38(2), 87–96.
Bozhydarnyk, V. V, & Sulym, H. T. (1999). Elementy teorii plastychnosti ta mitsnosti. Lviv: Svit.
Kushnir, R. M., Nykolyshyn, M. M., & Osadchuk, V. A. (2003). Pruzhnyi ta pruzhno-plastychnyi hranychnyi stan obolonok z defektamy. Lviv: SPOLOM.
Makhorkin, M., & Sulym, H. (2007). Asymptotyky i polia napruzhen u klynovii systemi za umov antyploskoi deformatsii. Mashynoznavstvo, 1, 8–13.
Savruk, M. P. (1988). Koefitsienty intensivnosti napriazhenii v telakh s treshchinami. In V. V. Panasiuk (Ed.). Mekhanika razrusheniia i prochnost materialov: sprav. posobie (Vol 2, 620). Kyev: Nauk. dumka.
Savruk, M. P. (2002). Pozdovzhnii zsuv pruzhnoho klyna z trishchynamy ta vyrizamy. Fiz.–khim. mekhanika materialiv, 5, 57–65.
Sulym, H. T., & Makhorkin, M. I. (2007). Antyploska deformatsiia klynovoi systemy z tonkymy radialnymy neodnoridnostiamy. Aktualni aspekty fizyko-mekhanichnykh doslidzhen. Mekhanika: Zb. nauk. prats (рр. 295–304). Kyiv: Nauk. dumka.
Ufliand, Ya. S. (1963). Intehralnye preobrazovaniia v zadachakh teorii upruhosti. Moskva: Nauka.
Carpinteri, A., & Paggi, M. (2005). On the asymptotic stress field in angularly non-homogeneous materials. Int. J. Fract, 135(4), 267–283.
Carpinteri, A., & Paggi, M. (2011). Singular harmonic problems at a wedge vertex: mathematical analogies between elasticity, diffusion, electromagnetism, and fluid dynamics. Journal of Mechanics of Materials and Structures, 6(1-4), 113–125.
Jiménez-Alfaro, S., Villalba, V., & Mantič, V. (2020). Singular elastic solutions in corners with spring boundary conditions under anti-plane shear. International Journal of Fracture, 223(1-2), 197–220. doi: 10.1007/s10704-020-00443-5.
Linkov, A., & Rybarska-Rusinek, L. (2008). Numerical methods and models for anti-plane strain of a system with a thin elastic wedge. Archive of Applied Mechanics, 78(10), 821–831.
Makhorkin, M., & Makhorkina, T. (2017). Analytical determination of the order of stress field singularity in some configurations of multiwedge systems for the case of antiplane deformation. Econtechmod. An international quarterly journal, 6(3), 45–52.
Makhorkin, M. I., Skrypochka, T. A., & Torskyy, A. R. (2020). The stress singularity order in a composite wedge of functionally graded materials under antiplane deformation. Mathematical modeling and computing, 7(1), 3–47. doi: 10.23939/mmc2020.01.039.
Makhorkin, .M., & Sulym, H. (2010). On determination of the stress-strain state of a multi-wedge system with thin radial defects under antiplane deformation. Civil and environmental engineering reports, 5, 235–251.
Pageau, S. S., Josef, P. F., & Bigger, S. B. (1994). The order of stress singularities for bonded and disbonded three–material junctions. Int. J. Solids Struct, 31(21), 2979–2997.
Picu, C. R., & Gupta, V. (1996). Stress singularities at triple junctions with freely sliding grains. Int. J. Solids Struct, 33(11), 1535–1541.
Savruk, M., & Kazberuk, A. (2016). Stress concentration at notches. Springer International Publishing, AG.
Shahani, A. R. (2006). Mode III stress intensity factors in an interfacial crack in dissimilar bonded materials. Arch. Appl. Mech. (Ing. Ar.), 75(6 7), 405–411.
Shahani, A. R., & Adibnazari, S. (2000). Analysis of perfectly bonded wedges and bonded wedges with an interfacial crack under antiplane shear loading. Int. J. Solids Struct., 37(19), 2639–2650.
Wieghardt, K. (1907). Über das Spalten und Zerreissen elastischer Körper. Z. Math. Phys, 55, 60–103.
