Hardening Models and Their Predictions of Material Response

Several hardening models are investigated in this paper to examine how they predict material behavior under closed-loop loading paths. The linear Prager's kinematic hardening rule and a new kinematic hardening model proposed in a previous paper are first used to solve a thin-walled tube problem subjected to combined internal pressure and axial loads. Closed-form transient and steady-state solutions are achieved for closed-loop loading paths, and the corresponding yield center loci and plastic strain trajectories are illustrated. This paper then shows that Phillips's kinematic hardening rule and the two-surface plasticity theory all predict an unreasonable material response. A conclusion is finally reached that the newly proposed kinematic hardening model has more potential than the other models, and further theoretical and experimental investigations are suggested to probe the optimum form of the plastic modulus to make this new model qualitatively, and also quantitatively, describe well material behavior.     

Use of Exact Solutions of Wave Propagation Problems to Guide Implementation of Nonlinear Seismic Ground Response Analysis Procedures

One-dimensional nonlinear ground response analyses provide a more accurate characterization of the true nonlinear soil behavior than equivalent-linear procedures, but the application of nonlinear codes in practice has been limited, which results in part from poorly documented and unclear parameter selection and code usage protocols. In this article, exact (linear frequency-domain) solutions for body wave propagation through an elastic medium are used to establish guidelines for two issues that have long been a source of confusion for users of nonlinear codes. The first issue concerns the specification of input motion as “outcropping” (i.e., equivalent free-surface motions) versus “within” (i.e., motions occurring at depth within a site profile). When the input motion is recorded at the ground surface (e.g., at a rock site), the full outcropping (rock) motion should be used along with an elastic base having a stiffness appropriate for the underlying rock. The second issue concerns the specification of viscous damping (used in most nonlinear codes) or small-strain hysteretic damping (used by one code considered herein), either of which is needed for a stable solution at small strains. For a viscous damping formulation, critical issues include the target value of the viscous damping ratio and the frequencies for which the viscous damping produced by the model matches the target. For codes that allow the use of “full” Rayleigh damping (which has two target frequencies), the target damping ratio should be the small-strain material damping, and the target frequencies should be established through a process by which linear time domain and frequency domain solutions are matched. As a first approximation, the first-mode site frequency and five times that frequency can be used. For codes with different damping models, alternative recommendations are developed.     

Equivalent Viscous Damping for a Bilinear Hysteretic Oscillator

A bilinear hysteretic model is commonly used to study elastoplastic structures. In this paper, a damped, bilinear hysteretic oscillator is studied under harmonic loading. We show the existence of an equivalent viscous damping for small values of a loading parameter such that the associated linear structure and the hysteretic structure have the same frequency response curves. We use the Kryloff-Bogoliuboff method of averaging to find the equivalent viscous damping as a function of the steady state amplitude. We present a model of a bilinear elastic oscillator which captures the steady-state dynamics of the hysteretic oscillator for low values of the loading parameter. We also study the nature of the dependence of the equivalent viscous damping on the kinematic hardening parameter.     

Two-Surface Plasticity Model for Cyclic Undrained Behavior of Clays

Based on a new type of kinematic hardening and the theory of critical state soil mechanics, a two-surface model is herein developed for predicting the undrained behavior of saturated cohesive soils under cyclic loads. The anisotropic hardening rule works in two steps: (1) introducing a new concept, memory center, to take into account the memory of particular loading history; and (2) regulating the movement of the bounding and loading surfaces according to the direction of loading paths in stress space. Conventional triaxial tests have been performed on reconstituted clay samples in the laboratory. The proposed model is verified with respect to the observed behavior of soil samples. It is shown that like a multisurface model, this model can realistically describe some important responses of clays subjected to both monotonic and cyclic loading, while incorporating the memory of particular loading events.     

1D Time-Domain Solution for Seismic Ground Motion Prediction

A full time-domain solution for predicting earthquake ground motion based on the 1D viscoelastic shear-wave equation is presented. The derivation results in a time-domain equation in the form of an infinite impulse response filter. A solution in the time domain has several advantages including causality, direct modeling of impulsive and transient processes, and ease of inclusion of nonlinear soil behavior. The method is applicable to any arbitrarily layered silhouette presented as SH-wave velocity, damping coefficient, and mass density profiles for designated soil intervals. For nonlinear evaluations, an equivalent-linear formulation is incorporated and the standard modulus and damping degradation curves become part of the input set. Input motion can be either rock-outcrop or body-wave motions measured or estimated at the bottom of the geologic profile, and the output is the estimated ground motion time history. Application of the method to vertical array strong motion records from Garner Valley, and Wildlife Site, Calif., shows that predicted surface (and interval) ground motion is virtually identical to that measured. The differences between the results of linear and nonlinear analyses are negligible for most cases. A comparison of the time-domain model with SHAKE shows that SHAKE fails to accurately predict time histories in some situations, whereas the time-domain solution always yields satisfactory predicted surface ground motions.     

Quantifying thermomechanical fatigue of light-metal-matrix composites by mechanical spectroscopy

This article reviews recent progress in understanding the stress-relaxation mechanisms in metal-matrix composites (MMCs) subjected to thermomechanical fatigue. Mechanical loss, dynamic shear modulus, and permanent torsional-strain measurements have been performed with forced oscillations during thermal cycling. A transient mechanical-loss maximum, which is absent in the monolithic material, appears during cooling. It has been attributed to the development of plastic zones around the reinforcements by dislocation generation and motion, which result from the differential thermal contraction of the matrix and reinforcement. This damping maximum is strongly dependent on both measurement and material parameters. The reversible shear-modulus evolution during thermal cycling suggests that no interfacial debonding occurs. In unalloyed matrices, extended thermal-stress-induced pleasicity occurs, leading to a plateau in the shear modulus, which is recovered at low temperatures by plastic-zone overlapping and matrix strain hardening. Simultaneously measured strain-temperature loops exhibit both reversible and permanent plasticity during thermal cycling (strain ratcheting).     

Seismic Simulation of Inelastic Soils via Frequency-Dependent Moduli and Damping

Although soils are known to exhibit nonlinear behavior even at small strains, evaluations of the response of sedimentary basins to strong seismic motions are almost always based on linear, elastic solutions incorporating frequency-independent damping. The principal reasons for this relate to the robustness of the linear algorithm and the ease with which the required parameters can be determined experimentally in engineering practice. Most often, but not always, attempts are made in these analyses to compensate for the inelastic behavior by adjusting the material parameters for the representative levels of strain by means of an iterative method. However, both the standard iterative method and the direct linear solution without iterations suffer from two important shortcomings. First, they do not account for the effect of high confining pressures on inelastic behavior. However, it is known from experiments with sands subjected to cyclic shearing strains under confining pressures of up to 5 Mpa, that in highly confined samples, the material remains nearly elastic for a larger range of strains than do those samples subjected to a lesser pressure. Second, the amplification analyses disregard the fact that small-amplitude, high-frequency components of deformation involve hysteresis loops with little modulus degradation or damping (i.e., nearly elastic secondary loops). Thus, motions computed at the surface of the basin with the standard method usually exhibit unrealistically low amplitudes at high frequencies. This article presents the results obtained with a series of “true” nonlinear numerical analyses with inelastic (Masing-type) soils and layered profiles subjected to broadband earthquake motions, taking into account the effect of the confining pressure. These show that it is possible to simulate closely the actual inelastic behavior of rate-independent soils by means of linear analyses in which the soil moduli and damping change with frequency. It is emphasized that the variation in the linear model of the material parameters with frequency arises solely because the strains have broad frequency content, and not because the materials exhibit any rate dependence when tested cyclically. The proposed new model is successfully applied to a 1-km-deep model for the Mississippi embayment near Memphis, Tenn. The seismograms computed at the surface not only satisfy causality (which cannot be taken for granted when using frequency-dependent parameters), but their spectra contain the full band of frequencies expected.     

Frictional Dissipation in Axially Loaded Simple Straight Strands

In this paper, an analytical investigation is made of the frictional damping properties of axially loaded metallic cables made from one layer of wires helically wrapped around a central wire. Our efforts are focused on the quantity of energy dissipated through friction due to the motions between wires when a cable is loaded. Although the local interwire pivoting drives the response of the cables studied, a first linear model is built where pivoting is allowed, but friction is not taken into account. Then, a law of friction is established and linearized to extend the linear model into a tractable piecewise linear hysteretic one. Through a variety of examples, it appears that the energy dissipated in friction over a load cycle is very small compared to other sources of dissipation, because axially loaded simple straight strands do not experience fretting-induced failures, except close to terminations. It is also shown that modifying the design of such cables is not expected to significantly improve their damping properties.     

Rate-Independent and Rate-Dependent Models for Hysteretic Behavior of Elastomers

Rate-independent and rate-dependent models are presented for the hysteretic shear stress-strain behavior of elastomeric damping materials. A rate-independent hysteretic model, called the general asymptote and power function (GAPF) model, is presented that simulates different types of hysteretic behavior depending on the selected asymptote function. A rate-dependent hysteretic model, formed from a parallel combination of the GAPF model and a dashpot, is also presented which simulates loading frequency dependent behavior in addition to strain amplitude dependent behavior. Closed-form expressions for the shear stress as a function of shear strain are provided for each model. The models are calibrated for three different damping materials, and good correlation between experimental and analytical hysteretic behavior is observed. The models are investigated under variable cyclic loading. To prevent unrealistic stress values (overshooting) after a small strain reversal followed by reloading, a sequential asymptote model is introduced, based on the GAPF model. The hysteretic models were incorporated into a finite-element program within an elastomeric damper element, and the results of nonlinear time history analyses of a building structure with elastomeric dampers under simulated earthquake loading are presented to illustrate behavior of the hysteretic models under several loading histories.     

Elastoplastic Model for the Long-Term Behavior Modeling of Unbound Granular Materials in Flexible Pavements

The constitutive modeling of cyclic plasticity of soils has made great progress, especially in the area of sands liquefaction modeling. Nowadays, the problem of rutting of flexible pavements linked to permanent deformations occurring in the unbound layers is taken into account only by empirical formulas. This paper presents an elastoplastic model with both isotropic and kinematic hardening. The yield surface, plastic potential, and isotropic hardening are based on a model for sands, which takes into account the influence of the initial void ratio and of the mean stress on the mechanical behavior. A kinematic hardening has been added in order to take into account the mechanical behavior of the material for large cycle numbers. A complete model is then developed, simulations are presented, and comparisons with repeated load triaxial tests carried out on a subgrade soil (clayey sand), have been made. These comparisons underline the capabilities of the model to take into account the monotonic, cyclic, and ratchetting behavior of unbound materials for roads.     

Nonlinear Dynamical Model for Hysteresis Based on Nonconvex Potential Energy

A model for hysteretic dynamics is proposed in the current paper. Hysteresis is treated as an input–output relation for a dynamic system. The Lagrangian of the dynamic system is constructed on the basis of a nonconvex potential energy and governing equations of the system dynamics are obtained using the Lagrangian equation. Bifurcations will be induced in the nonlinear dynamics due to the nonconvexity of the potential energy. It is shown that when the coefficients are chosen appropriately, the bifurcation diagram will lead to hysteretic behavior. Both the third- and fifth-order nonlinear terms are investigated and it is shown that the fifth-order nonlinearity is able to give a perfect prediction of experimental hysteretic behaviors. Hysteretic damping force of a magneto-rheological fluid damper and polarization hysteresis in piezoelectric materials are modeled successfully using the current model. The parameter identification for the model is also presented.     

Uniaxial Time-Dependent Ratcheting of SS304 Stainless Steel at High Temperatures

The uniaxial time-dependent strain cyclic behaviors and ratcheting of SS304 stainless steel were studied at high temperatures （350 ℃ and 700 ℃）. The effects of straining and stressing rates, holding time at the peak and/or valley of each cycle in addition to ambient temperature on the cyclic softening/hardening behavior and ratcheting of the material were discussed. It can be seen from experimental results that the material presents remarkable time dependence at 700 ℃, and the ratcheting strain depends greatly on the stressing rate, holding time and ambient temperature. Some significant conclusions are obtained, which are useful to build a constitutive model describiog the time-dependent cyclic deformation of the material.     

Learning of Dynamic Soil Behavior from Downhole Arrays

An increasing number of downhole arrays are deployed to measure motions at the ground surface and within the soil profile. Measurements from these arrays provide an opportunity to improve site response models and to better understand underlying dynamic soil behavior. Parametric inverse analysis approaches have been used to identify constitutive model parameters to achieve a better match with field observations. However, they are limited by the selected material model. Nonparametric inverse analysis approaches identify averaged soil behavior between measurement locations. A novel inverse analysis framework, self-learning simulations (SelfSim), is employed to reproduce the measured downhole array response while extracting the underlying soil behavior of individual soil layers unconstrained by prior assumptions of soil behavior. SelfSim is successfully applied to recordings from Lotung and La Cienega. The extracted soil behavior from few events can be used to reliably predict the measured response for other events. The field extracted soil behavior shows dependencies of shear modulus and damping on cyclic shear strain level, number of loading cycles, and strain rate that are similar qualitatively to those reported from laboratory studies but differ quantitatively.     

Three-Dimensional Nonlinear Seismic Analysis of Single Piles Using Finite Element Model: Effects of Plasticity of Soil

Much of the reported research on the dynamic analysis of pile foundations assumes linear behavior of soil that may not be valid for strong excitations. In this paper, material nonlinearity of the soil caused by plasticity and work hardening is considered in the dynamic analysis of pile foundations. An advanced plasticity based soil model, HiSS, is incorporated in a finite element technique. To simulate radiation effects, proper boundary conditions are used. The model and algorithm are verified with analytical results that are available for elastic and elastoplastic soil models. Analyses are carried out for free-field response and pile head response of end-bearing single piles. Both harmonic and transient excitations are considered in the analyses. Effects of frequency of excitation and stiffness of soil are investigated. It was found that the nonlinearity of soil has significant effects on the pile response for lower and moderate frequencies of excitations (a0<0.6) while at higher frequencies its effects are not as significant.     

Light-Based Motion Tracking of Equipment Subjected to Earthquake Motions

Traditional sensors, such as accelerometers and displacement transducers, are widely used in laboratory and field experiments in earthquake engineering to measure the motions of both structural and nonstructural components. Such sensors, however, must be physically attached to the structure and require cumbersome cabling and configurations and substantial time for setup. For reduced-scale experiments, these conventional sensors may substantially alter the dynamic properties of the system by changing the mass, stiffness, and damping properties of the specimen. Moreover, it is very difficult with traditional sensors to capture the three-dimensional motions of light or oddly shaped components such as microscopes, computers,?or other building contents. In this paper, the methodology of light-based motion tracking is applied to the measurement of the three-dimensional motions of various types of equipment and building contents commonly found in biological and chemical science laboratories. The system is comprised of six high-speed, high-resolution charge-coupled-device (CCD) cameras outfitted with a cluster of red-light emitting diodes (LEDs). Retroreflective (passive) spherical markers discretely located in a scene are tracked in time and used to describe the behavior of various types of equipment and contents subjected to a range of earthquake motions. Results from this study show that the nonintrusive, light-based approach is extremely promising in terms of its ability to capture relative displacements in three orthogonal directions and complementary rotations.     

Tissue motion assessment from 3D echographic speckle tracking

The potential of ultrasonic image speckle tracking to characterize tissue dynamics has been illustrated and validated elsewhere. In this paper we wish to extend this speckle tracking methodology to 3D. To investigate the feasibility of such an approach we first model the image formation process and simulate the 3D speckle motion inherent to tissue linear transformations (translation, rotation and deformation). It is shown that tissue axial rotation and translation are perfectly correlated with the tissue speckle motion while tissue deformation and non-axial rotations corrupt the speckle pattern with a motion-induced noise and are therefore more difficult to track when large motions are concerned. Furthermore, in the framework of our model, our results indicate that short ultrasound pulses with low frequencies and small beamwidths are more desirable for a speckle tracking methodology. The feasibility of speckle tracking is illustrated with an optical flow algorithm. A theoretical study of the correlation between various linear transformations of the tissue and the corresponding ultrasonic speckle motions is also performed.     

Finite element simulation of shot peening of an aluminum alloy considering hardening models

Shot peening is a surface engineering process acknowledged for its potential to develop fatigue strength and erosion-corrosion resistance of metallic materials. In the present study, a 3-D finite element model is employed to predict the effective parameters through a single shot impact and the accuracy of the simulation is validated using previous literatures. In order to induce uniform compressive residual stress patterns across the specimen, processing parameters such as shot velocity, impact angle and friction coefficient should be controlled. It is observed that by increasing the shot velocity and the friction coefficient, the depth of compressive residual stress increases. Moreover, a comparative study between isotropic and kinematic hardening models is performed to evaluate the significant role of the hardening models on the compressive residual stress. It is observed that the kinematic hardening model shows better compatibility with the experimental results compared to the isotropic hardening.     

Dynamics of SDOF Systems with Nonlinear Viscous Damping

The effects of nonlinear viscous damping on the dynamic response of single-degree-of-freedom (SDOF) structural systems are analyzed. This kind of damping characterizes a special class of fluid viscous dampers recently utilized in the field of vibration control as base-isolation devices or viscoelastic elements included in steel braces of framed structures. The analytical relationship adopted to reproduce the mechanical behavior of the fluid viscous dampers is a fractional power-law of the velocity, the exponent of which ranges between 0.1 and 0.2. This function had been previously calibrated on the results of a special experimental survey carried out at the University of Florence. The dynamics of the classical linear-viscous SDOF oscillator is herein reformulated on the basis of the above-mentioned fractional viscous damping (FrVD) relationship. In particular, the transient and steady-state responses are examined in both free and forced vibration conditions. The magnification and transmissibility factors are analytically determined for different damping levels. Moreover, the relation between the viscous damping coefficient and the frequency ratio (i.e., the ratio of the dynamic load to the oscillator frequencies) is defined. The diagrams describing these functions provide direct correlations between the damping as well as the elastic properties of the system and the frequency content of the dynamic action.     

Optimal Control of Earthquake Response Using Semiactive Isolation

The responses of two, low-rise, 2-degree-of-freedom base isolated structures with different isolation periods to a set of near-field earthquake ground motions are investigated under passive linear and nonlinear viscous damping, two pseudoskyhook semiactive control methods, and optimal semiactive control. The structures are isolated with a low damping elastic isolation system in parallel with a controllable damper. The optimal semiactive control strategy minimizes an integral norm of superstructure absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied and is determined through a numerical solution to the Euler–Lagrange equations. The optimal closed-loop performance is evaluated for a controllable damper and is compared to passive viscous damping and causal pseudoskyhook control rules. Results obtained from eight different earthquake records illustrate the type of ground motions and structures for which semiactive damping is most promising.