Defining equations of deformation of materials with double anisotropy

The mechanical properties of composite and polymer materials widely used in engineering are analyzed. It is confirmed that the absolute majority of them have structural anisotropy of different classes. In addition, it is shown that these structural materials often exhibit a sensitivity of the deformation characteristics to the type of stress state. Due to the fact that classical mathematical models describing the states of such materials lead to gross errors in the calculation of structural elements, and the well-known, specially developed theories for them are quite contradictory and have significant drawbacks, the authors propose an energy model of the determining relations for media with structural and deformation anisotropies. This model is
based on the use of the normalized stress tensor space, which has an undoubted advantage over the singular functions and parameters having an infinite interval of change, which are used in the known versions of the theories of deformation of materials with double anisotropy. As a specific class of structural anisotropy, orthotropic materials are accepted, for which the strain potential defined in the main structural axes is postulated. By differentiating the formulated potential according to the recommendations of the Castigliano rules, the equations of connection of two tensors of the second rank - strains and stresses - are established. It is shown that these equations
have a nonlinear form, which aggravates the problem of uniqueness of solutions to boundary value problems. To identify the resulting model of the defining equations, we recommend an
experimental program that includes mechanical tests for uniaxial tension and compression along the main axes of the anisotropy of the material, as well as for a net shift in the three planes
of orthotropy. The main technical constants of a number of composite and polymer materials widely used in engineering are given. On the basis of the use of the postulate about the positive
certainty of the energy surface, the consistency of the proposed strain potential is verified. Using this test, we prove the uniqueness theorem for solving boundary value problems in the mechanics of a deformable solid. Taking into account the rules of transformation of the components of the second-rank tensors when the axes of the selected coordinate system are rotated, it is shown that the stresses calculated in the main axes of orthotropy are recalculated in the new system according to traditional formulas.

1. Treschev A. A. 2020, Theory of deformation and strength of different resistant materials. Tula: TulSU. 359 p.

2. Schmueser D. W. 1983, «Nonlinear Stress-Strain and Strength Response of Axisymmetric Bimodulus Composite Material Shells», AIAA Journal, Vol. 21. № 12. рр. 1742–1747.

3. Reddy L. N., Bert C. W. 1982, «On the Behavior of Plates Laminated of Bimodulus Composite Materials», ZAMM, Vol. 62. № 6. рр. 213–219.

4. Jones R. M. 1977, «A Nonsymmetric Compliance Matrix Approach to Nonlinear Multimodulus Ortotropic Materials», AIAA Journal, Vol. 15. № 10. рр. 1436–1443.

5. Jones R. M. 1980, «Modeling Nonlinear Deformation of Carbon-Carbon Composite Material», AIAA Journal, Vol. 18. № 8. рр. 995–1001.

6. Jones R. M. 1971, «Bucling of Stiffened Multilayered Circular Shells with Different Ortotropic Moduli in Tension and Compression», AIAA Journal, Vol. 9. № 5. рр. 917–923.

7. Kregers A. F., Maksimov R. D., Turtsinysh R. P. 1973, «Nonlinear creep of fabric fiberglass under some types of complex stress state», Mechanics of polymers, № 2. pp. 212–218.

8. Amelina E. V. 2015, «On nonlinear deformation of carbon fiber plastics: experiment, model, calculation», IVT SB RAS: Computational Technologies, Vol. 20. № 5. pp. 27–52.

9. Kayumov R. A., Lukankin S. A., Paimushin V. N., Kholmogorov S. A. 2015, «Identification of mechanical characteristics of fiber-reinforced composites», Scientific Notes of the Kazan University. Physical and mathematical sciences, Vol. 157. Book 4. pp. 112–132.

10. Shafigullin L. N., Bobrishev A. A., Erofeev V. T., Treshchev A. A., Shafig-ullina A. N. 2015, «Development of the recommendations on selection of glass-fiber reiforced polyurethanes for vehicle parts», International Journal of Applied Engineering Research, Vol. 10. № 23. рр. 43758–43762.

11. Green A., Adkins J. 1965, Large elastic deformations and nonlinear mechanics of a continuous medium, M.: Mir. 456 p.

12. Treschev A. A., Bobrishev A. A., Shafigullin L. N. 2019, «Constitutive rela-tions for isotropic materials allowing quasilinearapproximation of the defor-mation law», IOP Conference Series: Materials Science and Engineering, Vol. 481. UNSP012014. Doi: 10.1088/ 1757-899X/481/1/

13. рр. 1–7.

14. Treschev A. A., Romashin D. A. 2011, «Defining relations for nonlinear anisotropic materials sensitive to the type of stress state», Vestnik Nizhegorodskogo university, N.Novgorod: Publishing House of the Lobachevsky National Research University. № 4. Part 4. pp. 1740–1742.

15. Treschev A. A. 2017, «Potential dependence between deformations and stresses for orthotropic physically nonlinear materials», Fundamental and applied Problems of Engineering and Technology, № 4-1 (324). pp. 71–74.

16. Kauderer G. 1961, Nonlinear mechanics, Moscow: Publishing house of foreign literature. 779 p.

17. Tursunov B. S. 1970, «On the properties of the stress potential of elastic bod-ies», AMM, Vol. 34. Issue 1. pp. 15–22.

18. Rose A. V., Zhigun I. G., Dushin M. N. 1970, «Three-reinforced woven mate-rials», Mechanics of polymers, № 3. pp. 471–476.

19. Jones R. M., Nelson D.A.R. 1976, «Theoretical-experimental correlation of material models for non-linear deformation of graphite», AIAA Journal, Vol. 14. № 10. pp. 1427–1435.

20. Jones R. M. 1977, «Stress-Strain Relations for Materials with Different Moduli in Tension and Compression», AIAA Journal. Vol. 15. № 1. pp. 16–25.

21. Zolochevsky A. A., Kuznetsov V. N. 1989, «Calculation of anisotropic shells made of differentmodulus materials under non-axisymmetric loading», Dynamics and strength of heavy machines. Dnepropetrovsk: DSU. pp. 84–92.