Differential Equations Of Equilibrium Research Articles

A new cracking model for concrete ties reinforced with bars having different diameters and bond laws

This paper presents a model to predict the cracking behavior of concrete ties, reinforced with ribbed and plain bars having different diameters, subjected to a statically applied axial force. The model is based on linear elastic stress-strain (σ-ε) laws for both types of bars as well as for concrete and takes into account the effects of curing on concrete mechanical properties, as well as the effects of tensile creep and concrete restrained shrinkage strains. Different slip-bond relationships for ribbed and plain bars are considered. The non-linear differential equilibrium equations of the ties are solved numerically. Model accuracy has been checked for predictions of the crack widths and spacings, considering 52 experimental data. The proposed model is then applied to the concrete footing of the NPS®Cls, in order to show the cracking behavior of this element at the start-up.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ ИДЕАЛЬНО УПРУГОПЛАСТИЧЕСКОЙ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ДЕФОРМАЦИИ В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХ ПРИ БИЛИНЕЙНОЙ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ

Abstract. The article considers the construction of differential equations of equilibrium in displacements for plane deformation of elastic perfectly plastic regarding shear deformations continuous medium and nonlinearly elastic continuous medium with respect to volumetric deformations with bilinear approximation of the closing equations, both regarding and regardless geometrical nonlinearity in a cylindrical coordinate system. Nonlinear diagrams of volumetric and shear deformation are approximated by bilinear functions. Proceeding from the assumption of independence, generally speaking, of volume and shear deformation from each other, five main cases of physical dependencies are considered, depending on the relative position of the break points of bilinear diagrams of volume and shear deformation. The construction of bilinear physical dependencies is based on the calculation of the secant moduli of volumetric and shear deformation. In this case, in the first section of the diagrams, the secant modulus of both volumetric and shear deformation is constant, while in the second section of the diagrams, the secant modulus of volumetric deformation is a function of volumetric deformation, and the secant shear modulus is a function of the intensity of shear deformations. Substituting the corresponding bilinear physical relations into the differential equations of equilibrium of a continuous medium, written both regardless and regarding geometrical nonlinearity, the resolving differential equations of equilibrium in displacements for plane deformation in a cylindrical coordinate system are received. The received differential equations of equilibrium in displacements in cylindrical coordinates can be applied in determining the stress-strain state of elastic perfectly plastic with respect to shear deformations continuous medium and nonlinearly elastic with respect to volumetric deformations continuous medium under conditions of plane deformation, both regarding and regardless geometrical nonlinearity, physical relations for which are approximated by bilinear functions.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ ИДЕАЛЬНО УПРУГОПЛАСТИЧЕСКОЙ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ОДНОМЕРНОЙ ДЕФОРМАЦИИ ПРИ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ БИКВАДРАТ ИЧНЫМИ ФУНКЦИЯМИ

The present article considers the construction of differential equations of equilibrium of geometrically and physically nonlinear ideally elastoplastic in relation to shear deformations of continuous medium under conditions of one-dimensional plane deformation, when the diagrams of volumetric and shear deformation are approximated by biquadratic 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, five 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. On the basis of received physical equations, differential equations of equilibrium in displacements for continuous medium are derived under conditions of plane one-dimensional deformation. Differential equations of equilibrium in displacements constructed in the present article can be applied in determining stress and strain state of geometrically and physically nonlinear ideally elastoplastic in relation to shear deformations of continuous medium under conditions of plane one-dimensional deformation, closing equations of physical relations for which, based on experimental data, are approximated by biquadratic functions.

Повторное знакопеременное нагружение упругопластической трехслойной пластины в температурном поле

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

ОБОБЩЕННАЯ МОДЕЛЬ ДВУХКАТУШЕЧНОЙ СИНХРОННОЙ ЭЛЕКТРОМАГНИТНОЙ МАШИНЫ ДЛЯ ТЕХНОЛОГИЧЕСКИХ СИСТЕМ ВИБРОУДАРНОГО ДЕЙСТВИЯ

Актуальность исследования обусловлена тем, что силовые электрические импульсные системы, созданные на основе линейных электромагнитных машин ударного и виброударного действия, являются перспективными для применения в технологиях по интенсификации процессов, связанных с поиском, добычей и переработкой георесурсов. В вопросах энергосбережения особый интерес вызывают линейные синхронные электромагнитные машины возвратно-поступательного движения ударного и виброударного действия. Частота ударов бойка в данных машинах синхронно связана с частотой источника промышленной сети переменного тока. К главным достоинствам оборудования, использующего линейные синхронные электромагнитные машины ударного и виброударного действия, относится снижение потребления электрической энергии при работе в резонансных режимах или близких к резонансным режимам. Исследование данных режимов связано с построением точных динамических моделей различных модификаций синхронных электромагнитных машин ударного и виброударного действия, работающих исключительно в переходных режимах. Цель: разработать обобщенную математическую модель двухкатушечной синхронной машины с электромагнитным приводом для технологических систем виброударного действия, упрощающей процесс математического описания известных вариантов схем машин при проведении динамических расчетов. Методы: дифференциальные уравнения электрического равновесия и механического взаимодействия, поступательно движущихся масс, уравнение Лагранжа второго рода, методы конечно-элементного моделирования двухмерного магнитного поля и структурного моделирования в программе Matlab Simulink, сопоставление расчетных и экспериментальных характеристик. Результаты. Предложена обобщенная расчетная схема построения многомассовой двухкатушечной синхронной электромагнитной машины, включающая известные модификации вариантов схем машин ударного действия. На основе расчетной схемы образована система дифференциальных уравнений обобщенной модели, позволяющая выполнять построение математических моделей, отражающих динамическое состояние известных вариантов схем синхронных ударных машин с электромагнитным приводом. Приведен пример реализации динамической модели четырехмассовой двухкатушечной синхронной электромагнитной машины с упругим реверсом бойка. Приведены результаты имитационного моделирования рабочего режима в виде диаграмм тока катушек, перемещения и скорости бойка, отражающие процессы включения и выхода на квазиустановившийся режим работы. На основе сопоставления значений интегральных характеристик показано, что расхождение результатов расчета и эксперимента составляет 4…6 %, что не превышает общепринятой погрешности для инженерных расчетов. Эффективность применения предложенной обобщенной модели заключается в упрощении математического описания вариантов схем синхронных машин с электромагнитным приводом и сокращении сроков реализации моделей.

Free Vibration of Thick FGM Plates under TSDT and Thermal Environment

Three parameters of thermal environment, varied calculated shear correction, and third-order shear deformation theory (TSDT) of displacement are important in the frequency study. These three effects have been studied on the non-dimensional and dimensional frequencies of thick FGM plates. An additional c1 displacement term in nonlinear coefficient of TSDT is used to present the frequency of vibration into the simply homogeneous equation of thick FGM plates. The determinant of the coefficient matrix containing the c1 displacement term in dynamic differential equilibrium equations can be derived into the five degree polynomial free vibration equation. The non-dimensional and dimensional of natural frequency can be obtained. The effects of plate thickness, temperature of environment and power law index of FGM on the non-dimensional and dimensional frequency of FGM plates are investigated.

Minimum thrust and minimum thickness of spherical masonry domes: A semi-analytical approach

A semi-analytical approach is proposed for addressing the minimum thrust and minimum thickness analysis of spherical masonry domes under their self-weight. The classical differential equilibrium equations of axially symmetric shells are resorted to, calling for the exact computation of self-weight and center of mass of an infinitesimal dome voussoir. The minimum thrust analysis is accomplished by determining a statically admissible stress state in the dome, that is associated to a kinematically admissible settlement mechanism of minimum thrust. Such a stress state entails vanishing hoop stresses in the lower part of the dome, where the opening of meridional cracks occurs. An original numerical scheme is proposed for automatically computing a distribution of hoop stresses in the dome cap, producing a statically admissible stress state. A suitable modification of the semi-analytical procedure for the minimum thrust analysis is then adopted to perform the minimum thickness analysis. Numerical results are presented for spherical domes with parameterized oculus and embrace angles, complemented by the comparison with alternative equilibrium models, and with approximate treatments of the dome self-weight adopted in the literature. The merit of the proposed procedure is finally proved in application to a real case. As an iconic result, a refined estimate of the minimum thickness of a hemispherical dome is derived, resulting in 0.04284 times the radius of the mid-surface.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ДЕФОРМАЦИИ В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХ ПРИ БИЛИНЕЙНОЙ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ

Problems of the formulation of differential equations of equilibrium in terms of displacements for a plane strain of continuous media at bilinear approximation of closing equations are considered leaving out of account geometric nonlinearity in the cylindrical coordinate system. Based on the assumption that the curves of volumetric and shear strain are independent from each other, six main cases of physical dependencies are considered, which are the functions of the relative position of break points on the bilinear curves of the volumetric and shear strain. Obtaining of bilinear physical dependencies is based on the calculation of secant moduli of the volumetric and shear strain. On the first line of the curves, secant moduli are constant for both volumetric and shear strain, while on the second line, the secant modulus of the volumetric strain is a function of the volumetric strain, and the secant modulus of the shear strain is a function of the shear strain intensity. Putting the corresponding bilinear physical equations into differential equations of continuum equilibrium, which disregard geometrical nonlinearity, the resulting differential equations of equilibrium are obtained in terms of displacements for a one-dimensional plane strain of continuum in the cylindrical coordinate system. These equations can be used when determining stress-strain state of continuous media under one-dimensional plane strains with no regard for geometrical nonlinearity, and whose physical relations are approximated by bilinear functions.

ANALYTICAL SOLUTION OF THE PROBLEM ON THE THERMALLY STRESSED STATE OF FUNCTIONALLY GRADED PLATES BASED ON THE 3D ELASTICITY THEORY

Two approaches to studying the thermally stressed state of layered functionally graded plates in the 3D formulation have been developed. In the first approach, using the Reissner variational principle, a system of integrodifferential equilibrium equations and the corresponding boundary conditions are obtained without introducing an approximation. In the second approach, to construct differential equations of 3D thermoelastic equilibrium, a polynomial approximation of sought-for functions across the structure thickness is used. Its salient feature is assignment of the functions to the outer surfaces of layers, which allows splitting the layers into sublayers with a corresponding increase in the accuracy of calculation results. For the particular case of hinge-supported plates with a thermal load distributed according to a trigonometric law, the system of integrodifferential equilibrium equations obtained for the first approach and the differential equations of the second approach allow an analytical implementation. Then, equations of the first approach are transformed into a system of ordinary differential equations for the distribution of required functions across the plate thickness, with an analytical search for the roots of characteristic equations and the corresponding eigenvectors. In the second approach, the system of differential equations is converted to a system of algebraic equations. The assignment of sought-for functions to the outer surfaces of layers allows one to split the layers into sublayers and thus to reduce the approximation error. The same result obtained by the two methods may point to its reliability.

Solution of the problem of rotation of a homogeneous disk by the variational method, the apparatus of vector analysis and the finite element method

This work considers two global approaches to obtaining solutions to problems in mechanics: analytical and numerical, their division into various methods, allowing a more flexible approach to the choice of optimal methods for obtaining solutions to problems of continuum mechanics. Analyzed the effectiveness of various analytical methods for obtaining resolving continuum mechanics. In particular, the derivation of differential equations of equilibrium from the condition of minimality of the potential energy of elastic deformation of a solid, as well as the mathematical apparatus of vector analysis of classical field theory. In addition to the analytical solution, a numerical solution of this problem was presented by the finite element method, performed in the ANSYS Mechanical APDL software package, followed by verification of the obtained simulation results with the results of the analytical solution.

The Stress State in the Boundary Region of a Conical Shell according to a Refined Theory

In this article, on the basis of a refined theory, we study the stress state of a conical shell of the “boundary layer” type. In comparison with the classical theory, the sought displacements of the shell are approximated by polynomials along the coordinate normal to the median surface with a degree two units higher. Based on the equations of the three-dimensional elasticity theory and the Lagrange variational principle, a system of differential equations of equilibrium in displacements with variable coefficients is obtained. The solution of the formulated boundary value problem is carried out by using the methods of finite differences and matrix sweep. The results can be used when assessing the strength and durability of shell structures.

Static analysis of cutout microstructures incorporating the microstructure and surface effects

This article develops a nonclassical model to analyze bending response of squared perforated microbeams considering the coupled effect of microstructure and surface stress under different loading and boundary conditions, those are not be studied before. The corresponding material and geometrical characteristics of regularly squared perforated beams relative to fully filled beam are obtained analytically. The modified couple stress and the modified Gurtin-Murdoch surface elasticity models are adopted to incorporate the microstructure as well as the surface energy effects. The differential equations of equilibrium including the Poisson's effect are derived based on minimum potential energy. Exact closed form solution is obtained for bending behavior of the proposed model considering the classical and nonclassical boundary conditions for both uniformly distributed and concentrated loads. The proposed model is verified with results available in the literature. Influences of the microstructure length scale parameter, surface energy, beam thickness, boundary and loading conditions on the bending behavior of perforated microbeams are investigated. It is observed that microstructure and surface parameters are vital in investigation of the bending behavior of perforated microbeams. The obtained results are supportive for the design, analysis and manufacturing of perforated nanobeams that commonly used in nanoactuators, nanoswitches, MEMS and NEMS systems.

Stress distribution in a rubber-metal block under compression by static load

Abstract. The problem of determining the stress-strain state of a prismatic rubber-metal block when it is compressed by a static load is considered. A mathematical model is proposed, which assumes that when a block is compressed by a vertical load, its horizontal sections remain horizontal, the metal components of the block are rigid, non-deformable, points located on a vertical line during loading remain on a parabola, and rubber is considered incompressible. Accepted assumptions provided an opportunity to obtain expressions for movements along the axes of symmetry, which provide the expected parabolicity during compression of the rubber-metal block and satisfy the condition of incompressibility. The formulas for normal and tangential stresses arising during block compression are derived, which satisfy the differential equation of equilibrium of the elementary volume, Hooke’s law, and take into account the boundary conditions on the sides of the rubber body. Given the geometric parameters of the block and the mechanical characteristics of the rubber, it was possible to calculate its drawdown depending on the degree of static load, as well as to analyze the distribution of normal and tangential stresses that occur in the block. The developed mathematical model of the compression process allowed, as an example, to analyze the stress-strain state of a rubber-metal block of the BRM‑102 type, for the manufacture of which rubber of the brand 51‑1562 was used. As a result of solving the problem, distribution diagrams of normal, tangent, and total stresses arising in dangerous sections (on the upper or lower sides of the rubber body) under loading are constructed. It was found that the maximum normal stresses and maximum total arise on the vertical axis of symmetry of the block; maximum tangential stresses are realized at corner points. The ability to receive this type of illustration allows you to visually determine which is more rational to use: one block with a horizontal section of 200(100 mm or two blocks with a size of 100(100 mm. The proposed mathematical model allows us to study the stress-strain state of a rubber-metal block under compression, and can also be used in the design of a block with predetermined characteristics. Keywords: rubber-metal block, stress-strain state, displacements, stresses, relative deformation, mathematical model.

Dynamic Analysis of Axially Loaded Beams Partially Supported by an Elastic Foundation Under a Moving Harmonic Load

In this paper, a first-order shear deformable beam element is formulated and employed to study the dynamic behavior of beams partially resting on a Pasternak elastic foundation subjected to an axial force and a moving harmonic load. The beam element is derived by using the solution of differential equilibrium equations of a beam segment to interpolate the transverse displacement and rotation. The dynamic response of the beams is computed with the aid of the Newmark method. A parametric study is carried out to highlight the effects of the moving load parameters, the axial force amplitude, and the foundation support on the dynamic characteristics of the beams. The obtained result reveals that the axial force and the moving velocity have an important role in the dynamic behavior of the beams, and the influence of the axial force on the dynamic behavior of the beam is dependent on the excitation frequency.

Elastic-viscoplastic solution of a circular tunnel in Hoek-Brown rock mass

A closed-form analytical solution to rheological phenomena is obtained based on Hoek-Brown (HB) yield criterion and the generalized Bingham model. Once the circular tunnel is excavated, an initial plastic region is formed. When the total stresses fulfil the yield condition, the initial plastic region steps into the viscoplastic region, in which the displacement develops with time. The one-dimensional constitutive equation of the generalized Bingham model is transformed into a three-dimensional expression in polar coordinates by combining the plane strain condition. With the help of equilibrium differential equation, geometric equation and boundary condition, the stress and displacement in the initial plastic region can be expressed. At the moment of t, the stresses and displacements are described similar to the solutions of the elasto-plastic state. The stresses, strains and displacements after transformation can be expressed in the viscoplastic region. The proposed solution was verified by the closed-form solution of circular tunnels in elastic-brittle-plastic rock mass.

Free vibration of simply supported steel I-girders with trapezoidal web corrugations

Natural frequency is essential information required to perform the dynamic analysis. In this study, the straight I-girder element with trapezoidal web corrugations and singly symmetric cross-section is derived based on Kang and Yoo’s thin-walled curved beam with doubly symmetric cross-section theory. Each node of the element has seven degrees of freedom including one warping degree of freedom. Using Hamilton’s principle, governing linear dynamic differential equations of equilibrium, elastic element stiffness matrix, and consistent element mass matrix are established. Matlab code is employed for natural frequency analysis of simply supported steel I-girders with trapezoidal web corrugations. The results are compared with the natural frequencies obtained with ABAQUS. Finally, it is found that the proposed equations provide a good prediction of natural frequency for the first three lateral bending modes and the first five torsional modes.

Nonlocal elasticity theory for lateral stability analysis of tapered thin-walled nanobeams with axially varying materials

The lateral-torsional buckling behavior of functionally graded (FG) non-local beams with a tapered I-section is here investigated using an innovative methodology. The material properties are supposed to vary continuously along the longitudinal direction according to a homogenization procedure, based on a power-law function, whereas the nanobeam is modeled within the framework of a Vlasov thin-walled beam theory. The flexural-torsional governing equations of the problem are derived based on the Eringen's nonlocal elasticity theory and the energy method. The system of lateral stability equations is, thus, reduced to a fourth-order differential equation in terms of the twist angle by uncoupling the equilibrium differential equations. The buckling loads are finally determined using the differential quadrature method (DQM), which is here applied as numerical tool to solve directly the differential equations of the problem in a strong form. A systematic investigation checks for the influence of some parameters such as the power-law index, tapering ratios, loading height parameter, boundary conditions and non-local parameter, on the lateral stability resistance of the tapered I-nanobeams. The numerical outcomes of this paper can be used as benchmarks for further studies on nanoscale tapered thin-walled beams.

Three-point bending of beams with consideration of the shear effect

The subject of the paper pertains to simply supported beams with bisymmetrical cross sections under three-point bending with consideration of the shear effect. The deformation of a planar cross section of the beam is described taking into account the assumed nonlinear hypothesis-theory. Two differential equations of equilibrium are obtained based on the principle of stationary potential energy. This system is analytically solved and the shear coefficients and deflections of the beams are derived. Moreover, the Young's modules of the materials and deflections of the beams are experimentally determined on a test stand. The results of the studies are specified in tables and compared.

A size-dependent model for shear deformable laminated micro-nano plates based on couple stress theory

In this paper, a modified first order shear deformation theory is proposed for laminated micro-nano plates. Couple stress components are involved in the differential equations of equilibrium for shear deformable micro-nano plates, which are established by considering the deformation mechanism of an element cut from the micro-nano plate. Based on the mechanics of composite materials and Koiter’s theory, the constitutive equations for laminated micro-nano plates are established to characterize the relationships between Cauchy stress tensor and the strain tensor, the couple-stress tensor and the curvature tensor. Governing equations with size-dependent effect are derived and are expressed in terms of the displacements and the rotation functions. The proposed theory is applied to investigate the eigenvalue buckling and free vibration of laminated micro-nano plates simply supported at the four sides. Matlab program based on the analytical solutions is developed and is testified by comparing the solutions for isotropic micro-nano plates from the proposed theory against those from an available model and from ABAQUS analysis. Various examples are studied to investigate the influence of scale ratio, aspect ratio, length-to-thickness ratio, material properties and stacking sequence on the buckling and free vibration response of the laminated micro-nano plates. The size-dependent effect is captured for all examples, particularly for micro-nano isotropic laminated plate with geometry size similar to its material length scale parameter, where the buckling load may be nearly up to 10 times larger than that given by the FSDT theory. A good agreement among solutions from the Matlab program, the existing literature and ABAQUS analysis demonstrates the efficiency and accuracy of the proposed theory.

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

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.