Dynamic magnification factors of SDOF oscillators under harmonic loading

The magnification factor for the steady-state response of a SDOF system under harmonic loading is described in many structural dynamics textbooks; the well-known analytical solution is easily obtained from the solution to the damped equation of motion for harmonic loading. The complete and steady-state solutions can differ significantly. An analytical expression for the maximum response to the complete solution (steady state plus transient) remains elusive; however, a simple analytical expression is identified herein for the undamped case. Differences in the magnification factors obtained for the two solutions are discussed.     

Micro-mechanical modeling of anisotropic inelastic response of soft tissues under supra-physiological loading

Under supra-physiological loading, soft tissues exhibit many inelastic phenomena, such as stress softening, hysteresis and permanent set [1]. Knowledge of the mechanical response of soft tissues under such a large range of deformation is vital for optimizations of vascular medical devices or improvement of injury prevention techniques. In this work, a micro-mechanical model is proposed for soft collagenous tissues. Besides the anisotropy the model can also describe the aforementioned inelastic effects under extremal loadings. To this end, the dispersion of collagen fibers in the soft tissues is captured by a probability distribution of fiber orientations around preferred directions. The deformation induced damage inside the tissue is assumed to take place between collagen fibrils and is included within a statistical mechanical framework. Finally, the accuracy of the model is assessed by comparison with experimental data available in the literature. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deflection analysis of micro-scale microstretch thermoelastic beam resonators under harmonic loading

In this paper, the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler–Bernoulli beam resonators, due to time harmonic load have been investigated. The axial ends of the beam are assumed to be at either clamped-clamped, simply supported-simply supported or clamped-free conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection. The numerical simulation has been carried out with the help of MATLAB software for magnesium like material. The graphical representations and observations have been discussed for deflection of beam under various boundary conditions and for distinct considered values of time and space as well.     

Mathematical modeling of flows through inelastic porous solids

We propose a framework, based on classical mixture theory, to describe the isothermal flow of an incompressible fluid through a deformable inelastic porous solid. The modeling of the behavior of the inelastic solid takes into account changes in the elastic response due to evolution in the microstructure of the material. We apply the model to a compression layer problem. The mathematical problem generated by the model is a free boundary problem.     

A three-dimensional discrete element model for heterogeneous solids under mechanical loading

The Discrete Element Method (DEM) is used to model solids under quasi-static and dynamic loading. In order to model elastic bodies, a microscopic model must be able to represent the macroscopic properties of the material. An energy-based approach to determine the model parameters is presented for an unit cell assemblage of 13 particles in the hexagonal close packing of spheres. The stored strain energy in the unit cell is added up and the specific strain energy expression is derived with respect to the macroscopic strains. The resulting stress-strain relations can be compared to the constitutive equations of the elastic continuum. To avoid cubic anisotropy for Poisson's ratios above zero, an advanced octahedrongap-filled hexagonal close packing of spheres is presented and validated. This approach for regular lattices can be transferred to heterogeneous materials with the goal of describing porous media such as cement stone. Therefore it is possible to use the presented regular lattices with statistically distributed properties or to investigate irregular distributions of particles. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary integral equations for an interface linear crack under harmonic loading

The present study is devoted to application of boundary integral equations to the problem of a linear crack located on the bimaterial interface under time-harmonic loading. Using the Somigliana dynamic identity the system of boundary integral equations for displacements and tractions at the interface is derived. For the numerical solution the collocation method with piecewise constant approximation on each linear continuous boundary elements is used. The distributions of the displacements are computed for different values of the frequency of the incident tension-compression wave. Results are compared with static ones.     

Variational approach for strain-driven,incremental homogenization of inelastic solids

This study is devoted to finite element modeling of phenomenological rate independent elastoplasticity coupled to damage. The new aspect concerns the treatment of both types of inelasticities. Consistent with physical observations these are interpreted as two pseudophases characterized by specific incremental quasihyperelastic potentials. On this basis, the governing macroscopic energetics is derived as a quasiconvex potential for macro-stresses and corresponding variational formulation is discretized and solved for two limit cases: pure plastic response and pure scalar damage response. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of physical nonlinearity on resonance vibrations and dissipative heating of inelastic rectangular plates

On the basis of a concept of complex characteristics, we present a statement of nonlinear conjugate problems of forced resonance vibrations and dissipative heating of inelastic plates of physically nonlinear materials, for which the real and imaginary parts of mechanical complex characteristics depend on the amplitude of deformations. Numerical-analytical methods for the solution of the indicated nonlinear problems are presented. An analysis of the influence of physical nonlinearity on the amplitude–frequency and temperature–frequency characteristics and on the coefficient of damping of vibrations of rectangular plates is performed.     

An implicit three-dimensional model for describing the inelastic response of solids undergoing finite deformation

The aim of this paper is to develop a new unified class of 3D nonlinear anisotropic finite deformation inelasticity model that (1) exhibits rate-independent or dependent hysteretic response (i.e., response wherein reversal of the external stimuli does not cause reversal of the path in state space) with or without yield surfaces. The hysteresis persists with quasistatic loading. (2) Encompasses a wide range of different types of inelasticity models (such as Mullins effect in rubber, rock and soil mechanics, traditional metal plasticity, hysteretic behavior of shape memory materials) into a simple unified framework that is relatively easy to implement in computational schemes and (3) does not require any a priori particular notion of plastic strain or yield function. The core idea behind the approach is the development of an system of implicit rate equations that allow for the continuity of the response but with different rates along different directions. The theory, which is in purely mechanical setting, subsumes and generalizes many commonly used approaches for hypoelasticity and rate-independent plasticity. We illustrate its capability by modeling the Mullins effect which is the inelastic behavior of certain rubbery materials. We are able to simulate the entire cyclic response without the use of additional internal variables, i.e., the entire response is modeled by using an implicit function of stress and strain measures and their rates.     

Vibrations of a beam-string complex system under a moving force

In this paper, the dynamic response of an Euler-Bernoulli beam and string system traversed by a constant moving force is considered. The force is moving with a constant velocity on the top beam. The complex system is finite, simply supported, parallel one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a beam-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case the aperiodic part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations. The dynamic influence lines of complex systems may be used for the analysis the complex models of moving load. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A theory of heating of Voigt solids and fluids by external energy sources and flame theory

The purpose of this paper is to develop

1. a theory of laser stimulated vaporization of droplets,

2. a theory of internal heating resulting from vibration waves in linearly responding elastic material, and

3. flame theory.

There are applications to sending information through clouds on laser beams and to the control of temperature in ultrasonic welding, and improvement of the design of aircraft engines and the processes used for the destruction of toxic chemicals.

We develop a theory of thermal excursions resulting from ultrasonic welding in 3 and 7 dimensions, and interpret it as an elastic interaction with damping in a Voigt solid. It is hypothesized that with good control of temperature, one could achieve strong and uniform welds by this process and greatly reduce the cost of manufacturing aircraft, and other aluminum structures. We consider equations describing the conservation of mass, momentum, and energy coupled by an equation of state, and consider general mass, momentum, and energy transfer relationships in a compressible body subjected to external stimuli. For the Voigt solid theory, a linear elastic theory with damping forces, we show how some simple local time averaging gives us a dovetailed system consisting of the elastic wave equations whose solution provides the source term for an otherwise uncoupled heat equation. For the more general theory of droplet vaporization, we illustrate a general nonlinear energy equation which includes a radiation energy conductivity term. We get a class of exact solutions for a nonlinear flame front boundary value problem.     

Thermo-mechanical computation of inelastic pavement structures under rolling tire loads

A realistic and numerically efficient computation of the tire-pavement interaction is essential for the investigation of the structural behavior of pavements under rolling tire load as base of the development of more durable pavement structures. The paper presents a thermo-mechanical coupled simulation model based on an Arbitrary Lagrangian Eulerian (ALE) formulation that considers the inelastic and temperature dependent material properties of asphalt. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deformation of polymers under friction loading

A study has been made of the laws of distribution of elastic and residual deformations in the surface layers of polymers subjected to friction. The dependence of these deformations on normal pressure, sliding velocity, and duration of loading has been determined. A relation between deformation and antifriction characteristics has been established, and the relative effect of normal and tangential loads on surface-layer deformation determined.Mekhanika Polimerov, Vol. 3, No. 3, pp. 539–543, 1967     

Self-heating and failure of plastics under cyclic loading

It has been shown that plastics heated by cyclic deformation have two characteristic temperatures—a critical temperature T_hc, at which the most heavily stressed part of the test piece is intensely heated, and a temperature T_h, at which the test piece fails. The values T_hc and T_h are determined not only by the physicomechanical properties of the material, but also by the state of stress and strain, by the scale factor, and by the heat transfer conditions. It has likewise been shown that the form of the fatigue fracture surface of plastics is determined by the type of deformation and the temperature T_h.Mekhanika Polimerov, Vol. 3, No. 3, pp. 483–492, 1967     

Plastic behavior of copper under impulsive heating

Zusammenfassung Es wird eine Methode zur Bestimmung der dynamischen Fließspannung in Abhängigkeit von der Temperatur beschrieben. Stossartig erzeugte Elektronenstrahlen werden benutzt, welche simultan Wärme und Stossenergie produzieren. Dabei wird kein hydrodynamisches Modell für das Material benützt. Die Methode wird auf die Bestimmung der Abhängigkeit von Fließspannung und Temperatur für Kupfer angewandt.     

Vibrations of systems with distributed parameters under the influence of random disturbances

Ukrainian Mathematical Journal -     

Effect of loading regime on the heating of plastics subjected to cyclic deformation

In cyclic loading in the W = const and ₀ = const regimes, as in the ₀ = const regime, both a low-temperature and a high-temperature thermal regime may exist, with an abrupt transition from the former to the latter. The effect of the loading regime on the temperature dependence of the hysteresis temperature rise T, which depends on the elastohysteresis properties of the material, results in the following relationship between the critical temperature rises: T ₀ ^c T _W ^c T ^c .Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 465–469, May–June, 1969.     

Strength of reinforced plastics under static loading

The question of the static fatigue of reinforced plastics is considered for plane stress. Relations that predict the static fatigue are given for very simple types of loading. Experimental data are presented for thin-walled tubes of SVAM (5:1) glass-reinforced plastic subjected to long-time tests at constant internal pressure.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 265–273, March–April, 1969.     

Behavior of rigid plastics under cyclic loading

Research progress on the dynamic fatigue of plastics is briefly reviewed. Attention is concentrated on the problems of damage accumulation and self-heating. The effect of various factors on the fatigue of plastics is considered. The possibility of predicting the cyclic life-time from the results of long-time static strength tests is examined. The prospects for the construction of a theory of fatigue strength in complex states of stress are weighed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 97–107, January–February, 1969.