Relaxation Kernel Research Articles

Stability assessment of the phase-field lattice Boltzmann model and its application to Taylor bubbles in annular piping geometries

This study parametrically assessed the stability of the phase-field lattice Boltzmann model (PFLBM) before applying it to analyze the effect of annular piping geometry on the flow of Taylor bubbles. The impacts of both eccentricity and pipe diameter ratio were examined, providing an insight into the behavior of these bubbles as well as the applicability and shortcomings in current prediction methodologies. A recently developed PFLBM was implemented into the open-source simulation framework, waLBerla, for this analysis. The stability properties of the code were investigated in detail by assessing various lattice discretizations and relaxation kernels applied to the Rayleigh–Taylor benchmark problem and a Rayleigh–Taylor instability in a tubular geometry, with gravitational Reynolds numbers of up to 30 000 and 10 000, respectively. This paper makes three contributions relating to the stability and usage of the PFLBM as well as the flow of Taylor bubbles in annular pipes. First, the work numerically explored the stability properties of the velocity-based, PFLBM and concluded the impact of various collision models and lattice discretizations on simulation results. Second, it provided a flexible open-source code that the interested researcher can use interactively for practical flow problems as well as the analysis of numerical properties of various lattice Boltzmann algorithms. Finally, it quantified the effect of pipe eccentricity and diameter ratio on the propagation of a Taylor bubble inside a water-filled annular pipe, concluding that a previously defined closure model captured the diameter ratio for the cases examined. To extend this work, future studies aim to analytically investigate the stability properties parametrically observed in this study and apply the findings to simulate the interaction of multiple Taylor bubbles.

Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness

The research object of this work is an orthotropic viscoelastic plate with an arbitrarily varying thickness. The plate was subjected to dynamic periodic load. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation, taking into account the tangential forces of inertia. The Bubnov–Galerkin method, based on a polynomial approximation of the deflection and displacement, was used. The problem was reduced to solving systems of nonlinear integrodifferential equations. The solution of the system was obtained for an arbitrarily varying thickness of the plate. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The computational algorithm was developed and implemented in the Delphi algorithmic language. The plate’s dynamic stability was investigated depending on the plate’s geometric parameters and viscoelastic and inhomogeneous material properties. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel almost coincide with the results of the elastic problem. Using the Koltunov–Rzhanitsyn kernel, the differences between elastic and viscoelastic problems are significant and amount to more than 40%. The proposed method can be used for various viscoelastic thin-walled structures such as plates, panels, and shells of variable thickness.

Vibrations of a viscoelastic isotropic plate under periodic load without considering the tangential forces of inertia

A mathematical model of the problem of viscoelastic isotropic plate vibrations based on the Kirchhoff-Love hypothesis in a geometrically nonlinear formulation was presented. The mathematical model was built without considering the tangential forces of inertia. To describe the viscoelastic properties of the plate material, a weakly singular Koltunov-Rzhanitsyn kernel with three different rheological parameters was chosen. To solve the problem of parametric vibrations of a viscoelastic plate with a weakly singular relaxation kernel, a numerical method based on the use of quadrature formulas was applied. A discrete model of this problem was first constructed using the Bubnov-Galerkin method; i.e., a system of integro-differential equations with variable coefficients was obtained, and then, using a numerical method based on the elimination of a singularity of the kernel, the problem of parametric vibrations of viscoelastic rectangular plates was solved. The influence of the viscoelastic properties of the material and the variability of the plate thickness on the oscillatory process was shown.

Transverse Forced Vibrations of the Plates, the Dissipative Properties of Which are Described Memory Functions

The fundamentals of theoretical methods for engineering calculations of bending vibrations of thin plates, the material of which has hereditary properties, are presented. The manifestation of the hereditary properties of the material of the plate under consideration at a given stress-strain state can be judged by the relaxation core of the material. The dependence of the creep and relaxation kernels on the time difference corresponds to the fact that the “memory” of the material about the force effect produced at a given moment is determined by the elapsed time interval. In particular, this means that if the force action on the elastic-viscous body is cyclic, then the deformation of the body will also be cyclic with some phase shift. A technique for solving forced vibrations of plates under the influence of harmonic loads is proposed. An exact solution to the problem of forced vibrations of a supported rectangular plate, whose dissipative properties are described by memory functions, is constructed.

Present problems of vibration isolation in heavy mining machines at long-term cyclic loads

Abstract. An integrated approach was used, including the construction of a mathematical model and the results of lengthy industrial tests of heavy mining machines with rubber elastic bonds. The issues of vibration isolation of heavy machines operating in extreme conditions, i.e. with prolonged cyclic loads and the influence of an external aggressive environment inherent in mining and processing enterprises. As an example, we con-sider vortex mixers with a vibration isolation system containing rubber elements. During long-term operation, the mass of the mixer does not remain constant; its increase is associated with the sticking of the initial product on the moving parts of the ma-chine. The location of such machines in the sinter factory at + 10.4 m, as well as the change in time of rotational symmetry of the drum rotation imposes certain requirements on the vibration isolation system: the system must be "soft" and stable in time, i.e. during 9-10 years of operation, its stiffness and dissipative parameters should not go beyond the permissible values. In order to reduce the aging effect, type 2959 rubber based on natural caoutchouc with a reinforced protective group was used, while the instability of the main mechanical parameters was taken into account in the developed theory of vibration isolation of machines. On the basis of the developed simulation model and the Boltzmann-Volterra integral relations with kernels of relaxation and aftereffect, an equation was worked out, which made it possible to take into account the rubber viscoelastic properties in full volume; in this equation, stiffness operator of elastic suspension in the machine is written by using fractional exponential function of the Yu. Rabotnov's type; on the basis of the mathematical model, the basic parameters of the machine under the study were calculated; in particular, for the vortex mixer, the time dependences of amplitude of the mixer housing vibrations and coefficient of vibration isolation efficiency were calculated with taking into account aging of elastic link material in the ma-chines; the calculation results were compared with the results of industrial tests of the vortex mixer operation lasting for 16 years. The theory and method for calculating vibration isolation systems with rubber elastic links for heavy mining machines were developed with taking into account material structure changes due to the effects of aging. The paper considers an example of calculating the mixer vibration isolation system taking into account rubber aging; the magnitude of the change in time of the main mechanical parameters was obtained experimentally over 16 years. This made it possible to determine temporary changes in the amplitude of oscillations of the mixer body and the coefficient of efficiency of the vibration isolation system. It was shown that the mixer vibration isolation system remained effective for 9-10 years, after which the mechanical characteristics of the rubber went beyond the permissible values and the system lost its functional pur-pose, and the amplitude of the mixer body exceeded the existing sanitary standards.

Calculation of elastic-creeping characteristics of a beam made of a layered composite material

Multilayer composite materials are often used in building structures. The direct calculation of layered structures requires large expenditures of computer time. Therefore, the homogenization method is used. This method reduces the problem of a layered material with isotropic layers to the problem of a homogeneous transversely isotropic medium. The material considered in the article is also elastic-creeping. In the equations of state of such a material, terms of the convolution type with difference creep (relaxation) kernels are added to the terms of the usual theory of elasticity. The creep (relaxation) kernels are represented by decreasing exponential functions depending on two parameters. This problem becomes a problem of the theory of elasticity with a parameter after applying the Laplace transform in time to it. The inverse Laplace transform can be done in a computer algebra package, for example, Wolfram Mathematica, Wolfram-alpha. The obtained characteristics of the material are used to solve the problem of a layered elastic-creeping beam with hinge support. Formulas are given for determining displacements in the case of layers parallel to the beam axis.

Vibrations of a geometrically nonlinear viscoelastic shallow shell with concentrated masses

Shell structures are widely used in various fields of technology and construction. Often, they play the role of a bearing surface with assemblies, overlays, and aggregates installed on them. At the same time, in solving various problems, such attached elements are considered as the elements concentrated at the points and rigidly connected. Vibrations of an orthotropic viscoelastic shallow shell with concentrated masses in a geometrically nonlinear setting are considered. The equation of motion for a shallow shell is derived based on the Kirchhoff-Love theory. The traditional Boltzmann-Volterra theory is used to describe the viscoelastic properties of a shallow shell. The effect of concentrated masses is taken into account using the Dirac delta function. Using the polynomial approximation of the deflections of the Bubnov-Galerkin method, the problem is reduced to solving a system of ordinary nonlinear integro-differential equations with variable coefficients. In the calculations, the three-parameter Koltunov-Rzhanitsyn kernel was used as a weakly singular relaxation kernel. A numerical method was used to solve the resulting system that eliminates the singularity in the relaxation kernel. The problem of nonlinear vibrations of an orthotropic viscoelastic shallow shell with concentrated masses is solved. The influence of concentrated masses and location, properties of the shell material, and other parameters on the amplitude-frequency response of the shallow shell vibrations is investigated.

Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history

<abstract><p>The purpose of this paper is to establish a general stability result for a one-dimensional linear swelling porous-elastic system with past history, irrespective of the wave speeds of the system. First, we establish an explicit and general decay result under a wider class of the relaxation (kernel) functions. The kernel in our memory term is more general and of a broader class. Further, we get a better decay rate without imposing some assumptions on the boundedness of the history data considered in many earlier results in the literature. We also perform several numerical tests to illustrate our theoretical results. Our output extends and improves some of the available results on swelling porous media in the literature.</p></abstract>

Numerical study of nonlinear problems in the dynamics of thin-walled structural elements

Mathematical model of the problem of vibration of thin-walled structural elements has been constructed based on Kirchhoff-Love theory. The problem is reduced, using the Bubnov-Galerkin method, to the solution of a set of nonlinear integro-differential Volterra type equations with weakly-singular kernels of relaxation. A numerical method based on the use of quadrature formulae being used for their solution. The influence of rheological parameters of the material on the values of critical velocity and amplitude-frequency characteristics of viscoelastic thin-walled structural elements is analyzed. It is shown that tacking account viscoelastic properties of the material of thin-walled structures lead to a decrease in the critical rate of gas flow.

Dynamic analysis of an orthotropic viscoelastic cylindrical panel of variable thickness

The intensive development of the modern industry is associated with the emergence of a variety of new composite materials. Plates, panels, and shells of variable thickness made of such materials are widely used in engineering and machine building. Modern technology for the manufacture of thin-walled structures of any configuration makes it possible to obtain structures with a given thickness variation law. Such thin-walled structures are subjected to various loads, including periodic ones. Nonlinear parametric vibrations of an orthotropic viscoelastic cylindrical panel of variable thickness are investigated without considering the elastic wave propagation. To derive a mathematical model of the problem, the Kirchhoff-Love theory is used in a geometrically nonlinear setting. The viscoelastic properties of a cylindrical panel are described by the hereditary Boltzmann-Volterra theory with a three-parameter Koltunov-Rzhanitsyn relaxation kernel. The problem is solved by the Bubnov-Galerkin method in combination with the numerical method. For the numerical implementation of the problem, an algorithm and a computer program were developed in the Delphi algorithmic language. Nonlinear parametric vibrations of orthotropic viscoelastic cylindrical panels under external periodic load were investigated. The influence of various physical, mechanical, and geometric parameters on the panel behavior, such as the thickness, viscoelastic and inhomogeneous properties of the material, external periodic load, were studied.

Щодо розрахунку релаксації напружень в тонкостінних циліндричних оболонках із лінійно-в'язкопружних матеріалів

The problems of stress relaxation analysis in thin-walled cylindrical shells made of linear viscoelastic materials under uniaxial and biaxial loading have been solved. The analysis is based on a there-dimensional model of viscoelasticity starting from the hypothesis of the deviators proportionality. The viscoelastic properties of a material are given with relationships that establish the dependence between stress and strain intensities as well as between the mean stress and volumetric strain by the Bolzmann-Volterra type equation. The kernels of relaxation intensity and volumetric relaxation are given with the Rabotnov exponential-fractional functions. The parameters of relaxation kernels are determined from creep test result using the relationships between creep kernels under the complex stress state and creep kernels under the one- dimensional stress state. The problems of the analysis of normal and tangential stresses relaxation in thin-walled cylindrical shells made of high density polyethylene “ПЭВП” under uniaxial tension, pure torsion and combined tension with torsion loading have been solved and experimentally approved.

Mathematical analysis of memory effects and thermal relaxation in nonlinear sound waves on unbounded domains

Motivated by the propagation of nonlinear sound waves through relaxing hereditary media, we study a nonlocal third-order Jordan–Moore–Gibson–Thompson acoustic wave equation. Under the assumption that the relaxation kernel decays exponentially, we prove local well-posedness in unbounded two- and three-dimensional domains. In addition, we show that the solution of the three-dimensional model exists globally in time for small and smooth data, while the energy of the system decays polynomially.

Mathematical modeling parametric vibrations of the pipeline with pulsating fluid flow

A mathematical model of the dynamics of a straight viscoelastic pipe conveying pulsating fluid has been developed in the paper. Using the Bubnov-Galerkin method, a mathematical model of the problem is reduced to solving a system of ordinary integro-differential equations, where time is an independent variable. Solution of integro-differential equations is determined by a numerical method based on the elimination of a singularity in the relaxation kernel of an integral operator. A system of algebraic equations is obtained according to the numerical method for unknowns. To solve the system of algebraic equations, the Gauss method is used. A computational algorithm is developed for solving the problems of the dynamics of viscoelastic pipelines conveying fluid.

The weakly nonlinear response and nonaffine interpretation of the Johnson–Segalman/Gordon–Schowalter model

We derive new analytical solutions for the nonaffine Johnson–Segalman/Gordon–Schowalter (JS/GS) constitutive equation with a general relaxation kernel in medium-amplitude oscillatory shear (MAOS) deformation. The results show time-strain separable (TSS) nonlinearity, therefore providing new physically meaningful interpretation to the heuristic TSS nonlinear parameter in MAOS [Martinetti and Ewoldt, Phys. Fluids 31, 021213 (2019)]. The upper-convected, lower-convected, and corotational Maxwell models are all subsets of the results presented here. The model assumes that the microscale elements causing stress in the material slip compared to the continuum deformation. We introduce a visualization of the nonaffine deformation field that acts on stress-generating elements to reinforce the physical interpretation of the JS/GS class of models. Finally, a case study is presented where previously published results, from fitting TSS models to MAOS data, can be reinterpreted based on the concept of nonaffine motion of the JS/GS framework.

Decay rates for the energy of a singular nonlocal viscoelastic system

This work deals with decay rates for the energy of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. We prove the decay rates for the energy of a singular one‐dimensional viscoelastic system with a nonlinear source term and nonlocal boundary condition of relaxation kernels described by the inequality for all t ≥ 0, with H convex.

Vertical nonlinear oscillations of viscoelastic systems with multiple degrees of freedom

Vertical oscillations of three loads of different masses connected by nonlinear viscoelastic springs (suspensions) are considered in the paper. An account for rheological properties of the suspension, an integral model with the Koltunov-Rzhanitsyn relaxation kernel is used. Effective computational algorithms have been developed for solving problems based on the use of quadrature formulas. The effect of rheological properties of suspension on the mass displacement from the position of static equilibrium is investigated as well as the influence of nonlinear properties of the suspension on the mode of vibration and frequency.

Dynamics of viscoelastic orthotropic shallow shells of variable thickness

Thin-walled structural elements such as plates, panels, and shells of variable thickness are widely used at present in engineering, machine-building, and construction. Modern technologies allow creating any structural elements of a given shape, material, and the law of thickness variation. Therefore, the solution to the problems of the statics and dynamics of plates, panels, and shells of variable thickness, considering the real properties of the material, is relevant. Nonlinear parametric oscillations of viscoelastic orthotropic shallow shells of variable thickness are considered in the paper. Using the Kirchhoff-Love hypothesis, a mathematical model of the problem is constructed in a geometrically nonlinear statement. To describe the viscoelastic properties of a shallow shell, the hereditary Boltzmann-Volterra theory with the Koltunov-Rzhanitsyn relaxation kernel is used. To obtain resolving equations of the problem, the Bubnov-Galerkin method was used in combination with the numerical method. The effects of various physico-mechanical and geometrical parameters of a shallow shell of variable thickness were investigated.

Nonlinear thermomechanical vibration mitigation analysis in rotating fractional-order viscoelastic bidirectional FG annular disks under nonuniform shocks

The current article is concerned with the investigation of the formation of the time-dependent three-dimensional distributions and redistributions of the stress and displacement fields of the rotating annular and circular plates/disks under the nonuniform distributions of the dynamic thermomechanical loads or shocks. Redistribution of the stress and displacement fields happens due to the Gerasimov-Caputo-type relaxation kernel of the fractional viscoelastic model of the material in addition to the dynamic nature of the loading. All the mechanical and thermal material properties of the plate or disk may be tailored in both radial and transverse directions. The 3D thermoviscoelasticity theory is employed to develop the governing equations of motion (3D vibration) and heat transfer. Different thermal and mechanical boundary conditions are implemented. The resulting nonlinear time-dependent fractional-order thermoviscoelastic integro-differential equations are solved by proposing and implementing a special procedure that uses numerical evaluation of the singular-kernel Caputo-integral and second-order forward, backward, and central discretization of the spatial and time domains. Eventually, the influences of the bidirectional distribution of the material properties, 2D temperature distribution, fractional order and parameters of the viscoelasticity model, and thermomechanical boundary conditions are investigated on the distributions of the displacement and stress components rigorously and accompanied by 3D demonstrations.

Analysis of Creep Strains and Stress Relaxation in Thin-Walled Tubular Members Made of Linear Viscoelastic Materials. 1. Superposition of Shear and Volume Creep

The problem of creep strains and stress relaxation analysis in thin-walled tubular members made of linear viscoelastic materials under combined tension and torsion loading is solved. The solution is obtained using viscoelastic models in the form of superposition of shear and volume creep. The creep and relaxation kernels are represented by fractional-exponential functions. The solutions are validated experimentally by analyzing longitudinal, transverse, and shear creep strains as well as the relaxation of normal and tangential stresses in thin-walled tubular members made of organic glass and high-density polyethylene.

Present problems of vibration isolation in heavy mining machines at long-term cyclic loads

On the basis of the developed simulation model and the Boltzmann-Volterra integral relations with kernels of relaxation and aftereffect, an equation was worked out, which made it possible to take into account the rubber viscoelastic properties in full volume; in this equation, stiffness operator of elastic suspension in the machine is written by using fractional exponential function of the Yu. Rabotnov’s type; on the basis of the mathematical model, the basic parameters of the machine under the study were calculated; in particular, for the vortex mixer, the time dependences of amplitude of the mixer housing vibrations and coefficient of vibration isolation efficiency were calculated with taking into account aging of elastic link material in the machines; the calculation results were compared with the results of industrial tests of the vortex mixer operation lasting for 16 years. The theory and method for calculating vibration isolation systems with rubber elastic links for heavy mining machines were developed with taking into account material structure changes due to the effects of aging.