On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Steady-state distribution of charged particles in the diffusion approximation

The steady-state distribution of charged particles in a weakly ionized plasma is examined for the case in which volume ionization, recombination, and diffusion in a space-charge field occur. A joint solution is obtained for the equation for charged-particle balance and the Poisson equations for the case of planar and cylindrical plasma configurations satisfying the Schottky condition at the boundaries of the region. A solution is also found for the case in which the ionization is localized in a spherically symmetric volume and in which the Schottky condition is satisfied at infinity. The condition for the existence of a steady-state solution is given and analyzed.     

Overall damage and elastoplastic deformation in fibrous metal matrix composites

A phenomenological constitutive model for fibrous composite materials with a ductile matrix is postulated incorporating damage mechanics with micromechanical behavior. The model is first formulated in an undamaged composite system and then transformed consistently achieved in terms of an overall damage tensor M for the whole composite. In the process of formulating this model, interesting results are obtained demonstrating the necessity of using a non-associated flow rule for plasticity in the damaged composite system together with a Hill's type yield criterion. It is also shown that using a Ziegler-Prager kinematic hardening rule for the ductile matrix leads to a general kinematic hardening rule for the composite that is a combination of a generalized Ziegler-Prager model and a Phillips-type model. Finally, an explicit expression for the elastoplastic stiffness tensor for the damaged composite is obtained.     

T-stress evaluation for slightly curved crack using perturbation method

A general formulation for evaluating the T-stress at crack tips in a curved crack is introduced. In the formulation, a singular integral equation with the distribution of dislocation along the curve is suggested. For a slightly curved crack, a small parameter is generally assumed for the crack configuration. By using the assumption for the small parameter, the perturbation method is suggested and it reduces the singular integral equation into many successive singular integral equations. If the cracked plate has a remote loading and the curve configuration is a quadratic function, the mentioned successive singular integral equations can be solved in a closed form. Therefore, the solution for the T-stress in a closed form is obtained. The obtained results for T-stress are shown by figures. It is found that if the involved parameter is not too small, the influence of the curve configuration is significant. Comparison for T-stresses obtained from a quadratic-shaped curved crack and an arc crack is presented.     

The edge dislocation in a three-quarter plane. Part II: Application to an edge crack

This paper presents the solution for the crack tip stress intensity factors for a short crack emanating from the corner of a three-quarter plane. The solution employs the influence functions for a dislocation along one of the projection lines of a three-quarter plane together with the Williams semi-infinite wedge solutions, and the result is valid for any remote loading configuration. The sole proviso is that the crack is short enough to be considered well within the region in which the asymptotes govern the stress state. This general solution is then applied to a specific geometry and is compared with a calibration for the stress intensity factors for a similar geometry.     

FLUTTER OF AN AIRFOIL WITH A CUBIC RESTORING FORCE

In this paper, the effect of a cubic structural restoring force on the flutter characteristics of a two-dimensional airfoil placed in an incompressible flow is investigated. The aeroelastic equations of motion are written as a system of eight first-order ordinary differential equations. Given the initial values of plunge and pitch displacements and their velocities, the system of equations is integrated numerically using a fourth order Runge-Kutta scheme. Results for soft and hard springs are presented for a pitch degree-of-freedom nonlinearity. The study shows the dependence of the divergence flutter boundary on initial conditions for a soft spring. For a hard spring, the nonlinear flutter boundary is independent of initial conditions for the spring constants considered. The flutter speed is identical to that for a linear spring. Divergent flutter is not encountered, but instead limit-cycle oscillation occurs for velocities greater than the flutter speed. The behaviour of the airfoil is also analysed using analytical techniques developed for nonlinear dynamical systems. The Hopf bifurcation point is determined analytically and the amplitude of the limit-cycle oscillation in post-Hopf bifurcation for a hard spring is predicted using an asymptotic theory. The frequency of the limit-cycle oscillation is estimated from an approximate method. Comparisons with numerical simulations are carried out and the accuracy of the approximate method is discussed. The analysis can readily be extended to study limit-cycle oscillation of airfoils with nonlinear polynomial spring forces in both plunge and pitch degrees of freedom.     

Shock wave diffraction on multi-facetted and curved walls

The two-dimensional diffraction of a shock wave over a wall made up of a series of plane and/or curved sections is considered. The analysis is based on the theory presented by, for the interaction of an originally plane shock wave with a corner. A method is presented by which the shock profile may be determined for a wall of any shape and for any incident Mach number, in regions where the characteristics form a simple wave. Comparisons are made between experimental measurements and theoretical predictions for convex walls consisting of a number of facets, and for circular arcs, for a range of incident shock wave Mach numbers. It is shown that the theory gives a satisfactory prediction of the wave shape, which improves as the Mach number increases. Modifications in the flow field behind the shock, compared to that for a simple corner made up of two plane walls is discussed, particularly relating to flow separation. For circular arc concave walls a inverse Mach reflection results experimentally, leading to regular reflection, for which the theory is of no use. PACS 47.40.Nm     

On correct account of finite rotations in finite plasticity theory

The non-uniqueness of the trantition from nonobjective constitutive relations to objective ones with the use of the principle of material frame-indifference (PMFI) is shown. To eliminate it, the concept of finite strain without rotations (FSWR) for a given material type and each strain component (elastic, plastic) is introduced. In FSWR the rotation is excluded with respect to the natural preferred configuration for a given material. Considered are a simple solid, a liquid, a monocrystal, a polycrystal and a composite. The proecedure is proposed for consistent generalization of known infinitesimal relations for finite strains and rotations. The structure of constitutive relations is derived for anisotropic elasto-plastic mono- and polycrystalline materials.     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

Generalized Airy Stress Functions

The classical Airy stress function in planar elastostatics cannot, in general, be a smooth function for multiply connected domains. Moreover, if a non-null body force field is active the classical Airy representation for the stress is not complete. Here, a generalized Airy representation for the stress is presented which preserves smoothness and which is complete. The generalized form identifies the explicit additional pieces that are needed for completeness in multiply connected domains and when a body force field is present.     

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh-Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.     

Diffusion velocity for a three-dimensional vorticity field

A discussion is presented on the existence of a diffusion velocity for the vorticity vector that satisfies extensions of the Helmholtz vortex laws in a three-dimensional, incompressible, viscous fluid flow. A general form for the diffusion velocity is derived for a complex-lamellar vorticity field that satisfies the property that circulation is invariant about a region that is advected with the sum of the fluid velocity and the diffusion velocity. A consequence of this property is that vortex lines will be material lines with respect to this combined velocity field. The question of existence of diffusion velocity for a general three-dimensional vorticity field is shown to be equivalent to the question of existence of solutions of a certain Fredholm equation of the first kind. An example is given for which it is shown that a diffusion velocity satisfying this property does not, in general, exist. Properties of the simple expression for diffusion velocity for a complex-lamellar vorticity field are examined when applied to the more general case of an arbitrary three-dimensional flow. It is found that this form of diffusion velocity, while not satisfying the condition of circulation invariance, nevertheless has certain desirable properties for computation of viscous flows using Lagrangian vortex methods. The significance and structure of the noncomplex-lamellar part of the viscous diffusion term is examined for the special case of decaying homogeneous turbulence.     

The bipore model in solid/liquid extraction: the continuous process

This paper presents the exact analytical solution for the general case of continuous mass transfer between a solid with a biporous structure (micro and macroporosity) and a countercurrent flowing fluid phase. The transport inside the solid is by molecular diffusion and outside of it the convective film resistance is included. A general expression is given which is valid for the infinite plate, for the infinite cylinder and for the sphere. The standard monopore case is obtained as a particular solution.     

Periodic oscillations of a class of non-autonomous non-linear elastic continua

A modified perturbation method for obtaining periodic solutions to a class of non-autonomous non-linear partial differential equations is developed. The classic small divisor is discussed in detail and a general method for its elimination is presented. New terminology is introduced for the purpose of discussing forcing functions that produce in a system a response that is of the same form as a non-linear periodic mode for the same system. Specific examples are examined to verify the results of this work.     

Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.     

On a moving interface crack with a contact zone in a piezoelectric bimaterial

An inplane problem for a crack moving with constant subsonic speed along the interface of two piezoelectric materials is considered. A mechanically frictionless and electrically permeable contact zone is assumed at the right crack tip whilst for the open part of the crack both electrically permeable and electrically insulated conditions are considered. In the first case a moving concentrated loading is prescribed at the crack faces and in the second case an additional electrical charge at the crack faces is prescribed as well. The main attention is devoted to electrically permeable crack faces. Introducing a moving coordinate system at the leading crack tip the corresponding inhomogeneous combined Dirichlet–Riemann problem is formulated and solved exactly for this case. All electromechanical characteristics at the interface are presented in a closed form for arbitrary contact zone lengths, and further, the transcendental equation for the determination of the real contact zone length is derived. As a particular case of the obtained solution a semi-infinite crack with a contact zone is considered. The numerical analysis performed for a certain piezoelectric bimaterial showed an essential increase of the contact zone length and the associated stress intensity factor especially for the near-critical speed region. Similar investigations have been performed for an electrically insulated crack and the same behavior of the above mentioned parameters is observed.     

Equivalent linearizations for practical hysteretic systems

Experimental testing of a friction damped base isolation system has indicated a need for a new model of friction damping and for an appropriate equivalent linearization technique. The model for the damping adopted is a combination of viscous damping, constant Coulomb friction and linear Coulomb friction.This model is incorporated into the equation of motion for a single-degree-of-freedom system and the exact solutions are given for free vibrations and for steady-state vibrations excited by a harmonic force. The exact solution is taken as a basis for an equivalent linearization technique that can be used in conjunction with conventional design spectra for a practical design of such a system.     

Poisson limit theorem for countable Markov chains in Markovian environments

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Modeling and dynamic analysis of a magnetically actuated butterfly valve

In this work, a magnetically actuated butterfly valve is considered and a novel and accurate mathematical model is derived. The equilibrium of the system is investigated and the effects of the inlet velocity and direct current voltage (DC) on the stable rotation angle of the valve are presented. Considering a time periodic perturbation arising from electric circuit, the effects of the operating angle, inlet velocity, and driving parameters on the periodic and chaotic dynamics of the system are investigated. It is observed that, for an opening angle less than the cut-off angle, there exists a unique DC voltage for a stable equilibrium. The stability of this equilibrium depends nonlinearly on the inlet velocity and the seating torque. An expression is derived for the threshold value for the stability of the valve. Under periodic voltage, the inlet velocity and stable angle induce a backward shift on the resonant frequency, and jump phenomena and subharmonics are observed for some values of the driving amplitude. The highest amplitudes of vibration are detected for a fully open valve, for an almost closed valve, and for a valve with large inlet velocity. Using bifurcation diagrams and Lyapunov exponents, it is shown that the system exhibits a route to chaos with windows of period doubling and unbounded motion. Some guidance for design of magnetically actuated butterfly valves is proposed as well as recommendations for future work.     

Indentation by nominally flat or conical indenters with rounded corners

Axisymmetric indentation of a flat surface is considered: specifically, the case of flat-ended indenter with rounded edges, and the case of a shallow cone with a rounded tip. Analytical solutions are obtained for the normal and sequential tangential loading, in both full or partial slip conditions (with the Cattaneofn9Mindlin approximation) , and for the complete interior stress field in all these conditions.Implications for strength of the contact are discussed with reference to metallic or brittle materials, with the intention to shed more light in particular to the understanding of common fretting fatigue or indentation testings with nominally flat or conical indenters. It is found that the strength of the contact, which is nominally zero for perfectly sharp flat or conical indenters, is well defined even for a small radius of curvature. This is particularly true for the flat indenter, for which the strength is even significantly higher than for the classical Hertzian indenter for a wide range of geometrical and loading conditions, rendering it very attractive for design purposes.