 # Differential equations of equilibrium of continuous medium for plane one-dimensional deformation at closing equations approximation by biquadratic functions

## Abstract

Problems of differential equations construction of equilibrium of a geometrically and physically nonlinear continuous medium under conditions of one-dimensional plane deformation are considered, when the diagrams of volumetric and shear deformation are approximated by quadratic functions. The construction of physical dependencies is based on calculating the secant moduli of volumetric and shear deformation. When approximating the graphs of the volumetric and shear deformation diagrams using two segments of parabolas, the secant shear modulus in the first segment is a linear function of the intensity of shear deformations, the secant modulus of volumetric expansion - contraction is a linear function of the first invariant of the strain tensor. In the second section of the diagrams of both volumetric and shear deformation, the secant shear modulus is a fractional (rational) function of the shear strain intensity, the secant modulus of volumetric expansion - compression is a fractional (rational) function of the first invariant of the strain tensor. Based on the assumption of independence, generally speaking, from each other of the volumetric and shear deformation diagrams, six main cases of physical dependences are considered, depending on the relative position of the break points of the graphs of the diagrams volumetric and shear deformation, each approximated by two parabolas. The differential equations of equilibrium in displacements constructed in the article can be applied in determining the stressed and deformed state of a continuous medium under conditions of one-dimensional plane deformation, the closing equations of physical relations for which, constructed on the basis of experimental data, are approximated by biquadratic functions.

## About the authors

### Sergey V. Bakushev

Penza State University of Architecture and Construction

Author for correspondence.
Email: bakuchsv@mail.ru

Professor of the Department of Mechanics, Dr. Sci. Tech.

28 Germana Titova St, Penza, 440028, Russian Federation

## References

1. Bakushev S.V. Approximations of warp diagrams using bilinear functions. Structural Mechanics and Structural Analysis. 2019;2(283):2-11. (In Russ.)
2. Bakushev S.V. Approximation of deformation diagrams by quadratic functions. Structural Mechanics and Structural Analysis. 2020;3(290):2-14. (In Russ.)
3. Bakushev S.V. Differencial'nye uravneniya i kraevye zadachi mekhaniki deformiruemogo tvyordogo tela [Differential equations and boundary problems in the mechanics of a deformable solid]. Moscow: LENAND Publ.; 2020. (In Russ.)
4. Novozhilov V.V. Teoriya uprugosti [The theory of elasticity]. Leningrad: Sudpromgiz Publ.; 1958. (In Russ.)
5. Lyapichev Y.P. Choice of mathematic models of soils in static and seismic analyses of embankment dams. Structural mechanics of engineering constructions and buildings. 2020;16(4):261–270. https://doi.org/10.22363/1815-5235-2020-16-4-261-270 (In Russ.)
6. Duncan J.M., Chang Y.Y. Nonlinear analysis of stress and strain in soils. Journal of the Soil Mechanics and Foundations Division. 1970;96(5):1629–1653.
7. Mroz Z., Norris V., Zienkiewicz O. Anisotropic hardening model for soils and its application to cyclic loading. Int. J. Num. & Anal. Methods in Geomechanics. 1978;2:203–221.
8. Roscoe K.H., Burland J.B. On the generalized stress-strain behaviour of ‘wet clay'. In: Heyman J., Leckie F.A. (eds.) Engineering Plasticity (p. 535–609). Cambridge: Cambridge University Press; 1968.
9. Prevost J.H. Anisotropic undrained stress-strain behavior of clays. Journal of the Geotechnical Engineering Division. 1978;104(8):1075–1090.
10. Aziz H.Y., Maula B.H. Estimation of negative skin friction in deep pile foundation using the practical and theoretically approaches. Journal of Engineering and Applied Sciences. 2018;13(10):3340–3349.
11. Miller T., Lee C. Novel pile design for multi-level car park above twin rail tunnels. Australian Geomechanics. 2017;52(44):15–27.
12. Zhang Y. Estimation of Free Penetration of Steel Pile and Numerical Simulation. Ship Building of China. 2017;(58):547–556.
13. Egorov V.V., Abu-Khasan M.S., Isachenkova K.E. Consideration of nonlinearity in the calculations of building structures. BST: Byulleten' stroitel'noj tehniki [BEB: Building Equipment Bulletin]. 2020;4(1028):62-64. (In Russ.)
14. Shutov V.A., Mirenkov V.E. Deformable solid body and postulates mechanics. Creativity and modernity. 2019; 1(9):125-128. (In Russ.)
15. Perelmuter A.V., Tur V.V. Whether we are ready to proceed to a nonlinear analysis at designing? International Journal of Civil and Construction. 2017;13(3):86-102. (In Russ.)
16. Protosenya A.G., Iovlev G.A. Stress-strain state prediction surrounding underground structure, constructed in nonlinear deformed medium-soft soils. Izvestiya Tula State University. Earth Sciences. 2020;(2):215–228. (In Russ.)
17. Fedorovsky V.G., Bobyr G.A., Bokov I.A., Iliyn S.V. Application of finite element method to the geotechnical ULS analysis. Bulletin of SRC “Stroitelstvo”. 2019;1(20):102–112. (In Russ.)
18. Khristich D.V., Astapov Y.V., Artyukh E.V., Sokolova M.Y. Numerical modeling of stresses in the massif of a clay soil under the foundation. Izvestiya Tula State University. Earth Sciences. 2019;(4):312-319. (In Russ.)
19. Bukotas G., Kačianauskas R. Analysis of axisymmetric bore-type foundation in respect of plastic deformation. Journal of Civil Engineering and Management. 1997;3(10):24–31. doi: 10.3846/13921525.1997.10531680.
20. Kositsyn S.B., Than Huan Linh. The analysis of the stress-strain state of the intersecting cylindrical shells for the elasto-plastic deformations with a view of geometrical nonlinearity. Structural Mechanics of Engineering Constructions and Buildings. 2013;(1):3-9. (In Russ.)
21. Agapov V.P., Vasilev A.V. Account for geometrical nonlinearity in the analysis of reinforced concrete columns of rectangular section by finite element method. Vestnik MGSU (Monthly Journal on Construction and Architecture). 2014; (4):7-43. (In Russ.)
22. Agapov V.P., Aidemirov K.R. Application of finite element method taking into account physical and geometric nonlinearity for the calculation of prestressed reinforced concrete beams. Herald of Dagestan State Technical University. Technical Sciences. 2017;44(1):127-137. doi: 10.21822/2073-6185-2017-44-1-127-137. (In Russ.)
23. Sokolov S.A., Kachaun A.N., Skudalov P.O., Cheremnykh S.V. Analysis of the “Molodechno” type truss beam nodes, taking into account the physical and geometric nonlinearity. Vestnik of Tver State Technical University. Series: Building. Electrical Engineering and Chemical Technology. 2019;2(2):36-42. (In Russ.)
24. Bahaz A., Amara S., Jaspart J.P., Demonceau J.F. Analysis of the Behaviour of Semi Rigid Steel End Plate Connections. MATEC Web of Conferences. 2018;149:02058. doi: 10.1051/matecconf/201814902058.
25. Popov O.N., Malinovskii A.P., Moiseenko M.O., Treputneva T.A. Current state of stress-strain analysis of inhomogeneous construction members strained beyond the elastic limit. Vestnik Tomskogo gosudarstvennogo arkhitekturno-stroitel'nogo universiteta. Journal of Construction and Architecture. 2013;4(41):127–142. (In Russ.) Copyright (c) 2020 Bakushev S.V. 