Inhomogeneous Rod Research Articles

Об идентификации характеристик неоднородных вязкоупругих тел в рамках модели дробного порядка

Nowadays, the development of models of viscoelastic materials with complex inhomogeneous structure is one of the hottest problems of continuum mechanics. Along with classical models, fractional-differential models of viscoelasticity have become increasingly popular. In this paper, we present a model for describing steady-state oscillations of inhomogeneous viscoelastic bodies in terms of fractional-order differential operators. Taking into account the fractional order of the model operators, the corresponding form of the complex module describing the material properties is constructed. The module includes four characteristics: instantaneous and long-term elastic moduli (in the case of material inhomogeneity, they are the functions of coordinates), relaxation time and fractionality parameter. The properties of the complex module are studied, and the ranges of the model parameters, at which the rheological properties are most pronounced, are identified. A general formulation of the inverse problem (IP) on the identification of function-parameters of the model based on the acoustic sounding data is proposed. In the framework of this formulation, the inverse problems for specific objects , namely, an inhomogeneous rod and a round plate, are considered. In both model problems, the influence of the fractionalization parameter on the amplitude-frequency characteristics is analyzed. It has been found that in the vicinity of viscoelastic resonances the fractionality parameter affects the parameters of the oscillatory process most significantly, which is typical for such problems. A solution to the nonlinear IPs under consideration is constructed based on the linearization method, which also serve as basis for the iterative processes supplemented with the elements of the projection approach. This allows one to determine corrections to the required functions in the specified classes of functions using regularization. For both IPs, a series of computational experiments were carried out. The analysis of the experimental results made it possible to formulate recommendations for the selection of optimal sensing modes.

Deformation, shape transformations, and stability of elastic rod loops within spherical confinement

Mechanical insight into the packing of slender objects within confinement is essential for understanding how polymers, filaments, or wires organize and rearrange in limited space. Here we combine theoretical modeling, numerical optimization, and experimental studies to reveal spherical packing behavior of thin elastic rod loops of homogeneous or inhomogeneous stiffness. Across varying loop lengths, a rich array of configurations including circle, saddle, figure-eight, and more intricate patterns are identified. A theoretical framework rooted in the local equilibrium of force and moment is proposed for the rod loop deformation, facilitating the determination of internal and contact forces experienced by the rods during deformation. For the confined homogeneous rod loops, their stable and metastable configurations are well described using proposed Euler rotation curves, which offer a concise and effective approach for configuration prediction. Moreover, formulated analysis on the stability and critical force for homogeneous rod loops on great circles of the spherical confinement are performed. For inhomogeneous rod loops with two segments of differing stiffness, the stiffer segment exhibits less deviation from the great circle, while the softer segment undergoes more pronounced deformation. These findings not only enhance our understanding of buckling and post-buckling phenomena but also offer insights into filament patterning within confining environments.

On models of longitudinal vibrations of inhomogeneous rods

The main provisions of the model of longitudinal vibrations of rods are presented, taking into account certain phenomena. To construct a model of longitudinal vibrations of rods from the system, the method of successive approximation was used. Models are based on three components: the basic law or principle of dynamics, a hypothesis or basic statement, and a construction algorithm method.

Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions

Functionally graded materials have broad engineering applications including mechanical engineering, electronics, chemistry, and biomedical engineering. One notable advantage of such materials is that their stiffness distribution can be optimized to avoid stress concentration. A novel approach for solving the equations describing the longitudinal vibration of functionally graded rods with viscous and elastic boundary conditions is proposed. The characteristic equation of the system is derived for the solution of the undamped case for the constant stiffness rod. Then, a homotopy method is applied to compute the eigenvalues and mode shapes of graded rods for viscoelastic boundary conditions. The changes of the eigenvalues and mode shapes as function of the damping parameters are investigated. The optimal damping of the system is computed. It is shown that the qualitative behavior depends on the relation between the actual damping and the optimal damping of the system. The energy density distribution of graded rods is also discussed. An energy measure, the mean scaled energy density distribution is introduced to characterize the energy distribution along the rod in the asymptotic time limit. The significance of such a measure is that it reveals how the energy tends to distribute along the rod. It is shown that the energy distribution can be manipulated by changing the damping parameters. Qualitative changes depending on the relation between the actual damping and the optimal damping are highlighted.

VIBRATIONS OF A CYLINDRICAL SHELL STIFFENED WITH INHOMOGENEOUS LONGITUDINAL AND ANNULAR RIBES WITH LIQUID

Free vibrations of an inhomogeneous anisotropic, fluid-contacting cylindrical shell stiffened with inhomogeneous rods and rings are consider. The Hamilton – Ostrogradsky variation principle was used when solving the problem. It was accepted that the nonhomogeneity of rods and rings used in the strengthening change by the exponential law. The inhomogeneous of the cylindrical shell change by the linear law in the direction of the thickness. The fluid was accepted as ideal. Rigid contact condition between the rods and the cylindrical shell was considered. Using the contact conditions, the frequency equation was structured, the roots were found implemented by the numerical method and characteristically curves were built.

Efficient Numerical Methods of Inverse Coefficient Problem Solution for One Inhomogeneous Body

In the present paper, the problems of longitudinal and flexural vibrations of an inhomogeneous rod are considered. The Young’s modulus and density are variable in longitudinal coordinate. Vibrations are caused by a load applied at the right end. The proposed method allows us to consider a wider class of inhomogeneity laws in comparison with other numerical solutions. Sensitivity analysis is carried out. A new inverse problem related to the simultaneous identification of the variation laws of Young’s modulus and density from amplitude–frequency data, which are measured in given frequency ranges, is considered. Its solution is based on an iterative process: at each step, a system of two Fredholm integral equations of the first kind with smooth kernels is solved numerically. The analysis of the kernels is carried out for different frequency values. To find the initial approximation, several approaches are proposed: a genetic algorithm, minimization of the residual functional on a compact set, and additional information about the values of the sought-for functions at the ends of the rod. The Tikhonov regularization and the LSQR method are proposed. Examples of reconstruction of monotonic and non-monotonic functions are presented.

Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

Parametric vibrations of a viscous-elastic medium-contacting, damaged, orthotropic cylindrical shell stiffened with inhomogeneous rods

Parametric vibrations of a viscous-elastic medium-contacting, damaged, orthotropic cylindrical shell stiffened with inhomogeneous rods

Longitudinal-transverse thermal- force bending and stability layered inhomogeneous profiled rod

The solution to the problem of the stress-strain state of an inhomogeneous profiled rod is based on the use of nonlinear equilibrium conditions and physical relations of a layered thermo elastic thin rod. A differential equation of bifurcation inhomogeneous rod stability of variable cross-section is obtained. The equation has variable functional coefficients. In the initial state, the rod is subjected to bending with the implementation of one of the asymmetric shapes. The critical state occurs under the action of a longitudinal load corresponding to one of the lowest symmetrical shapes, orthogonal to the initial shape. In the first series, numerical calculations of an inhomogeneous I-rod with a variable cross section height are performed. Shelves and wall I-rod are made of steel, aluminum and titanium alloys. The graphs of maximum deflection and normal stresses acting at the calculate points at the boundaries of the layers are plotted depending on the longitudinal load at the given levels of transverse loads and thermal field. A significant influence of the rod physical structure, the profiling its form and the factor of nonlinearity of static relations on the stress fields has been established. A homogeneous temperature field with a nominal value of 80°C creates fields of self-balanced stresses in an inhomogeneous rod. The components of normal stresses in this case reach 20-40% of the materials permissible resistance level. The presence of rod parts with a significant difference in the coefficients of thermal expansion in the composition enhances this effect. In the second, the stability analysis of an inhomogeneous I-rod with a variable width cross section was performed. The transition of the initial S-shaped bend to an unstable state is shown.

Influence of Creep on Longitudinal Fracture of Inhomogeneous Rod Loaded in Torsion and Bending

The present paper analyzes the influence of creep on longitudinal fracture in continuously inhomogeneous rod of circular cross-section loaded in torsion and bending. The rod exhibits continuous material inhomogeneity in both radial and longitudinal directions. The creep is described by using non-linear time-dependent relations between the principle stresses and strains. A time-dependent solution to the strain energy release rate is derived by analyzing the complementary strain energy. The time-dependent strain energy release rate is found also by considering the energy balance for verification. The solutions are applied to perform a parametric study of the strain energy release rate under creep.

Numerical solution of the inverse boundary value heat transfer problem for an inhomogeneous rod

The article is devoted to solving an inverse boundary value problem for a rod consisting of composite materials. In the inverse problem, it is required, using information about the temperature of the heat flow in the media section, to determine the temperature at one of the ends of the rod. The paper presents a method of projection regularization, which made it possible to approximately estimate the error of the obtained solution to the inverse problem. To check the computational efficiency of this method, test calculations were carried out.

The integral formula in problems of the stability of inhomogeneous rods

A heterogeneous in length bar with a variable cross-section is considered. The bar is compressed by a variable longitudinal force which is distributed along its axis. The article describes the case of stability loss of the straight form of equilibrium of a bar, when both, linear and curved forms are possible. The critical combination of rigidity and longitudinal force is the result of an integral representation for the solution of the given stability equation with variable coefficients by the aid of the solution of similar equation but with constant coefficients. The integral representation includes the Green function of the given equation which can be obtained by the method of perturbations. The example of compiling of the equation for critical loading is reduced.

Torsion of Non-Uniform Cylindrical and Prismatic Rods Made of Ideally Plastic Material under Linearized Yield Criterion

The general relations of the torsion theory of inhomogeneous rods made of an ideal rigid-plastic material are considered. In the case of the linearized yield criterion, integrals are obtained that determine the stressed and deformed states of an ideal rigid-plastic inhomogeneous rod during torsion. The field of characteristics of the basic relations is constructed, the lines of stress rupture are found.

Spatially Structure Spatial Problem of the Stressed-Deformed State of a Structural Inhomogeneous Rod

The basic design relations for thermal-force spatial bending with tension, transverse shear and torsion were obtained for a spatially hospismatic rod of rectangular cross section, composed of quasi-homogeneous parts (phases), which were made of various structural materials. Approximations of the transverse shears functions and a membrane analogy were used for shear deformations during torsion, using the Tymoshenko hypotheses. As a sesult, obtained relations allow one to perform approximate formulations and solutions of various boundary-value direct and inverse problems, including: identifying the stress-strain state of a composite rod under thermal power, evaluating its strength and stiffness, identifying rational geometric and structural parameters of the inhomogeneous structure of the rod, and optimization problems. Expressions were obtained for the stiffness characteristics of the zeroth, first, and second orders in bending with tension, shear and tensional stiffness of the section, which allowed us to formulate the boundary value problems of spatial deformation of composite rods.

Analysis of internal delamination in multilayered rod in torsion

A multilayered inhomogeneous rod with an internal delamination crack loaded in torsion is analyzed. The crack has a circular cross-section. The delamiantion is located symmetrically with respect to the middle of the rod. The layers exhibit continuous material inhomogeneity in radial and length directions. The material properties are distributed symmetrically with respect to the middle of the rod. In order to obtain the torsion moments in the crack arms, the rod is treated as a structure with one degree of internal static indeterminacy. The delamination is studied in terms of the strain energy release rate by using the compliance method. Effects of material inhomogeneity, length and location of the crack on the delamination behaviour are investigated.

About torsion of inhomogeneous rods made of ideal rigid plastic material under linearized condition of plasticity

The torsion of inhomogeneous rods made of ideal rigid plastic material is investigated in this work. The integrals determining the stress and strain states of a rod under linearized plasticity condition are obtained. The field of characteristics of the basic relations is constructed, the rupture lines of stresses are found.

Inhomogeneous Rods in Torsion: Longitudinal Fracture Analysis by Considering the Energy Balance

Fracture of inhomogeneous non-linear elastic rods with a longitudinal circular cylindrical crack is analyzed. Rods have circular cross-section and are loaded in torsion. General approach for analyzing the strain energy release rate is developed by considering the balance of the energy. The rods exhibit continuous material inhomogeneity in both radial and length directions. The general approach is applied to analyze the strain energy release rate for a longitudinal crack in a clamped inhomogeneous rod loaded in torsion. The strain energy release rate is derived also by considering the complementary strain energy for verification.

Mathematical Theory of Normal Waves in Radially Inhomogenous Dielectric Rod

The problem on normal waves in a radially inhomogeneous dielectric rod is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found.

Soft metamaterials with dynamic viscoelastic functionality tuned by pre-deformation.

The small amplitude dynamic response of materials can be tuned by employing inhomogeneous materials capable of large deformation. However, soft materials generally exhibit viscoelastic behaviour, i.e. loss and frequency-dependent effective properties. This is the case for inhomogeneous materials even in the homogenization limit when propagating wavelengths are much longer than phase lengthscales, since soft materials can possess long relaxation times. These media, possessing rich frequency-dependent behaviour over a wide range of low frequencies, can be termed metamaterials in modern terminology. The sub-class that are periodic are frequently termed soft phononic crystals although their strong dynamic behaviour usually depends on wavelengths being of the same order as the microstructure. In this paper we describe how the effective loss and storage moduli associated with longitudinal waves in thin inhomogeneous rods are tuned by pre-stress. Phases are assumed to be quasi-linearly viscoelastic, thus exhibiting time-deformation separability in their constitutive response. We illustrate however that the effective incremental response of the inhomogeneous medium does not exhibit time-deformation separability. For a range of nonlinear materials it is shown that there is strong coupling between the frequency of the small amplitude longitudinal wave and initial large deformation.This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

Controlling the One-Dimensional Motion of Hybrid Vibrational Rod Systems

A mathematical model of the one-dimensional controlled motion of a hybrid vibrational system is proposed. The plant includes an inhomogeneous elastic rod with loads concentrated at its endpoints. An analytical-numerical procedure for finding the eigenvalues (eigenfrequencies) and eigenfunctions is developed. A novel procedure to take into account the nonuniform distributed forces and forces concentrated at the endpoints using a modification of Grinberg’s approach is presented. Efficient statements and constructive approximate solutions of the problems for controlling the motion of a countably dimensional vibrational system with a complex distribution of eigenfrequencies are proposed.