Axisymmetric Flow in an Oil Reservoir of Finite Depth Caused by a Point sink above an Oil-Water Interface

The flow of a stratified fluid (e.g., oil/water) withdrawn from a vertically confined porous medium through a point sink is considered. The withdrawal tends to cause the oil-water interface to move upwards. So long as the interface is below the well, the less dense fluid (oil) is pumped into the well without the denser fluid (water) until a critical flow rate is reached. The flow is considered to be axisymmetric, and involves a nonlinear boundary condition along the free surface. A boundary-integral equation method (BIEM) is used to find the interface position for different pumping rates. For small flow rates, a small-parameter expansion is derived and the results are compared with numerical solutions to the problem. There exists a critical withdrawal rate beneath which the water does not break through into the sink, this rate depending on the sink location and bottom geometry.     

Light propagation in two-layer tissues with an irregular interface

We study light propagation in a half-space composed of two homogeneous layers each having different optical properties from the other. This problem is a model for light propagation in tissues composed of a thin epithelial layer supported from below by a thick stromal layer. The interface between the two layers is irregular. Assuming that this interface is a small perturbation of a plane that is parallel to the boundary surface, we obtain an asymptotic approximation to the solution. We give a numerical method to compute this asymptotic approximation. Finally, we show how to recover this irregular interface surface from boundary measurements when the optical properties of the two layers are known.     

An eccentric crack of mode III at the interface between two different media in a rectangular sheet

Abandoning the traditional assumption that the cracked-plate is infinite, the general solution to the stress intensity factor of an eccentric crack at the interface between two dissimilar layers in a finite rectangular sheet under an arbitrary anti-plane shear stress is found by use of the Fourier transform and Fourier series in this paper. It is easily verified that the stress intensity factor of this problem is independent of material constants of the cracked-plate bonded or welded by two different layers. We may also prove that all results of an eccentric crack or central crack at the interface between two layers in a strip are the special cases of the general solution in this study.     

An interface crack in a functionally graded piezoelectric bi-layer under anti-plane shear impact

The transient response of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear impact loading is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness, and the two layers are connected weak-discontinuously. Laplace transform and Fourier transform are used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate are presented for the FGPM to show the effects on the electric loading, variation and gradient of material properties, and thickness of layers. Following things are helpful to increase the resistance of transient fracture of interface crack in FGPMs: (a) increase of the material properties from the interface to the upper or lower free surface; (b) decrease of weak discontinuity at the interface; (c) increase of the gradient of material properties; (d) certain direction and magnitude of the electric loading; and (e) increase of the thickness of the FGPM layer.     

Coning during withdrawal from two fluids of different density in a porous medium

The steady response of the interface between two fluids with different density in a porous medium is considered during extraction through a line sink. Supercritical withdrawal, or coning as it is often called, in which both fluids are being withdrawn, is investigated using a coupled integral equation formulation. It is shown that for each entry angle of the interface into the sink there is a range of supercritical solutions that depend on the flow rate, and that as the flow rate decreases the cone narrows. As the magnitude of the entry angle increases this range of flow-rate values decreases to a narrow range as the entry becomes vertical. Only one branch of solutions (that with horizontal entry) has the property that the interface levels off at a finite height, and this is investigated as a separate branch of solution.     

Two-scale homogenization of elastic layered composites with interfaces oscillating in two directions

In a variety of situations of practical interest, the interface between two phases in a composite cannot be reasonably assumed to be smooth but has to be taken as being rough at the microscopic scale. How to determine the effective properties of such a composite remains a largely open problem in micromechanics. The present work is concerned with layered composites in which the interface between two neighboring layers oscillates quickly and periodically along two directions in the plane normal to the layering direction. In this case, the classical homogenization theory of layered composites is no longer applicable, since the interfacial oscillations prevent the layered composite in question from being homogeneous in the plane perpendicular to the layering direction. To overcome this difficulty, a two-scale homogenization method is proposed in the present work. First, at the mesoscopic scale, each zone in which an interface oscillates is homogenized as an interphase by an asymptotic analysis. The effective elastic properties of this interphase are determined by using a numerical method based on the fast Fourier transform (FFT) or estimated by applying the generalized self-consistent scheme (GSCS). Then, at the macroscopic scale, the effective elastic moduli of the composite made of the resulting plane layers and interphases are calculated with the help of the classical homogenization theory of layered composites. Finally, numerical examples are provided to illustrate the results for the effective elastic moduli of a layered composite obtained by the two-scale homogenization method proposed and to compare them with the corresponding numerical results given by the finite element method (FEM).     

Effect of a graded layer on the plastic dissipation in mixed-mode fatigue crack growth along plastically mismatched interfaces

Prior work by the authors has proposed a dissipated energy theory of fatigue crack growth in ductile solids under mode I loading based on the total plastic dissipation per cycle ahead of the crack. The approach has since been extended to a general bimaterial interface geometry under mixed-mode I/II loading, with application to fatigue debonding of layered materials. An inherent assumption of this prior work is that a perfect crack exists along the interface between the two materials. The current work extends the approach to incorporate a grading of material properties between the two layers, as may occur in a variety of welding, soldering or layered manufacturing applications. Attention is restricted to elastic perfectly-plastic layers with identical elastic properties and a mismatch in yield strength across a linearly graded interface, with the crack on the boundary of the weaker material. A dimensionless plastic dissipation is extracted from 2-D plane strain finite element models over the full range of yield strength mismatches, graded layer thicknesses and mixed-mode loading conditions. Results reveal that for all modes of loading, the effect of a graded layer is to increase the total plastic dissipation per cycle, which is bounded by the extremes in plastic mismatch for a perfect crack interface. While the graded layer has a measurable effect, the plastic dissipation for all strength mismatches is dominated by the mode of loading.     

A fracture mechanics problem of a functionally graded layered structure with an arbitrarily oriented crack crossing the interface

In this paper, the fracture mechanics problem for an arbitrarily oriented crack crossing the interface in a functionally graded layered structure is investigated. The elastic modulus is assumed to be continuous at the interface, but its derivative may be discontinuous. Applying the superposition principle and Fourier integral transform, the stress fields and displacement fields are derived. A group of auxiliary functions defined in both layers are introduced and then the mixed-mode crack problem is turned into solving a group of singular integral equations. The mixed-mode stress intensity factors (SIFs) are obtained by solving the singular integral equations. The influences of the material nonhomogeneity parameter, normalized crack length and crack angle on the SIFs are investigated. It is found that the mixed-mode SIFs can be affected greatly by the crack angle. Moreover, the mixed-mode SIFs usually attain their extremum when the crack tips get to the interface during one crack moves from one layer into another layer. The present work may form the basic work for establishing a multi-layered fracture mechanics model of FGMs with an arbitrarily oriented crack and general mechanical properties.     

Mixed‐mode delamination in 2D layered beam finite elements

In this work, a 2D finite element (FE) formulation for a multi‐layer beam with arbitrary number of layers with interconnection that allows for mixed‐mode delamination is presented. The layers are modelled as linear beams, while interface elements with embedded cohesive‐zone model are used for the interconnection. Because the interface elements are sandwiched between beam FEs and attached to their nodes, the only basic unknown functions of the system are two components of the displacement vector and a cross‐sectional rotation per layer. Damage in the interface is modelled via a bi‐linear constitutive law for a single delamination mode and a mixed‐mode damage evolution law. Because in a numerical integration procedure, the damage occurs only in discrete integration points (i.e. not continuously), the solution procedure experiences sharp snap backs in the force‐displacements diagram. A modified arc‐length method is used to solve this problem. The present model is verified against commonly used models, which use 2D plane‐strain FEs for the bulk material. Various numerical examples show that the multi‐layer beam model presented gives accurate results using significantly less degrees of freedom in comparison with standard models from the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Cusp flows due to an extended sink in two dimensions

The steady two-dimensioanl potential flow of a finite-depth fluid into an extended or distributed sink, in which the free surface dips to form a cusp above the centre of the sink, is examined. The extended sink is a region where the vertical outflow velocity V is constant and uniform. Numerical solutions for the free-surface profiles are obtained by use of a boundary-integral technique. Solutions are only found for the supercritical case where the Froude numbers are greater than one. In the limiting case where the extended sink width tends to zero, the problem reduces to that of a line sink beneath the free surface, and comparisons are made to existing results for this type of flow.     

Characterization of the interface in a carbon-epoxy composite using transmission electron microscopy

Two commercial kinds of unsized pan-based carbon fibres (high strength and high modulus) were subjected to electrochemical treatments (oxidative and non-oxidative) and to a nitrogen plasma. After embedding in an epoxy resin, they were sectioned by ultramicrotomy (diamond knife), and the interface between the resin matrix and the fibre was studied by the lattice fringes method (TEM). The adhesion of the fibre to the matrix depends on the surface treatment and the external microtexture of the fibre. High-modulus fibres (plasma etching) show a good adhesion only when the carbon layers are perpendicular to the interface. High-strength fibres exhibit two types of adhesion. Some treatments yield a surface which mainly presents carbon layers parallel to the interface with a rugosity of about 1–2 nm. The second type of adhesion consists of fibre-matrix interpenetration. In this case carbon layers are slightly exfoliated and perpendicular to the interface. For most treatments, interfacial shear stress was determined by the pull-out test. We found no correlation between TEM observations and shear stress data. Consequently, we were unable to discuss adhesion from the transmission electron microscopy results. It would probably be more reliable to consider the problem with a concept based on interface toughness. However, as a first step, we have defined a new parameter, the contact index, and have shown the relations between the contact index, the morphology of the interface and the interfacial shear stress determined by the pull-out test.     

Withdrawal of fluid through a line sink beneath a free surface above a sloping boundary

A series truncation method is used to compute the flow into a line sink from a region of fluid with a free surface and a sloping wall beneath the sink. The method admits a well-known exact solution for a particular value of the slope. Solutions with a cusp above the sink, and with a stagnation point above the sink are computed for all values of the slope, and compared with results at both ends of the range, i.e. with results for both a vertical wall and a horizontal bottom, with good agreement.     

Torsional impact response of a penny-shaped interface crack in a layered composite

The torsional impact response of a penny-shaped interface crack in a layered composite is considered in this study. The geometry of the composite consists of two bonded dissimilar elastic layers which are sandwiched between two half-spaces made of a different material. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometry parameters is discussed.     

Scattering of antiplane shear waves in a fiber-reinforced composite medium with interfacial layers

Summary This paper deals with the scattering of antiplane shear waves in a metal matrix composite reinforced by fibers with interfacial layers. We assume same-size cylindrical inclusions and same-thickness interface layers with nonhomogeneous elastic properties. The effective complex wave numbers follow from the coherent wave equation which depends only upon the scattering amplitude of the single scattering problem. Effective elastic constants can be obtained from phase velocities of coherent waves. Numerical calculations for an SiC-fiber-reinforced Al composite are carried out, and the effect of interface properties on scattering cross section, phase velocity, attenuation of coherent plane wave, and effective elastic constant is shown graphically.     

A front remeshing technique for a Lagrangian description of moving interfaces in two‐fluid flows

The numerical analysis of two‐fluid flows involves the treatment of a discontinuity that appears at the separating interface. Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation is called in this work as the moving Lagrangian interface remeshing technique (MLIRT). Copyright © 2005 John Wiley & Sons, Ltd.     

Deflection versus penetration of a wedge-loaded crack: effects of branch-crack length and penetrated-layer width

A crack intersecting an interface between two dissimilar material layers may advance by either deflecting along or penetrating through the interface. The criterion of deflection versus penetration can be established by the comparison of two ratios, the energy release rate ratio and the fracture energy ratio, of deflection to penetration. The effects of: (1) a finite length of the branch crack emanating from the main crack tip; and (2) a finite width of the layer subjected to crack penetration were examined in the present study. The results reveal that the above two factors have profound effects on the criterion of deflection versus penetration for a wedge-loaded crack.     

A moving Griffith crack at the interface of two dissimilar elastic layers

The anti-plane shear problem of a Griffith crack traveling with a constant velocity at the interface of two dissimilar isotropic elastic layers is considered. Integral transform method is used to reduce the problem to the solution of a singular integral equation which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations. The results for the particular cases of a moving Griffith crack at the interface of a layer and a half-space and two half-spaces are derived. Numerical results for the stress intensity factor are displayed graphically.     

Thermal stresses in a viscoelastic trimaterial with a combination of a point heat source and a point heat sink

A general solution for a thermoviscoelastic trimaterial combined with a point heat source and a point heat sink is presented in this work. Based on the method of analytic continuation associated with the alternation technique, the solutions to the heat-conduction and thermoelastic problems for three dissimilar, sandwiched media are derived. A rapidly convergent series solution for both the temperature and stress field, expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties, while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking the inverse Laplace transform. A typical example concerning the interfacial stresses generated from a combined arrangement of a heat source and sink are discussed in detail. The corresponding thin-film problem is also discussed.     

Convection in a Two-Layer System with a Deformable Interface Under Low Gravity Conditions

The onset of thermal convection in a system of two horizontal layers of immiscible liquids of similar densities is studied under low gravity conditions. A constant heat flux is prescribed at both rigid boundaries. A generalized Boussinesq approach that allows correct accounting for the interface deformation is used. The long-wave perturbations emerge under low-gravity conditions; either monotonic or oscillatory modes are critical depending on the problem. Moreover, two different modes of the monotonic instability exist. For the first instability mode, the convection dominates, whereas the interface remains nearly undeformable. The second monotonic instability mode is substantially related to interface deformations. The system of non-linear amplitude equations describing the behavior of long-wave regimes at finite-amplitude interface deflection and finite supercriticalities is obtained. The analytical and numerical investigations of these equations show that the stable non-trivial stationary solutions are absent, and after a transient at least one of the layers is split into the areas not connected to each other. The nonlinear regimes of cellular convection are studied numerically by the Level Set method.     

Fabrication and microstructure evolution of Al/Mg bimetal using a near-net forming process

The Al/Mg bimetal was fabricated by using the lost-foam casting method that is considered a near-net forming process, and the microstructure evolution of the Al/Mg bimetal from different cross-section locations of the bimetal was investigated in the present work. An obvious interface layer generated in the middle of the aluminum and magnesium, which showed a variation for different cross-section locations of the bimetal. The interface layer at the bottom cross-section location had a maximum thickness, wherein a big gap was observed. The thicknesses of the interface layer gradually decreased from the bottom location to the top location. At the middle cross-section location, the interface layer was uniform and compact. In contrast, the non-uniform interface layer was obtained at the top cross-section location, and the partial location had no the interface layer. The interface layers at the bottom and middle locations consisted of three reaction layers including the Al₁₂Mg₁₇?+?δ-Mg, the Al₁₂Mg₁₇?+?Mg₂Si, and the Al₃Mg₂?+?Mg₂Si layers. However, the partial location at the top cross-section location only had the Al₁₂Mg₁₇?+?δ-Mg layer.