Semi-infinite Half-space Research Articles

Determination of thermal conductivity of phase pure 10H-SiC thin films by non-destructive Raman thermometry

The 10 H SiC thin films are potential candidates for devices that can be used in high temperature and high radiation environment. Measurement of thermal conductivity of thin films by a non-invasive method is very useful for such device fabrication. Micro-Raman method serves as an important tool in this aspect and is known as Raman thermometry. It utilises a steady-state heat transfer model in a semi-infinite half space and provides for an effective technique to measure thermal conductivity of films as a function of film thickness and laser spot size. This method has two limiting conditions i.e. thick film limit and thin film limit. The limiting conditions of this model was explored by simulating the model for different film thicknesses at constant laser spot size. 10H SiC films of three different thicknesses i.e. 104, 135 and 156 nm were chosen to validate the thin film limiting condition. Thermal conductivity of these thin films varied from 0.60 – 4.80 (Wm−1K−1).

Elastic and Elastoplastic Contact Mechanics of Concentrated Coated Contacts

Machines operate under increasingly harsher contact conditions, causing significant wear and contact fatigue. Sub-surface stresses are responsible for the premature contact fatigue of rolling element bearings, meshing gears, and cam–follower pairs. Surface protection measures include hard, wear-resistant coatings. Traditionally, contact integrity has been predicted using classical Hertzian contact mechanics. However, the theory is only applicable when the contact between a pair of ellipsoidal solids of revolution may be considered as a rigid indenter penetrating a semi-infinite elastic half-space. Many coatings act as thin bonded elastic layers that undergo considerably higher pressures than those predicted by the classical theory. Furthermore, inelastic deformation of bonded solids can cause plastic flow, work-hardening, and elastoplastic behaviour. This paper presents a comprehensive, integrated contact mechanics analysis that includes induced sub-surface stresses in concentrated counterformal finite line contacts for all the aforementioned cases. Generated pressures and deformation are predicted for hard coated surfaces, for which there is a dearth of relevant analysis. The contact characteristics, which are of particular practical significance, of many hard, wear-resistant advanced coatings are also studied. The paper clearly demonstrates the importance of using efficient semi-analytical, detailed holistic contact mechanics rather than the classical idealised methods or empirical numerical ones such as FEA. The novel approach presented for the finite line contact of thin-layered bonded solids has not hitherto been reported in the open literature.

BDFLUX: A new bounded domain model for quantifying vertical fluxes in streambeds using heat as a tracer

The use of heat as a tracer has become a widely used approach to estimate vertical streambed fluxes (VSFs). We present BDFLUX, a new bounded domain model that can be used as a rapid-screening tool to quantify VSFs and thermal properties of streambed sediments by analyzing on-site temperature–time series. BDFLUX uses a closed-form analytical solution that takes thermal advection and thermal conduction into account. BDFLUX, unlike other one-dimensional closed-form analytical solutions, does not assume a semi-infinite half-space. The initial and boundary conditions are described by arbitrary functions that can be directly determined by fitting the measured temperature–time series in the streambed at various depths. The particle swarm optimization (PSO) method and Markov Chain Monte Carlo (MCMC) method are used to estimate VSFs using temperature–time series observed at three depths. BDFLUX is validated using synthetic temperature–time series, demonstrating that it can perform well when the top and bottom boundary conditions are determined directly by the measured data. The capabilities of BDFLUX are further evaluated using laboratory and field data. The simulation results show that BDFLUX performed well in terms of interpreting laboratory experiments and field data, with a root mean square error as low as 0.059. The simulation of laboratory experiments shows that the MCMC method has a lower computational efficiency than PSO method. The magnitude of nonvertical flow components at a field site were quantified by analyzing the results of estimated VSFs in different vertical subdomains. Overall, BDFLUX improves one-dimensional heat analytical models in terms of assessing the magnitude of the nonvertical flow component in heterogeneous streambed sediments.

Dispersion Curves of Love Waves in Elastic Waveguides Loaded with a Newtonian Liquid Layer of Finite Thickness

In this paper, the authors analyse the propagation of surface Love waves in an elastic layered waveguide (elastic guiding layer deposited on an elastic substrate) covered on its surface with a Newtonian liquid layer of finite thickness. By solving the equations of motion in the constituent regions (elastic substrate, elastic surface layer and Newtonian liquid) and imposing the appropriate boundary conditions, the authors established an analytical form of the complex dispersion equation for Love surface waves. Further, decomposition of the complex dispersion equation into its real and imaginary part, enabled for evaluation of the phase velocity and attenuation dispersion curves of the Love wave. Subsequently, the influence of the finite thickness of a Newtonian liquid on the dispersion curves was evaluated. Theoretical (numerical) analysis shows that when the thickness of the Newtonian liquid layer exceeds approximately four penetration depths 4δ of the wave in a Newtonian liquid, then this Newtonian liquid layer can be regarded as a semi-infinite half-space. The results obtained in this paper can be important in the design and optimization of ultrasonic Love wave sensors such as: biosensors, chemosensors and viscosity sensors. Love wave viscosity sensors can be used to assess the viscosity of various liquids, e.g. liquid polymers.

Pullout characteristics of functionally graded and degraded adhesive anchors

Three-dimensional axisymmetric elasticity solutions for pull-out stresses in bonded anchors with spatially stiffness-varying adhesive are presented. A stiff rod embedded in a semi-infinite rigid half-space through an adhesive bondlayer, representing a general anchor problem is analyzed. The adhesive layer is considered to have a smoothly varying stiffness over embedded length. Two cases of particular engineering relevance are considered: (i) stiffness grading of the bondlayer to enhance performance while retaining the critical length characteristics of bonded anchors, and (ii) modulus reduction of the bondlayer representing adhesive degradation proximal to the loaded-end. Theoretical solutions are developed adopting a stress function approach in conjunction with a variational method that compare well with 3D axisymmetric finite element (FE) results. Both theoretical and FE results indicate that the maximum shear stress in the adhesive decreases over 60% for a graded bondlayer for the parameters considered here without warranting a longer embedment length. In contrast, the degraded bondlayer reduces shear stress peaks significantly but warrants a larger embedment length to enable shear-dominated stress-transfer, disadvantageously loading the embedded-end in tension. A design map showing the critical embedment length required for degraded bondlines as a function of fractional embedment length over which bondline is regarded to have degraded is presented. In addition, interfacial fracture behaviors of the tailored and degraded adhesive anchors were examined through FE analyses finding that the tailoring reduces the energy release rate and has the potential for enhancing damage tolerance. The findings of the study indicate that the stiffness-tailored and -degraded bondlayers significantly redistribute the stress field with concomitant influence on stress-transfer and interfacial debonding characteristics of bonded anchors.

A novel intelligent displacement prediction model of karst tunnels

Karst is a common engineering environment in the process of tunnel construction, which poses a serious threat to the construction and operation, and the theory on calculating the settlement without the assumption of semi-infinite half-space is lack. Meanwhile, due to the limitation of test conditions or field measurement, the settlement of high-speed railway tunnel in Karst region is difficult to control and predict effectively. In this study, a novel intelligent displacement prediction model, following the machine learning (ML) incorporated with the finite difference method, is developed to evaluate the settlement of the tunnel floor. A back propagation neural network (BPNN) algorithm and a random forest (RF) algorithm are used herein, while the Bayesian regularization is applied to improve the BPNN and the Bayesian optimization is adopted for tuning the hyperparameters of RF. The newly proposed model is employed to predict the settlement of Changqingpo tunnel floor, located in the southeast of Yunnan Guizhou Plateau, China. Numerical simulations have been performed on the Changqingpo tunnel in terms of variety of karst size, and locations. Validations of the numerical simulations have been validated by the field data. A data set of 456 samples based on the numerical results is constructed to evaluate the accuracy of models’ predictions. The correlation coefficients of the optimum BPNN and BR model in testing set are 0.987 and 0.925, respectively, indicating that the proposed BPNN model has more great potential to predict the settlement of tunnels located in karst areas. The case study of Changqingpo tunnel in karst region has demonstrated capability of the intelligent displacement prediction model to well predict the settlement of tunnel floor in Karst region.

3D structure-soil-structure interaction in an arbitrary layered half-space

The aim of the study is to evaluate the 3D dynamic response of a finite geological region containing two structures and rested on a semi-infinite elastic layered half-space with a dynamic source radiating transient waves. The hybrid modelling approach is applied. It is based on the decomposition of the whole domain under consideration into two sub-regions: a finite-sized near-field elastic isotropic zone with two containment structures and the open semi-infinite far-field layered region. The far-field zone is a semi-infinite elastic isotropic arbitrary layered medium where the near-field finite geological region is located on. The 3D hybrid computational tool is based on the boundary element method (BEM) for the far-field layered zone and the finite element method (FEM) for the finite near-field domain. The model for the semi-infinite layered zone is further extended by the incorporation of a new condensation algorithm which makes it possible to handle 3D wave propagation through arbitrary layered half-space. The condensation algorithm is developed to avoid high computational memory cost while retaining the compatibility with the hybrid FEM-hosted procedure which facilitates the useful solution for the practical three-dimensional engineering problems. The BEM model of the dynamically active far-field zone is inserted as a macro-finite element (MFE) in the FEM commercial program ABAQUS. The accuracy and convergence study of the hybrid numerical scheme is presented. Numerical simulations convincingly illustrate that the dynamic response of structure-soil-structure system depends on different key factors and their mutual interplay. These factors are arbitrary layering of the far-field geological zone, the characteristics of the dynamic source, the site effects phenomena, the structure-soil-structure dynamic interaction, the type and geometrical disposition of foundations and structures and the 3D features of the dynamic motion.

Shear shock formation in incompressible viscoelastic solids

Experiments have shown that shear waves induced in brain tissue can develop into shock waves, thus providing a possible explanation of deep traumatic brain injuries. Here, we study the formation of shock waves in soft viscoelastic solids subject to an imposed velocity at their boundary. We consider the plane shearing motion of a semi-infinite half-space, which corresponds to a spatially one-dimensional problem. Incompressible soft solids whose behaviour is described by the Fung–Simo quasi-linear viscoelasticity theory (QLV) are considered, where the elastic response is either exponential or polynomial of Mooney–Rivlin–Yeoh type. Waveform breaking can occur at the blow-up of acceleration waves, leading to one sufficient condition for the formation of shocks. A slow scale analysis based on a small amplitude parameter yields a damped Burgers-like equation, thus leading to another shock formation condition. Numerical experiments performed using a dedicated finite volume scheme show that these estimates have limited accuracy. Their validity is restricted in the elastic limit too, where exact shock formation conditions are known.

Interaction of shear waves with semi-infinite moving crack inside of a orthotropic media

The present work investigates the motion of a semi-infinite moving crack inside a semi-infinite half-space of orthotropic medium subjected to anti-plane shear wave. The crack is located at a finite depth from the surface of semi-infinite orthotropic medium. Our aim is to examine how such anisotropy and geometric parameters can be adjusted to reduce the magnitude of stress intensity factor (SIF) to control the crack propagation near the crack tip region. As mathematical tools, Fourier transformation and inverse Fourier transformation techniques are employed to convert the governing mixed boundary value problem to the well-known Weiner-Hopf equation with suitable boundary conditions. Some physical quantities such as SIF at the crack tip and crack opening displacement (COD) around the crack tip have been derived. Graphical exhibition has been carried out to show the impact of relevant parameters such as crack velocity, layer depth from the surface to crack and orthotropic material properties on SIF and COD. The numerical results show that SIF decay with crack depth from the layer. It is also observed that SIF decreases with an increase in crack velocity and finally tends to zero as crack velocity approches near SH-wave velocity. Also, the value of COD decays as we move along the damage near the crack tip along negative x-axis and finally tends to zero at the crack tip. This behavior of COD is consistent with the physical nature of the semi-infinite crack of the problem. The results are validated for isotropic material with some reported work and are well in agreement. The study of these physical quantities (SIF, COD) ensures the arrest of onset of crack expansion by monitoring geometric parameters and wave velocity to avoid fracture.

An attempt to calculate analytical elastic fields in two joined quarter spaces due to an interior point load

The problem of two joined quarter spaces under a body force can be useful in many applications, such as welded materials, where the free surface is in close proximity. Due to the complexity of the domain, no analytical solutions have been presented for calculating elastic fields within the two regions. In this study, we address the problem of a point load acting at the interior of one of the two joined isotropic quarter spaces. Elastic fields for the two regions are presented in terms of Papkovich-Neuber functions. Traction-free boundary conditions and perfect bonding at the interface were both considered. Green's analysis and proposed potential Green's function of two joined quarter spaces were combined with the integral image method to overcome the boundary conditions. Stress distributions between the two regions and displacement on the free surfaces of the two-quarter spaces for different material combinations are illustrated. The continuity of stresses and displacements at the interface were preserved, however, disturbance of stresses on the free surface was observed. This stress disturbance may be due to the complexity of the domain and/or the proposed harmonic functions. The solution leads back to the solution of point load at the interior of two joined semi-infinite half-spaces (Rongved's solutions) and of point loading at the interior of a semi-infinite space (Mindlin's solutions).

Cure monitoring and damage identification of CFRP using embedded piezoelectric sensors network

Because of the advantages of high specific strength and high specific stiffness, carbon fiber reinforced plastics (CFRP) have been the most ideal materials in the field of civil aviation. Cure monitoring in manufacturing process and damage identification in service stage of CFRP are always hot topics. The Semi-Analytical Finite Element (SAFE) method and micromechanical model are employed to analyse the propagation characteristics of the Lamb-like waves in a continuous flat aluminium plate attached to a viscoelastic unidirectional CFRP in semi-infinite half-space. Then the vacuum bag moulding process of prepregs is monitored using Fiber Bragg Grating (FBG) and piezoelectric sensors encapsulated in Stanford Multiactuator-Receiver Transduction (SMART) Layer. The calculated energy velocities and attenuations of guided waves show the same trends with the numerical results while curing. After the CFRP is demoulded, the damage identification experiments are carried out. By continuing to use the sensor network embedded in the manufacturing phase, the artificial damages can be precisely located. The results demonstrated that the life-cycle monitoring of CFRP can be achieved effectively by the piezoelectric sensors network.

Seismic response of monopiles to vertical excitation in offshore engineering

This paper studies the dynamic response of a large diameter and thin-walled monopile, embedded in a partially saturated porous seabed with saturation degrees larger than 95%, under a seawater layer, subjected to vertically incident P wave excitations. The ground is a semi-infinite homogenous half-space with a horizontal surface. The solution of the coupled water-pile-soil vibration problem is constructed using a rigorous semi-analytical method. By considering the coupled boundary conditions at the pile-soil and water-soil interfaces, the earthquake induced response is reduced to integral equations and solved numerically. Results for the free field soil displacement and kinematic interaction factor are obtained, which are different from the results reported in the literature for onshore earthquake engineering applications as these do not have the effects of the water layer and the vibration induced dynamic water pressures. The existence of a water layer will always decrease the vertical response of the seabed, but may amplify the monopile's displacement near some resonant frequencies, compared to the case of no water. These results are useful in understanding the kinematic interaction of soil and monopile in offshore engineering applications.

Analytical Elastostatic Contact Mechanics of Highly-Loaded Contacts of Varying Conformity

In applications requiring high load carrying capacity, conforming contacting pairs with a relatively large contact footprint are used. These include circular arc, Novikov, and Wildhaber gears found, for example, in helicopter rotors. Closely conforming contacts also occur in many natural endo-articular joints, such as hips, or their replacement arthroplasty. The main determining factors in contact fatigue are the sub-surface shear stresses. For highly loaded contacts, classical Hertzian contact mechanics is used for many gears, bearings, and joints. However, the theory is essentially for concentrated counterforming contacts, where the problem is reduced to a rigid ellipsoidal solid penetrating an equivalent semi-infinite elastic half-space. Applicability is limited though, and the theory is often used inappropriately for contacts of varying degrees of conformity. This paper presents a generic contact mechanics approach for the determination of sub-surface stresses, which is applicable to both highly conforming as well as concentrated counterforming contacts. It is shown that sub-surface shear stresses alter in magnitude and disposition according to contact conformity, and lead to the different modes of fatigue failure noted in practice.

Influence of irregular geologies and inhomogeneity on SH-wave propagation

In the present paper, a study is performed in an irregular earth crust, layered over a semi-infinite half-space under the effect of gravity. The irregularities at the interface are possible combinations of geometric shapes such as rectangular, paraboic and triangular notches. The aim of the study is to come up with the influence of these irregularities on the phase velocity of shear horizontal waves. The current work also explores how inhomogeneities affect SH-wave propagation. The medium is assumed to exhibit inhomogeneities as a function of depth. These functions are the product of a linear algebraic function and an exponential function of depth. By means of separation of variables and the substitution method, the equation of motion is reduced to the hypergeometric equation. Suitable boundary conditions are employed to derive a closed form of the dispersion equation. Numerical computations are performed to visualize the impact of irregularity and inhomogeneity. It is observed that the irregular interfaces and the inhomogeneity involved in the medium have a significant effect on SH-wave propagation.

Analytical approximations for the dispersion of electromagnetic modes in slabs of biaxial crystals

Anisotropic crystals have recently attracted considerable attention because of their ability to support polaritons with a variety of unique properties, such as hyperbolic dispersion, negative phase velocity, or extreme confinement. Particularly, the biaxial crystal $\alpha$-MoO$_3$ has been demonstrated to support phonon polaritons, light coupled to lattice vibrations, with in-plane anisotropic propagation and unusually long lifetime. However, the lack of theoretical studies on electromagnetic modes in biaxial crystal slabs impedes a complete interpretation of the experimental data, as well as an efficient design of nanostructures supporting such highly anisotropic polaritons. Here, we derive the dispersion relation of electromagnetic modes in biaxial slabs surrounded by semi-infinite isotropic dielectric half-spaces with arbitrary dielectric permittivities. Apart from a general dispersion relation, we provide very simple analytical expressions in typical experiments in nano-optics: the limits of short polaritonic wavelength and/or very thin slabs. The results of our study will allow for an in-depth analysis of anisotropic polaritons in novel biaxial van der Waals materials.

The Cylindrical Wave Approach for the Electromagnetic Scattering by Targets behind a Wall

An overview of the cylindrical wave approach in the modeling of through-wall radar problems with targets hidden behind a dielectric wall is reported. The cylindrical wave approach is a technique for the solution of the two-dimensional scattering by buried circular cross-section cylinders in a semi-analytical way, through expansion of the scattered fields into cylindrical waves. In a through-wall radar application, the scattering environment is made by a dielectric layer between two semi-infinite half-spaces filled by air. For this layout, two possible implementations of the cylindrical wave approach have been developed in the case of plane-wave excitation. The first was an iterative scheme with multiple-reflection scattered fields, and the second was a fast and non-iterative solution, through suitable basis functions (i.e., reflected and transmitted cylindrical waves). Such waves take into account all the interactions of the source field with the interfaces bounding the dielectric layers and the targets. The non-iterative approach was also extended for excitation from the radiated field by a line source. A final system was derived for the computation of the scattered field by PEC or dielectric targets. Numerical results show the potentialities of the cylindrical wave approach in the modeling of through-wall radar, in particular in the evaluation of the scattered fields by human targets in a building’s interior, modeled with a two-dimensional approach.

Study of Love Waves in a Clamped Viscoelastic Medium with Irregular Boundaries

A mathematical model is presented to investigate the effects of sandiness, irregular boundary interfaces, heterogeneity and viscoelasticity on the phase velocity of Love waves. Geometry of the problem is consisting of an initially stressed viscoelastic layer with corrugated irregular boundaries, which is sandwiched between heterogeneous orthotropic semi-infinite half-space with initial stress and pre-stressed dry sandy half-space. Heterogeneity arises in the upper half-space is due to trigonometric variation in elastic parameters of the orthotropic medium. Inclusion of the concept of corrugated irregular viscoelastic layer clamped between two dissimilar half-spaces under different physical circumstances such as initial stress and heterogeneity brings a novelty to the existing literature related to the study of Love wave. Dispersion equation for Love wave is obtained in closed form. The obtained dispersion relation is found to be in well agreement with classical Love wave equation. Numerical example and graphical illustrations are made to demonstrate notable effect of initial stress, internal friction, wave number and amplitude of corrugations on the phase velocity of Love waves.

Random vibration of train-track-ground system with a poroelastic interlayer in the subsoil

A semi-analytical train-track-ground model considering rail irregularity is developed to study random vibration of the system. The ground is modeled as a layered half-space comprising two elastic layers, a poroelastic interlayer and a semi-infinite elastic half-space from top to bottom. Solution to the ground is derived adopting ‘adapted stiffness matrix method’. The ground is coupled with the track according to continuity conditions at the ballast-ground interface. The track-ground system is coupled with the train adopting Hertzian contact spring model. Pseudo-excitation method (PEM) is introduced to solve the random vibration induced by rail irregularity. The method is compared with existing research. Dynamic responses of the system such as amplitude, power spectral density (PSD) and standard deviation (SD) of displacement, velocity and acceleration are investigated. The existence of groundwater significantly influences vertical displacement for locations far from track center and long wavelength of rail irregularity, while it mainly affects the vertical velocity and acceleration for locations far from track center but short wavelength of rail irregularity. The maximum SD of vertical displacement increases with increasing speed, and no ‘critical speed’ exists for it when 0 m/s <c<200 m/s, which is different from the variation of vertical displacement with speed due to axle loads. High frequency components of the PSD of maximum vertical displacement, velocity and acceleration attenuate faster from the rail to the ground and with the distance from the track center on ground surface.

Variational approach for settlement analysis of circular plate on multilayered soil

• A semi-analytical method is developed for the analysis of a uniformly-loaded circular plate on a multilayered soil. • The method can analyze a thin circular plate's flexural behavior on any number of soil layers. • A parametric study on two- and three-layered soils highlights important insights about plate-layered soil interactions. • Charts that can facilitate the analysis of differential settlement of concrete tanks are provided. A continuum-based analysis method is developed to analyze a uniformly-loaded circular plate founded on a multilayered soil deposit. The multilayered soil deposit forms a semi-infinite half space, and the behavior of each soil layer is assumed to be linear elastic. The plate is considered to be isotropic, homogeneous, and linear elastic. We apply variational principles to obtain the governing differential equations of the plate-layered soil system and solve them semi-analytically. We then compare results from the present analyses with those from finite element analysis. Using the proposed analysis method we perform a parametric study, results from which provide important insights into the effects of soil layering – such as the thicknesses and stiffness ratios of multiple soil layers – on the flexural behavior of the circular plate. Analyses for wide ranges of plate-to-soil stiffness ratios provide further insights on plate-soil interactions.

A Perturbation Method for Scattering by a Dielectric Cylinder Buried in a Half-Space

A simple method is presented for transverse magnetic plane-wave scattering by a dielectric cylinder of arbitrary cross section buried in a half-space. The interface between two semi-infinite dissimilar half-spaces is considered as an infinite aperture, which is represented by a perturbed equivalent magnetic current. Coupled integral equations for the aperture electric field (or equivalent magnetic current) and the equivalent currents on the buried cylinder are obtained from surface equivalence principles and are solved numerically by the method of moments. Data for the electric field at the interface and far field are presented for cases of interest and are compared to the results available in the literature.