Determination of stress in composite plates with cracks on the basis of the method of integral equations and solutions by Green

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Authors:

O.V.Maksymovych, Dr. Sc. (Tech.), Prof., orcid.org/0000-0002-2892-7735, University of Science and Technology, Bydgoszcz, Poland, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

A.R.Dzyubyk, Cand. Sc. (Tech.), Assoc. Prof., orcid.org/0000-0003-2091-171X, Lviv Polytechnic National University, Lviv, Ukraine, e‑mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

Kh.A.Barvinska, Cand. Sc. (Tech.), orcid.org/0000-0003-3464-3728, Lviv Polytechnic National University, Lviv, Ukraine, e‑mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

L.V.Dzyubyk, Cand. Sc. (Tech.), orcid.org/0000-0001-6942-9436, Lviv Polytechnic National University, Lviv, Ukraine, e‑mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

Abstract:

Purpose. To construct singular integral equations based on solutions of the Green problem and to develop a method for calculating stresses in the vicinity of cracks in anisotropic plates of complex form.

Methodology. The investigation was carried out using the theory of elasticity of anisotropic bodies and the method of integral equations. The numerical algorithm for solving the constructed equations was based on Lobatto quadrature formulas.

Findings. The method of determination of stresses near cracks in plates of complex form was developed with the help of the apparatus of integral equations. The kernel of the equations were constructed on Green’s solutions, which makes it possible to satisfy automatically the boundary conditions on the given contours. Green’s solutions were presented and the integral equations were constructed on their basis for the half-plane with a free or rigidly fixed boundary and plates with an elliptical hole, which are weakened by cracks. Also the investigation of stress intensity factors was performed for a system of cracks placed near a straight-line boundary in a half-plane or near an elliptical hole in a plate.

Originality. In the article a method for calculating the stress-strain state in anisotropic plates of complex form based on a device of singular integral equations was proposed, the kernels of which are constructed on Green’s solutions. Its efficiency is illustrated while considering the plates of various forms with cracks.

Practical value. The results of the research presented in this work can be used in calculations of the strength of composite plates of complex form with cracks. The results of the performed calculations give an opportunity to predict the limit equilibrium of composite plates with cracks located close to its straight-line and elliptic boundaries.

References.

1. Tran, H. D., & Mear, M. E. (2018). Calculation of T-stress for cracks in two-dimensional anisotropic elastic media by boundary integral equation method. International Journal of Fracture, 211(1-2), 149-162.

2. Ang, W. T. (2014). Hypersingular integral equations in fracture analysis. Elsevier.

3. Zhang, J., Lin, W., & Dong, Y. (2019). A dual interpolation boundary face method for elasticity problems. European Journal of Mechanics-A/Solids, 73, 500-511.

4. Bej, M., Hota, M. K., & Mohanty, P. K. (2016). On the Approximate Evaluation of Real Singular and Strongly Singular Integrals. Applied Mathematical Sciences, 10(3), 119-126.

5. Savruk, M. P., & Kazberuk, A. (2016). Stress Concentration at Notches. Springer.

6. Kryvyy, O. F. (2014). Tunnel Internal Crack in a Piecewise Homogeneous Anisotropic Space. Journal of Mathematical Sciences, 52(1), 89-98.

7. Savruk, M. P., & Chornen’kyi, А. V. (2015). Plane Problem of the Theory of Elasticity for a Quasiorthotropic Body with Cracks. Materials Science, 51(3), 311-321.

8. Maksymovych, O., Pasternak, Ia., Sulym, H., & Kutsyk, S. (2014). Doubly periodic cracks in the anisotropic medium with the account of contact of their faces. Acta mechanica et automatic, 8(3), 160-164.

9. Maksymovych, O., & Illiushyn, O. (2017). Stress calculation and optimization in composite plates with holes based on the modified integral equation method. Engineering Analysis with Boundary Elements, 83, 180-187.

10. Savruk, M. P., & Kazberuk, А. (2014). Curvilinear Cracks in the Anisotropic Plane and the Limit Transition to the Degenerate Material. Materials Science, 50(2), 189-200.

11. Dzyubyk, A. R., Nykolyshyn, T. M., & Porokhovs’kyi, Y. V. (2016). Influence of Residual Stresses on the Limit Equilibrium of a Pipeline with Internal Crack of Arbitrary Configuration. Materials Science, 52(1), 89-98.

12. Dreus, A., Коzhevnikov, А., Sudakov, А., & Lysenko, K. (2016). Investigation of heating of the drilling bits and definition of the energy efficient drilling modes. Eastern-European Journal of Enterprise Technologies, 3(7(81)), 41-46.

13. Murakami, Yu. (Ed.) (1990). Stress intensity factor refe­rence book. Vol. 1. Moscow: Mir.

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ISSN (print) 2071-2227,
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