Semi-infinite Solids Research Articles

A new analytical method for solving the problem of one-dimensional transient heat conduction in a slab by using the temperature solutions of semi-infinite solids

A new analytical method for solving the problem of one-dimensional transient heat conduction in a slab by using the temperature solutions of semi-infinite solids

Water droplets evaporating on horizontal semi-infinite solids at room temperature

A global approach to the evaporating droplets on solid substrates in stagnant air is presented. The focus is on simplifying the fundamental physics and mathematics involved in order to develop a sound engineering method for determining droplet temperatures and evaporation rates of sessile water droplets on semi-infinite substrates. A mathematically based factor, γ, is introduced to describe the droplet shape, where γ takes the value of 1 for flat (pan cake shaped) droplets and 2 for half sphere shaped droplets, and a representative value in the range of 1–2 between these extremes. It is shown that droplets on thermally insulating substrates gets colder and evaporates at about half the rate of similar droplets on high thermal conductivity substrates. Heat radiation dominates the evaporation rate for insulating substrates while substrate thermal conductivities above 1 W/m K do not significantly increase the evaporation rate. The theoretical findings were verified by recording temperature and drying time of droplets evaporating on semi-infinite substrates of XPS foam, PMMA and stainless steel at 24 °C and about 30% relative humidity. The convective water vapour flow from the cold droplets was directed upwards due to the lower molar mass of the water vapour. For the 10–13 mm radius droplets studied in the present work, the average convective radius to ambient air was about 15 mm ± 5 mm larger than the droplet radius. It was largest for droplets on stainless steel and smallest for droplets on the XPS-foam.

Dynamic response of a solid with multiple inclusions under anti-plane strain conditions by the BEM

In this work, a 2D problem in elastodynamics that involves the finite elastic solid containing multiple cylindrical and elliptical inclusions and/or cavities of arbitrary size, number, material properties and geometrical configuration is solved. More specifically, anti-plane strain conditions are assumed to hold and all external loads have a time harmonic variation. The numerical method used for this purpose is the direct, displacement-based boundary element method (BEM) with sub-structuring capabilities. The BEM employs frequency-dependent fundamental solutions for a point load in the unbounded continuum, and requires discretization of all external and internal surfaces and interfaces only. The method is well suited for the computation of stress concentration factors, while solution at points inside the domain of interest can be expressed directly in terms of boundary data without recourse to domain discretization. Thus, an effective numerical scheme is developed, verified and subsequently used for extensive parametric studies. The numerical results obtained here show a marked dependence of the stress and displacement wave fields on the shape of the inclusions, their material properties, number and position, as well as their mutual interaction with the incoming horizontally polarized shear (SH) waves. The potential of the BEM to efficiently produce accurate results for the dynamic response of both finite and semi-infinite solids that are strengthened and/or weakened by multiple inclusions, as compared to other domain-type methods, makes it a valuable computational tool for solving certain categories of engineering problems. These range from fields as diverse as material science and non-destructive testing evaluation, to seismology and exploratory geophysics.

Effect of thermal sensitivity of materials of tribojoint on friction temperature

An analytical solution of a friction-heat problem for two semi-infinite solids taking into account the temperature dependence of their thermal characteristics has been obtained. It is assumed that the specific power of friction is constant and the thermal contact of the solids is imperfect. Because of the latter assumption, the linearization of the corresponding boundary heat conduction problem carried out using the Kirchhoff transform has turned out to be incomplete. Two methods for the final linearization of this problem have been proposed, i.e., the expansion of the nonlinear function in a power series, followed by the retention of only two first terms in it, and the use of linearizing multipliers. A numerical analysis for the materials with linear dependences of the thermal conductivity and the specific heat on the temperature is carried out.

Power Transfer Coefficients between Two 3D Semi-Infinite Solids in EFEA Formulation

The energy finite element analysis (EFEA) is more effective for analyzing high frequency vibration problem compared to the traditional finite element method (FEM). When applying the EFEA to complicated structures, it is necessary to obtain power transfer coefficients at structural joints. This paper proposes the theoretical formulations of computing power transfer coefficients between two 3D semi-infinite solids, and demonstrates the validation of the developed theoretical formulations through two examples with different incident waves: dilatational wave and distortional wave.

Finite-size effects on the quasistatic displacement pulse in a solid specimen with quadratic nonlinearity

There is an unresolved debate in the scientific community about the shape of the quasistatic displacement pulse produced by nonlinear acoustic wave propagation in an elastic solid with quadratic nonlinearity. Early analytical and experimental studies suggested that the quasistatic pulse exhibits a right-triangular shape with the peak displacement of the leading edge being proportional to the length of the tone burst. In contrast, more recent theoretical, analytical, numerical, and experimental studies suggested that the quasistatic displacement pulse has a flat-top shape where the peak displacement is proportional to the propagation distance. This study presents rigorous mathematical analyses and numerical simulations of the quasistatic displacement pulse. In the case of semi-infinite solids, it is confirmed that the time-domain shape of the quasistatic pulse generated by a longitudinal plane wave is not a right-angle triangle. In the case of finite-size solids, the finite axial dimension of the specimen cannot simply be modeled with a linear reflection coefficient that neglects the nonlinear interaction between the combined incident and reflected fields. More profoundly, the quasistatic pulse generated by a transducer of finite aperture suffers more severe divergence than both the fundamental and second order harmonic pulses generated by the same transducer.

Two Ellipsoidal Inhomogeneous Inclusions in Dissimilar Media by the Modified Equivalent Inclusion Method

The elastic stress field for multiple ellipsoidal inhomogeneous inclusions in one of two perfectly bonded semi-infinite solids subjected to external stress is investigated by the modified equivalent inclusion method. Using the general solution for the inhomogeneous inclusion in bimaterial expressed in terms of Garlekin stress vectors, the potential functions necessary to solve the problem are obtained. The elastic field induced by multiple inclusions in dissimilar media could be found from the superposition of the separate solutions of individual precipitate. The effects of the planner interface with parameters of depth from the interface and both pairs of elastic moduli and also shapes of the inclusion on the solution are given as numerical example, which is of great significance in physical applications.

On the molecular forces acting between macroscopic bodies

The eigenfrequencies are identified for two electromagnetically coupled semi-infinite solids with plane-parallel surfaces (two half-spaces) separated by a third, polarizable body. The corresponding van der Waals–London and Casimir forces are calculated from the zero-point energy (vacuum fluctuations) of the normal modes. It is shown how the results can be extended to bodies of any shape; in particular, the force is given for a sphere interacting with a half-space. The calculations are performed using the well-known Drude–Lorentz (plasma) model of (non-magnetic) polarizable matter. The polarization degrees of freedom are explicitly introduced. It is shown that the polarization dynamical variables for the two bodies are coupled through the electromagnetic field, very similar with two infinite sets of coupled harmonic oscillators.

Stationary Theory of Heat-Conductivity for an Axi-Symmetrical Piece Homogeneous Space with Circular Inclusion

This paper discusses the theory of thermo-conductivity due steady-state heat flow of semi-infinite homogeneous solids containing a perfectly rigid circular inclusion located on the plane where the two half-solids are joint. The half-spaces have different thermo-physical characteristics and a steady heat flow is applied to the solids. The effects of selected spatial heat sources are also investigated. Mathematical model and general solution is presented along with selected particular cases and simulations, followed by pertinent discussions and conclusions.

Penetration of rigid objects into semi-infinite compressible solids

Penetration of rigid object into semi-infinite compressible solid is investigated in the present study. First a detailed numerical analysis of the penetration process is performed via smooth particle hydrodynamics. Based on the numerical results, estimate for the resistive force that the target exerts on the penetrating object is obtained. It is shown that in the computation of the resistive force the quasi-static coefficient can accurately be obtained through the spherical cavity expansion theory. However for a given target material the coefficient associated with the hydro-dynamic term significantly depends on the impact velocity unlike commonly assumed constant values. Penetration equations for rigid object with arbitrary nose geometries are derived. Developed analytical model is then verified through some experimental and analytical results reported in the literature.

Dynamic stress in a semi-infinite solid with a cylindrical nano-inhomogeneity considering nanoscale microstructure

In this paper, the dynamic stress around a cylindrical nano-inhomogeneity embedded in a semi-infinite solid under anti-plane shear waves is investigated. The surface/interface stress effects around the nano-inhomogeneity and at the straight edge of the semi-infinite solid are both considered. The boundary condition at the straight edge of the semi-infinite solid with surface/interface effects is satisfied by the image method. The incident, scattered and refracted displacement fields in the nano-sized composites are expressed by employing the wave function expansion method. The addition theorem for a cylindrical wave function is applied to accomplish the superposition of wave fields in the two semi-infinite solids. Analyzes show that the effect of interface properties, especially that at the straight edge, on the dynamic stress is significant, and the effect increases noticeably due to the nanoscale of the structure. The incident frequency and angle of waves and the shear modulus ratio of the nano-inhomogeneity to matrix also show a pronounced effect on the dynamic stress distribution if the semi-infinite solid shrinks to nanoscale.

Cryscor: a program for the post-Hartree–Fock treatment of periodic systems

Cryscor is a periodic post-Hartree-Fock program based on local functions in direct space, i.e., Wannier functions and projected atomic orbitals. It uses atom centered Gaussians as basis functions. The Hartree-Fock reference, as well as symmetry information, is provided by the Crystal program. Cryscor presently features an efficient and parallel implementation of periodic local second order Møller-Plesset perturbation theory (MP2), which allows us to study 1D-, 2D- and 3D-periodic systems beyond 1000 basis functions per unit cell. Apart from the correlation energy also the MP2 density matrix, and from that the Compton profile, are available. Very recently, a new module for calculating excitonic band gaps at the uncorrelated Configuration-Interaction-Singles (CIS) level has been added. Other advancements include new extrapolation techniques for calculating surface adsorption on semi-infinite solids. In this paper the diverse features and recent advances of the present Cryscor version are illustrated by exemplary applications to various systems: the adsorption of an argon monolayer on the MgO (100) surface, the rolling energy of a boron nitride nanoscroll, the relative stability of different aluminosilicates, the inclusion energy of methane in methane-ice-clathrates, and the effect of electron correlation on charge and momentum density of α-quartz. Furthermore, we present some first tentative CIS results for excitonic band gaps of simple 3D-crystals, and their dependence on the diffuseness of the basis set.

Oswald-Kasper-Gaukler model for reflection electron energy loss spectroscopy

The Oswald-Kasper-Gaukler (OKG) model for elastic electron backscattering [J. Electr. Spectrosc. Rel. Phen. 61(1993)251] has been extended within the partial-intensity approach to take inelastic collisions into account. Analytical expressions have been derived for the path-length distribution and the partial intensities, achieving good agreement with results of Monte Carlo (MC) calculations of these quantities. A criterion is given to predict the validity of the model for a given material, geometry, and kinetic energy. Experimental reflection electron energy loss spectroscopy (REELS) spectra have been compared with REELS spectra calculated using the modified OKG model, obtaining good agreement between them. The proposed model is also applied in a quantitative analysis of experimental REELS. In all investigated cases, the single-scattering loss distributions retrieved from this analysis agree with results from previous analyses---based on MC calculations---within 5%. The presented model can therefore be employed in quantitative analyses of REELS of semi-infinite solids, while it is both numerically simpler and conceptually clearer than related approaches.

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.

Thermally induced local slip of contacting solids in vicinity of surface groove

The thermoelasticity stick–slip contact problem for semi-infinite solids, one of which possesses a surface groove, is considered. The local slip of boundaries occurs in the vicinity of arisen intersurface gap due to different thermal deformations of the opposite faces caused by the temperature jump. By making use of complex potentials method, the corresponding plane contact problem is reduced to a set of three singular integral equations. The analytical solution is constructed for certain shape of the groove. The contact normal and tangential stresses are analyzed along with the parameters of the gap and slip zones, and their dependencies on mechanical and thermal loading.

COMPRESSION INSTABILITIES OF TISSUES WITH LOCALIZED STRAIN SOFTENING

The stress–strain relationship of biological soft tissues affected by Marfan's syndrome is believed to be nonconvex. More specifically, Haughton and Merodio recently proposed a strain energy density leading to localized strain-softening, in order to model the unusual mechanical behavior of these isotropic, incompressible tissues. Here we investigate how this choice of strain energy affects the results of some instabilities studies, such as those concerned with the compression of infinite and semi-infinite solids, slabs, and cylinders, or with the bending of blocks, and draw comparisons with known results established previously for the case of a classical neo-Hookean solid. We find that the localized strain-softening effect leads to early instability only when instability occurs at severe compression ratios for neo-Hookean solids, as is the case for bulk, surface, and bending instabilities.

Contact strength of two elastic half spaces with circular groove

We present the analysis of the stress-and-strain state of a contact couple formed by two isotropic semi-infinite solids one of which has a small surface groove. Based on the classical fracture criteria, namely, on the criterion of maximal principal stresses and the criterion of maximal shear stresses, we determine the regions of the most possible formation of the zones of crack initiation and plastic zones. Brittle cracking can be induced both by tensile and compressive stresses arising on the interface. For some shapes of the groove considered as an example, our analysis reveals that the fracture process starts from the boundary of contacting solids.

Theory of near-field radiative heat transfer for stratified magnetic media

We report a general model to deal with near-field radiative heat transfer for the layered magnetic media, which is also applicable for layered nonmagnetic media. In the derivation, a method using the reciprocity theorem with dyadic Green’s function (DGF) and transfer matrix is introduced, which make the expressions simple and compact. By means of multiple reflection diagrams, it is clear to understand the physical interpretation of DGF and it is easy to evaluate it with physical picture for simple construction. Finally, the expression for the near-field radiative heat transfer between two semi-infinite solids separated by a medium is given as an example. Some numerical computations are conducted for both non-magnetic and magnetic media, which reveals the validity of the derived expressions and the difference between the two types of materials from the point of view of near-field radiative heat transfer. Furthermore, the results indicate that the near-field radiative flux will be further enhanced by excitation of surface polaritons in TE polarization for magnetic media.

Harvesting Nanoscale Thermal Radiation Using Pyroelectric Materials

Pyroelectric energy conversion offers a way to convert waste heat directly into electricity. It makes use of the pyroelectric effect to create a flow of charge to or from the surface of a material as a result of heating or cooling. However, an existing pyroelectric energy converter can only operate at low frequencies due to a relatively small convective heat transfer rate between the pyroelectric materials and the working fluid. On the other hand, energy transfer by thermal radiation between two semi-infinite solids is nearly instantaneous and can be enhanced by several orders of magnitude from the conventional Stefan–Boltzmann law as the gap separating them becomes smaller than Wien’s displacement wavelength. This paper explores a novel way to harvest waste heat by combining pyroelectric energy conversion and nanoscale thermal radiation. A new device was investigated numerically by accurately modeling nanoscale radiative heat transfer between a pyroelectric element and hot and cold plates. Silica absorbing layers on top of every surface were used to further increase the net radiative heat fluxes. Temperature oscillations with time and performances of the pyroelectric converter were predicted at various frequencies. The device using 60/40 porous poly(vinylidene fluoride–trifluoroethylene) achieved a 0.2% efficiency and a 0.84 mW/cm2 electrical power output for the cold and hot sources at 273 K and 388 K, respectively. Better performances could be achieved with 0.9Pb(Mg1/3Nb2/3)–0.1PbTiO3 (0.9PMN-PT), namely, an efficiency of 1.3% and a power output of 6.5 mW/cm2 between the cold and hot sources at 283 K and 383 K, respectively. These results are compared with alternative technologies, and suggestions are made to further improve the device.

Singular integral transforms for heat transfer problems in semi-infinite solids with coatings

This paper is devoted to singular integral transform method for non-stationary one-dimensional heat transfer problems in semi-infinite solids. Different geometries (semi-space, cylindrical channel and spherical cavity) are studied, and resulting temperature fields are compared. Special attention is given to multi-layer surfacing of solid as a means of heat defense. Kernel splitting method for these problems with non-stationary heat exchange conditions on free boundary is developed.