Elastic Half-space Research Articles

On the effective medium model of the interaction of Rayleigh waves with mass–spring oscillators on the surface

The interaction of Rayleigh surface waves with a random arrangement of oscillators on the surface of an elastic half-space is analyzed with the help of the surface elastodynamic Green’s function formalism and the effective medium model valid in the long-wavelength limit is derived. It is shown that in order to apply the central assumption of the model, namely, that the local surface displacement at the contact point of an oscillator is equal to the average effective medium displacement, the spring constant of the oscillators must be corrected to include the stiffness of the contact calculated in the quasistatic approximation. The previous results pertaining to the interaction of Rayleigh and Lamb waves with microspheres are shown to be correct despite the apparent inconsistencies in the assumptions.

Solving transient problems in soft Elasto-Hydrodynamic lubrication

Many important problems in elasto-hydrodynamic lubrication arise when one of the contacting bodies is elastically soft. An example is the squeezing of a thin liquid film between a rigid sphere/cylinder and a soft elastic substrate, in which the sphere may not be sliding and be subjected only to normal loads (stationary contact), or it can be sheared at the same time and undergo steady or non-steady sliding. These problems are notoriously difficult to solve for soft solids under large loads and thin lubricant films. For this reason, indentation and sliding problems have usually been analyzed separately, and solutions for transient sliding problems are few. Here we develop a numerical method that avoids difficult-to-converge iterative methods and is able to solve both normal indentation and/or transient sliding lubrication problems. The scheme is fully automatic, stable, and efficient and requires only a linear matrix equation to be solved at every time step. We present solutions of two transient sliding problems: in the first of which, a rigid sphere undergoes transient normal lubricated contact followed by transient sliding on an elastic half space. In the second, a rigid cylinder undergoes transient lubricated sliding on an elastic foundation.

Torsional wave propagation on a penny-shaped crack in an orthotopic layer sandwiched between two rigid discs bonded by an orthotropic elastic half-space

The mechanical stability of the interface of two materials determines the stress behavior of the interface. So, observation of failure analysis in orthotropic composites with penny-shaped crack and circular disc in orthotropic materials due to the presence of torsional waves play a major role in structural design. The present article concerns the study of the torsional wave propagation of a penny-shaped crack in an orthotropic layer and two circular discs bonded between the layer and half-spaces. A general solution for the system is presented as a set of dual integral equations using the Hankel transform technique. Using Abel's transform method, the equations have been transformed into Fredholm integral equations of the second kind, which have been solved numerically to compute the stress intensity factors (SIFs) near the rims of crack and discs. Numerical results are obtained using material constants of two orthotropic mediums to demonstrate the impact of material non-homogeneity, normalized disc radius, and layer depth on SIFs and portrayed by virtue of graphs. The analysis of the physical quantity SIF in the present model leads to speculation about the stability of composites against the propagation of cracks in layered engineering solids by surveilling geometric parameters of orthotropic materials and layer depth.

A unified approach for relationships among Green's function, normal modes and dispersion spectrum in layered elastic half-space, with corrected misconceptions on surface wave dispersion and testing

SUMMARY In this paper, the global stiffness matrix [K] and the Fourier–Bessel series methods are proposed to derive the accurate Green's function and dynamic response in a form that is directly related to the dispersion curve and experimental dispersion spectrum. Detailed analyses are carried out for the two-layered half-space with different velocity profiles, including the homogeneous half-space as a special case. Our studies indicate that, in Rayleigh wave analysis, the original Rayleigh equation, instead of the rationalized Rayleigh equation as previously derived and used, should be used since the latter would contain extra non-physical roots. We further reveal and characterize three distinct types of leaky waves: the intrinsic surface leaky wave, the apparent Rayleigh mode with a frequency gap associated with a low-velocity half-space and the fast-guided P–SV wave in the layered medium with a high VS contrast between the upper layer and the lower half-space. All leaky modes can be captured by local minima of |det[K]| instead of tracing complex roots in other existing approaches. In the experimental estimation of dispersion curves for practical applications, we have observed that the truncation effect is the major source of uncertainty regardless of the wavefield transformation method utilized. Furthermore, the truncation effect is both location- and model-dependent, without a unique optimal near offset. As such, in order to reduce the uncertainty from the truncation effect, the receiver layout should be considered in the inversion of dynamic response, instead of relying on ensuring a minimum near offset. This becomes possible with the present fast and accurate complete dynamic Green's function by which all wave phenomena (including different types of leaky waves) and receiver locations can be considered in the wavefield transformation.

Mechanics of shape-locking-governed R2G adhesion with shape memory polymers

Shape memory polymers (SMPs), with unique properties such as tunable elastic modulus, temporary shape locking and shape recovery upon external stimulation, are emerging as a new class of smart materials with switchable adhesion capabilities. A prominent feature of the adhesion between SMP and a spherical indenter is the so-called R2G adhesion, defined as making contact in the rubbery state to a certain indentation depth followed by detachment in the glassy state. While it has been demonstrated that the R2G adhesion with SMPs can achieve orders of magnitude higher adhesive strength compared to conventional elastic adhesive systems with similar geometrical and adhesion parameters, the fundamental mechanics of R2G adhesion and why it leads to such tremendous adhesion enhancement remain largely mysterious at this point. Here, combined experimental testing, theoretical analysis, and finite element analysis (FEA) based on a thermomechanical constitutive model of SMP are performed to investigate the mechanics of R2G adhesion with a rigid spherical indenter. It is shown that the orders of magnitude enhancement of R2G adhesion over conventional elastic adhesion systems is mostly governed by the shape locking effect during the transition from the rubbery to glassy states. The shape locking effect freezes the deformed configuration of the SMP substrate, resulting in nearly conformal contact between the spherical indenter and the glassy-state SMP substrate, thus greatly increases the effective radius of curvature of the contact surface. Our experimental measurements and FEA simulations demonstrate that the net effect of shape locking leads to a pull-off force of a sphere nearly the same as that of a flat punch on an elastic half-space with the same contact radius. An explicit expression of the pull-off force for R2G adhesion is proposed based on flat-punch adhesion. Our results from combined experimentation, modeling and simulations reveal the fundamental mechanics of R2G adhesion. Such understanding provides guidance for the design of SMP smart adhesives.

High efficiency generation of S-wave via a transmissive binary coding metasurface based on machine learning approach

The well-known Lamb’s problem deals with the generation of surface and cylindrical body waves in an elastic half-space due to vertical point excitations on its surface. Owing to the potentials of S-waves in various engineering applications, the physical response of Lamb’s problem has inspired the authors to innovate a new idea for the generation of such S-waves by applying vertical excitations on an elastic half-space. An analytical model based on Lamb’s problem is established to study the elastic half-space wave motions under a vertical point load. Due to mathematical complexities in attaining the corresponding displacement fields, a machine learning approach is proposed to analyze the entire wavefield of the elastic half-space. Through a trained neural network, it is discovered that exertion of multiple vertical excitations with opposite magnitudes can induce beam splitting of transmitted S-waves. By optimizing the parameters, low R-waves and P-waves but high S-waves are observed in the wavefields predicted by the trained neural network. Analogous to excitations with opposite magnitudes, “0” and “1” coding elements that have out-of-phase response are introduced as a binary coding metasurface to investigate the feasibility of a binary coding metasurface in realizing beam splitting of transmitted S-waves via an input of P-waves. The beam splitting of transmitted S-waves is observed in both aluminum and elastomer thin plates. The outcome is significant and it proves that wave mode conversion of P-waves to S-waves is actualizable in both hard and soft materials through a transmissive binary coding metasurface, thus providing an avenue for various shear wave applications.

Closed-form formulation for the response of single floating piles to lateral dynamic loads

This paper presents a mathematical model that provides the response to lateral dynamic loads of single floating piles embedded in a homogeneous viscoelastic soil layer. The formulation results in closed-form expressions of the pile’s swaying, rocking and cross swaying-rocking compliance components, that do not require solving complex integral equations or iterative calculations. The model exploits Tajimi’s elastodynamic theory to compute the dynamic soil resistance to pile movements. One of its main features is allowing for the effect of the thickness of the soil layer on the response of the soil-pile system, by properly accounting for resonance effects associated with the formation of standing waves. The model is validated by comparing its results for the special case of a floating pile embedded in elastic half space against published analytical and numerical results. The paper concludes with a parametric study, focused on exploring under which conditions standing waves may result in “softer” response of the soil-pile system, a phenomenon that may be important for the serviceability analysis of foundation systems sensitive to resonance effects.

Effect of Partially Imperfect Interface on Dynamic Response of a Circular Lined Tunnel in an Elastic Half-Space Subjected to Plane SH Waves

Effect of Partially Imperfect Interface on Dynamic Response of a Circular Lined Tunnel in an Elastic Half-Space Subjected to Plane SH Waves

Quantifying the Error Associated with the Elastic Halfspace Assumption in Site Response Analysis

Quantifying the Error Associated with the Elastic Halfspace Assumption in Site Response Analysis

Elastodynamic response of an orthotropic layered elastic half-space to time-harmonic loading.

The elastodynamics of an orthotropic half-space coated by a thin orthotropic layer is theoretically investigated in this article. We newly propose explicit expressions of free Rayleigh waves in a layered half-space that are dependent on only one unknown constant representing amplitude. The main contribution is on deriving, in a simple manner, the theoretical predictions of far-field Rayleigh wave motion arising from time-harmonic loads using elastodynamic reciprocity theorems. These are the very first closed-form exact solutions found for the forced motion of Rayleigh waves in a layered half-space of orthotropic materials. To demonstrate the theoretical results, computation of Rayleigh wave motion in a jointed rock, including a layer of quartz-schist and a half-space of soil, is considered. We present the phase and group dispersion curves superimposed with the amplitude spectra that provide useful information on wave modes, frequencies, and displacement amplitudes. The inclusion of the amplitude spectra in the dispersion curves is a significant improvement over other dispersion curves currently available in the literature. The analytical predictions are compared with numerical results found by finite element analysis, and they show excellent agreement for the cases of a uniform distributed load and a varying distributed load both applied over a strip on the layer surface. The calculations obtained in the current study could generally be very useful for applications in seismology and materials characterization of coated structures.

Accuracy of a DtN Finite Element Approach for the Elastic Half-Space

In this work, the accuracy of a DtN finite element approach is numerically assessed. This procedure allows an efficient and accurate numerical solution of boundary-value problems of axisymmetric elasticity in semi-infinite domains. For this kind of problems, no explicit closed-form expression for the associated DtN map exists, so a suitable semi-analytical approximation of it, in series form, is used to impose exact boundary conditions on a semi-spherical artificial boundary. The procedure is tested by solving the classical Boussinesq problem, whose exact solution is known analytically. The computed numerical solution is compared to the analytical solution, achieving an excellent agreement between both solutions, both for displacements and stresses. The convergence of the analytical solution to the numerical solution is numerically demonstrated, in terms of artificial boundary location, series truncation order and finite element mesh size.

Adhesion of graded elastic materials: A full self-consistent model and its application

A full self-consistent model (FSCM) of the axisymmetric adhesive contact between a rigid punch with an arbitrary surface shape and a power-law graded elastic half-space is developed. The self-consistent equation between the surface gap and the surface interaction (e.g., the Lennard–Jones force law) involves a nonlinear singular integral, posing a great challenge to numerical calculations. By applying the properties of Gauss’s hypergeometric function, the integral singularity is eliminated in the numerical calculation through Riemann–Stieltjes integral. Case studies for power-law punch profiles are performed and the self-consistent equation can be expressed in a dimensionless form with three dimensionless parameters, namely a shape index, a gradient exponent, and a new generalized Tabor number. The FSCM results are obtained by solving the self-consistent equation through the surface central gap control method and Newton–Raphson iterative method. For large generalized Tabor numbers, the force–displacement curves are ‘S-shaped’ and condense to the extended JKR limit in the high-load branch. As the generalized Tabor number decreases, a continuous transition from the extended JKR model to the Bradley model for the adhesion of power-law graded materials is obtained. It is found that the pull-off force of a graded material usually depends on the three dimensionless parameters, but for some cases of the shape index, it is not sensitive to the gradient exponent when the generalized Tabor number is fixed. Asymptotic solutions are derived to predict the unstable jump points, which coincide well with the FSCM predictions. The FSCM is applied to validate the extended Maugis–Dugdale (M–D) model of graded materials and it is found that the accuracy of the original M–D-n-k model using the maximum strength condition to determine the cohesive stress is limited. By introducing the rigid-limit-consistency condition of the pull-off force to determine the cohesive stress, the M–D-n-k model is improved and its predictions show good consistency with the FSCM results.

Contact stress induced by the rotation of a rigid bolt bonded to an elastic body

Torsional deformation of an elastic half-space containing a vertical cylindrical cavity caused by a rotation of a bonded rigid bolt with a long shank is studied in this article by employing Weber integral transforms and integral equations. In the limiting case of the infinite shank, we suggest two different techniques reducing the problem to the regular integral equations of the second kind, which permit efficient solutions in distinct ranges of geometric parameters and allow us to derive a highly accurate approximate analytic solution of the problem for any values of the parameters. Almost exact approximations are suggested for the stress concentration at the edge of the bolt head. In the case of a bolt with a long finite shank, relationships between the torque and the angle of twist of the bolt are derived by utilizing solutions obtained for the bolt with the infinite shank as asymptotic approximations. The effect of the geometric parameters is investigated providing insight into the behavior of the bolt and contact stress in dependence on the bolt head radius and the length of the shank.

Effect of an Ideal Liquid Layer on the Surface Instability of an Incompressible Elastic Half-Space with Finite Prestrains

Effect of an Ideal Liquid Layer on the Surface Instability of an Incompressible Elastic Half-Space with Finite Prestrains

Tensile Fault Dislocation in an Irregular-Layered Elastic Half-Space

In the present paper, an analytical solution for the static deformation of a two dimensional model consisting of an infinite homogeneous isotropic elastic layer of uniform thickness placed over an irregular isotropic elastic half-space due to movement of a long tensile fault has been obtained. The rectangular shaped irregularity is assumed to be present in the lower half-space and assuming that the fault lies in the elastic layer at a finite depth say ’<i>h</i>’ to the upper surface of the layer. For numerical computation, the expressions of displacements and stresses are calculated by using Sneddon’s method and the effect of source depth and irregularity on the displacements and stresses has been investigated graphically.

An equivalent beam approach for assessing tunnelling-induced distortions of frames with infills

This paper presents an approach to evaluate the response of low- and medium-rise frames with continuous foundations, either with or without infills, to tunnelling employing an equivalent beam with a behaviour dominated by shear deformations. Simplified soil-structure interaction models, consisting of a beam resting on an elastic continuum half-space, are compared against advanced three-dimensional analyses in which the tunnel, the soil, and the building are explicitly modelled. In the simplified approach, the frame is schematised as a Timoshenko beam and reliable procedures to estimate both bending and shear stiffness are discussed. In the refined modelling strategy, an advanced elastoplastic constitute law is employed, capable of reproducing fairly well the soil response to the excavation for increasing values of volume loss, while the full geometry of the structure is considered. First, the results of the proposed numerical approaches are compared in terms of tunnelling-induced foundation displacements, bay deformations and maximum tensile strains in the infills. Then, for the infill panels, the reliability of estimating the maximum tensile strain from the angular distortion of the frame bays is assessed. Finally, a meta-model is proposed to predict the maximum angular distortion based on greenfield settlements, eccentricity, and relative soil-structure stiffness.

Joint inversion of tectonic stress and magma pressures using dyke trajectories

AbstractIn volcano-tectonic regions, dyke propagation from shallow magmatic chambers is often controlled by the interaction of the local and regional stress fields. The variations of the stress fields result from a combination of factors including the regional tectonic stress, the geometry of pressurized magma chambers, the layering and the pre-existing discontinuities (e.g. fractures). In this contribution, we describe and apply a new multiparametric inversion technique based on geomechanics that can invert for both the far field stress attributes and the internal pressure of magma chambers or stocks, constrained by observed dyke or eruptive fissure orientations. This technique is based on the superposition principle and uses linear elastic models that can be solved using many types of numerical methods. For practical reasons, we chose a 3D boundary element method (BEM) for a heterogeneous elastic half-space, where magma chambers are modelled as pressurized cavities. To verify this approach, the BEM solution has been validated against the known 3D analytical solution of a pressurized cylindrical cavity. Then the effectiveness of this technique and its practical use is demonstrated through application to natural examples of dyke network development around two different volcanic systems, the Spanish Peaks (USA) and the Galapagos Islands (Ecuador). Results demonstrate that regional stress characteristics as well as the internal pressure of magma chambers can be estimated from observed radial and circumferential dyke patterns and some knowledge of magma chamber geometry.

Seismometer Records of Ground Tilt Induced by Debris Flows

ABSTRACT A change in surface loading causes the Earth’s surface to deform. Mass movements, such as debris flows, can cause a tilt large enough to be recorded by nearby instruments, but the signal is strongly dependent on the mass loading and subsurface parameters. Specifically designed sensors for such measurements (tiltmeters) are cumbersome to install. Alternatively, broadband seismometers record translational motion and also tilt signals, often at periods of tens to hundreds of seconds. Their horizontal components are thereby the most sensitive to tilt. In this study, we show how to obtain tilt caused by the passing by of debris flows from seismic measurements recorded within tens of meters of the flow and investigate the usefulness of this signal for flow characterization. We investigate the problem on three scales (1) large-scale laboratory experiments at the U.S. Geological Survey debris-flow flume, where broadband seismometers and tiltmeters were installed for six 8–10 m3 experiments, (2) the Illgraben torrent in Switzerland, one of the most active mass wasting sites in the European Alps, where a broadband seismometer placed within a few meters of the channel recorded 15 debris-flow events with volumes up to 105 m3, and (3) Volcán de Fuego, Guatemala, where a broadband seismometer recorded two lahars. We investigate how the tilt signals compare to debris-flow parameters such as mean normal stresses, usually measured by expensive force plates, and debris-flow height. We model the elastic ground deformation as the response of an elastic half-space to a moving surface load. In addition, we use the model with some simplifications to determine the maximum debris-flow heights of Volcán de Fuego events, where no force plate measurements are available. Finally, we address how and under what assumptions the relatively affordable and straightforward tilt measurements may be utilized to infer debris-flow parameters, as opposed to force plates and other complicated instrument setups.

The effect of contact conditions on the performance of flexural seismic metasurfaces

Plane-strain motion of a flexural seismic metasurface in the form of a regular array of thin Kirchhoff plates attached to the surface of an elastic half-space is analysed. Two types of contact conditions, including simply supported plates and plates moving along horizontal rails are studied. Dispersion of time harmonic waves is investigated both asymptotically and numerically. A major effect of the contact conditions on metasurface behaviour is discovered. In particular, it is shown that frequency band gaps are not the feature of the array composed of simply supported plates. It is also demonstrated that the scaling laws, expressed through geometric and material problem parameters, drastically differ from each other for two considered setups.

Inferring Mass Loss by Measuring Contemporaneous Deformation around the Helheim Glacier, Southeastern Greenland, Using Sentinel-1 InSAR

The elastic response of solid earth to glacier and ice sheet melting, the most important consequences of climate change, is a contemporaneous uplift. Here, we use interferometric synthetic aperture radar (InSAR) measurements to detect crustal deformation and mass loss near the Helheim glacier, one of the largest glaciers in southeastern Greenland. The InSAR time series of Sentinel-1 data between April 2016 and July 2020 suggest that there is a maximum cumulative displacement of ~6 cm in the line of sight (LOS) direction from the satellite to the ground near Helheim. We use an exponentially decreasing model of the thinning rate, which assumes that the mass loss starts at the lower-elevation terminal region of the glacier and continues to the higher-elevation interior. A linear inversion of the derived crustal uplift in the vicinity of bedrock using this model for surface loading in an elastic half-space suggests a mass loss of 8.33 Gt/year, which agrees with the results from other studies.