Asymptotic method in two-dimensional problems of electroelasticity

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Authors:

A.Shporta, orcid.org/0000-0002-1260-7358, Dnipro University of Technology, Dnipro, Ukrainе, 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.

T.Kagadiy, Dr. Sc. (Phys.-Math.), Assoc. Prof., orcid.org/0000-0001-6116-4971, Dnipro University of Technology, Dnipro, Ukrainе, 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.

O.Onopriienko, orcid.org/0000-0002-3127-4616, Dnipro State Agrarian and Economic University, Dnipro, Ukrainе, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 2020, (1):130-134
https://doi.org/10.33271/nvngu/2020-1/130

 повний текст / full article



Abstract:

Purpose. Generalization of the asymptotic method for solving two-dimensional problems of electroelasticity. Accounting for electric charges arising from deformation on the surfaces of piezoelectric materials. Checking the possibility of taking into account the magnetic field and the opposite effect when exposed to an electric field.

Methodology. The mathematical model of the piezoelectric material is described using the equilibrium equations, the electroelastic state, and the Cauchy relations. A small parameter is introduced as a ratio of the physical characteristics of the material. Transformations of coordinates and desired functions depending on the specified parameter are proposed.

Findings. The introduction of these transformations allowed splitting the initial boundary-value problem into two components with different properties. Each of them contains both mechanical and electrical components. The solution is sought as a superposition of solutions of both types. Each of the types of stress-strain states contains the main function and an auxiliary one. The expansion of the desired functions in rows by parameter eand the construction of asymptotic sequences lead to the fact that in each approximation the main functions are sought from the Laplace or Poisson equations. Auxiliary ones are found by integration. The analysis of the boundary conditions is given. It is shown that they can almost always be formulated for basic functions.

Originality. The method proposed earlier by the authors for reducing the boundary value problems of linear and nonlinear elasticity theory to the sequential solution of potential theory problems is generalized for the case of modern piezoelectric materials described by electroelasticity equations.

Practical value. With the help of the proposed approach, analytical solutions of practically important problems of electroelasticity can be obtained; estimates of the stress-strain state of products from piezoelectric materials are carried out. The results can be used as null approximations in numerical calculations.

References.

1. Kaloerov, S. A., & Samodurov, A. A. (2014). The problem of electro-elasticity for multiply-connected plates. Mathematical Methods and Physicomechanical Fields, (3), 62-77.

2. Wan, Y., Yue, Y., & Zhong, Z. (2012). Multilayered piezomagnetic/piezoelectric composite with periodic interface cracks under magnetic or electric field. Eng. Fract. Mech., 84, 132-145.

3. Onopriienko, O., & Loboda, V. (2015). Electrical crack between dissimilar piezoelectric materials with contacting faces. Journal of Zaporizhzhya National University of Physical and Mathematical Sciences, (1), 117-127.

4. Wang, X., & Zhou, K. (2017). A crack with surface effects in a piezoelectric material. Mathematics and Mechanics of Solids, 3-19.

5. Huang, C., & Hu, C. (2013). Three-dimensional analyses of stress singularities at the vertex of a piezoelectric wedge. Applied Mathematical Modelling, 37, 4517-4537.

6. Yang, Y., Pang, J., Dai, H.-L., Xu, X.-M., Li, X.-Q., & Mei, C. (2019). Prediction of the tensile strength of polymer composites filled with aligned short fibers. Journal of Reinforced Plastics and Composites, 38(14), 658-668.https://doi.org/10.1177/0731684419839223.

7. Kirilyuk, V. S., Levchuk, O. I., & Gavrilenko, E. V. (2017). Mathematical modeling and analysis of the stress state in an orthotropic piezoelectric medium with a circular crack. Systematic information and information technologies, (3), 117-126. https://doi.org/10.20535/SRIT.2308-8893.2017.3.11.

8. Kaloerov, S., & Glushankov, E. (2018). Determining the Thermo-Electro-Magneto-Elastic State of Multiply Connected Piecewise-Homogeneous Piezoelectric Plates. Journal of Applied Mechanics and Technical Physics, 59, 1036-1048. https://doi.org/10.1134/S0021894418060093.

9. Jie Su, Liao-Liang Ke, & Yue-Sheng Wang (2016). Axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space under a conducting punch. International Journal of Solids and Structures, 90, 45-59. https://doi.org/10.1016/j.ijsolstr.2016.04.011.

10. Kagadiy, T. S., & Shporta, A. H. (2015). The asymptotic method in problems of the linear and nonlinear elasticity theory. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, (3), 76-81.

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ISSN (print) 2071-2227,
ISSN (online) 2223-2362.
Journal was registered by Ministry of Justice of Ukraine.
Registration number КВ No.17742-6592PR dated April 27, 2011.

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