The effect of strain gradient on the determination of the stress intensity factor is studied for plane crack problems using a simplified model of Toupin-Mindlin gradient elasticity theory combined with a mixed finite element method. A variational statement of the boundary value problem is proposed with respect to displacements, strains, strain gradients, Cauchy stresses, and double stresses, requiring only the first derivatives of displacements and thus avoiding the higher-order derivatives inherent in classical statements. The elastic energy release rate at the crack tip is evaluated through an energy balance concept that accounts for the strain gradient. Numerical results for mode I plane crack problems show that strain gradient effects reduce the stress intensity factor compared with classical elasticity solutions.
A method is presented for constructing explicit analytical solutions to three-dimensional thermoelastic problems for a homogeneous isotropic space, half-space, and layer using direct integration and Vihak key functions. Unlike the case with classical potential functions, the stress tensor components are expressed through the Vihak functions in integral rather than differential form. This prevents an unjustified increase in the differential order of the governing equations for the key functions and the associated difficulties in constructing solutions and satisfying boundary and asymptotic conditions. It is shown that, for the domains considered, the three Vihak functions used in each problem can be expressed through a single function obtained in explicit form. This is particularly convenient in solving problems of controlling the thermally stressed state, where the control function is linked to a single key function. It is demonstrated that only three of the six compatibility equations in terms of stresses are sufficient to determine the Vihak functions. The remaining equations can then be used to uniquely determine the displacement vector components.
The stress-strain state of the metal rod induced by the action of a laser or thermal pulse on its end is investigated taking into account the microstructural transformations accompanying heating and subsequent cooling of the material. The investigation is carried out within the framework of a dynamic formulation of a coupled thermomechanical problem. Method for evaluating the residual stress-strain state of the rod during thermal pulse irradiation of the end is developed. The axisymmetric problem is solved numerically using the thermodynamically consistent theory of inelastic behavior of the material with application of the finite element method and allowance for the dependence of the physical and mechanical properties of the material on temperature. The influence of microstructural transformations on the dynamic and quasi-static response of the material and the residual stress-strain state in the irradiation zone is studied. It is established that microstructural transformations can significantly affect the residual tensile stresses in the surface layer of the material and lead to their significant reduction, or even change them to compressive ones.
A class of three-dimensional problems of thermoelasticity and thermomagnetoelectroelasticity for anisotropic solids with constant volumetric heat generation has been considered. It has been shown that the volume integral of the influence function associated with distributed thermal loading can be reduced to the surface integral by applying the Ostrogradsky–Gauss theorem. The analytical form of the corresponding integrand has been derived using the Radon integral transform. For efficient evaluation of the resulting kernel, it has been proposed to expand it into series of spherical harmonics. Based on the Funk–Hecke theorem, explicit expressions for the coefficients of this expansion have been obtained. The proposed approach provides the analytical tools for investigating the effect of constant volumetric heat generation both throughout the entire body and within its selected subregions. It also enables the analysis of piecewise-constant volumetric heat generation distributions.
The present study is aimed to investigate the effect of temperature, especially in dynamic strain ageing (DSA) regime, on the fatigue crack growth rate (FCGR) behaviour of low C‑Mn steel. The FCGR tests have been performed on standard compact tension specimens with crack surrounded by small-scale-yielding and largescale-yielding conditions. The scanning electron microscopic studies have also been performed to correlate the fatigued surface features with continuum length scale observations in terms of fatigue crack growth rate and fracture mechanics-based crack driving force parameter. The test fatigue crack growth rate behaviour nearly remains unaffected by temperature and crack tip yield conditions.
The free vibrations of circular cylindrical shells with various thickness-to-midsurface radius ratios and different combinations of clamped and hinged edges of the shell are studied using the finite element method. The dependence of the natural frequency vibrations on the increase in the shell thickness is analyzed. The symmetric and asymmetric modes of free vibrations, frequencies, and modes of the vibrations corresponding to bending, torsion, tension-compression, and shear deformations are considered. Significant attention is paid to validation of the reliability of the numerical results.
The sufficient conditions for global robust synchronization for two non-identical single machine infinite bus power systems under impulsive perturbations have been obtained. The problem is considered for the delayed master-slave linear state-error feedback control, where chaos can occur. The problem has been solved under the assumption that there is a positive significant lower limit of the set of intervals between impulse perturbations which must be greater than double delay value. The results obtained have been demonstrated graphically with examples.
An analytical solution has been obtained for the plane problem describing a layer of ideal compressible liquid resting on a rigid surface subjected to a suddenly applied velocity perturbation. Laplace and Fourier integral transforms have been used to derive expressions for the velocity and pressure fields in the frequency domain. The original functions have been recovered using tabulated relations and the convolution theorem. As a result, the exact analytical expressions for pressure and velocity have been obtained in the time domain in the form of series, where each successive term describes a subsequent reflected wave. These expressions have made it possible to evaluate the characteristics of the wave field at the arbitrary point of the liquid layer for various types of non-stationary load. In the case where the excitation suddenly occurs on the fixed region of the bottom, the velocities in the near-field and far-field zones have been calculated as an example.
An approximate solution for the constrained variational problem describing the start-up mode of the tower crane slewing mechanism during steady load hoisting has been developed. The root mean square value of the slewing drive torque has been selected as optimization criterion. The drive torque, its rate of change, and the drive power have been constrained. A dynamic model describing the coupled motion of the hoisting and slewing mechanisms has been formulated in the form of a system of ordinary differential equations. The approximate solution to the optimization problem has been represented by two polynomial functions: the first one ensures the satisfaction of the prescribed boundary conditions of motion, while the second one minimizes the optimization criterion. A modified particle swarm optimization algorithm has been employed to determine the optimal solution. As a result, pendulum-type load oscillations as well as oscillations in the structural elements of the crane have been significantly reduced.
Numerical method for solving the problems of nonstationary vibrations for structural members made of electroelastic functionally graded material is presented taking into account electromechanical losses and interaction with the acoustic medium. Universal numerical approach is used for studying the vibrations of plane layers, cylinders, and spheres. It is based on finite-difference expressions. To take into account the dissipative characteristics of the material, the Kelvin–Voigt viscoelasticity model is used for the case of electroelasticity. Similar to complex moduli, the mechanical, dielectric, and piezoelectric damping coefficients are introduced to account for mechanical and electrical losses. Numerical studies of the behavior of functionally inhomogeneous cylinders and spheres under the action of nonstationary electrical load have shown that mechanical losses have the greatest effect on the damping of vibrations. It is concluded that taking into account energy dissipation according to the viscoelastic Kelvin–Voigt model with experimentally substantiated damping coefficients allows modeling the vibrations of piezoelectric elements with sufficient accuracy. The dependence of the logarithmic decrements of vibrations on the polymer fraction and the geometric dimensions of the cylinder is studied. The damping of vibrations of a piezoelectric sphere immersed in the fluid under electrical load of a step-like profile is analyzed. It is established that the damping of vibrations occurs in this case more than ten times faster than when accounting for internal mechanical losses without the acoustic medium.
Within the framework of the three-dimensional linearized theory of stability of deformable bodies, a plane static problem of compression of a semibounded body (foundation) with a thin coating layer along a crack located in the foundation material parallel to the straight interface between two different materials for elastic potentials corresponding to the case of equal roots of the respective characteristic equations has been investigated. Analytical-numerical approach is proposed that allows reducing the original boundary-value problem stated in terms of potential harmonic functions to the eigenvalue problem for the system of the Fredholm integral equations of the first kind and the additional condition obtained in general form for a wide class of combinations of two different materials. For compressible materials with a harmonic-type potential, the critical values of the load parameter have been determined, and their dependence on the geometric, physical and mechanical parameters of the problem has been analyzed.
The application of variational methods in mechanics to the principles of formation of functional systems of living objects is described. In particular, it is important to search for the mathematical formalism of such principles. The research is carried out using methods of system analysis and variational calculation. The details of the structure of functioning biological objects are established. A new principle of coordinated optimum is proposed as a form of the principle of minimum energy dissipation during the control processes in living nature.
The problems of calculating the number of cycles to fatigue of prismatic bars under uniaxial asymmetrical bending have been solved. The solutions are based on the concept of equivalent stresses, the structure of which is given by the hypothesis of the existence of single isochronous ultimate stress diagrams using power transcendental functions. The calculation results are approved experimentally.
The formulation and methodology for solving the problem of electrothermomechanical behavior and predicting the durability of inelastic shear-sensitive flexible cylindrical shells with piezoelectric sensors and actuators under axisymmetric resonant vibrations are presented. The numerical experiments for a shell with a piezosensor are used to investigate the influence of geometric nonlinearity on the frequency dependence of deflection amplitudes, the vibration heating temperature and the electrical index of the sensor. Based on the criterion for assessing local durability by the permissible values of the maximum heating temperature, it is studied how the operating time of the system depends on the extreme amplitudes of mechanical load and heat dissipation conditions from its surfaces.
The mechanism of forced generation and the structure of cross-waves on the free liquid surface in a partially filled cylindrical container have been investigated by the superposition method. For the mathematical description of the cross-waves, single-mode and three-mode nonlinear models have been developed. The single-mode model has described the generation of the cross-waves resulting from the excitation of a resonant mode when the frequency of the cylindrical wall coincides with the corresponding natural frequency. The three-mode model has shown that the cross-waves arise as a superposition of three natural modes, which has enabled an accurate approximation of experimentally observed wave patterns on the free liquid surface. Graphical representations of the free-surface profile obtained with three natural modes have demonstrated the principal features of the wave structures observed near the container wall in the partially filled cylindrical container with a vibrating surface.
New numerical-analytical method for solving problems of physically nonlinear deformation of thin shallow shells has been developed. The shells have a complex-shaped plan and are made of materials with different resistance in tension and compression. The method is based on the combined use of the R-functions, the Ritz method and the Runge–Kutta–Merson method. The R-functions method makes it possible to present an approximate solution in the form that exactly satisfies the boundary conditions and is invariant with respect to the domain shape where the approximate solution is sought. The problem of nonlinear-elastic deformation of the shallow shell of complex shape with combined boundary conditions has been solved. The influence of the direction of external loading, the geometric shape, and the boundary conditions on the stress–strain state has been studied.
A new methodological approach to the statement and solution of the variational problem of the brachistochrone on an inclined plane is proposed. This approach takes into account the joint minimization of the time of motion of a material point on the inclined plane without friction and initial velocity in a vertical uniform gravitational field, and the length of the curve along which its brachistochrone motion is carried out. Therefore, a multiplicative criterion is created which allows for optimizing the product of two separate criteria for minimizing the time and length of the curve of brachistochrone motion of the material point. In the process of solving, a corresponding isoperimetric problem is stated and solved which allows for an analytical solution in closed form as the systems of parametric equations that algebraically describe the brachistochrone curves on the inclined plane. All possible types and variants of curves that arise in this problem depending on the boundary conditions and parameters of the inclined plane are considered. Based on the solution of the auxiliary problem and using a new multiplicative criterion, an optimal solution to the brachistochrone problem on the inclined plane is obtained.
A sliding mode controller for wheel mobile robots to follow a predefined trajectory on the basis of studies of kinematic and dynamic modeling is designed. The chattering issue of the sliding mode controller is always a critical weakness affecting both the system and the stability of the robot during trajectory tracking. This paper also focuses on constructing a kinematic model to provide velocity robot input into the sliding mode controller. The model follows the reference velocity and tracks a reference trajectory derived from the dynamic model synchronously. Addressing both velocity and trajectory enables the optimization of path length and robot stability. To achieve this, the controller must handle environmental disturbances and uncertainties in the system, such as unexpected errors in the design or mechanical fabrication of the system. The stability of the control system is initially evaluated via Lyapunov stability theory. The effectiveness of the model is evaluated through MATLAB/Simulink simulations which consider various levels of trajectory complexity and noise disturbances.
The influence of the Winkler elastic foundation on the dynamics of three-layer conical shells with a discretely inhomogeneous lightweight rib-reinforced core under impulse load has been analyzed for various boundary conditions. In the lightweight core, the rib spacing greatly exceeds the rib cross-sectional dimensions; therefore, the theory of independent static and kinematic hypotheses for each layer has been applied. Based on the Hamilton–Ostrogradsky variational principle, the equations of motion and the natural boundary and initial conditions for the three-layer conical shell on the Winkler elastic foundation have been derived using the Timoshenko shell-and-beam theory. This approach has made it possible to construct a physically consistent finite element model for the three-layer conical shells composed of different materials and to obtain numerical results describing the dynamic response of the structure resting on the elastic foundation. Through specific examples, the study has analyzed the influence of geometric parameters, taper angle, boundary conditions, and elastic-medium stiffness on the dynamics and natural frequencies of the three-layer conical structure subjected to impulse load. New mechanical effects have been established.
The free vibration behavior of beams with both variable cross-sectional geometry and material inhomogeneity along the beam axial direction is studied. The finite element method is employed to analyze the influence of cross-sectional variations on the dynamic characteristics of beams with polynomially graded material properties along the beam length. Both one- and three-dimensional finite elements are developed incorporating axial material gradients via user-defined material subroutines (UMAT) in ABAQUS package. The free vibration analysis is performed using both finite element models to determine natural frequencies and mode shapes across a wide range of tapered beam configurations. The accuracy and efficiency of the models are validated by comparing the numerical results with data available in the literature. The findings reveal that variations in cross-sectional geometry have a significant impact on the natural frequencies highlighting their critical role in the dynamic response of axially inhomogeneous tapered beams.
