This paper analyzes delamination of multilayered inhomogeneous beam structure by utilization a non-linear viscoelastic mechanical model. The non-linear time-dependent response is treated by a non-linear spring and a non-linear dashpot connected in series to a system of two linear springs and two linear dashpots. The model is under stress that is a linear function of time. The constitutive law of the model representing a non-linear dependence between stress, strain and time is derived. The main goal of the paper is to obtain a solution of the strain energy release rate for the delamination in the multilayered inhomogeneous beam by applying the non-linear viscoelastic model. Solutions are derived by using the complementary strain energy and by analyzing the balance of the energy with taking into account the non-linear viscoelastic behaviour. For this purpose, the constitutive law of the non-linear viscoelastic mechanical model is used. The solutions are applied to obtain results for multilayered beams with non-linear variation of material properties in longitudinal direction. The influence of different parameters on the time-dependent strain energy release rate is assessed. The study indicates the effectiveness of the viscoelastic mechanical models with non-linear springs and dashpots in time-dependent delamination analyses of multilayered inhomogeneous beam structures.

This work is devoted to the numerical analysis of vertical elastic coaxial cylindrical shells completely or partially filled with a quiescent compressible fluid with account for sloshing of its free surface. The problem is solved in the axisymmetric formulation using a semi-analytical version of the finite element method. The fluid medium is described by the wave equation, which together with the conditions prescribed at its boundaries is reduced to the weak form by the Bubnov - Galerkin method. A mathematical formulation of the dynamic problem for thin-walled structures is developed using the variational principle of virtual displacements and linear theory of thin shells based on Kirchhoff - Love hypotheses. The fluid pressure on the walls of the structure is calculated according to Bernoulli's equation. Sloshing modes caused by gravitational effects on free surface of liquid medium are excluded from the resulting system of equations through the use of the iteration dynamic condensation method. The numerical model has been verified by comparing it with the known data for the case of a single shell partially filled with fluid. The influence of the fluid level on the lowest natural frequencies of the system vibrations at different variants of kinematic boundary conditions for shells (rigid clamping at both edges, cantilever support) and different widths of the annular gap between them has been evaluated. It has been found that for the examined configurations, the level of the fluid in the annular channel has a stronger influence on the frequency spectrum compared to the level of fluid in the cavity of the inner shell due to the fact that vibration frequencies change in a wider range.

As is known, the calculation of the static strength of elastic composite bodies (CB) is reduced to finding the maximum equivalent stresses for these bodies. The finite element method (FEM) is widely used for the analysis of the stress state of CB. The basic discrete models (BM), which take into account the inhomogeneous structure of bodies in the framework of a micro-approach, have a high dimension. To reduce the dimension of discrete models, multigrid finite elements (MgFE) are effectively used. However, there are BM CB (for example, BM bodies with a micro-homogeneous structure), which have such a high dimension that the implementation of FEM for such BM using MgFE, due to limited computer resources, is difficult. To solve this problem, it is proposed to use fictitious discrete models whose dimensions are less than the dimension of the BM CB. In this paper, we propose a method of fictitious discrete models (MFDM) for calculating the strength of elastic bodies with an inhomogeneous, micro-homogeneous regular structure. The proposed method is implemented using FEM with the use of MgFE and adjusted strength conditions that take into account the error of approximate solutions. The method is based on the position that the solutions that meet the BM CB differ little from the exact ones. The calculation of CB according to MFDM is reduced to the construction and calculation of the strength of fictitious discrete models (FM), which have the following properties. The FM reflects: the shape, characteristic dimensions, attachment, loading and type of the inhomogeneous structure of the CB, and the distribution of elastic modulus corresponding to the BM CB. The FM dimension is less than the BM dimension of the CB. The sequence consisting of FM converges to BM, i.e. the limiting FM coincides with BM. The convergence of such a sequence ensures uniform convergence of the maximum equivalent voltages of the FM to the maximum equivalent voltage of the BM. Two types of FM are considered. The first type of FM consists of scaled discrete models, the second type consists of FM with variable characteristic dimensions. Calculations show that the implementation of FEM for FM using MgFE leads to a large saving of computer resources, which allows the use of MFDM for bodies with a micro-homogeneous regular structure. The calculation of the strength of CB according to MFDM requires times less computer memory than a similar calculation using BM CB, and does not contain a procedure for grinding BM. The use of adjusted strength conditions allows us to use approximate solutions with a large error in the calculations of CB for strength, which leads to an increase in the efficiency of MFDM. The given example of calculating the strength of a beam with an inhomogeneous regular fibrous structure according to MFDM shows its high efficiency.

We consider a thin oxide film on the surface of a molten metal during induction melting. The alternating electromagnetic field excites eddy currents in the metal volume, which heat it, and Lorentz force, which causes the forced convection of the melt, the heating and the flows in the metal are discussed briefly. The contribution to the mechanical stresses in the film that gives the alternating electromagnetic field, the thermal expansion of the film, and the melt motion are considered in detail. The equations system describing magnetic field diffusion, equations of motion and heat transfer in the melt are written in the axisymmetric formulation. The corresponding boundary conditions are described, the governing dimensionless criteria, which determine the structure and intensity of the melt flow, including those at the surface where the film is located, are given. The film elastic deformation equation is derived from Hooke's law and written in terms of displacement in dimensional and dimensionless forms. On the base of literature review, the values of physical characteristics of the film, which are not available for direct measurement, are proposed. The verification of the mathematical model is given. Possible flows in the melt are calculated, taking into account the action of dynamic and thermal action of the film on the surface. The unambiguous relation of the film stress state with these flows is shown. The influence of the magnetic field diffusion parameter and the Hartmann number, which determine, respectively, the structure and intensity of the forced flow, on the film deformations is demonstrated. The mode map of regimes is constructed that relates the integral deformation of the film to the parameters of the magnetic field and the initial size of the film. It is found that the situations are possible when the film in the stress-strain state does not change its total size and remains in stable equilibrium on the surface of the moving melt. Recommendations for the usage of the presented results are given.

The paper considers a cylindrical segmented shell - a pressure cylinder made by winding from unidirectional roving. The shell has a different thickness and reinforcement scheme for segments, and metal embedded elements on the front and rear ends of the shell for fixing the covers. The problem of calculating the stress-strain state and estimating the strength of the shell under loading by internal pressure is posed. To solve the problem, a two-level numerical mathematical model of the shell is developed. At the first level, the shell is modeled using the effective characteristics of the composite material by segments. At the second level, the deformation of individual segments of the shell is modeled - the front and rear joint nodes with metal embedded elements and a pin-pin connection with an explicit description of the segment reinforcement scheme. The boundary conditions for the segment deformation problem are determined as a result of calculating the shell VAT using the first-level model. The developed model is compared with the results of bench tests of the shell for the effect of internal pressure. Based on the results of solving the problem for the shell segments, the analysis of interlayer stresses in the contact zone of the composite shell - metal embedded was performed and estimates of the structural strength were obtained.

Products made of metals and alloys with a fine-grained structure, which have high physical, mechanical and operational characteristics, are becoming increasingly in demand in many technical and technological fields. The most common and efficient technologies for the production of parts from this class of materials are various processes of severe plastic deformation (SPD) (in general practice, at low homologous temperatures). At the same time, to achieve high degrees of deformation, a significant part of metals and alloys require increased processing temperatures, undergo significant heating during deformation processes, which may be accompanied by changes in the grain and subgrain structure due to recovery and recrystallization processes. An empirical approach to the development of SPD modes that ensure the formation of the necessary grain structure requires huge time and financial costs, and therefore considerable attention is paid by researchers in the field of solid mechanics and metal forming to approaches and methods of mathematical modeling. In connection with the foregoing, the number of publications on this subject has been intensively growing in recent years. The currently known models differ significantly in their approaches, the depth of penetration into the physics of processes, and the scope of consideration. The proposed brief review is focused on a qualitative analysis of works on this topic, a preliminary classification of existing models according to their purpose, versatility, and functionality. It seems possible to single out the two most common approaches to describing the change in the grain structure in the processes of thermomechanical processing of metals and alloys: continual (in most cases, single-level) and multilevel, based on the introduction of internal variables and physical theories. This review is devoted to the consideration of publications focused on the first of these approaches. Until now, the most common are macrophenomenological continuum models based on the analysis of experimental data conducted both in laboratories on macrosamples and in real production conditions. Models of this class are usually formulated in the form of operator relations over field quantities, they are relatively easy to implement due to their easy “embedding” into widely used commercial software packages, but they require significant costs for experiments to identify models, they are characterized by a low degree of versatility. Continuous theories are relatively less common, but still often used. These theories are based on the description of physical mechanisms and the evolution of the structure of metals and alloys in terms of continuum variables.

In this paper, the general formulation of the problem of coupled deformation of a porous deformable medium with a fluid flowing through the pores is formulated, mathematically investigated and numerically implemented within the framework of physical and geometric nonlinearity. We present the formulation of the problem in velocities of solid phase displacements and the rate of pore pressure change in differential and variational forms. A phenomenological approach was used to formulate the mechanical model. The equations of the coupled consolidation model were derived from the general conservation laws of continuum mechanics using spatial averaging over a representative volume element. The consolidation model took into account the change in the porosity and permeability of the medium during deformation. The equations of filtration and porosity change, originally presented in Euler approach, were reformulated in Lagrangian coordinates of the solid phase using the relative fluid velocity according to ALE (Arbitrary Lagrangian - Eulerian) approach. The Gâteaux differentiation technique was used to linearize the variational equilibrium equations. For spatial discretization of the saddle system of equations, the finite element method (FEM) was used: quadratic serendipity elements for approximating the equilibrium equations and Brick type elements for approximating the filtration equation. To solve the system of equilibrium and filtration equations, a generalization of the implicit scheme with internal iterations at each time step by the Uzawa method was used. The results of numerical simulation of elastoplastic deformation of a water-saturated soil under load with fluid outflow are presented. To simulate the constitutive relations of elastoplastic deformation of soil under short-term loads, a generalization of S.S. Grigoryan's model to large deformations is proposed. The calculations were carried out in our own program code. The developed consolidation model can be used to simulate the formation of tracking ruts and unevenness of natural roads, as well as to calculate the uneven settlement of engineering structures.