#### 539.3 Modeling the bending of beams made of rubber-like materials

##### Firsanov V. V. (Moscow Aviation Institute (National Research University))

###### INCOMPRESSIBILITY, BENDING, ELASTICITY, MODEL, HYPOTHESES, DEFORMATIONS, DISPLACEMENTS, STRESSES, LOAD, BOUNDARY CONDITIONS

doi: 10.18698/2309-3684-2021-4-316

Since the classical hypotheses of Bernoulli for beams and Kirchhoff for thin plates contradict the additional condition of incompressibility for rubber-like (incompressible) materials (invariability of the volume during deformation), a calculation model for a bending beam is proposed, which does not lead to a serious complication of the problem in comparison with the classical solution. The invariability of the volume is manifested under the action of a power load; in the case of a temperature load, the deformation of the volume change is not zero. The absence of volumetric deformations for rubber-like materials is a consequence of Hooke's law for such materials. Summing the linear deformations expressed in terms of stresses and taking Poisson's ratio 0.5, we obtain the equality of the indicated sum to zero. Many rubber-like materials are incompressible and low-modulus, which means their weak resistance to tension and shear, but the resistance of the material to change in volume tends to infinity, therefore the physical relations of the generalized Hooke's law are transformed into the so-called "neo- Hooke " equations of the relationship between stresses and strains. Of the two independent physical characteristics (modules) for incompressible materials, only one module remains, which characterizes the resistance of the medium to change in shape. In physical relations for an incompressible material, the product of an infinitely large volumetric modulus by the deformation of a change in volume equal to zero is an uncertainty that is replaced by some force function that has the dimension of stresses and is an additional unknown. At the same time, the system of governing equations of the mechanics of incompressible media is supplemented by the equation of invariability of volume. The scheme for solving the problem in displacements for traditional structural materials turns into a mixed scheme for rubber-like materials, since for them not only displacements but also the mentioned force function act as the main unknown sought function.

Timoshenko S.P., Vojnovskij-Kriger S., Plastiny i obolochki [Plates and Shells]. Moscow, Nauka Publ., 1966, 636 p.
Vasil'ev V.V., Lur'e S.A. K probleme postroeniya neklassicheskih teorij plastin [On the problem of constructing non-classical plate theories]. Mechanics of Solids, 1990, pp. 158–167.
Vasil'ev V.V., S.A. Lur'e S. A. Variant utochnennoj teorii izgiba balok iz sloistyh plastmass [A variant of the refined theory of bending a laminar plastic beam]. Polymer Mechanics, 1972, no. 4, pp. 577–768.
Carrera E., Giunta G., Petrolo M. Beam Structures: Classical and Advanced Theories. Wiley, 2011, 204 p.
Biderman V.L., Mart'yanova G.V. Variatsionnyy metod raschota detaley iz neszhimayemogo materiala [Variational method for calculating parts made of incompressible material]. Raschoty na prochnost' [Strength calculations],1977, iss. 18, pp. 3–27.
Dimitriyenko Yu.I., Gubareva E.A., Kol'zhanova D.Yu., Karimov S.B. Incompressible layered composites with finite deformations on the basis of the asymptotic averaging method. Маthematical Modeling and Coтputational Methods, 2017, no. 1, pp. 32–54.
Firsanov V.V. Bending beams made of a material with an unchangeable volume. Mekhanika kompozitsionnykh materialov i konstruktsii, 2020, vol. 26, no. 2, pp. 200–211.
Pobedrya B. E. Equations of state of viscoelastic isotropic media. Mechanics of Composite Materials, 1967, no. 3, 427 p.
Treloar L.R.G. The Physics of Rubber Elasticity. OUP Oxford, 2005, 324 p.
Herakovich C.T. A Concise introduction to elastic solids: an overview of the mechanics of elastic materials and structures. Springer, 2017, 136 p

Фирсанов В.В. Моделирование изгиба балок из резиноподобных материалов. Математическое моделирование и численные методы, 2021, № 4, с. 3–16.