The stress singularity order in a composite wedge of functionally graded materials under antiplane deformation

2020;
: pp. 39–47
https://doi.org/10.23939/mmc2020.01.039
Received: October 22, 2019
Revised: January 27, 2020
Accepted: January 28, 2020
1
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine
2
Lviv National Agrarian University
3
Centre of Mathematical Modelling of IAPMM NASU named after Ya. S. Pidstryhach

In this paper, finding the order of singularity in multi-wedge systems containing elements made of functionally gradient material (FGM) with an angular gradient under antiplane deformation is studied.  These elements are proposed to be modeled by means of multi-wedge composite, where the shear modulus changes from wedge to wedge according to a certain functional dependence (in this article we consider the linear, quadratic, and exponential dependencies).  It is found that the model region composed of 20 elements provides a relative error in the calculation of the stress field singularity order, which does not exceed 5%.  Using the simulation of FGM by a multi-wedge system, the influence of an insert made of functionally graded material with an angular gradient on the singularity order in a three-component composite wedge has been studied. A number of regularities have been established.

  1. Wieghardt K.  Über das Spalten und zerreissen elastischer Körper.  Z. Math. Phys. 55, 60–103 (1907).
  2. Williams M. L.  Stress singularities resulting from various boundary conditions in angular corners of plates in extension.  Journal of Applied Mechanics. 19 (4), 526–528 (1952).
  3. Savruk M., Kazberuk A.  Stress concentration at notches.  Springer International Publishing AG (2016).
  4. Paggi M., Carpinteri A., Orta R.  A mathematical analogy and a unified asymptotic formulation for singular elastic and electromagnetic fields at multimaterial wedges.  Journal of Elasticity. 99 (2), 131–146 (2010).
  5. Makhorkin M., Makhorkina T.  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 (2017).
  6. Makhorkin M., Sulym H.  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 (2010).
  7. Carpinteri A., Paggi M.  On the asymptotic stress field in angularly nonhomogeneous materials.  Int. J. Fract. 135 (4), 267–283 (2005).
  8. Carpinteri A. Paggi M.  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 (2011).
  9. Fedorov A. Yu., Matveenko V. P.  Investigation of stress behavior in the vicinity of singular points of elastic bodies made of functionally graded materials.  J. Appl. Mech. 85 (6), 061008-1–061008-13 (2018).
  10. Hu X. F., Yao W. A., Yang S. T.  A symplectic analytical singular element for steady-state thermal conduction with singularities in anisotropic material.  J. of heat transfer. 140 (9), 091301-1–091301-13 (2018).
  11. Xiaofei H.  Stress singularity analysis of multi-material wedges under antiplane deformation.  Acta Mech. Solida Sinica. 26 (2), 151–160 (2013).
  12. Tranter C. J.  The use of the Mellin transform in finding the stress distribution in an infinite wedge.  Quarterly Journal of Mechanics and Applied Mathematics. 1, 125–130 (1948).
  13. Marur P. R., Tippur H. V.  Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient.  Int. J. Solids Struct. 37, 5353–5370 (2000).
  14. Linkov A., Rybarska-Rusinek L.  Evaluation of stress concentration in multi-wedge systems with functionally graded wedges.  International Journal of Engineering Science. 61, 87–93 (2012).
  15. Linkov A. M., Koshelev V. F.  Multi-wedge points and multi-wedge elements in computational mechanics: evaluation of exponents and angular distribution.  Int. J. Solids and Structures. 43, 5909–5930 (2006).
  16. Tikhomirov V. V.  Stress singularity in a top of composite wedge with internal functionally graded material.  St. Petersburg Polytechnical University J.: Physics and Mathematics. 1 (3), 278–286 (2015).
  17. Lomakyn V. A.  Theory of elasticity of inhomogeneous bodies.  Moscow, МGU (1976), (in Russian).
  18. Parton V., Perlin P.  Mathematical methods of the theory of elasticity. Two volumes. Mir (1984).
  19. Makhorkin M. I., Skrypochka T. A.  Stress singularity in a multiwedge system with interconnected elastic characteristics of its elements, under antiplane deformation.  Visnyk of Zaporizhzhya National University. Physical and mathematical Sciences. 2, 170–179 (2017), (in Ukrainian).
  20. Makhorkin M.  Effect of a wedge type insert of the functionally graded materialon the stress singularity in a composite wedge structure under antiplane deformation.  Applied problems of mechanics and mathematics. Scientific proceeding. 16, 112–118 (2018), (in Ukrainian).
Mathematical Modeling and Computing, Vol. 7, No. 1, pp. 39–47 (2020)