Interface conditions simulating influence of a thin elastic wedge with smooth contacts

The paper presents a method for deriving interface conditions simulating the influence of a thin wedge in a multi-wedge system with smooth contacts. It consists in successive (i) employing the Mellin’s transform, (ii) separation of the symmetric and anti-symmetric parts of a solution, (iii) distinguishing terms tending to infinity, when the wedge angle tends to zero, (iv) appropriate re-arrangement of the terms to avoid degeneration, (v) using truncated power series in equations for the thin wedge and (vi) inspection of the characteristic determinant and finding models simulating the influence of the thin wedge for various combinations of parameters. The paper extends and improves the results previously obtained by the authors for a harmonic problem. The analysis leads to three physical models of contact interaction, which cover all the ratios of shear modules of a thin wedge and neighbour wedges. Numerical examples illustrate the accuracy provided by the method employed and the models derived.     

Plane elasticity problem for a multi-wedge system with a thin wedge

The paper presents a method for studying a system of elastic wedges containing a thin wedge with the angle Θ₀, which may be arbitrary small. An analysis shows that the considered problem, involving 2-D vectors of tractions and displacements, cannot be solved by straight-forward extension of the method previously worked out by the authors for analogous scalar problems. The difficulty arises because of the disclosed feature of the dependences between the Mellin transformed displacements and tractions at the boundaries of a thin wedge: they are linearly dependent when their Taylor’s expansions in Θ₀ are represented by the first terms only. The difficulty is removed by using the consequences of the linear dependence and by an appropriate re-arrangement of variables. Then simple physical models, simulating the influence of a thin wedge on a multi-wedge system, become available. The models cover the cases of a very rigid and very compliant thin wedge and also intermediate cases. The ranges of the models applicability are studied analytically and illustrated by numerical results.     

Unsteady boundary layer on a circular cylinder embedded to a wedge

The boundary layer growth on a circular cylinder embedded to a wedge, when the motion is started impulsively from rest, is discussed using the method of inner and outer expansions. The equation for the time of separation involving the Reynolds number and the wedge angle is obtained. A uniformly valid solution is also found for the entire flow field. It is found that separation first occurs at the points where the cylinder meets the wedge. Also, the time of earliest separation decreases with increase in the Reynolds number and with increase in the wedge angle.     

Wave propagation along a non-principal direction in a compressible pre-stressed elastic layer

The dispersive behavior of small amplitude waves propagating along a non-principal direction in a pre-stressed, compressible elastic layer is considered. One of the principal axes of stretch is normal to the elastic layer and the direction of propagation makes an angle θ with one of the in-plane principal axes. The dispersion relations which relate wave speed and wavenumber are obtained for both symmetric and anti-symmetric motions by formulating the incremental boundary value problem for a general strain energy function. The behavior of the dispersion curves for symmetric waves is for the most part similar to that of the anti-symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress and propagation angle, it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds, while other higher modes have an infinite phase speed. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the Rayleigh surface wave speed and the limiting wave speeds of the layer, respectively. Numerical results are presented for a Blatz–Ko material and the effect of the propagation angle is clearly illustrated.     

Hydroelastic analysis of water impact of flexible asymmetric wedge with an oblique speed

Current paper deals with hydroelastic impact of asymmetric and symmetric wedge sections with oblique speed into calm water. It is aimed to provide a better insight regarding fluid–structure interaction of the wedge sections of a high-speed craft into water in more realistic condition, in the presence of heel angle and oblique speeds. The defined problem is numerically investigated by coupled Finite Volume Method and Finite Element Method under two-way approach consideration. Accuracy of the proposed model is assessed in different steps. The results of current method are compared against previous experimental, numerical and theoretical methods and good agreement is displayed in these comparisons. Subsequently, the method is used in order to examine the fluid and structure behavior during the elastic impact of the wedge into water. Accordingly, four different physical situations are simulated. In the first part, symmetric impact with no oblique speed is simulated. The results of this part show fluctuations in vertical force and pressure of the midpoint during the impact time. Also, the relation of deadrise with deflection and pressure is observed in this part. In the second part, heel angle is also taken into consideration. It is concluded that the pressure and deflections at the right side of the wedge reduce, but these parameters increase at the left side. Moreover, it is observed that, the pressure at the midpoint of the left side of the wedge with deadrise angle of 10°, becomes negative, when the wall of the flexible wedge reaches its largest deflection. It is also observed that, the pressure at left side of the wedge with deadrise angle of 20°, reaches zero. Such behavior does not occur for the wedges of 30° and 45° deadrise angles. In the third part of simulations, oblique water entry of a flexible wedge of 20° deadrise angle is simulated, and no heel angle is considered. Harmonic behavior is observed for the vertical force, horizontal force, pressure of the midpoint and its deflection. First peaks of all of these variables are larger than the second peak. The obtained results lead us to conclude that an increase in oblique speed yields larger deflection and pressure at the right side. Meanwhile, no significant effect is observed for the left side of the wedge. Also, larger oblique speed is found to yield larger forces and angular moment. Final part of simulations involves the oblique water entry of a flexible wedge of 5° heel angle. Comparison of the results in the final part with that of third part, show that heel angle affects the pressure and deflection at both sides of the wedge. It is also observed that pressure and deflections of the left side increase, while those of right side increase. It is also seen that, similar as in the case of no heel angle, an increase in oblique speed leads to an increase of pressure and deflection at the starboard. It also leads to an increase in frequency of the vibration at right side.     

Theoretical modeling of guided wave propagation in a sandwich plate subjected to transient surface excitations

A fast and efficient two-dimensional (2D) semi-analytical model based on global matrix method is developed to study the general characteristics of guided wave propagation in a honeycomb composite sandwich structure (HCSS) subjected to time-dependent transient surface excitations. The HCSS used in this study has an extremely lightweight and thick nomex honeycomb core, which is sandwiched between two thin graphite woven composite skins. The homogenized material properties of the skin and the core are considered to be elastic and quasi-isotropic in nature. Far-field time history of surface displacements are calculated for vertical and horizontal tone-burst surface excitations that are representative of thickness and radial mode of vibrations of piezoelectric transducers, respectively. Results are compared with those obtained from Finite element modeling (FEM) in LS-DYNA showing good agreement. Wavelet transform is then performed on the time-domain signal to obtain the group velocities of the propagating modes for their accurate identification on the basis of the theoretical dispersion curve. It is found that the response signal is dominated by the first anti-symmetric mode for a vertical excitation, whereas, the signal characteristics are multimodal in nature with dominating higher order symmetric and anti-symmetric modes for a horizontal excitation. The model is expected to be helpful for appropriate guided wave mode tuning and rapid analysis of data for experimental detection of disbonds in these novel structures.     

Stability of plane poiseuille and couette flow of a Maxwell fluid

The classical Orr-Sommerfeld analysis is extended to a Maxwell fluid in fully developed Poiseuille flow between two flat plates and Couette flow between two flat plates. For the Poiseuille flow problem eigenmodes that are anti-symmetric in position are considered to augment the literature results for the symmetric eigenmodes. A shooting method with a stiff integrator, orthonormalization, and Newton-Raphson iterations on the eigenvalue are used to find the eigenvalues. The most dangerous mode is the anti-symmetric one, and both symmetric and anti-symmetric modes are more dangerous when the wave number and the Weissenberg number are large. No unstable eigenvalues are found.     

Some Properties of a New Model for Slow Flow of Granular Materials

Some properties of a new continuum model for the bulk flow of a dense granular material in which neighbouring grains are in contact for a finite duration of time and in which the contact force is non-impulsive – the so called slow flow regime – are presented. The model generalises both the plastic potential and double-shearing models and contains an additional kinematic quantity – the intrinsic spin. The stress tensor is, in general, non-symmetric and separate yield conditions govern translational and rotational yield. We consider homogeneous, quasi-static loadings for the symmetric part of the stress and dynamic loading for the anti-symmetric part of the stress. A solution for the stress state in terms of a single parameter, namely the major principal direction of the symmetric part of the stress, is presented. This direction itself is determined by a consideration of the flow equations in the context both dilatant and isochoric simple shear flows. These simple flows are used to complete the characterisation of the relationship between the anti-symmetric part of the stress and the intrinsic spin.     

Anti-symmetric waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface on the dispersive behavior of in-plane time-harmonic symmetric waves in a pre-stressed incompressible symmetric layered composite, was analyzed recently by Leungvichcharoen and Wijeyewickrema (2003). In the present paper the corresponding case for time harmonic anti-symmetric waves is considered. The bi-material composite consists of incompressible isotropic elastic materials. The imperfect interface is simulated by a shear-spring type resistance model, which can also accommodate the extreme cases of perfectly bonded and fully slipping interfaces. The dispersion relation is obtained by formulating the incremental boundary-value problem and using the propagator matrix technique. The dispersion relations for anti-symmetric and symmetric waves differ from each other only through the elements of the propagator matrix associated with the inner layer. The behavior of the dispersion curves for anti-symmetric waves is for the most part similar to that of symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress for perfectly bonded and imperfect interface cases, a finite phase speed may exist only for the fundamental mode while other higher modes have an infinite phase speed. However, for a fully slipping interface in the low wavenumber region it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the phase speeds of the surface wave or the interfacial wave or the limiting phase speed of the composite. The bifurcation equation obtained from the dispersion relation yields neutral curves that separate the stable and unstable regions associated with the fundamental mode or the next lowest mode. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of imperfect interfaces on anti-symmetric waves is clearly evident in the numerical results.     

Two fundamental modes of disturbances and their development in a plane plume above a horizontal heat source

Experimental investigations are conducted on development of the disturbances which appear in the transition to turbulence in a natural convection plume above a horizontal line heat source in air. Both the power spectra of velocity and temperature in natural transition show that there seem to be two fundamental modes of disturbances. One is an outstanding peak about 0.8 Hz and the other a small one about 1.1 Hz in the spectra. The disturbances of these fundamental frequencies are observed as anti-symmetric modes around the entrance to the transition region. The disturbance of the first fundamental frequency is a selectively amplified anti-symmetric mode in that area. In contrast, the disturbance of the second fundamental frequency is thought to be originated from a symmetric mode and then transformed into an anti-symmetric mode of the same frequency during its growth.     

Nonlinear coupling and instability in the forced dynamics of a non-shallow arch: theory and experiments

Dynamic instability of a non-shallow circular arch, under harmonic time-depending load, is investigated in this paper both in analytical and experimental ways. The analytical model is a 2-d.o.f.?reduced model obtained by using a Galerkin projection of a mono-dimensional curved polar continuum. The determination of the regions of instability of the symmetric periodic solution and the discussion of the post-critical behavior are carried out, comparing the results with the experimental evidence on a companion laboratory steel prototype. During post-critical evolution, both periodic and non-periodic solutions are obtained varying the excitation control parameters. The theoretical and experimental models are analyzed around the primary external resonance condition of the first symmetric mode, in the case of a nearly 2:1 internal resonance condition between the first symmetric and anti-symmetric modes. When the motion loses regularity, synthetic complexity indicators are used to describe, in quantitative sense, the nonlinear response.     

Scattering of guided waves by circumferential cracks in composite cylinders

A somewhat generalized numerical procedure is used in this paper to study the problem of wave scattering by circumferential cracks in composite pipes. The study is motivated by the need to develop a model for the quantitative, ultrasonic non-destructive evaluation of cracks in pipes. For this purpose, a stiffness-based Rayleigh–Ritz type approach is employed first to obtain the approximate wave numbers and wave modes. Using the wave function expansions of the incident and scattered fields in the axial direction and decomposing the problem into separate symmetric and anti-symmetric problems, a three-dimensional wave scattering problem is reduced to two, independent two-dimensional problems over the circular cross-section. Both these problems can be reduced further to quasi-one-dimensions by discretizing the cross-section into finite elements and using a transfer matrix approach in the circumferential direction. This simplification greatly reduces the computational time. A comparison of the results for an isotropic pipe demonstrates the reliability and accuracy of the modified numerical procedure. Numerical results for the reflection and transmission coefficients of different incident wave modes are also presented for a 2-ply composite pipe with a crack. The crack may have an arbitrary circumferential length and radial depth. Simple extrapolations from one wave to another wave, separately incident on a crack, are demonstrated to be impossible due to different mode conversions by the crack.     

橡胶楔体与刚性缺口接触的大变形分析

利用一种新的橡胶材料应变能函数，对橡胶楔体与刚性缺口接触大变形问题进行了分析。得到了接触尖点附近变形的奇异性特征，给出了奇异性指数与材料常数、橡胶楔体角度、刚性缺口角度之间的关系式。同时编制了大变形有限元程序，计算得到了与理论解一致的结论。     

The critical angle of the anisotropic elastic wedge subject to uniform tractions

The classical solution for an isotropic elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2₀ satisfies the equation tan 2₀ = 2₀. This is the critical wedge angle which also renders infinite solutions for other types of loadings. In this paper, we study the associated problem for the anisotropic elastic wedge. We first present uniform stress solutions which are possible for symmetric loadings. For antisymmetric loadings, a uniform stress solution is in general not possible and we present a non-uniform stress solution in which the stress depends on but not on r. The non-uniform stress solution breaks down at a critical angle. We present an equation for the critical angle which depends on the elastic constants. The Stroh formalism is employed in the analysis. An integral representation of the solution is obtained by using new identities which are derived in the paper.     

Analysis of bonded anisotropic wedges with interface crack under anti-plane shear loading

The antiplane stress analysis of two anisotropic finite wedges with arbitrary radii and apex angles that are bonded together along a common edge is investigated. The wedge radial boundaries can be subjected to displacement-displacement boundary condi- tions, and the circular boundary of the wedge is free from any traction. The new finite complex transforms are employed to solve the problem. These finite complex transforms have complex analogies to both kinds of standard finite Mellin transforms. The traction free condition on the crack faces is expressed as a singular integral equation by using the exact analytical method. The explicit terms for the strength of singularity are extracted, showing the dependence of the order of the stress singularity on the wedge angle, material constants, and boundary conditions. A numerical method is used for solving the resul- tant singular integral equations. The displacement boundary condition may be a general term of the Taylor series expansion for the displacement prescribed on the radial edge of the wedge. Thus, the analysis of every kind of displacement boundary conditions can be obtained by the achieved results from the foregoing general displacement boundary condition. The obtained stress intensity factors （SIFs） at the crack tips are plotted and compared with those obtained by the finite element analysis （FEA）.     

Determination of the length of a short crack at a v-notch from a full field measurement

Determination of the length of a short crack at the root of a v-notch, from a full kinematic field measurement, is performed using a direct method. It is based on a matched asymptotic expansions procedure together with the theory of singularities. The first corrective term of the outer expansion can be straightforwardly expressed as a function of the crack length. Its extraction is achieved through the calculation of the associated generalized stress intensity factors for elastic homogeneous materials as well as bimaterials. Numerical simulations are carried out on a finite element solution disturbed by a random noise. In addition, the method used to compute the generalized stress intensity factors proved accurate and robust.     

Ecoulement visqueux entre deux cones coaxiaux

A numerical method of resolution of laminar incompressible flows in cones of revolution is proposed by asymptotic expansions in powers of 1/r (r radius vector). Remarks on linearity allow to calculate all wanted terms, function after function, by fourth-order Runge-Kutta process. Two examples are selected: the flow between two symmetric cones and one between a cone and a plane. The study of the flow between two symmetric cones as a function of the aperture angle reveals the existence of two patterns separated by a discontinuity at approximately 156°.     

Bending of a radially prestressed annular plate by tilting a central rigid inclusion

A thin annular plate contains a rigid, circular, central inclusion. The plate is subjected to a large axisymmetric radial load at its outer edge, where it is also restrained against transverse displacement and rotation. A couple applied to the rigid inclusion causes it to rotate about its diameter out of the plane of the plate. We use the method of matched asymptotic expansions to find an approximate expression for the applied couple as a function of the angle of rotation of the rigid inclusion. If the outer radius of the annulus is very large compared to the inner radius, then the couple required to rotate a truly rigid inclusion is 25% higher than the couple required to rotate an inclusion whose membrane strain stiffness is the same as that of the plate (cf. ref. [3]) through the same small angle.     

Quasi-static compression and spreading of an asymptotically thin nonlinear viscoplastic layer

The problem of quasi-static compression and spreading (squeezing) of a thin viscoplastic layer between approaching absolutely rigid parallel-arranged plates is solved using asymptotic integration methods rapidly developed in recent years in the mechanics of deformable thin bodies. A solution symmetric about the coordinate axes is sought in the same region of the layer as in the classical Prandtl problem. The layer material is characterized by a yield point and a hardening function relating the intensities of the stress and strain rate tensors. The conditions of no-flow and reaching certain values by tangential stresses are imposed on the plate surfaces. The coefficients at the terms of the asymptotic expansions corresponding to the minus first and zero powers of the small geometrical parameter are obtained. An approximate analytical solution in the case of power hardening and large Saint-Venant numbers is given. The physical meaning of the roughness coefficient characterizing the cohesion between the plates and viscoplastic material is discussed.     

Rotary aerodynamic derivatives in the asymptotic theory of a high-aspect-ratio wing

The linear problem of inviscid incompressible flow around a high-aspect-ratio wing at an angle of attack and in the presence of steady pitching and rolling rotation is considered. The main integral equation of the problem is reduced to a sequence of one-dimensional integral equations without use of the matched asymptotic expansions method. The first few terms of the series for the circulation distribution over the wing surface are calculated. For an elliptic high-aspect-ratio wing the corresponding aerodynamic forces are calculated. The derivatives of the aerodynamic coefficients of the wing with respect to the angle of attack and the angular velocities are determined. The asymptotic expressions obtained are compared with the results of numerical calculations of the corresponding derivatives using the discrete vortex method.