Inversion of a full-range stress-strain relation for stainless steel alloys

This paper presents an approximate inversion of the stress-strain relation for stainless steel alloys. Using currently available stress-strain relations based on a modified Ramberg-Osgood equation, a new expression for the stress σ as an explicit function of the total strain ε is obtained. The new expression is valid over the full-range of the stress well beyond the 0.2% proof stress σ_0.2, defined as the stress level corresponding to the plastic strain value of 0.2%. The validity of the inverted expression is tested over a wide range of material parameters. The tests show that the new expression results in stress-strain curves which are both qualitatively and quantitatively consistent with the fully iterated numerical solution of the full-range stress-strain relation.     

On the Iwan models for lap-type bolted joints

The paper presents mathematical modeling of the non-linear constitutive relation for bolted joints in the framework of the Kragelsky-Demkin theory of rough contact. It is shown that this approach, which maintains the tribology-related features of bolted joint interfaces, leads to a singular Iwan distribution density. In particular, we show that the Iwan density is expressed in terms of the height distribution density of the surface asperities, whereas its singular exponent is determined by the shape exponent of the surface asperities. Following this, constitutive relations for lap joints and the corresponding backbone (force-deflection) curves are obtained. Finally, Masing's hypothesis is applied and Goodman's relation for energy dissipation is recovered in order to describe the effects of cyclic loading. The two cases of a rough surface in contact with a flat surface and of two contacting rough surfaces are treated separately.     

Inversion of Ramberg–Osgood equation and description of hysteresis loops

The Ramberg–Osgood equation has been approximately inverted. Four orders of approximations providing progressively more accurate inversions are considered. The second order inversion is used to develop closed-form relations for stress in terms of strain and for cyclic stress in terms of cyclic strain. Explicit relations between the cyclic-strength coefficient and the cyclic strain-hardening exponent and between the cyclic strain-hardening exponent and the elastic modulus are developed. Application to nine different engineering metals shows that the proposed approximate inversion provides a powerful tool for describing stress–strain relations. Examples of hysteresis loops for a number of strain time-histories are presented. Also, it is shown that augmenting the strain time-history with fictitious strain segments to close all the hysteresis loops yields the same stress–strain relation as that obtained through the rain-flow counting method.     

Trapping vibratory energy of main linear structures by coupling light systems with geometrical and material non-linearities

Cubic potential and hysteresis behavior (Bouc–Wen type) of a non-linear energy sink are used to localize the vibratory energy of a linear structure. A general methodology is presented to deal with time evolutionary energy exchanges between two oscillators. Invariant manifold of the system and its stability borders are detected at fast time scale while traced equilibrium and singular points at slow time scale let us predict possible behaviors of the system during its pseudo-stationary regime(s). The paper is followed by an example that considers the Dahl model for representing the hysteresis behavior of the non-linear energy sink. All analytical developments and results are compared with those obtained by direct integration of system equations. Obtained analytical developments can be endowed for designing non-linear energy sink devices with hysteresis behavior to localize vibratory energy of main structures for the aim of passive control, energy harvesting and/or both of them.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

Development of data-based model-free representation of non-conservative dissipative systems

A general procedure is presented for developing data-based, non-parametric models of non-linear multi-degree-of-freedom, non-conservative, dissipative systems. Two broad classes of methods are discussed: one relying on the representation of the system restoring forces in a polynomial-basis format, and the other using artificial neural networks to map the complex transformations relating the system state variables to the needed system outputs. A non-linear two-degree-of-freedom system is used to formulate the approach under discussion and to generate synthetic data for calibrating the efficiency of the two methods in capturing complex non-linear phenomena (such as dry friction, hysteresis, dead-space non-linearities, and polynomial-type non-linearities) that are widely encountered in the applied mechanics field. Subsequently, a reconfigurable test apparatus was used to generate experimental measurements from a physical non-linear “joint” involving two-dimensional motion (translation and rotation) and complicated interaction forces between the different motion axes, among its internal elements. Both the polynomial-basis approach and the neural network method were used to develop high-fidelity, non-parametric models of the physical test article. The ability of the identified models to accurately “generalize” the essential features of the non-linear system was verified by comparing the predictions of the models with experimental measurements from data sets corresponding to different excitations than those used for identification purposes. It is shown that the identification techniques under discussion can be useful tools for developing accurate simulation models of complex multi-dimensional non-linear systems under broadband excitation.     

Characterization of strain rate effects in sheet laser forming

This work presents numerical simulations and experimental validation of sheet laser forming processes using a single-step straight path with different laser beam powers (four levels ranging from 30 W to 120 W) and scanning speeds (four levels ranging from 5 mm/s to 20 mm/s) in graphite-coated AISI 304 stainless steel 0.6-mm-thick sheets. The numerical simulations of these cases are performed via a coupled thermomechanical finite element formulation accounting for large strains, temperature-dependent material properties and convection–radiation phenomena. Firstly, a rate-independent plastic model is used. Although this model adequately predicts the final bending angle for the cases achieving relatively low maximum temperatures, i.e. cases with low laser beam powers and high scanning speeds, it fails in describing the deformation pattern for the cases with higher maximum temperatures, i.e. cases with high laser beam powers and low scanning speeds. Secondly, in order to overcome this drawback, a rate-dependent viscoplastic model including a stress-dependent viscosity law is proposed to simulate the same cases. The final bending angles provided by this model are found to be in good agreement with the experimental measurements for the whole ranges of laser beam power and scanning speed studied in this work. Therefore, the use of this viscoplastic model in the simulation of sheet laser forming allows us to conclude that the strain rate effects, which mainly play a relevant role at high temperatures, can be adequately characterized.     

Characterization of hysteretic stress-strain behavior using the integrated Preisach density

In this paper, we introduce the concept of Integrated Preisach-Mayergoyz (IPM) density to analyze static uniaxial compression tests at values well below the critical strength, and to characterize the elasticity of materials with hysteresis in their stress-strain relationship. The IPM density can be deduced from a particular force protocol following basic data treatment. The advantage of the IPM density over prior approaches is that no second order differentiation of the data is required which reduces the errors and uncertainties typical for past practice in the specific context of rock elasticity using scanning curves and PM density analysis. The characterization of the elasticity of the material is established in terms of a non-hysteretic strain contribution in the form of a non-linear but reversible equation of state, and a hysteretic contribution represented by the IPM density. The IPM inversion procedure is tested for simulated stress-strain data subjected to additive noise, and the results are compared to the traditional methodology. In addition, we analyze the hysteretic and non-hysteretic characteristics of five natural building stones, and show evidence for a classification based on the inferred properties.     

Hysteresis models of dynamic mode atomic force microscopes: analysis and identification via harmonic balance

A new class of models based on hysteresis functions is developed to describe the operation of dynamic mode atomic force microscopy. Such models can account for dissipative phenomena affecting the interaction between the probe and the sample. The model analysis, which is developed using frequency domain techniques, provides a insights into experimentally observed behavior. Experimental data corroborates the models developed.     

On non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities

This paper examines the validity of non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities. As an example, we treat a hinged-hinged Euler-Bernoulli beam resting on a non-linear elastic foundation with distributed quadratic and cubic non-linearities, and investigate the primary (Ω≈ω_n) and subharmonic (Ω≈2ω_n) resonances, in which Ω and ω_n are the driving and natural frequencies, respectively. The steady-state responses are found by using two different approaches. In the first approach, the method of multiple scales is applied directly to the governing equation that is a non-linear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure, and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained numerically by the finite difference method.     

Symmetry classification and optimal systems of a non-linear wave equation

In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.     

Control of primary and subharmonic resonances of a Duffing oscillator via non-linear energy sink

The use of non-linear energy sink to passively control vibrations of a non-linear main structure under the effect of bi-frequency harmonic excitation is addressed here. The excitation is assumed to induce both 1:1 and 1:3 resonance, and the response of the system is studied after using the Multiple Scale/Harmonic Balance Method, applied to obtain amplitude modulation equations in the slow time scale. The efficiency of the non-linear energy sink to reduce or suppress vibrations of the main structure is finally discussed.     

Non-linear output frequency response functions of MDOF systems with multiple non-linear components

In engineering practice, most mechanical and structural systems are modelled as multi-degree-of-freedom (MDOF) systems such as, e.g., the periodic structures. When some components within the systems have non-linear characteristics, the whole system will behave non-linearly. The concept of non-linear output frequency response functions (NOFRFs) was proposed by the authors recently and provides a simple way to investigate non-linear systems in the frequency domain. The present study is concerned with investigating the inherent relationships between the NOFRFs for any two masses of non-linear MDOF systems with multiple non-linear components. The results reveal very important properties of the non-linear systems. These properties clearly indicate how the system linear characteristic parameters govern the propagation of the non-linear effect induced by non-linear components in the system. One potential application of the results is to detect and locate faults in engineering structures which make the structures behave non-linearly.     

Shape synchronization of drive-response for a class of two-dimensional chaotic systems via continuous controllers

This paper investigates the drive-response synchronization in shape for a class of two-dimensional continuous systems of chaos. The shape of the chaotic attractor of the drive chaotic system is considered in this paper. Using the signed curvatures of plane curves to describe the shapes of trajectories for drive and response systems, the continuous controller for shape synchronization is synthesized based on the fundamental theorem on plane curves in classical differential geometry. The continuous controller synthesized can guarantee that the response system is synchronized with the drive chaotic system in shape. The shape synchronization is obtained in spite of different dimensions in drive and response systems. Finally, the Duffing oscillator is utilized as an illustrative example. Simulation results show that the method proposed in this paper is effective for the application of secure communication.     

Multiscale mass-spring models of carbon nanotube foams

This article is concerned with the mechanical properties of dense, vertically aligned CNT foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition) which represents a finite portion of the foam thickness. An upper-scale model formed by a chain of non-uniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are fitted with excellent agreement.     

An analytic approximate approach for free oscillations of self-excited systems

A new analytic approximate technique for non-linear problems, namely the homotopy analysis method, is employed to propose an approach for free oscillations of self-excited systems. Different from perturbation methods on this topic, this approach does not depend upon any small/large parameters at all and therefore is valid for free oscillations of all self-excited systems. Besides, unlike other analytic techniques, this approach provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. Two examples are employed to illustrate the validity and flexibility of this approach.     

An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude

The currently developed analytic technique known as the homotopy analysis method is employed to propose a new approach for free oscillations of positively damped systems with algebraically decaying amplitude. In contrast to perturbation techniques, this approach is valid even for damped systems without any small/large parameters. Besides, unlike other analytic techniques, this approach itself provides us with a convenient way to adjust and control convergence of approximation series. Some typical examples are employed to illustrate its validity, effectiveness and flexibility.     

On the identification of non-linear mechanical systems using orthogonal functions

Real world mechanical systems present non-linear behavior and in many cases simple linearization in modeling the system would not lead to satisfactory results. Coulomb damping and cubic stiffness are typical examples of system parameters currently used in non-linear models of mechanical systems. This paper uses orthogonal functions to represent input and output signals. These functions are easily integrated by using a so-called operational matrix of integration. Consequently, it is possible to transform the non-linear differential equations of motion into algebraic equations. After mathematical manipulation the unknown linear and non-linear parameters are determined. Numerical simulations, involving single and two degree-of-freedom mechanical systems, confirm the efficiency of the above methodology.     

Effect of non-linearities in the identification and fault detection of gear-pair systems

This study investigates issues related to parametric identification and health monitoring of dynamical systems with non-linear characteristics. In the first part, a gear-pair system supported on bearings with rolling elements is selected as an example mechanical model and the corresponding equations of motion are set up. This model possesses strongly non-linear characteristics, accounting for gear backlash and bearing stiffness non-linearities. Then, the basic steps of the parametric identification and fault detection procedure employed are outlined briefly. In particular, a Bayesian statistical framework is adopted in order to estimate the optimal values of the gear and bearing model parameters. This is achieved by combining experimental information from vibration measurements with theoretical information built into a parametric mathematical model of the system. In the second part of the study, characteristic numerical results are presented. First, based on the effect of the system parameters on its dynamics, a solid basis is created for explaining some of the peculiar results obtained by applying classical gradient-based optimization methodologies for the strongly non-linear system examined. Some serious difficulties, associated with the existence of irregular response or the coexistence of multiple motions, are first pointed out. A solution to some of these problems, through the application of a suitable genetic algorithm, is then presented. Special problems, related to more classical identification issues associated with the presence of measurement noise and model error, are also investigated.     

Vibrations of rod-rigid element systems with a local non-linearity

The paper deals with vibrations of systems consisting of non-coaxial rods connected by rigid bodies and of a local non-linearity. The motion of the rods is described by classical wave equation and the solution of the d’Alembert type is applied in the study. This leads to solving ordinary differential equations with a retarded argument. The local non-linearity is described through irrational functions and in a special case it includes the polynomial of the third degree. Detailed considerations are given for a system consisting of three rods and two rigid bodies. In numerical analysis non-linear effects are discussed. The results concerning harmonic vibrations are presented for the local non-linearities having characteristics of a soft type as well as of a hard type.