The energy of a growing elastic surface

The potential energy of the elastic surface of an elastic body which is growing by the coherent addition of material is derived. Several equivalent expressions are presented for the energy required to add a single atom, also known as the chemical potential. The simplest involves the Eshelby stress tensors for the bulk medium and for the surface. Dual Lagrangian/Eulerian expressions are obtained which are formally similar to each other. The analysis employs two distinct types of variations to derive the governing bulk and surface equations for an accreting elastic solid. The total energy of the system is assumed to comprise bulk and surface energies, while the presence of an external medium can be taken into account through an applied surface forcing. A detailed account is given of the various formulations possible in material and current coordinates, using four types of bulk and surface stresses: the Piola-Kirchhoff stress, the Cauchy stress, the Eshelby stress and a fourth, called the nominal energy-momentum stress. It is shown that inhomogeneity surface forces arise naturally if the surface energy density is allowed to be position dependent.     

Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.     

Foam rheology: Relation between extensional and shear deformations in high gas fraction foams

A two-dimensional hexagonal foam cell model is used to derive analytic expressions for the bulk stress tensor and foam microstructure for any small homogeneous deformation. We show that calculations done for deformations where the principal axes of stress and strain coincide, such as in extension, are sufficient to provide all information about shear deformation. The stresses and foam structure for any given strain and initial cell orientation in shear bears a unique relation to a different strain and orientation in extension. Such a mapping is obtained using the assumption that the principal axes of strain and stress corotate with each other. This in turn implies that high gas fraction foams follow the Lodge-Meissner relation, i.e. the ratio of the normal-stress difference to the shear stress equals the shear strain. The spatially periodic structure of foam along with the fact that the cell centers move affinely with the bulk, makes the above assumption a justifiable one.     

Analytic test cases for three-dimensional hydrodynamic models

Exact periodic solutions are generated for the 3-D hydrodynamic equations in linearized form. A linear slip condition is enforced at the bottom, based on the velocity at the bottom. It is shown that the bottom stress can be equivalently expressed in terms of the vertically averaged velocity, and expressions for this bottom stress coefficient are derived in terms of the primary parameters of the problem. As a result, the three-dimensional structure may be assembled from conventional solutions to (a) the 1-D vertical diffusion equation; and (b) the 2-D vertically averaged shallow water equations. In the latter, the bottom stress effects are shown to be complex and frequency-dependent, and an additional rotational term is required for their representation.     

Bounded asymptotic solutions for incomplete contacts in partial slip

Asymptotic solutions for the contact pressure and shearing tractions present adjacent to the edge of an incomplete contact are derived, and for each a dimensional scaling factor (akin to a generalised stress intensity factor), is defined. These two quantities may therefore be used to define the complete local stress state, for which purpose the Muskhelishvili potentials are derived. The application of this technique to a comparison of the Hertz–Cattaneo contact, the flat and rounded contact and the tiled punch is described, and the extent of the domain of validity of the asymptote found.     

On co-rotational and other rate type constitutive equations

We derive expressions for the dilatational properties of suspensions of gas bubbles in incompressible fluids, using a cell model for the suspension. A cell, consisting of a gas bubble centered in a spherical shell of incompressible fluid, is subjected to a purely dilatational boundary motion and the resulting stress at the cell boundary is obtained. The same dilatational boundary motion is prescribed at the boundary of an “equivalent” cell composed of a one-phase, uniformly compressible fluid with unknown dilatational properties. By specifying that the stress at the boundary of the one-phase cell is equal to the stress at the boundary of the two-phase suspension cell, we obtain expressions for the unknown dilatational properties as a function of observable properties of the suspension. The dilatational viscosity of a suspension with a Newtonian continuous phase and the analogous properties for suspensions with non-Newtonian continuous phases are obtained as functions of the boundary motion, volume fraction of gas, and properties of the incompressible continuous phase. Results are presented for continuous phases which are Newtonian fluids, second-order fluids, and Goddard—Miller model fluids.     

Linear stability diagrams for the shear flow of an Oldroyd-B fluid with slip along the fixed wall

We consider the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. Slip is allowed by means of a generic slip equation predicting that the shear stress is a non-monotonic function of the velocity at the wall. The complete one-dimensional stability analysis to one-dimensional disturbances is carried out and the corresponding neutral stability diagrams are constructed. Asymptotic results for large values of the elasticity number and finite element calculations are also presented. The instability regimes are within or coincide with the negative-slope regime of the slip equation. The numerical calculations agree with the linear stability results when the size of the initial perturbation is small. Large perturbations may destabilize a linearly stable steady state, leading to a periodic solution. The period and the amplitude of the periodic solutions increase with elasticity. Received: 19 June 1997 Accepted: 22 September 1997     

Rheology of a suspension of particles in viscoelastic fluids

In this paper we study the bulk stress of a suspension of rigid particles in viscoelastic fluids. We first apply the theoretical framework provided by Batchelor [J. Fluid Mech. 41 (1970) 545] to derive an analytical expression for the bulk stress of a suspension of rigid particles in a second-order fluid under the limit of dilute and creeping flow conditions. The application of the suspension balance model using this analytical expression leads to the prediction of the migration of particles towards the centerline of the channel in pressure-driven flows. This is in agreement with experimental observations. We next examine the effects of inertia (or flow Reynolds number) on the rheology of dilute suspensions in Oldroyd-B fluids by two-dimensional direct numerical simulations. Simulation results are verified by comparing them with the analytical expression in the creeping flow limit. It is seen that the particle contribution to the first normal stress difference is positive and increases with the elasticity of the fluid and the Reynolds number. The ratio of the first normal stress coefficient of the suspension and the suspending fluid decreases as the Reynolds number is increased. The effective viscosity of the suspension shows a shear-thinning behavior (in spite of a non-shear-thinning suspending fluid) which becomes more pronounced as the fluid elasticity increases.     

Nonlinear dynamics of idler gear systems

This work examines the nonlinear, parametrically excited dynamics of idler gearsets. The two gear tooth meshes provide two interacting parametric excitation sources and two possible tooth separations. The periodic steady state solutions are obtained using analytical and numerical approaches. Asymptotic perturbation analysis gives the solution branches and their stabilities near primary, secondary, and subharmonic resonances. The ratio of mesh stiffness variation to its mean value is the small parameter. The time of tooth separation is assumed to be a small fraction of the mesh period. With these stipulations, the nonsmooth separation function that determines contact loss and the variable mesh stiffness are reformulated into a form suitable for perturbation. Perturbation yields closed-form expressions that expose the impact of key parameters on the nonlinear response. The asymptotic analysis for this strongly nonlinear system compares well to separate harmonic balance/arclength continuation and numerical integration solutions. The expressions in terms of fundamental design quantities have natural practical application.     

Balance equations and structural models for phase interfaces

The general balance equations are developed for an interface represented by a dividing surface and for a moving common line represented as an intersection of dividing surfaces. The surface excess variables associated with a dividing surface are expressed both in terms of those variables describing the three-dimensional interfacial region of finite thickness and in terms of those variables describing bulk phases that extend up to the dividing surface.A structural model for the interface is suggested in which a suspension of solid bodies representing surfactant molecules is distributed about a singular surface separating two adjacent bulk solvent phases. The suspension is required to have the same average behavior as the interfacial region. This is interpreted as meaning that the general jump balance for a continuum dividing surface represented by an interfacial suspension is a local area average. Specific results are derived for two structural models, each in the same simple shear field. One consists of a dilute suspension of neutrally buoyant spheres floating with their centers restricted to the dividing surface. The other is a dilute suspension of chains of neutrally buoyant spheres with the sphere at one end of the chain floating in the dividing surface.     

Estimates for the elastic moduli of random trigonal polycrystals and their macroscopic uncertainty

Explicit expressions of the upper and lower estimates on the macroscopic elastic moduli of random trigonal polycrystals are derived and calculated for a number of aggregates, which correct our earlier results given in Pham [Pham, D.C., 2003. Asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals. Int. J. Solids Struct. 40, 4911–4924]. The estimates are expected to predict the scatter ranges for the elastic moduli of the polycrystalline materials. The concept of effective moduli is reconsidered regarding the macroscopic uncertainty of the moduli of randomly inhomogeneous materials.     

On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid

The complete solution for the pressure and the velocity field up to O(De) of a dilute suspension of neutrally buoyant, non-Brownian rigid spheres suspended in an unbounded, weakly viscoelastic matrix fluid, where is the solid volume fraction and De is the Deborah number of the matrix fluid, is presented. The spheres are subjected to an arbitrary linear velocity profile at infinity. The analytical solution is used for the prediction of the bulk stress, and specifically for the calculation of the first and the second normal stress differences in simple shear and uniaxial elongational flows. A comparison of the results with available values reported in the literature is also offered. The final expressions for the bulk normal stress differences in shear and uniaxial elongational flow fully agree with those reported earlier by Greco et al., J. Non-Newton. Fluid Mech., 147 (2007) 1–10.     

Stress and strain field near tip of mode III growing crack in materials with creep behavior

Using a proposed constitutive relation for materials with creep behavior, the stress and strain distribution near the tip of a Mode III growing crack is examined. Asymptotic equations of the crack tip field are derived and solved numerically. The stresses remain finite at the crack tip. Obtained qualitatively is the crack tip velocity and the local autonomy of the near tip field solution is discussed.     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

Asymptotic solution of the three-dimensional multicomponent laminar boundary-layer equations with large suction

The flow of a multicomponent compressible gas in a three-dimensional laminar boundary layer is investigated for large values of the suction parameter. Asymptotic expressions are derived for the profiles of the velocities, temperatures, and concentration of the components across the boundary layer, as well as for the friction, heat-, and mass-transfer coefficients on the surface of the body.The authors wish to thank G. A. Tirskii for a discussion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 41–45, July–August, 1975.     

Some Exactly Solvable Problems of the Radiation of Three–Dimensional Periodic Internal Waves

An exact solution of the problem of the generation of three–dimensional periodic internal waves in an exponentially stratified, viscous fluid is constructed in a linear approximation. The wave source is an arbitrary part of the surface of a vertical circular cylinder which moves in radial, azimuthal, and vertical directions. Solutions satisfying exact boundary conditions, describe both the beam of outgoing waves and wave boundary layers of two types: internal boundary layers, whose thickness depends on the buoyancy frequency and the geometry of the problem, and viscous boundary layers, which, as in a homogeneous fluid, are determined by kinematic viscosity and frequency. Asymptotic solutions are derived in explicit form for cylinders of large, intermediate, and small dimensions relative to the natural scales of the problem.     

Localized solutions to the nonlinear wave equation for flexural motions of an elastic beam traveling in an air-filled tube

Steady-progressive-wave solutions are sought to the nonlinear wave equation derived previously [J. Fluids Struct. 16 (2002) 597] for flexural motions of an elastic beam traveling in an air-filled tube along its center axis at a subsonic speed. Fluid-structure interactions are taken into account through aerodynamic loading on the lateral surface of the beam subjected to small but finite deflection but end effects and viscous effects are neglected. Linear dispersion characteristics are first examined by exploiting the small ratio of the induced mass to the mass of the beam per unit length. Centered around the traveling speed of the beam, there exists such a narrow range of propagation velocity that the linear steady propagation is prohibited. In this range, it is revealed that some interesting nonlinear solutions exist. The periodic wavetrain is found to exist as the exact solution. Asymptotic analysis is then made by applying the method of multiple scales and the stationary nonlinear Schrödinger equation is derived for a complex amplitude. A monochromatic solution to this equation corresponds to the exact periodic solution. Imposing undisturbed boundary conditions at infinity, it is revealed that the localized solution exists as a result of balance between the linear instability and the nonlinearity. This solution is checked by solving the nonlinear equation numerically. It is further revealed that the amplitude-modulated wavetrain exists not only in the range of the velocity mentioned above but also outside of it.     

The hydrodynamic wall effect for a disperse system

The flow of a highly dilute suspension of spheres (radius α) between two parallel ridid planes (distance L) in slow shearing motion is studied. Even for the limiting situation, (α/L) small but finite, there is a layer-one sphere diameter thick—immediately adjacent to the wall in which bulk quantities are so complicated functionals of the parameters of the microstructure that evaluating them seems out of the question. Nevertheless, it is still simple to obtain average bulk quantifies (e.g. apparent viscosity) and even the evaluation of local bulk quantities far away from the wall poses no problem. The reason being that the customary continuum constitutive equation for the bulk stress can be utilized, although a slip velocity has to supplement it. This applies to any disperse system and can be applied to different flows, too. For the spherical suspension at hand an explicit expression for this slip velocity is obtained.     

Transverse radiation pressure and torque exerted by a plane electromagnetic wave on a rotating circular cylinder of isotropic material

Relativistically correct expressions are derived for the time-averaged total force and torque acting on a rotating conducting circular cylinder, immersed in an incident plane electromagnetic wave. With the help of Maxwell's stress tensor analytical expressions can be obtained for the time-averaged force and torque.