Showing papers in "Quarterly Journal of Mechanics and Applied Mathematics in 2006"

Instabilities in elastomers and in soft tissues

Abstract: Summary Biological soft tissues exhibit elastic properties that can be dramatically different from rubber-type materials (elastomers). To gain a better understanding of the role of constitutive relationships in determining material responses under loads we compare three different types of instabilities (two in compression, one in extension) in hyperelasticity for various forms of strain-energy functions typically used for elastomers and for soft tissues. Surprisingly, we find that the strain-hardening property of soft tissues does not always stabilize the material. In particular we show that the stability analyses for a compressed half-space and for a compressed spherical thick shell can lead to opposite conclusions: a soft tissue material is more stable than an elastomer in the former case and less stable in the latter case.

Universal relations in isotropic nonlinear magnetoelasticity

Abstract: In this paper we first summarize the basic constitutive equations for (nonlinear) magnetoelastic solids capable of large deformations. Equivalent formulations are given using either the magnetic induction vector or the magnetic field vector as the independent magnetic variable in addition to the deformation gradient. The constitutive equations are then specialized to incompressible, isotropic magnetoelastic materials in order to determine universal relations. A universal relation, in this context, is an equation that relates the components of the stress tensor and the components of the magnetic field and/or the components of the magnetic induction that holds independently of the specific choice of constitutive law for the considered class or subclass of materials. As has been shown previously for the case in which the magnetic induction is the independent magnetic variable, in the general case there exists only one possible universal relation. We show that this is also the case if the magnetic field is taken as the independent variable and that the universal relations resulting from the two cases are equivalent. A number of special cases are found for certain specializations of the constitutive equations. These include some connections between the deformation, the magnetic field and magnetic induction that do not involve the components of the stress tensor. Universal relations are then examined for some representative homogeneous and inhomogeneous universal solutions.

Flow around nanospheres and nanocylinders

Abstract: For micro- and nanoscale problems, boundary surface roughness often means that the usual no-slip boundary condition of fluid mechanics does not apply. Here we examine the steady low-Reynolds-number flow past a nanosphere and a circular nanocylinder in a Newtonian fluid, with the no-slip boundary condition replaced by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. We apply the so-called Navier boundary condition and use the method of matched asymptotic expansions. This model possesses a single parameter to account for the slip, the slip length l, which is made dimensionless with respect to the corresponding radius, which is assumed to be of the same order of magnitude as the slip length. Numerical results are presented for the two extreme cases, l = 0 corresponding to classical theory, and l → ∞ corresponding to complete slip. The streamlines for l > 0 are closer to the body than for l = 0, while the frictional drag for l > 0 is reduced below the values for l = 0, as might be expected. For the circular cylinder, results corresponding to l → ∞ are in complete accord with certain low-Reynolds-number experimental results, and this excellent agreement is much better than that predicted by the no-slip boundary condition.

Steady streaming around a spherical drop displaced from the velocity antinode in an acoustic levitation field

Abstract: The steady (acoustic) streaming associated with a spherical drop displaced from the velocity antinode of a standing wave is studied. The ratio of the particle size to the acoustic wavelength is treated as small but non-zero, and the solution is developed in the form of a two-term expansion in terms of the corresponding smallness parameter. The drop viscosity is assumed to be much higher than that of the surrounding fluid, which is the case for a drop in a gas medium. There are essentially three distinct regions where the steady streaming flow is analysed: inside the drop (internal circulation), in the Stokes shear-wave layer at the surface on the gas side, and the gas outside the Stokes layer (the outer streaming region). Solutions for the internal circulation and the outer streaming are obtained in the limit of small Reynolds number. Despite the gas-to-liquid viscosity ratio being small, the outer streaming may be dramatically affected by the fact that the sphere is liquid as opposed to solid. The parameter that measures the effect of liquidity is essentially the viscosity ratio divided by the relative (to the particle size) thickness of the Stokes layer. The case of a solid sphere is recovered by letting this parameter go to zero.

Transient elastohydrodynamic drag on a particle moving near a deformable wall

Abstract: We examine the prescribed time-dependent motion of a rigid particle (a sphere or a cylinder) moving in a viscous fluid close to a deformable wall. The fluid motion is described by a nonlinear evolution equation, derived using lubrication theory, which is solved using numerical and asymptotic methods; a local linear pressure-displacement model describes the wall. When the particle moves from rest towards the wall, fluid trapping beneath the particle leads to an overshoot in the normal force on the particle; a similarity solution is used to describe trapping at early times and a multiregion asymptotic structure describes fluid draining at late times. When the particle is pulled from rest away from the wall, a peeling process (described by a quasi-steady travelling wave) determines the rate at which fluid can enter the growing gap between the particle and the wall, leading to a transient adhesive normal force. When a cylinder moves from rest transversely over the wall, transient peeling motion is again observed (especially when the wall is initially indented), giving rise to an overshoot in the transverse drag. Simulations for a translating sphere show highly nonlinear wall deformations characterized by a localized crescent-shaped ridge. Despite generating sharp transient deformations, we found no numerical evidence of finite-time choking events.

On the scattering of spherical electromagnetic waves by a layered sphere

Abstract: A spherical electromagnetic wave is scattered by a layered sphere. The exact expressions of the scattered and interior fields are obtained by solving the corresponding boundary-value problem, by means of a combination of Sommerfeld's and T-matrix methods. A recursive algorithm with respect to the number of layers is extracted for the computation of the fields in every layer. The far-field pattern and the scattering cross-sections are determined in terms of the physical and geometrical characteristics of the scatterer. As the point-source tends to infinity, the known results for plane wave incidence are recovered. Numerical results are presented for several cases and various parameters of the layered spherical scatterer.

A sphere in a second degree polynomial creeping flow parallel to a wall

Abstract: Comprehensive results are provided for the creeping flow around a spherical particle in a viscous fluid close to a plane wall, when the external velocity is parallel to the wall and varies as a second degree polynomial in the coordinates. By linearity of Stokes equations, the solution is a sum of flows for typical unperturbed flows: a pure shear flow, a ‘modulated shear flow’, for which the rate of shear varies linearly in the direction normal to the wall, and a quadratic flow. Solutions considered here use the bipolar coordinates technique. They complement the accurate results of Chaoui and Feuillebois (2003) for the pure shear flow. The solution of Goren and O'Neill (1971) for the quadratic flow is reconsidered and a new analytical solution is derived for the ambient modulated shear flow. The perturbed flow fields for these two cases are presented in detail and discussed. Results for the force and torque friction factors are provided with a 5 × 10 −17 accuracy as a reference. For the quadratic flow, there is a force and a torque on a fixed sphere. A minimum value of the torque is found for a gap of about 0·18a, where a is the sphere radius. This minimum is interpreted in term of the corresponding flow structure. For the modulated shear flow, there is only a torque. The free motion of a sphere in an ambient quadratic flow is also determined.

Optimal orientation of anisotropic solids

Abstract: Results are presented for finding the optimal orientation of an anisotropic elastic material. The problem is formulated as minimizing the strain energy subject to rotation of the material axes, under a state of uniform stress. It is shown that a stationary value of the strain energy requires the stress and strain tensors to have a common set of principal axes. The new derivation of this well-known coaxiality condition uses the six-dimensional expression of the rotation tensor for the elastic moduli. Using this representation it is shown that the stationary condition is a minimum or a maximum if an explicit set of conditions is satisfied. Specific results are given for materials of cubic, transversely isotropic (TI) and tetragonal symmetries. In each case the existence of a minimum or maximum depends on the sign of a single elastic constant. The stationary (minimum or maximum) value of energy can always be achieved for cubic materials. Typically, the optimal orientation of a solid with cubic material symmetry is not aligned with the symmetry directions. Expressions are given for the optimal orientation of TI and tetragonal materials, and are in agreement with results of Rovati and Taliercio obtained by a different procedure. A new concept is introduced: the strain deviation angle, which defines the degree to which a state of stress or strain is not optimal. The strain deviation angle is zero for coaxial stress and strain. An approximate formula is given for the strain deviation angle which is valid for materials that are weakly anisotropic.

Steady-state motion of a mode-III crack on imperfect interfaces

Abstract: Mishuris, Gennady; Movchan, N.V.; Movchan, A.B., (2006) 'Steady-state motion of a Mode-III crack on imperfect interfaces', Quarterly Journal of Mechanics and Applied Mathematics 59(4) pp.487-516 RAE2008

On the development of rational approximations incorporating inertial effects in coating and rimming flows: a multiple-scales approach

Abstract: Analysis of thin-film rimming and coating flows is extended to create a more general model, which can be identified as a new distinguished limit that has the work of certain previous studies as specific limiting cases. Specifically, three-dimensional flows under the influence of gravity are considered in which the Reynolds number is large enough to necessitate consideration of both inertial and centrifugal effects, along with those of viscous, gravitational and surface-tension forces. Reduction of dimensionality is shown to be attainable within a systematic (multiple-timescale) asymptotic framework that encompasses all these effects, leading to a two-dimensional formulation that not only is more general than many of the existing ones but also represents a rational approximation in the sense that, in particular, the sizes of the dimensionless physical parameters to which it corresponds, together with the orders of magnitude of the errors resulting from the reduction, can be precisely characterized.

Trapped modes for an elastic strip with perturbation of the material properties

Abstract: The elasticity operator, for zero Poisson coefficient, with stress-free boundary conditions on a two-dimensional strip with local perturbation of Young's modulus, is considered. We prove the existence of embedded eigenvalues and describe their asymptotic behaviour.

Representation formulae of general solutions in the theory of hemitropic elasticity

Abstract: We consider the steady state oscillation equations of the theory of elasticity of hemitropic materials We derive general representation formulae for the displacement and microrotation vectors by means of six scalar metaharmonic functions These formulae are very convenient and useful in many particular problems for domains with concrete geometry Here we consider two canonical transmission problems for piecewise homogeneous bodies with spherical interfaces and with the help of the representation formulae construct explicit solutions in the form of absolutely and uniformly convergent series The representations can also be applied to multi-layered bodies with spherical and cylindrical interfaces

Inverse homogenization and design of microstructure for pointwise stress control

Abstract: Summary New higher order homogenization results are employed in an inverse homogenization procedure to identify graded microstructures that provide desirable structural response while ensuring stress control near joints or junctions between structural elements. The methodology is illustrated for long cylindrical shafts reinforced with sti cylindrical elastic b ers with generators parallel to the shaft. The local b er geometry can change across the shaft cross section. The methodology is implemented numerically for cross{sectional shapes that possesses reentrant corners typically seen in lap joints and junctions of struts. Graded locally layered microgeometies are identied that provide the required structural rigidity with respect to torsion loading while at the same time mitigating the inuence of stress concentrations at the reentrant corners. Modern design practice increasingly incorporates the use of load bearing components made from composite materials. Composites are now used in structural geometries that involve abrupt dimensional changes within structural components, such as skins connected to ribs, panel reinforcements and junctions of struts. Associated with these geometries are stress concentrations and the potential for failure. In this paper a design strategy is formulated for identifying graded microstructures that can be used to control the local uctuating stresses near stress concentrations induced by rivets, bolts or reentrant corners. Reentrant corners are are typically found in lap joints and near the junction between stieners and panels. The inverse homogenization design method is based upon the formulation of a homogenized design problem expressed in terms of suitable macroscopic quantities that satisfy two requirements: The rst requirement is that the homogenized design problem should be computationally tractable. The second requirement is that the solution of the homogenized design problem must provide the means to explicitly identify graded microstructures that engender suitable structural response while at the same time control local uctuating stresses in regions located near stress concentrations. It is now well known that eectiv e macroscopic constitutive properties relating average stress to average strain can be employed in the numerical design of composite structures

Asymptotic analysis of chattering oscillations for an impacting inverted pendulum

Abstract: The chattering oscillations for an inverted pendulum impacting between lateral walls, a prototype of a class of impact dampers, are analysed. Attention is focused on the periodic chattering appearing when the rest positions cease to be attracting, and the aim is that of computing the time r required by the micro-oscillations to come to rest. An algorithm is proposed to compute τ, and it is shown how this time depends on the excitation amplitude. When T becomes equal to the excitation period, the considered chattering is observed to lose attractivity. This occurs for a certain excitation amplitude threshold, which is computed and whose (very weak) dependence on the excitation frequency is illustrated. An asymptotic estimate of the chattering time with respect to a small parameter proportional to the excitation amplitude is given. This is obtained by an appropriate expansion in Taylor series of the relevant quantities involved in the analysis. It is shown that r is asymptotically proportional to the square root of the excitation amplitude.

A spectral approach for scattering by impedance polygons

Abstract: We study a new spectral approach for scattering by two-dimensional polygonal objects with arbitrary surface impedance conditions. In this delicate exterior problem, the Wiener-Hopf method cannot be applied, while asymptotic methods can only be used if corners are widely spaced compared to wavelength. A new method based on the Sommerfeld-Maliuzhinets integral representation is presented to reduce the problem to simple spectral equations in the complex plane. For this, we use an expression of the spectral function, where we can isolate the contribution of any element of an arbitrary surface. Considering polygons with impedance boundary conditions, it then becomes possible to derive functional equations on spectral functions of Maliuzhinets type for finite or infinite objects. We apply this approach to an important class of three-part impedance polygons composed of a finite segment attached to two semi-infinite planes, and reduce this problem to non-singular Fredholm integral equations, suitable for approximation or numerical inversion. In the particular cases of a three-part impedance plane or symmetric impedance polygon, we show that the system of integral equations in the spectral domain can be simply uncoupled.

Electromagnetic scattering from an anisotropic impedance half-plane at oblique incidence : The exact solution

Abstract: Scattering of a plane electromagnetic wave from an anisotropic impedance half-plane at skew incidence is considered. The two matrix surface impedances involved are assumed to be complex and different. The problem is solved in closed form. The boundary-value problem reduces to a system of two first-order difference equations with periodic coefficients subject to a symmetry condition. The main idea of the method developed is to convert the system of difference equations into a scalar Riemann-Hilbert problem on a finite contour of a hyperelliptic surface of genus 3. A constructive procedure for its solution and the solution of the associated Jacobi inversion problem is proposed and described in detail. Numerical results for the edge diffraction coefficients are reported.

Exact solutions to the unsteady two-phase Hele-Shaw problem

Abstract: Summary While many explicit solutions to the single-phase Hele-Shaw problem are known, solutions to the two-phase problem (also known as the ‘Muskat problem’) are scarce. This paper presents a new class of exact time-dependent solutions to the two-phase Hele-Shaw problem. It is demonstrated that an elliptical inclusion of one phase remains elliptical under evolution when immersed in any unsteady far-field linear flow of a second ambient phase. On the basis of this solution class, an ‘elliptical inclusion model’ for interactions in inhomogeneous porous media is outlined.

Accuracy of the multipole expansion applied to a sphere in a creeping flow parallel to a wall

Abstract: An example system is studied to discuss precision of the multipole expansion, applied to determine forces exerted on particles by a viscous low-Reynolds-number fluid flow. A single sphere in an ambient flow (pure shear, quadratic, and modulated shear) parallel to a close plane wall is considered. Forces and torques exerted by the ambient flow on a motionless sphere are evaluated. Their precision is determined and related to a multipole order of the truncation. Similar analysis is performed for a moving sphere with no ambient flow and for a freely moving sphere. Relative motion of the sphere with respect to the wall gives rise to strong lubrication interactions. It is analysed how these interactions affect accuracy of the pure multipole expansion, and what are the smallest distances where it becomes insufficient. An alternative precise method is applied, in which lubrication expressions are subtracted from the hydrodynamic forces and torques, and the residue is evaluated as a fast-convergent series of inverse powers of the distance between the sphere centre and the wall. The accuracy of this procedure is carefully analysed.

Pure shear axes and elastic strain energy

Abstract: It is well known that a state of pure shear has distinct sets of basis vectors or coordinate systems: the principal axes, in which the stress a is diagonal, and pure shear bases, in which diag a = 0. The latter is commonly taken as the definition of pure shear, although a state of pure shear is more generally defined by tr a = 0. New results are presented that characterize all possible pure shear bases. A pair of vector functions are derived which generate a set of pure shear basis vectors from any one member of the triad. The vector functions follow from a compatibility condition for the pure shear basis vectors, and are independent of the principal stress values. The complementary types of vector basis have implications for the strain energy of linearly elastic solids with cubic material symmetry: for a given state of stress or strain, the strain energy achieves its extreme values when the material cube axes are aligned with principal axes of stress or with a pure shear basis. This implies that the optimal orientation for a given state of stress is with one or the other vector basis, depending as the stress is to be minimized or maximized, which involves the sign of one material parameter.

Indentation of an incompressible inhomogeneous layer by a rigid circular indenter

Abstract: This work deals with an incompressible inhomogeneous layer bonded to a rigid substrate and indented without friction by a rigid circular indenter. The corresponding mixed boundary-value problem of elasticity is reduced to equivalent dual integral equations. It is shown that the pliability function in these equations may be found from a system of nonlinear differential equations and that its behaviour is peculiar when the elastic medium is incompressible. A novel technique taking into account this peculiarity is developed in order to reduce the dual integral equations to Fredholm integral equations of the second kind with symmetric strictly coercive operators. For a homogeneous layer and a flat indenter, the structure of the Fredholm integral equations permits an approximate analytical solution which is very accurate for any layer thickness. For an indenter of three-dimensional profile, leading asymptotic terms of the solution are derived in the case of a thin inhomogeneous layer.

Decomposition of elasticity tensors and tensors that are structurally invariant in three dimensions

Abstract: The elastic stiffness or compliance is a fourth-order tensor that can be expressed in terms of two second-order symmetric tensors A and B and a fourth-order completely symmetric and traceless tensor Z (or z) It is shown that the parts associated with A, B and Z (or z) are all structurally invariant under a three-dimensional transformation Thus a linear combination of the three parts gives a general expression for three-dimensional structural invariants All three-dimensional structural invariants available in the literature are shown to be special cases of this general expression Invariants that are inherited by each structural invariant are presented

The instability of the flow in a suddenly blocked pipe

Abstract: This paper considers the decay of Poiseuille flow within a suddenly blocked pipe. For small to moderate times the flow is shown to consist of an inviscid core flow coupled with a boundary layer at the pipe wall. A small-time asymptotic solution is developed and it is shown that this solution is valid for times up to the point at which the boundary layer fills the whole pipe. A small-time composite solution is used to initiate a numerical marching procedure which overcomes the small-time singularity that arises in the flow and so allows us to describe the ultimate decay of the flow within a blocked pipe. The stability of this flow is then considered using both a quasi-steady approximation and a transient-growth analysis based upon marching solutions of the linearized Navier-Stokes equations. Our transient stability analysis predicts a critical Reynolds number, for transition to turbulence, in the range 970 < Re < 1370.

Instabilities of a flexible surface in supersonic flow

Abstract: The stability of a supersonic boundary layer above flexible surface is considered in the limit of large Reynolds number and for Mach numbers O(1). Asymptotic theory of viscous-inviscid interaction has been used for this purpose. We found that for a simple elastic surface, for which deflections are proportional to local pressure differences, the boundary-layer flow remains stable as it is for a rigid wall. However, when either damping or surface inertia is included the flow becomes unstable. Moreover, in a certain range of wave numbers the boundary layer develops more then one unstable mode. It is interesting that these modes are connected to one another via saddle points in the complex,frequency plane. A more complex Kramer-type surface is also analysed and in some parameter ranges is found to permit the evolution of unstable Tollmien-Schlichting waves. The neutral curves are found for a variety of situations related to the parameters associated with the flexible surface.

Polynomial representations for initial-boundary-value problems involving the inviscid Proudman–Johnson equation

Abstract: The central aim of this paper is to show how two-point Hermite interpolation can be used to construct polynomial representations of solutions to some initial-boundary-value problems for the inviscid Proudman-Johnson equation. This classic equation of fluid dynamics can be regarded as first-order hyperbolic, and an important by-product of our analysis is an understanding of how Hermite interpolation can be utilized for such equations. Different types of boundary conditions may result in finite time blow-up and/or large time approach to the steady state depending on the value of a parameter appearing in the problem.

Thermoelastic waves in a constrained isotropic plate: Incompressibility at uniform temperature

Abstract: Thermoelastic waves are investigated which propagate through an isotropic plate which is constrained to be incompressible at uniform temperature. This is an example of a deformation-temperature constraint, also known as a strain-temperature constraint. The boundaries of the plate are taken to be traction free and either isothermal or insulated. Dispersion relations are derived and expanded asymptotically in the long-wave low-frequency limit. The short-wave limit is also discussed. The higher modes are investigated numerically. Graphical comparison with the unconstrained material is also presented.

Nonlinear Instability of Hypersonic Flow over a Cone

Abstract: This study investigates the nonlinear stability of hypersonic viscous flow over a sharp slender cone. The attached shock and the effects of curvature are taken into account. Asymptotic methods are used for large Reynolds number and large Mach number. A weakly nonlinear analysis is carried out allowing an equation for the amplitude of disturbances to be derived. The coefficients of the terms in the amplitude equation are evaluated for particular values of the parameters. The effect of the shock and curvature on the nonlinear stability of the flow are discussed.

On propagation of attenuated Rayleigh waves along a fluid-solid interface of arbitrary shape

Abstract: This paper is devoted to the quantitative description of high-frequency 'leaky' wave modes along an interface of arbitrary shape between a compressible fluid medium and an elastic solid, in the context of linearized elasticity. The interface is assumed to be smooth, with the typical radius of curvature much larger than the excitation wavelength. The employed technique consists of using a combination of the classical ray expansion of WKB-type and its boundary-layer version (see V. M. Babich and N.Ya. Kirpichnikova, The Boundary-Layer Method in Diffraction Problems, Springer, Berlin 1979). The main result of this work is explicit formulae for the leading-order term in the high-frequency approximation of the wave field.