Nonlinear Dynamic Elasticity Research Articles

Variability in the Natural Frequencies of a Nine-Story Concrete Building from Seconds to Decades

Abstract Since 2001, the Southern California Seismic Network has archived continuous waveform data from strong-motion station CI.MIK in Caltech Hall (formerly Millikan Library)—a nine-story reinforced concrete building. Simple spectral analysis of this 20 yr record reveals that the building’s fundamental frequencies have increased gradually by 5.1% (east–west) and 2.3% (north–south), with larger long-term variability up to 9.7% (east–west) and 4.4% (north–south). This finding is unexpected, as previous analysis of forced vibration tests and strong-motion records has shown that between 1968 and 2003 the fundamental frequencies decreased by 22% (east–west) and 12% (north–south), largely attributed to minor structural damage and soil–structure system changes from major earthquakes. Today, the building’s apparent structural stiffness is comparable to what it was in 1986, before the Whittier Narrows earthquake. Using data from earthquakes and forced vibrations, we also document the building’s nonlinear dynamic elasticity, which is characterized by a rapid softening (decrease in apparent frequencies) at the onset of strong motion, followed by a slower, log–linear recovery trend over the scale of minutes. This nonlinear behavior does not appear to have changed with time.

Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow

This study deals with the development of an accurate, efficient and robust method for the numerical solution of the interaction of compressible flow and nonlinear dynamic elasticity. This problem requires the reliable solution of flow in time-dependent domains and the solution of deformations of elastic bodies formed by several materials with complicated geometry depending on time. In this paper, the fluid–structure interaction (FSI) problem is solved numerically by the space-time discontinuous Galerkin method (STDGM). In the case of compressible flow, we use the compressible Navier–Stokes equations formulated by the arbitrary Lagrangian–Eulerian (ALE) method. The elasticity problem uses the non-stationary formulation of the dynamic system using the St. Venant–Kirchhoff and neo-Hookean models. The STDGM for the nonlinear elasticity is tested on the Hron–Turek benchmark. The main novelty of the study is the numerical simulation of the nonlinear vocal fold vibrations excited by the compressible airflow coming from the trachea to the simplified model of the vocal tract. The computations show that the nonlinear elasticity model of the vocal folds is needed in order to obtain substantially higher accuracy of the computed vocal folds deformation than for the linear elasticity model. Moreover, the numerical simulations showed that the differences between the two considered nonlinear material models are very small.

New wave patterns to the doubly dispersive equation in nonlinear dynamic elasticity

This study aims to obtain travelling wave solutions of the doubly dispersive equation in nonlinear dynamic elasticity by the sine-Gordon expansion method. We give physical explanation of the presented solutions under suitable parameters via the 3D, 2D and contour simulations.

Numerical Analysis of Propagation of Nonlinear Waves in Prestressed Solids

<p>The details of numerical algorithms implemented in CAE FIDESYS for the analysis of the propagation of nonlinear waves in elastic and viscoelastic bodies are discussed. It’s taken into account that waves propagation lead to new strains which superimpose on existing stresses (induced anisotropy) in the media. For the formulation of problem we used the theory of repeated superposition of large strains. The details of numerical algorithms for the analysis of the propagation of nonlinear waves in elastic and viscoelastic bodies are discussed. The implementation of the spectral element method for the nonlinear dynamic problems of elasticity under finite strains is considered. Some details of parallel computing on multicore and multiprocessor systems for the problems of nonlinear dynamic elasticity are presented. The results of numerical experiments obtained in CAE Fidesys are shown, in particular: the analysis of propagation of nonlinear shock wave; the analysis of propagation of surface waves; the dynamic processes related with the origination of a hole in a weakly compressible nonlinear-viscoelastic material.</p>

Observations and models of dynamic earthquake triggering

Seismologists have long accepted the idea that step-function perturbations to the deformation field acting on a fault can change its likelihood of, or 'trigger', failure as fault slip. We review the observations that have lead to the very recent recognition that transient perturbations (e.g., associated with seismic waves) also affect failure probabilities, and more broadly, observations of the spatial and temporal variations in both triggering deformations and triggered responses. Many of these cannot be explained by conventional models of earthquake nucleation, requiring consideration of ideas developed in other disciplines, such as those describing and explaining nonlinear dynamic elasticity from rock-mechanics. In addition to the scientific challenges, these observations and models significantly impact earthquake forecasts and hazard assessments. We focus on observations of natural earthquakes and from the rock-mechanics laboratory, and some of the explanatory models that we and others have proposed. While many of these observations and models are just being vetted now, even newer ones related to slow aseismic fault slip and non-volcanic tremor (seismic radiation that scales very differently from that from earthquakes) may lead to substantive modifications and advances. We conclude with a few tantalizing examples as a prelude to a companion presentation.

Nonlinear ultrasound can detect accumulated damage in human bone

Bone micro-damage is commonly accepted as a relevant parameter for fracture risk assessment, but there is no available technique for its non-invasive characterization. The objective of this work is to study the potential of nonlinear ultrasound for damage detection in human bone. Ultrasound is particularly desirable due to its non-invasive and non-ionizing characteristics. We show results illustrating the correlation of progressive fatigue of human bone samples to their nonlinear dynamical response. In our experiments, damage was induced in 30 samples of diaphyseal human femur using fatigue cycling. At intervals in the cycling, the nonlinear response of the samples was assessed applying Nonlinear Resonant Ultrasound Spectroscopy (NRUS). The nonlinear parameter α, which in other materials correlates with the quantity of damage, dramatically increased with the number of mechanical testing cycles. We find a large spread in α in the pristine samples and infer that the spread is due to damage differences in the sample population. As damage accumulates during cycling, we find that α is much more sensitive to damage than other quantities measured, including the slope and hysteresis of the load/displacement curve, and the dynamic wavespeed. To our knowledge, this study represents the first application of the concept of nonlinear dynamic elasticity to human bone. The results are promising, suggesting the value of further work on this topic. Ultimately, the approach may have merit for in vivo bone damage characterization.

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves.

Nonlinear and nonequilibrium dynamics in geomaterials.

The transition from linear to nonlinear dynamical elasticity in rocks is of considerable interest in seismic wave propagation as well as in understanding the basic dynamical processes in consolidated granular materials. We have carried out a careful experimental investigation of this transition for Berea and Fontainebleau sandstones. Below a well-characterized strain, the materials behave linearly, transitioning beyond that point to a nonlinear behavior which can be accurately captured by a simple macroscopic dynamical model. At even higher strains, effects due to a driven nonequilibrium state, and relaxation from it, complicate the characterization of the nonlinear behavior.

Hysteresis and the Dynamic Elasticity of Consolidated Granular Materials

Quasistatic elasticity measurements on rocks show them to be strikingly nonlinear and to have elastic hysteresis with end point memory. When the model for this quasistatic elasticity is extended to the description of nonlinear dynamic elasticity the elastic elements responsible for the hysteresis dominate the behavior. Consequently, in a resonant bar, driven to nonlinearity, the frequency shift and the attenuation are predicted to be nonanalytic functions of the strain field. A resonant bar experiment yielding results in substantial qualitative and quantitative accord with these predictions is reported. {copyright} {ital 1999} {ital The American Physical Society }

Asymptotic behavior of solutions in nonlinear dynamic elasticity

A Dynamic system of 2-D nonlinear elasticity with nonlinear interior dissipation is considered. It is assumed that the principal part of elastic operator is perturbed by the unstructured lower order linear terms. Asymptotic behavior of solutions when time $t \rightarrow 0$ is analyzed. It is shown that in the case of <em>zero load</em> applied to the plate, the arbitrarily large decay rates can be achieved provided that both the "damping" coefficient and the "traction" coefficient are suitably large. This result generalizes and extends, to the nonlinear and multidimensional context, the earlier results obtained only for the one-dimensional linear wave equation. In the case of a <em>loaded plate</em> the existence of compact global attractor attracting all solutions is established.

Third‐order effects in the propagation of finite amplitude stress waves

The propagation of a finite amplitude stress wave through a homogeneous, isotropic elastic solid is analyzed for cases of one‐dimensional dilatation and shear deformation. The strain energy function is expanded in invariant form as a fourth‐order polynomial in displacement gradients. The displacement equations of nonlinear dynamic elasticity are solved by perturbation techniques that yield a uniformly valid approximation. The result for a dilatational wave, which matches previous predictions, is essentially like a planar wave in a fluid, aside from redefinition of the coefficient of nonlinearity. The primary nonlinear effect for a dilatational wave is encountered at the second order. In contrast, the shear wave case, which was not fully analyzed in previous investigations, has no analog in inviscid fluids. The first‐order shear displacement leads to a second‐order dilatational displacement, which then influences the shear displacement at the third order. The overall effect of nonlinearity is to produce amplitude dispersion, as well as a small phase shift in the waveform. [Work supported by the NSF.]

Uniqueness theorems for finite elastodynamics

Described in this paper is a study of the uniqueness of solutions to the boundary-initial value problems of nonlinear dynamical elasticity. Particular consideration is given to the uniqueness implications of certain well-known a priori restrictions on the material response.