### №1-2020

https://elibrary.ru/contents.asp?id=42642469

**Chernova P.D., Goloveshkin V.A., Myagkov N.N.**

## Model of interaction of a rigid mesh with a deformable target

An analytical model of the high-velocity interaction of a rigid mesh with a semi-infinite deformable target, which is modeled by a rigid-plastic body, is proposed. We consider the so-called “normal” impact of the mesh on the target: we assume that at the initial moment and subsequent moments of time the mesh is parallel to the target surface, and the mesh velocity vector is perpendicular to the target surface. The model reproduces the most interesting case when the mesh aperture is comparable to or less than the diameter of the wire from which the mesh is woven. The dependence of the mesh penetration depth on the impact velocity and the geometric parameters of the mesh, which are characterized by one dimensionless parameter equal to the ratio of the wire diameter to the mesh period, is studied. Two versions of the model are considered: with and without taking into accounts the fragmentation of the ejected material of the target. The results obtained on the basis of the proposed model are compared with the numerical solutions based on the complete system of equations of the deformable solid mechanics. Numerical simulations were performed using the LS-DYNA package. The example of the penetration of a steel mesh into an aluminum-alloy target with impact velocities of 1-3 km/s is analyzed. It is shown that the model taking into account fragmentation agrees well with the numerical simulations for the mesh parameter interval , which the lower boundary decreases with increasing impact velocity: for km/s, respectively.

**Yankovskii A.P.**

## Refined model of viscoelastic-plastic deformation of flexible shallow shells with spatial reinforcement structures

A model of viscoelastic-plastic deformation of spatially reinforced flexible shallow shells is developed. The instant elastoplastic behavior of the components of the composition is determined by the theory of plastic flow with isotropic hardening. The viscoelastic deformation of these materials is described by the equations of the Maxwell – Boltzmann model. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The obtained relations make it possible to determine with varying degrees of accuracy the displacements of shell points and the stress-strain state in the components of the composition (including residual ones). In this case, the weakened resistance of the composite structure to transverse shear is modeled. In a first approximation, the obtained equations and boundary conditions correspond to the traditional non-classical Reddy theory. The solution of the formulated problem is constructed numerically using an explicit «cross» type scheme. The viscoelastic-plastic dynamic behavior of composite cylindrical rectangular panels under the action of a load generated by an air blast wave is investigated. Designs have a «flat»-cross or spatial reinforcement structure. It has been demonstrated that in some cases, even for relatively thin composite curved panels, the Reddy theory is unacceptable for adequate calculations of their dynamic viscoelastic-plastic deformation. It is shown that the size and shape of the residual deflections of the reinforced gently shallow shells substantially depend on which of their front surfaces (concave or convex) is subjected to an external load. It was found that in both cases of loading residual longitudinal folds are formed in a thin cylindrical shallow composite shell. It has been demonstrated that even for a relatively thin panel, replacing a «flat»-cross reinforcement structure with a spatial reinforcement structure can significantly reduce the amount of residual deflection and the intensity of residual strain in the binder. In the case of relatively thick shallow shells, the effect of such a replacement of the reinforcement structures is manifested even more.

**Grishanina T.V., Rybkina N.M.**

## To the calculation of a straight high aspect-ratio wing in an incompressible flow using a nonstationary aerodynamic theory

Bending-torsional vibrations of a straight high aspect-ratio wing in an incompressible flow of an ideal gas are considered. Linear aerodynamic loads (lift and torque) are determined by non-stationary and quasi-stationary theories of cross sections flat flow. The displacements and twisting angles of the wing console cross sections during bending-torsional vibrations are represented by the Ritz method as an expansion for given functions with unknown coefficients, which are considered as generalized coordinates. The equations of aeroelastic wing oscillations are composed as Lagrange equations and written in matrix form as first-order differential equations. The problem of determining eigenvalues is solved on the basis of the obtained equations. The main purpose of this work is to compare calculations for determining the dynamic stability boundary (flutter) obtained using non-stationary and quasi-stationary aerodynamic theories. Calculations are performed for a wing model with constant cross-section characteristics. As the set functions eigenmodes of bending and torsional vibrations of a cantilever beam of constant cross-section were used. Calculations are performed to determine the flutter boundary for a different number of approximating functions. The results obtained allow us to conclude that when using the quasi-stationary and refined quasi-stationary theories for determining aerodynamic loads, the values of the critical flutter velocity are obtained less than when calculating using the non-stationary theory. This makes it possible to use a simpler (from the point of view of labor intensity) quasi-stationary theory to determine flutter boundaries. It is also found that the influence of the attached air masses, which is taken into account in the non-stationary and refined quasi-stationary theories, is very small.

**Starovoitov E.I., Zakharchuk Yu.V.**

## The physically nonlinear deformation of circular sandwich plates with a compressible filler

Three-layer structural elements are used in aerospace and transport engineering, construction, production and transportation of hydrocarbons. The theory of deformation of three-layer plates with incompressible fillers is currently developed under external force influences quite well. Here is the formulation of the boundary value problem on the bending of an elastoplastic circular three-layer plate with a compressible filler. For thin bearing layers, the Kirchhoff hypothesis is accepted. In a relatively thick lightweight filler, the hypothesis of Tymoshenko is performed with a linear approximation of radial displacements and deflection along the layer thickness. The work of shear stresses and compression stresses is assumed to be small and is not taken into account. The contour is assumed to have a rigid diaphragm that prevents the relative shift of the layers. The physical equations of state in the bearing layers correspond to the theory of small elastic-plastic deformations of Ilyushin. The filler is nonlinear elastic. The inhomogeneous system of ordinary nonlinear differential equations of equilibrium is obtained by the Lagrange variational method. Boundary conditions are formulated. The solution of the boundary value problem is reduced to finding the four desired functions – the deflection of the lower layer; shear, radial displacement and compression function in the filler. The method of successive approximations based on the method of elastic solutions is applied for the solution. The General iterative analytical solution of the boundary value problem in Bessel functions is obtained. Its parametric analysis is carried out at uniformly distributed load and rigid sealing of the plate contour. The influence of the compressibility of the filler on the stress-strain state of the plate is numerically investigated. The comparison of the calculated deflection values obtained by the traditional model with incompressible filler and in the case of its compression is given.

**Bobok D.I.**

## Analytical solution of the problem of bending of a round plate made of shape memory alloy

In this paper, we consider the problem of the solids mechanics about the bending of a circular plate made of a shape-memory alloy (SMA) during a direct thermoelastic martensitic phase transformation under the action of a constant in magnitude and uniformly distributed transverse load radius. The problem of relaxation in a similar plate during direct phase transformation has also been solved. As the second problem, a normal load uniformly distributed over the radius is applied to the plate surface in the austenitic phase state. Next, the plate material is cooled through the temperature range of direct thermoelastic martensitic transformation. It is required to determine how the uniformly distributed load should decrease during such a transition so that the deflection of the plate remains unchanged. During the work, rigidly and articulated plates were investigated. The solution was obtained in the framework of the Kirchhoff-Love hypotheses. To describe the behavior of the plate material, we used the well-known model of linear deformation of SMA during phase transformations. The solution was obtained under the assumption that the phase composition parameter at each moment of the process under consideration is uniformly distributed over the plate material, which corresponds to non-coupled statement of the problem for the case of uniform distribution of temperature over the material. The possibility of structural transformation in the plate material is not taken into account. It neglects the variability of the elastic moduli during the phase transition, as well as the property of the SMA diversity resistance. To obtain an analytical solution to all the equations of the boundary value problem, the Laplace transform method with respect to martensite volume fraction parameter was used. After transformation in the space of images, an equivalent elastic problem is obtained. As a result of solving the equivalent elastic problem, the Laplace images of the desired quantities are obtained in the form of analytical expressions, which include operators that are Laplace images of elastic constants. These expressions are fractional rational functions of the Laplace image of the phase composition parameter. After returning to the original space, which is carried out analytically by decomposing the expressions for the desired quantities in the image space into simple factors, the desired analytical solutions are obtained.

**Firsanov Vic.V.**

## Computational models of beam bending taking into account shear deformation

The classical model of beam bending is based on Bernoulli’s hypotheses: there is no transverse linear strain, no shear strain in the plane, where is the longitudinal and is the transverse coordinates of the beam, and no transverse normal stress. At the same time, both the transverse normal and tangent stresses are preserved in the equilibrium equations, since without them the problem of bending the beam has no solution. Implementation of the corresponding physical relations is neglected. For isotropic and orthotropic linear elastic materials, the shear strain is determined by dividing the tangent stress by the shear modulus. The larger the shear modulus, for example, compared to the elastic modulus in tension and bending, the closer we are to the hypothesis of no shear deformations, and Vice versa, the smaller the shear modulus, the more problematic the use of this hypothesis. This is especially true for the problem of bending orthotropic plates that are not reinforced in the transverse direction. Then the shear modules in the transverse direction are mainly determined by the properties of the weak binder and can be significantly less than the physical characteristics of an orthotropic package with planar reinforcement. In a beam, the reinforcement is carried out in the plane, and if the beam cannot be reinforced in the transverse direction due to too small a normal transverse stress, then a small number of layers at angles must be applied, since the bent beam also works for shear. Therefore, the shear modulus is determined not only by the binder, but also by the reinforcing fibers, and can be commensurate with the elastic modulus, and be several times smaller, depending on the number of reinforcing fibers. The aim of the work is to assess the effect of shear deformation on the stress-strain state of the beam.

Pages: 98-107 Elibrary

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**Saganov E.B., Sharunov A.V.**

## Solution of problem on sphere of alloy with memory of shape under action of constant pressure, taking into account divergence of material

In the work, a numerical solution of the problem on the stress-strain state (SSS) of a thick-walled sphere made of a shape memory alloy (SMA), which is under the influence of constant internal or external pressure in the mode of martensitic inelasticity (MI) taking into account elastic deformations and the property of material tension-compression asymmetry. Under the property, tension-compression asymmetry refers to the dependence of the material constants of these alloys on the type parameter of the state of stress. The parameter associated with the third invariant of the stress deviator is used as a parameter of the type of stress state. The solution was obtained on the basis of the model of nonlinear deformation of SMA during phase and structural transformations. When solving the problem without taking into account elastic deformations, the provision on active processes of proportional loading is used. In the framework of the deformation process under consideration, the influence of the SMA diversity resistance as well as elastic deformations on the distribution of radial and ring stresses in the sphere cross section is demonstrated. It has been established that the distribution of radial and circular stresses over the sphere cross section is nonlinear, and the stresses themselves can vary nonmonotonously during loading. In the course of work, the module of the finite element complex Simulia Abaqus was verified, which was developed for the analysis of the SSS of structures from SMA in the MI mode. As a verification basis, the obtained numerical solution of the spatial three-dimensional boundary-value problem of SSS of a thick-walled spherical shell made of SMA under the loading of internal or external pressure, taking into account the different resistance of these alloys, was used. The obtained numerical solution converges to the analytical solution of the corresponding problem without taking into account elastic deformations with increasing of Young’s modulus.

Pages: 108-121 Elibrary

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**Artamonova N.B., Sheshenin S.V.**

## Coupling consolidation problem in a nonlinear formulation. theory and method of solution

The consolidation problems are related to the study of soil deformation under load in the presence of fluid outflow. In the process of joint deformation of the porous skeleton and the fluid contained in the pores, the solid and liquid phases of the soil interact. The filtration processes in the soil mass are described by a coupling system of differential equations with rapidly oscillating coefficients. To solve such equations, averaging over the representative volume element (RVE) is used. In the paper, the equations of the nonlinear consolidation model are written from the general laws of conservation of continuum mechanics (the equilibrium equation, the law of mass conservation of solid and liquid phases of the soil, and Darcy’s filtration law) using spatial averaging over the representative volume element. The following assumptions were made: the fluid fills the pores entirely, the fluid is Newtonian and homogeneous, the deformation of the fluid with a change in pore pressure obeys the law of barotropy, and the soil skeleton material is incompressible. To determine effective properties, an approach based on solving local problems in a representative volume element is possible. As a result, a coupled physically and geometrically nonlinear formulation of the boundary value problem was obtained using the Lagrange approach with adaptation for the solid phase and the ALE (Arbitrary Lagrangian-Eulerian) approach for the fluid under the assumption of quasistatic deformation of the rock skeleton. In the method of solving the coupled problem, linearization of variational equations is carried out in combination with internal iterations according to the Uzawa method for connecting at each time step. For spatial discretization, the finite element method is used: trilinear type elements for approximating the filtration equation and quadratic elements for approximating the equilibrium equation. An implicit time scheme can be used to take into account the inertia forces.

Pages: 122-138 Elibrary

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**Russkikh S.V., Shklyarchuk F.N.**

## Numerical solution of nonlinear motion equations of compound elastic systems with joins

The unsteady motion of two elastic systems described by nonlinear differential equations in generalized coordinates is considered. It is believed that in the initial state or during the transformation process, these two systems are connected to each other in a finite number of points by elastic or geometric holonomic bonds. Based on the principle of virtual displacements (D’Alembert-Lagrange), the equations of motion of the composite system in the same generalized coordinates are obtained taking into account the constraints. In this case, elastic bonds are taken into account by adding the potential energy of deformation of the connecting elements, which is expressed using the conditions of the connection through the generalized coordinates of the two systems. Geometric bonds are taken into account in the variational equation by adding variations to the work of unknown reactions of bond retention at their small possible changes and are expressed through variations of the generalized coordinates of the systems under consideration. From this extended variational equation, equations of the composite system are obtained, to which algebraic equations of geometric relationships are added. This approach is equivalent to the approach of obtaining equations in generalized coordinates with indefinite Lagrange multipliers representing reactions in bonds. As an example, we consider a system consisting of a bending elastic, inextensible cantilever beam that performs non-linear quadratic longitudinal-transverse vibrations, at the end of which a heavy rigid body is pivotally connected, which rotates through a finite angle. The beam bending is represented by the Ritz method by two generalized coordinates. Two linear constraints on the displacements of the beam and the body in the hinge are satisfied exactly, and the third nonlinear constraint, representing the condition of inextensibility of the beam, is added to the equations of motion of the system, including an unknown reaction to hold this constraint. Numerical solutions of the initial problem of forced nonlinear vibrations of a beam with an attached body are obtained in two versions with comparisons: 1) the nonlinear coupling is satisfied analytically accurately, and the unknown reaction is excluded from the vibration equations; 2) the connection is differentiated in time and is satisfied by numerical integration together with the differential equations of motion of the system.