Lateral Inertia Effect Research Articles

Longitudinal Seismic Response of a Pipe Pile Embedded in Saturated Thermoelastic Ground Subjected to Rayleigh Waves

This work investigates the longitudinal vibration characteristics of the pipe pile in the liquid-saturated ground due to Rayleigh waves propagation with considering the influence of the nonisothermal condition on the wave motion. Firstly, the Rayleigh wavefield in the liquid-saturated thermoelastic ground is deduced theoretically with taking advantage of the Biot theory and the non-Fourier law of heat conduction. Secondly, the longitudinal frequency domain solution of the pipe pile under Rayleigh waves is derived analytically. Finally, the dependence of the pipe pile response on the physical parameter is illustrated by utilizing several numerical examples. The result reveals that increase in the thermal expansion coefficient of saturated soil significantly augments the longitudinal displacement and side friction of pile body. Increase in the permeability of saturated soil will reduce the pile head displacement. When the pile length is less than its effective length, the rise of the pile length will reduce the pile head displacement. Augment of both the outer radius and wall thickness of the pipe pile will help to reduce the pile head displacement. However, the lateral inertia effect of the pipe pile has slight influence on its longitudinal behavior under Rayleigh waves.

Vertical dynamic response of pipe pile embedded in unsaturated soil based on fractional‐order standard linear solid model and Rayleigh‐Love rod model

AbstractConsidering the flow‐independent viscosity of soil skeleton and the lateral inertia effect of pipe pile, the vertical dynamic behavior of a pipe pile embedded in unsaturated soil is investigated in this work when the pipe pile is subjected to the vertical dynamic loading. In order to model this problem, the fractional‐order standard linear solid (FSLS) model is introduced to characterize the flow‐independent viscosity of the soil skeleton, the Rayleigh‐Love rod model is employed to specify the lateral inertia effect of pipe pile, while both vertical and radial soil displacement are entirely considered. The analytical solution of the pipe pile complex impedance is obtained through the rigorous theoretical derivation. Numerical results with known analytical solution are finally presented to discuss the influence of the model parameters for the FSLS model, lateral inertia effect and geometrical characteristic for the pipe pile, and saturation degree and intrinsic permeability for the unsaturated soil on the vertical complex impedance of the pipe pile. The main results indicate that enlarging the fractional‐order index, raising the strain relaxation time, or shortening the stress relaxation time will reduce the oscillation amplitude and natural frequency for the dynamic stiffness and damping of the pipe pile but will improve their mean value. The lateral inertia effect of pipe pile has effect on its complex impedance only in the high‐frequency region. The outer and inner radii of pipe pile, pile length, soil saturation degree, and soil intrinsic permeability are highly correlated with the pipe pile complex impedance.

Modelling the effect of lateral inertia on the dynamic response of aluminium square tube extrusions

Crashworthiness of automotives require the design of crush tubes installed as a frontal structure to absorb energy by deforming plastically during collision. The crushing behaviour of aluminium extrusions modelled as square tubes is investigated under quasi-static and dynamic loading conditions using the non-linear explicit finite element code LS DYNA. Previous works have attributed the dynamic increase of the peak and mean crushing load of aluminium thin-walled structures under axial compression, to inertia effects. Results of our investigation revealed that there is a considerable thickening of the shell side walls before and during progressive folding. This shell thickening is established as the cause of lateral inertia which consequently results in increased crush strength. In addition, a positive correlation was established between the shell thickening and crushing speeds. 

Dispersion relation for harmonic generation in nonlinear elastic waves

We present a theory for the dispersion of generated harmonics in a traveling nonlinear wave. The harmonics dispersion relation—derived by the theory—provides direct and exact prediction of the collective harmonics spectrum in the frequency-wavenumber domain and does so without prior knowledge of the spatial-temporal solution. The new relation is applicable to a family of initial wave functions characterized by an initial amplitude and wavenumber. We demonstrate the theory on nonlinear elastic waves in a homogeneous rod and demonstrate its extension to periodic rods. We investigate a thick elastic rod admitting longitudinal motion. In the linear limit, this rod is dispersive due to the effect of lateral inertia. The nonlinearity is introduced through either the stress–strain relation and/or the strain–displacementgradient relation. Using a theory we have developed earlier, we derive an exact general nonlinear dispersion relation for the thick rod. We then derive a special case of this relation and show that it provides an exact prediction of the generated harmonics spectrum, in frequency versus wavenumbers. Both relations are validated by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) for the general nonlinear dispersion relation and short-term, pre-breaking dispersion (by Fourier transformations) for both the general and specialized relation.

Effects of lateral inertia on the group velocities and specific damping capacities of viscoelastic polymeric rods

This analytical study addresses in detail the group velocities and the specific damping capacities of viscoelastic polymeric rods considering the lateral inertia effect. Determination of the group velocity and the specific damping capacity in the viscoelastic rods are crucial for the subsequent various evaluations. The group velocities and the specific damping capacities are evaluated using selected viscoelastic polymeric models published in the literature. So far, there is no analytical solution including the lateral inertia effect for the group velocity and damping capacity of the viscoelastic rods. The group velocities are calculated with help of the phase velocities by using Rayleigh's formula. The specific damping capacities are determined with the help of three different formulas. The analysis reveals generally that due to the lateral inertia effect the group velocities reduce. However, the specific damping capacities are observed to increase, for relatively high-frequency values. The dispersive characteristics that describe by the ratio of the group velocity to the phase velocity are also determined.

New Analytical Solution to Predict the Vertical Impedance of a Large-Diameter Pipe Pile in Soil Considering Wave Propagation in Visco-Elastic Continuum

According to the theories of wave propagation in visco-elastic continuum and Rayleigh-Love rod, a simplified model in axisymmetric conditions for the longitudinal vibration of large-diameter pipe pile (LDPP) in radially heterogeneous surrounding soil with viscous damping is presented. The relevant analytical solution for dynamic impedance at pile head is derived by using complex stiffness transfer method, which is also validated via independent comparisons with previous solutions. Besides, parametric analyses are carried out to reveal both the radial heterogeneity of surrounding soil and the lateral inertia effect of pile shaft on the dynamic impedance of LDPP. The obtained analytical solutions are suitable for the longitudinal vibration issues of floating LDPP in visco-elastic surrounding soil with radial heterogeneity, which can be conveniently degenerated to describe the longitudinal vibration of a floating solid pile in soil with radial heterogeneity as well as a floating LDPP or solid pile in radially homogenous soil.

Propagation of stress pulses in a Rayleigh-Love elastic rod

In this paper, the longitudinal wave propagation in an elastic rod is studied based on the 1D Rayleigh-Love rod theory considering the lateral inertia effect. The Laplace transform method is applied to solve the initial boundary value problem. After conducting the inverse transform, a kernel function in form of integral is obtained, which reveals the essential dispersion characteristics of the wave propagation in a Rayleigh-Love rod. Then the general solution of stress is expressed as the convolution of the kernel function and the boundary loading. Specific examples are given for the problems of typical boundary pulses, i.e. the rectangular, trapezoidal, triangular, and two-stage pulses. Moreover, the dispersive waveforms from our analysis compare nicely with those from the finite element simulation, which indicates that our analytical solution can be used for the dispersion correction in the Hopkinson bar tests.

Dynamics of nonlocal thick nano-bars

The thick bar model, accounting for the lateral deformation, shear stiffness, and lateral inertia effect, is the most comprehensive structural theory to study the axial deformation of carbon nanotubes. Physically motivated definition of the axial force field and associated higher order boundary conditions are determined applying a consistent variational framework. The effects of long-range interactions are suitably realized in the framework of the nonlocal integral elasticity. The integral convolutions of the nonlocal constitutive law are determined and suitably resorted with the equivalent nonlocal differential model equipped with non-standard boundary conditions. Preceding contributions on the elastodynamic analysis of the elastic thick bar are, therefore, amended by properly taking into account the higher order and non-standard boundary conditions. The established size-dependent thick bar model is demonstrated to be exempt from the inherent drawbacks of the nonlocal differential formulation and leads to well-posed elastodynamic problems. The wave desperation response and free vibrational behavior of elastic thick bars with kinematic constraints of nano-mechanics interest are rigorously investigated by making recourse to a viable solution approach. New numerical benchmarks are detected for the elastodynamic response of nonlocal thick nano-bars. A consistent approach for nanoscopic study of the field quantities in the nonlocal mechanics is proposed that is capable of properly confirming the smaller-is-softer phenomenon.

Dynamic compressive properties of Kalgoorlie basalt rock

In this study, the basalt rock extracted from the Kalgoorlie region of Western Australia is intensively studied on its compressive properties under both static and dynamic loads covering strain rate between 2.22 × 10-6/s to 408/s. The ultimate compressive strength and corresponding failure strain are quantified. The test results show that Kalgoorlie basalt rock exhibits high sensitivity to strain rate effect on its compressive strength especially above 100/s and dynamic increment factor up to 2.3 at strain rate 403/s. The failure strain also shows dependency to high strain rate. Discussion is made on fragment analysis which found the natural heterogeneous and anisotropic of WA basalt rocks cause variations on its compressive strength and dependent on the failure angle of the joints (layer formation). The dynamic increase mechanism on material compressive properties is observed to be correlated to the failure crack path formation, which can be explained through the fracture process captured from high-speed camera images analysis. Comparisons are also made on rock strengths with others’ test data. A novel method based on numerical modelling is introduced which removes the influence of lateral inertia effect and specimen end friction effect out of the laboratory testing results. The true dynamic increase factor (DIF) for Kalgoorlie basalt rock at different strain rates are derived for more accurate analysis and design.

Axial dynamics of functionally graded Rayleigh-Bishop nanorods

Present study considers the longitudinal dynamic analysis of carbon nanotubes which has been modeled as an axially functionally graded Rayleigh-Bishop rod with the help of nonlocal stress gradient elasticity theory. Governing equation of motion and boundary conditions for the axial dynamics of nanorod have been obtained with variational formulation. Ritz method has been used in the solution of the problem. Effects of lateral inertia and transverse deformation of carbon nanotubes, material composition properties and grading power-law index on the axial dynamics response have been investigated. Mode shapes of functionally graded Rayleigh-Bishop nanorod have been depicted in various cases. Boundary conditions and power-law index have important effect on dynamics of axially graded Rayleigh-Bishop nanorod. Present results could be useful at designing of nano-electromechanical devices.

Numerical investigation of lateral inertia effect in dynamic impact testing of UHPC using a Split-Hopkinson pressure bar

The paper focuses on investigating the lateral inertia effect in dynamic impact testing of ultra-high performance concrete (UHPC) using a Split-Hopkinson pressure bar (SHPB), based on numerical analysis. The assigned material parameters of the UHPC are obtained by calibrating the parameters of the Karagozian and Case concrete (KCC) model in LS-DYNA software. An impact analysis model of UHPC using SHPB technology is established and calibrated by the test results. A parametric study is performed to estimate the influences of the diameter and aspect ratio (AR) of the UHPC specimen, and the strain rate on the lateral inertia effect in the SHPB testing. The numerical simulation results show that: (1) the impact analysis model can predict the test results of UHPC specimens using the SHPB technique with reasonable accuracy; (2) to reduce the lateral inertia effect, the suggested diameter and AR of the UHPC specimen is 0.9–0.95 times the diameter of the bar, and 0.35–0.45 at a 100 s−1 strain rate or lower, respectively; (3) the lateral inertia effect of UHPC in the SHPB testing is more noticeable with an increase in strain rate, and the diameter of the UHPC specimen is recommended to be 0.95 times the diameter of the bar, as the strain rate is higher than 100 s−1.

Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

A model is proposed for investigating the size-dependent frequency response of arbitrarily oriented microscale frames used in the build-up of lattice structures with micro unit cells. The model employs the Rayleigh–Love, the Timoshenko and the modified couple stress theories to overcome the weaknesses of the conventional theories. Descriptions of the model and finite element implementation are presented. Predictions from the reduced forms of the model agree with published results. The frequency analyses of different microscale frames reveal the influence of material lengthscale, dead weight, lateral inertia and orientation angles. For small aspect ratios, neglecting the lateral inertia effect incurs a substantial error in predicting the frequencies of higher modes, but only marginally affects the lower modes. The resonant frequencies exhibit a sharp drop in the presence of dead weight, but it increases in the presence of material lengthscale.

Effect of magnetic field on viscoelastic wave characteristics

In this study, the longitudinal wave characteristics of magnetic field sensitive viscoelastic rods under the transverse magnetic field effect is addressed by including lateral inertia effect. The analysis is based on the Love rod theory. The polymeric rod is modeled as standard linear solid viscoelastic material. The obtained explicit solution is illustrated graphically. The comparative results of the analysis show that the transverse magnetic field has a remarkable effect on the wave phase velocity and attenuation coefficient. The transverse magnetic field especially leads to a significant reduction on attenuation coefficient for high frequencies values and large diameters. The analysis results presented here especially for low frequencies ranges can provide a reliable support for the similar experimental studies in related the magnetic field effects.

Longitudinal waves in carbon nanotubes in the presence of transverse magnetic field and elastic medium

In the present paper, the coupling effect of transverse magnetic field and elastic medium on the longitudinal wave propagation along a carbon nanotube (CNT) is studied. Based on the nonlocal elasticity theory and Hamilton's principle, a unified nonlocal rod theory which takes into account the effects of small size scale, lateral inertia and radial deformation is proposed. The existing rod theories including the classic rod theory, the Rayleigh-Love theory and Rayleigh-Bishop theory for macro solids can be treated as the special cases of the present model. A two-parameter foundation model (Pasternak-type model) is used to represent the elastic medium. The influence of transverse magnetic field, Pasternak-type elastic medium and small size scale on the longitudinal wave propagation behavior of the CNT is investigated in detail. It is shown that the influences of lateral inertia and radial deformation cannot be neglected in analyzing the longitudinal wave propagation characteristics of the CNT. The results also show that the elastic medium and the transverse magnetic field will also affect the longitudinal wave dispersion behavior of the CNT significantly. The results obtained in this paper are helpful for understanding the mechanical behaviors of nanostructures embedded in an elastic medium.

Numerical Investigation of the Effects of Pulse Shaper, Lateral Inertia, and Friction on the Calculated Strain-Rate Sensitivity of UHP-FRC Using a Split Hopkinson Pressure Bar

AbstractThis research investigates the effect of pulse shaper, lateral inertia and friction on the strain-rate sensitivity of ultra-high-performance fiber-reinforced concrete (UHP-FRC) using a split Hopkinson pressure bar (SHPB) by finite-element analysis. Assigned material parameters of UHP-FRC are provided by calibrating concrete material characteristics given in commercially available software. Following the material model calibration, a parametric study on the design of pulse shaper is carried out to satisfy the requirements of SHPB testing. Material confinement effects, including inertia effect and friction effect, are isolated from the strain-rate sensitivity of UHP-FRC under compressive impact loading. Numerical simulation results show that (1) the calibrated KCC model can well-predict the experimentally behavior of UHP-FRC; (2) an aluminum 6061-T6 disc with 1.8-mm thickness and 8-mm diameter is an ideal pulse shaper for SHPB testing of UHP-FRC at 20 m/s impact velocity; (3) at an impact velocity ...

Mathematical Modeling and Analysis of Drill String Longitudinal Vibration with Lateral Inertia Effect

Comparative analysis whether considering the lateral inertia or not, aiming at the longitudinal vibration of the drill string in drilling progress, is proposed. In the light of the actual condition, the mechanical model of the drill string about vibration is established on the basis of the theoretical analysis. Longitudinal vibration equation of the drill string is derived from the Rayleigh-Love model and one-dimensional viscoelastic model. According to the Laplace transform method and the relationships among parameters of the model, the solutions to complex impedance at the bottom of the drill string are obtained, and then the comparison results are analyzed, which is the lateral inertia effect on longitudinal vibration characteristics. The researches show that the smaller the length of the drill string, the greater the cross-sectional area of the drill string, the greater the damping coefficient of bottom hole on the bottom of the drill string, and the more evident the effect on the dynamic stiffness of the drill string with lateral inertia effect. The Poisson ratio of the drill string only has some effects on it taking account of the lateral inertia effect, and the influence is relatively small compared with the former three conditions.

Impact strength enhancement of aluminum tetrahedral lattice truss core structures

The present study aimed at understanding the behavior of tetrahedral lattice truss core structures under impact loading. Experimental investigations on tetrahedral lattice truss core structures made of aluminum 3003-O were performed using a standard testing machine and nylon split Hopkinson pressure bars. Results showed that the peak force increased about 23% from quasi-static loading to dynamic loading while the base material is hardly rate sensitive. In order to prove such a peak force increase was due to the lateral inertia effect, digital photography techniques were used to catch the deforming process of the core. A quantitive measurement of this process (length, shape) was extracted by a digital image analysis based on the edge detection algorithm. It was observed that the beams of the core were simply compressed at the early stage of loading before bending prevailing and this turning point took place much later under impact loading than the case of quasi-static loading. It was found that the beams were consequently more compressed before the occurrence of buckling during impact loading. This is the direct proof of the existence of the lateral inertia effect. Because of its important strain hardening capacity of base material (aluminum 3003-O), the observed peak force increase can be explained by this lateral inertia effect. Furthermore, numerical simulations using Abaqus code were also carried out to understand thoroughly the role of the lateral inertia effect played on the lattice truss core structures.

Influence of the concrete DIF model on the numerical predictions of RC wall responses to blast loadings

Reliable prediction of structural responses to blast loadings requires an accurate dynamic material model. The dynamic strength of concrete material is normally higher than the static strength. Based on extensive experimental data, a number of empirical DIF (dynamic increase factor) relations have been proposed to model concrete material strength increment at high strain rates. Most of these empirical relations are obtained by fitting the scattered dynamic testing data. It is commonly acknowledged that the induced structural effects such as lateral inertia confinement effect are inevitable in high-speed impact tests. Therefore directly fitting the laboratory testing data may not necessarily obtain the true dynamic concrete material properties. Some recent studies investigated the contributions of lateral inertia and end friction confinement effects on DIF of concrete materials in laboratory tests, and proposed relations to remove these influences to obtain the true DIF for concrete materials. The present study carries out numerical simulations of a reinforced concrete wall under different blast loadings. Different DIF relations including CEB defined DIF, DIF proposed in previous studies after removing structural effect in laboratory tests, and NO DIF, i.e., neglecting dynamic strength increment, are considered in numerical simulations. Verification of the numerical model is made through the comparisons of the numerical simulation results with field test data. The results demonstrate that using the new DIF model yields the best prediction of the structural responses. The responses of a typical RC wall under blast loads with a 5m stand-off distance but different TNT explosive charge weights are also simulated using different DIF relations. The responses of RC wall obtained from numerical models with different DIF relations of concrete material are compared. From the numerical simulation results, the range of scaled distance that DIF relation has significant influences on the numerical simulation results is identified.

Vibrational energy flow models for the Rayleigh–Love and Rayleigh–Bishop rods

Abstract Energy Flow Analysis (EFA) has been developed to predict the vibrational energy density of the system structures in the medium-to-high frequency range. The elementary longitudinal wave theory is often used to describe the longitudinal vibration of a slender rod. However, for relatively large diameter rods or high frequency ranges, the elementary longitudinal wave theory is inaccurate because the lateral motions are not taken into account. In this paper, vibrational energy flow models are developed to analyze the longitudinally vibrating Rayleigh–Love rod considering the effect of lateral inertia, and the Rayleigh–Bishop rod considering the effect not only of the lateral inertia but also of the shear stiffness. The derived energy governing equations are second-order differential equations which predict the time and space averaged energy density and active intensity distributions in a rod. To verify the accuracy of the developed energy flow models, various numerical analyses are performed for a rod and coupled rods. Also, the EFA results for the Rayleigh–Love and Rayleigh–Bishop rods are compared with the analytical solutions for these models, the traditional energy flow solutions, and the analytical solutions for the classical rod.

Simulation of Laser Shock Wave Propagation and Dispersion in SHPB

Laser shock wave oscillation caused by lateral inertia effects of Hopkinson bar in large-diameter SHPB apparatus, and the geometry dispersion effect, particularly, the rise time of the stress wave in different diameters and lengths of Hopkinson bar were investigated. The three-dimensional model of member bar is established by the finite element analysis software ABAQUS, and the different shapes pressure pulses including rectangular, triangular and Gaussian pulses induced by laser shock have been loaded on the end face of the bar, respectively. Results indicate that the triangle pressure pulse and Gaussian pressure pulse show less dispersion effect than rectangle stress pulse on wave shape, and Gaussian stress pulse can keep the morphology better and reduce the dispersion effect more effectively than triangle stress pulse in the propagation process. In addition, as the bar diameter increases and the distance of the propagating stress wave raises, wave oscillation enhances significantly in the bar, the same as the rise time of stress wave increases gradually and the maximum stress also has a certain degree of attenuation, which have influence on laser shock processing or forming.