Applied Mathematic Technologies in Nonlinear Mechanics of Thin-Walled Constructions

The problem of forced vibration of a hinged beam with piezoelectric layers is solved. Issues of mechanical and electric excitation of vibration and the possibility of damping mechanically induced vibration by applying a voltage to the electrodes of the piezolayers are studied. The effect of the physically nonlinear behavior of the passive layers on the response of the sensor layer and entire structure and the effect of geometric nonlinearity on the behavior of the structure and sensor layer are analyzed. The interaction of physical and geometrical nonlinearities for transient and stationary processes is studied

A mathematical model of flexible (according to the theories of T. von Karman and Green–Lagrange) physically nonlinear porous size-dependent Euler-Bernoulli beams subjected to transversal alternating loading is developed. The required differential equations are derived from the Hamilton–Ostrogradsky principle. Iterative algorithm is developed to compute chaotic and hyperchaotic vibrations in a mechanical system with “almost” infinite degrees of freedom. These algorithm include the Finite Difference Method (FDM) combined with Birger's Method of Variable Elasticity Parameters (MVEP), which takes into account physical nonlinearity. Chaos is analysed according to Gulick's definition. The instability of beam structures, including both metallic continuous and porous functionally graded Euler–Bernoulli beams, is studied within the framework of the Lavrentiev–Ishlinsky and Rayleigh–Taylor concepts.

This work provides a mathematical understanding of the elastic fingering provoked by a large axial extension of a soft solid cylinder bonded between rigid plates. In this prototypical system model, a topological transition from a ground axis-symmetric meniscus is quasi-statically controlled by the applied displacement, which acts as the order parameter of a pitchfork bifurcation. Since the isotropic elastic energy becomes nonconvex under finite strains, geometric nonlinearity is of paramount importance for the loss of uniqueness of the solution of the boundary value problem. Nonetheless, physical nonlinearity in the elastic energy is found to exert an opposite stabilizing effect. It indeed penalizes the local stretching at the free boundary that would arise as a consequence of any change of its Gaussian curvature. The theoretical and numerical results are in agreement with recent experimental observations, showing that elastic fingering is strongly affected by the aspect ratio of the disc and can be even suppressed in soft materials with physical nonlinearity.

In the article, in a geometrically and physically nonlinear formulation, the problem of the stress-strain state of shallow shells on a rectangular plane under pressure loading is solved. The solving equations of the theory of flexible shallow shells are obtained on the basis of the Henki-Ilyushin deformation theory of plasticity and the equations of the geometrically non-linear theory of shells, which contain quadratic terms with respect to the angles of rotation of the normals to the median surface. The geometrical parameters of the median surface are taken in the initial undeformed state. The equations are written in a form that makes it possible to carry out numerical calculations of problems with allowance for geometric and physical nonlinearities or only with geometric nonlinearity, to take into account unloading, compressibility of the material and secondary plastic deformations. The results of numerical calculations of critical loads, unloading zones and secondary plastic deformations are presented. Recommendations are given on the choice of geometrical parameters when designing shallow shells.

The scaled boundary finite element method (SBFEM) has been used in many fields of engineering to solve the governing equations in bounded and unbounded 2D as well as 3D domains. In solid mechanics, the semi‐analytical solution strategy of the SBFE formulation (numerical in circumferential direction, analytical in radial direction) is based on the assumption of linear elastic material behavior and only small geometrical changes. However, a large group of materials (e.g. rubber) shows geometrical and physical nonlinearity at mechanical loading. In this contribution, the extension of the SBFEM to geometrical and physical nonlinearity is examined. A plane finite element is developed which uses the concept of shape functions constructed by the SBFEM in the framework of a nonlinear finite element analysis. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Based on Timoshenko’s beam theory and Vlasov’s thin-walled member theory, a new model of spatial thin-walled beam element is developed for analyzing geometrical and physical nonlinearity, which incorporates an interior node and independent interpolations of bending angles and warp and takes diversified factors into consideration, such as traverse shear deformation, torsional shear deformation and their coupling, coupling of flexure and torsion, and the second shear stress. The geometrical nonlinear strain is formulated in updated Lagarange (UL) and the corresponding stiffness matrix is derived. The perfectly plastic model is used to account for physical nonlinearity, and the yield rule of von Mises and incremental relationship of Prandtle–Reuss are adopted. Elastoplastic stiffness matrix is obtained by numerical integration based on the finite segment method, and a finite element program is compiled. Numerical examples manifest that the proposed model is accurate and feasible in the analysis of thin-walled structures.

Objectives Modern building codes prescribe the calculation of building structures taking into account the nonlinearity of deformation. To achieve this goal, the task is to develop a methodology for calculating prestressed reinforced concrete beams, taking into account physical and geometric nonlinearity. Methods The methodology is based on nonlinear calculation algorithms implemented and tested in the computation complex PRINS (a program for calculating engineering constructions) for other types of construction. As a tool for solving this problem, the finite element method is used. Non-linear calculation of constructions is carried out by the PRINS computational complex using the stepwise iterative method. In this case, an equation is constructed and solved at the loading step, using modified Lagrangian coordinates. Results The basic formulas necessary for both the formation and the solution of a system of nonlinear algebraic equations by the stepwise iteration method are given, taking into account the loading, unloading and possible additional loading. A method for simulating prestressing is described by setting the temperature action on the reinforcement and stressing steel rod. Different approaches to accounting for physical and geometric nonlinearity of reinforced concrete beam rods are considered. A calculation example of a flat beam is given, in which the behaviour of the beam is analysed at various stages of its loading up to destruction. Conclusion A program is developed for the calculation of flat and spatially reinforced concrete beams taking into account the nonlinearity of deformation. The program is adapted to the computational complex PRINS and as part of this complex is available to a wide range of engineering, scientific and technical specialists.

In this paper, a new conservative high-order compact finite difference scheme is studied for the initial-boundary value problem of the generalized Rosenau-regularized long wave equation. We design new conservative nonlinear fourth-order compact finite difference schemes. It is proved by the discrete energy method that the compact scheme is uniquely solvable; we have the energy conservation and the mass conservation for this approach in discrete Sobolev spaces. The convergence and stability of the difference schemes are obtained, and its numerical convergence order is $O(\tau^{2}+h^{4})$ in the $L^{\infty}$ -norm. Furthermore, numerical results are given to support the theoretical analysis. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

Nonlinear dynamics of heterogeneous shells Part 1. Statics and dynamics of heterogeneous variable stiffness shells

High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics

The elastoplastic state of isotropic homogeneous cylindrical shells with elliptic holes and finite deflections under internal pressure is studied. Problems are formulated and numerically solved taking into account physical and geometrical nonlinearities. The distribution of stresses (displacements, strains) along the boundary of the hole and in the zone of their concentration is analyzed. The data obtained are compared with the numerical solutions of the physically nonlinear, geometrically nonlinear, and linear problems. The stress-strain state of cylindrical shells in the neighborhood of the elliptic hole is analyzed with allowance for nonlinear factors

A relevant problem in the development and improvement of numeric analytical methods for the research of structures, buildings and construction is studying the stress–strain state of structures and construction with boundaries that have complex shapes. Deformations and stresses arise in a domain with a geometrically non-linear shape of the boundary (cut-outs and cuts). These stresses and deformations have great values and gradients. Experiments carried out using the photoelasticity method show a change in the deformation order ratios for different subareas of the boundary cut-out area depending on proximity to the apex of the angular cut-out. Areas with minor deformations are observed, and areas where linear deformations and shears are more significant than rotations are also observed. In addition, areas where section rotations are more significant than linear and shear deformations are observed. According to the experimental data, the mathematical model of the SSS in the area of the apex of the cut-out of the domain boundary should take into account non-linear deformations. Hence, it is necessary to formulate the boundary value problem of the theory of elasticity, taking into account the geometrical non-linearity. The research aim of this paper is to formulate the problem of the elasticity theory taking into account the geometrical non-linearity in furtherance of the proposed mathematical model justified by the experimental data obtained using the photoelasticity method. The obtained formulation of the elasticity theory problem allows analyzing the form of the system of equations of the boundary value problem depending on the proximity of the considered area to the irregular point of the boundary, i.e., taking into account the difference in the effect of linear and shear deformations, rotations and forced deformations on the solution to the geometrically non-linear elastic problem dealing with forced deformations in the area of an angular cut-out of the boundary of the plane domain.

Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses

The Hamilton - Ostrogradsky principle is used in the paper to construct a mathematical model of vibrations of geometrically nonlinear nano-dimensional shallow axisymmetric spherical shells. The model is based on the following relations and assumptions: the body of the shell is elastic, homogeneous and isotropic, the Kirchhoff - Love hypothesis, a modified momentary theory of elasticity is used to account for the dependence of elastic behavior of the shell on the dimension-dependent parameter; the shallowness of the shell is determined based on the hypotheses of V.Z. Vlasov, the geometrical nonlinearity is determined using T. Karman's hypothesis. An approach for determining the “true” solution is proposed. The partial differential equation is reduced to a Cauchy problem with the help of the second-order accuracy finite difference method, which is solved using several Runge - Kutta-type methods: the 4th- and 2nd-order Runge - Kutta methods, the 4th-order Runge - Kutta - Fehlberg method, the 4th-order Cash - Karp method, the 8th-order Runge - Kutta - Prince - Dormand method, the implicit 2nd- and 4th-order Runge - Kutta method. An algorithm and a software complex for obtaining numerical results are developed. The convergence of the above methods for a space and time coordinate is studied. The study is based on the qualitative theory of nonlinear dynamics: signals, 2D and 3D phase-plane portraits, power spectrums, Morley's wavelets, deflection curves, Poincare sections and autocorrelation functions are analyzed, the sign of Lyapunov's index is analyzed using several methods: Wolf's, Kantz's, Rosenstein's. An example of analyzing plates and shallow shells is given. The analysis of the obtained results shows that with the increase of the dimension-dependent parameter the vibration character changes from chaotic to harmonic, and the value of the dynamic critical load increases.

Nonstandard finite difference scheme for a Tacoma Narrows Bridge model