Stress-free Surface Research Articles

The effect of dual-phase-lag model on reflection of thermoelastic waves in a solid half space with variable material properties

The present article represents an analysis of reflection of P-wave and SV-wave on the boundary of an isotropic and homogeneous generalized thermoelastic half-space when the boundary is stress-free as well as isothermal. The modulus of elasticity is taken as a linear function of reference temperature. The basic governing equations are applied under four theories of the generalized thermoelasticity: Lord-Shulman (L-S) theory with one relaxation time, Green-Naghdi (G-N) theory without energy dissipation and Tzou theory with dual-phase-lag (DPL), as well as the coupled thermoelasticity (CTE) theory. It is shown that there exist three plane waves, namely, a thermal wave, a P-wave and an SV-wave. The reflection from an isothermal stress-free surface is studied to obtain the reflection amplitude ratios of the reflected waves for the incidence of P-and SV-waves. The amplitude ratios variations with the angle of incident are shown graphically. Also the effects of reference temperature of the modulus of elasticity and dual-phase lags on the reflection amplitude ratios are discussed numerically.

Characterization of Ni–20Cr–5Al model alloy in supercritical water

MCrAlY is a class of coating materials that provide corrosion and oxidation resistance to many Ni and Fe based alloys by forming dense alumina layer on the surface. In order to assess its potential as corrosion resistant coatings on components in supercritical water cooled nuclear reactors (SWCR), a Ni–20Cr–5Al model alloy is tested in SCW (500°C and 25MPa) for over 6000h. The long term corrosion behavior of the samples with various surface preparations is evaluated by measuring weight change and examining surface microstructure and oxide formation. The results show that surface preparation alone can lead to changes in weight gain as great as an order of magnitude. Smooth and near stress free surface allows for more oxidation to take place in SCW, hence more weight change. Simple grinding with abrasive paper yields the least and most stable weight change while grit blasting has some effect in reducing weight gain. Comparing to other alloys tested under similar condition, Ni–20Cr–5Al has the lowest weight change. Although not detected, the formation of Al2O3 or an Al modified Cr2O3 superficial layer is likely the reason for such low weight change.

Effect of local fluid flow on reflection of plane elastic waves at the boundary of a double-porosity medium

This study solves a mathematical model for the propagation of harmonic plane waves in a double-porosity solid saturated by a viscous fluid. An eigen-system of order four implies the existence of three longitudinal waves and a transverse wave in the composite porous medium. Eigenvalues of this system define the four complex velocities, which are resolved further to define the phase velocities of four attenuated waves in the medium. Variation of the fluid content in pores due to the wave-induced flow is expressed in terms of dilatations of the constituents in porous aggregate. Reflection of plane waves is studied at the stress-free plane surface of the medium. A numerical example is solved to calculate the partition of the incident energy among the four reflected waves. Conservation of the incident energy flux at the boundary is obtained through the presence of an interaction energy share, even when the poroelastic solid is saturated with a non-viscid fluid. This interaction energy disappears in the absence of the wave-induced flow in non-dissipative porous medium. Else, this interaction energy represents the net dissipation effect of the wave-induced flow as well as the pore-fluid viscosity on the partition of the incident energy at the free surface.

Effect of couple-stresses on the stress intensity factors for a crack in an infinite elastic strip under tension

Stresses around a crack in an infinite elastic strip are evaluated based on linearized couple-stress theory under uniform tension. The crack is perpendicular to the stress-free edges of the strip. Solutions are constructed both for a half-plane and for a strip. First, the boundary conditions at the stress-free surfaces are satisfied using a Fourier transform. Next, the mixed boundary value conditions with respect to the crack are reduced to dual integral equations. To solve these equations, differences in the displacement and the rotation at the crack are expanded through a series of functions that are zero -valued outside the crack. The unknown coefficients in each series are solved in order to satisfy the boundary conditions inside the crack using the Schmidt method. The stresses are expressed in terms of infinite integrals, and the stress intensity factors are determined using the integrand characteristics for an infinite value of the variable of integration. Numerical calculations are conducted for selected crack configurations, and the effect of the couple-stresses on the stress intensity factors is revealed.

Bending and Free Vibration Analysis of Isotropic Plate Using Refined Plate Theory

In this paper, Trigonometric shear deformation theory is applied for bending and free vibration analysis of thick plate. In this theory in plane displacement field uses sinusoidal function in terms of thickness coordinate. It accounts for realistic variation of the transverse shear stress through the thickness and satisfies the shear stress free surface conditions at the top and bottom surfaces of the plate. The theory obviates the need of shear correction factor like other higher order or equivalent shear deformation theories. Simply supported thick isotropic plate is considered for detail numerical study. Navier's solution technique is employed for the analytical solution. The results are obtained for displacements, stresses and natural bending mode frequencies and compared with those of other refined theories and exact solution from theory of elasticity.

A New Higher Order Shear Deformation Model for Static Behavior of Functionally Graded Plates

AbstractIn this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concerned flexural behavior of FG plates with Metal-Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static behavior of functionally graded plates.

Motion of the particles due to viscoelastic surface waves of an assigned frequency

The motion of the material particles due to viscoelastic properties of the half-space is considered when the propagating surface wave has an assigned frequency. The detailed examination shows that for Rayleigh waves in classical elasticity as well as for the viscoelastic Rayleigh-type surface waves the sense of the particle path along the ellipse is retrograde on the surface of the half-space and changes into direct only once at depth about one seventh to one quarter of the wavelength. When there is a viscoelastic surface wave that does not satisfy the adopted four criteria for behaviour at infinity (the case of viscoelastic Rayleigh-type surface wave) but only two of them, this additional wave is direct on the surface of the half-space and does not change or may change many times the sense of the particle path along the ellipse with the distance from the stress-free surface of the half-space. In contrast to the elastic case, the ellipse axes in the viscoelastic case are not parallel and orthogonal to the surface of the half-space, respectively. Their orientation depends on the magnitude of the viscous part of the Lamé moduli and tends to a constant at great depth. The numerical computations in the paper refer to some typical values of the complex Lamé moduli and to some real materials.

Rotational and voids effect on the reflection of P waves from stress-free surface of an elastic half-space under magnetic field and initial stress without energy dissipation

The influence of rotation, magnetic field, voids and initial stress on the reflection of P waves in thermoelasticity without energy dissipation are studied. The basic governing equations for isotropic and homogeneous thermoelastic half-space with voids, rotation, are based on Green and Naghdi (GN) theory under the effect of initial stress, where there is an initial magnetic field parallel to the plane boundary of the half-space. The solution of the problem in xz-plane indicates the existence of four plane waves; P1;P2;P3 and P4 and an SV waves. The relevant boundary conditions at stress-free insulated surface are satisfied by appropriate solutions in the half-space to obtain the reflection coefficients for the incidence of P waves. The numerical values of the complex modulus of the reflection coefficients are visualized graphically to display the effect of the rotation, magnetic field, initial stress and voids parameters.

Flexure of thick orthotropic plates by exponential shear deformation theory

In the present paper, a variationally consistent exponential shear deformation theory taking into account transverse shear deformation effect is presented for the flexural analysis of thick orthotropic plates. The inplane displacement field uses exponential function in terms of thickness coordinate to include the shear deformation effect. The transverse shear stress can be obtained directly from the constitutive relations satisfying the shear stress free surface conditions on the top and bottom surfaces of the plate, hence the theory does not require shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results obtained for static flexure of simply supported orthotropic plates are compared with those of other refined theories and elasticity solution wherever applicable. The results obtained by present theory are in excellent agreement with those of exact results and other higher order theories. Thus the efficacy of the present refined theory is established.

Wave Propagation in a Green–Naghdi Thermoelastic Solid with Diffusion

The Green and Naghdi theory of thermoelasticity is applied to study plane-wave propagation in an elastic solid with thermo-diffusion. The governing equations of an elastic solid with generalized thermo-diffusion are solved to show the existence of three coupled longitudinal waves and a shear vertical (SV) wave in a two-dimensional model of the solid with thermo-diffusion. The reflection of plane waves from a thermally insulated stress-free surface of an elastic solid with thermo-diffusion is also studied. A non-homogeneous system of four equations in reflection coefficients is obtained. The speeds of the plane waves are computed numerically and plotted against frequency for a particular range. The complex absolute values of the reflection coefficients of all reflected waves are computed numerically and plotted against the angle of incidence of the striking wave at the free surface. The effects of diffusion parameters are shown graphically for speeds and reflection coefficients of plane waves.

Eulerian adaptive finite-difference method for high-velocity impact and penetration problems

Owing to the complex processes involved, faithful prediction of high-velocity impact events demands a simulation method delivering efficient calculations based on comprehensively formulated constitutive models. Such an approach is presented herein, employing a weighted essentially non-oscillatory (WENO) method within an adaptive mesh refinement (AMR) framework for the numerical solution of hyperbolic partial differential equations. Applied widely in computational fluid dynamics, these methods are well suited to the involved locally non-smooth finite deformations, circumventing any requirement for artificial viscosity functions for shock capturing. Application of the methods is facilitated through using a model of solid dynamics based upon hyper-elastic theory comprising kinematic evolution equations for the elastic distortion tensor. The model for finite inelastic deformations is phenomenologically equivalent to Maxwell’s model of tangential stress relaxation. Closure relations tailored to the expected high-pressure states are proposed and calibrated for the materials of interest. Sharp interface resolution is achieved by employing level-set functions to track boundary motion, along with a ghost material method to capture the necessary internal boundary conditions for material interactions and stress-free surfaces. The approach is demonstrated for the simulation of high velocity impacts of steel projectiles on aluminium target plates in two and three dimensions.

The synergistic effect between erosion and corrosion of API pipeline in CO2 and saline medium

Erosion–corrosion is common in oil and gas processing plants and pipelines where there is interaction between solid particles, corrosive fluid and target material. The objective of the present work is to assess the synergism between erosion and corrosion of API X-70 pipeline steel under liquid–solid impingement conditions in 2g/l NaCl solution purged with CO2 as the corrosive (acidic) media and aluminum oxide as the erodent. The total material loss rate, the components of erosion, corrosion and their synergistic interactions have been determined. The experimental results show that there is an immense correlation exists between erosion and corrosion. During erosion–corrosion, erosion enhances corrosion and corrosion enhances erosion resulting in higher material loss. Scanning electron microscopy analysis of the worn specimen reveals that plastic deformation and fracture are the dominant mechanisms during the erosion–corrosion process. Furthermore, corrosion removes the work hardened layer and exposes a stress free surface for erosion while erosion removes the passive film from the surface and creates a work hardened layer which is highly susceptible to corrosion.

Buckling analysis of functionally graded hybrid composite plates using a new four variable refined plate theory

In this research, mechanical buckling of hybrid functionally graded plates is considered using a new four variable refined plate theory. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. Governing equations are derived from the principle of minimum total potential energy. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. The effectiveness of the theories is brought out through illustrative examples.

Dispersion curves for 3D viscoelastic beams of solid circular cross section with fractional derivatives

The aim is to extend the theory to a viscoelastic beam that satisfies stress-free surface boundary conditions. A viscoelastic material (polyvinyl chloride) was used in the numerical calculation, and the phase and group velocity curves were derived for a viscoelastic beam from the case without damping to the case with damping proportional to the first-order derivative with respect to time. Based on the preliminary data, the phase and group velocity curves were derived for a beam of solid circular cross section. As a result, it was confirmed that, as earlier pointed out for elastic materials, these curves were controlled by the phase velocity inherent to the material. Finally, with the phase velocity and the group velocity of the beam, regularities were derived for the absolute value of the complex velocity on the complex plane.

Wave propagation in dual-phase-lag anisotropic thermoelasticity

The present paper is concerned with the propagation of plane waves in a transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The governing equations are solved in x–z plane to show the existence of three plane waves. Reflection of these plane waves from thermally insulated as well as isothermal stress-free surfaces is studied to obtain a system of three non-homogeneous equations in reflection coefficients of reflected waves. For numerical computations of speeds and reflection coefficients, a particular material is modeled as transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The speeds of plane waves are computed numerically for a certain range of the angle of propagation and are shown graphically against the angle of propagation for the cases of dual-phase-lag (DPL) thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity. Reflection coefficients of various reflected plane waves are computed numerically for thermally insulated as well as isothermal cases and are shown graphically against the angle of incidence for the cases of DPL thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity.

Analytical representation of the fault slip velocity from spontaneous dynamic earthquake models

We have analyzed the most relevant features of three different analytical representations of the time evolution of the cosesimic slip velocity derived from theoretical basis; the so‐called modified Yoffe function (MY), which pertains to a singular crack solution, the solution for a nonspontaneous crack obeying a position‐weakening governing equation (PR) and the solution for a 1‐D fault model subject to a linear slip‐weakening friction law (B). By considering the same input parameters, we quantitatively compare these slip velocity functions (SVF) and we found that the time evolutions of the velocity and the correspondent slip predicted by the MY and B functions are very similar, while the PR predicts a very sharp peak. Correspondingly, the PR SVF is richer in high frequency and the fall off of its spectrum at high frequencies goes roughly as ω–1.5, while those of MY and B more closely follow ω–2. Then we select two spontaneous, 3‐D, dynamic, subshear models, representing a crack‐like or a pulse‐like rupture and we account for both homogeneous and heterogeneous configurations. We then compare the three SVF in order to see how they are able to reproduce the 3‐D solutions; we also show how the input parameters of the SVF can be constrained from the results of the dynamic models. In the homogeneous cases our results indicate that the MY and the PR SVF reproduce adequately well the main features of a dynamic solution in the case of a crack‐like rupture. The PR function overestimates vpeak and the MY SVF predicts a too rapid deceleration. In the case of a pulse‐like rupture both the MY and the B SVF tend to underestimate vpeak , but all of them capture very well the final cumulated fault slip. Moreover, the B function fits better that the MY the overall behavior of the fault slip. The considered SVF are able to reproduce the spectral fall off of a 3‐D solution at intermediate frequencies (for ω < 20 Hz), the MY and the PR for a crack‐like rupture and the MY and the B SVF for a pulse‐like rupture. In particular, for ω < 10 Hz the spectral content of the B function is practically indistinguishable from that of the spontaneous pulse‐like solution. In the heterogeneous configurations the analytical functions cannot reproduce all the spectral details of the numerical solutions, but we see how it is possible to fit the overall behavior of a single pulse in fault the slip velocity time history. The thorough analysis performed in this work can contribute to the discussion about the debated choice of the source time function to be used in the kinematic models, which in turn is extremely important in the contest of hazard assessment and ground motions generation, although stress heterogeneities, geometrical irregularities, attenuation and free surface effects can definitively smear the details of the analytical functions.

Erratum: “Analyzing the growth of InxGa1−xN/GaN superlattices in self-induced GaN nanowires by x-ray diffraction” [Appl. Phys. Lett. 98, 261907 (2011)

an underestimation of the In content by approximately 30% of the given values, otherwise our findings remain valid. We derive here the correct expression that replaces Eq. (2) of the original paper. Symmetric Bragg x-ray diffraction probes the vertical (normal to the layers) lattice constants of strained layers in a heterostructure. For coherent epitaxial growth without plastic relaxation, all layers attain a common in-plane lattice parameter. In case of a planar heterostructure on a bulk substrate, all layers have the same lateral lattice parameter as the substrate. The resulting average vertical lattice parameter of the heterostructure is governed by the Poisson effect, and the well-known result is given below in Eq. (9) .I n a nanowire, however, the heterostructure is free to relax laterally, and the in-plane lattice parameter is obtained from a balance between layers. If the requirement of the stressfree side surface is satisfied on average, the Poisson effect is compensated and the average vertical lattice parameter is given by Eq. (11) below. To derive this not obvious result, let us consider a periodic superlattice stack made from two materials, 1 and 2. Material 1 is taken as a reference for all strain values and material 2 possesses an eigenstrain e0 with respect to it. We take the lateral and the vertical eigenstrain components equal to each other. Thus, the lattice constants c 0 and c 0 in the fully relaxed state are related by e0 ¼ð c 0 � c 0 Þ=c 0 . For the coherent growth considered here, the in-plane lattice parameter can be approximated to assume a constant value throughout the whole stack, so that the in-plane total strain e1xx ¼ e2xx ¼ exx. The same condition holds for the yy components, the axes x and y are in the lateral plane. For the lateral relaxation of the SL, we distinguish two cases

Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations

AbstractThe swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.

Reflection of Plane Waves from Free Surface of an Initially Stressed Rotating Orthotropic Dissipative Solid Half-Space

The governing equations of an initially stressed rotating orthotropic dissipative medium are solved analytically to obtain the velocity equation which indicates the existence of two quasi-planar waves. The appropriate particular solutions in the half-space satisfy the required boundary conditions at the stress-free surface to obtain the expressions of the reflec-tion coefficients of the reflected quasi-P (qP) and reflected quasi-SV (qSV) waves in closed form for the incidence of qP and qSV waves. A particular model is chosen for numerical computation of these reflection coefficients for a certain range of the angle of incidence. The numerical values of these reflection coefficients are shown graphically against the angle of incidence for different values of initial stress parameter and rotation parameter. The impact of initial stress and rotation parameters on the reflection coefficients is observed significantly.

A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates

A refined trigonometric shear deformation theory (RTSDT) taking into account transverse shear deformation effects is presented for the thermoelastic bending analysis of functionally graded sandwich plates. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, does not require shear correction factor, the displacement components are expressed by trigonometric series representation through the plate thickness to develop a two-dimensional theory and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The sandwich with homogeneous facesheet and FGM core is considered. Material properties of the present FGM core are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The influences played by the transverse shear deformation, thermal load, plate aspect ratio, and volume fraction distribution are studied. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated. It can be concluded that the proposed theory is accurate and simple in solving the thermoelastic bending behavior of functionally graded plates.