Showing papers in "Acta Mechanica in 2007"

Size-dependent elastic properties of unidirectional nano-composites with interface stresses

Abstract: Surfaces and interfaces in solids may behave differently from their bulk counterparts, particularly when the geometry is on the nanoscale. Our objective in this work is to assess the overall behavior of composites containing cylindrical inclusions with surface effects prevailing along the interfaces. In the formulation, we first decompose the loadings into three different deformation modes: the axisymmetric loadings, the transverse shear and the antiplane shear. For each deformation mode, we derive the energy potential incorporating the surface effects. Using a variational approach, we construct the Euler-Lagrange equation together with the natural transition (jump) conditions. The surface effects are represented by an interface of a membrane type, with in-plane moduli different from those of either phase. The overall elastic behavior of the composite is characterized by five constants. Four of them, except the transverse shear modulus, are derived in simple closed forms using an approach of neutral inclusion. For the transverse shear, we derive the value based on the generalized self-consistent method.

Erratum to: Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis

Abstract: In this paper, the change of the elastic fields induced by the interface energies and the interface stresses from the reference configuration to the current configuration is considered. It is emphasized that the governing equations taking into account the interface energy effect should be established within the framework of finite deformation in the first place, and then the approximations of governing equations for a finitely deformed multi-phase elastic medium by an infinitesimal strain analysis can be formulated. Hence it can be seen that the asymmetric interface stress has to be used in the Young-Laplace equation. According to the above mentioned formalism, analytical expressions of the size-dependent effective moduli of a particle-filled composite material with interface energy effect are derived. It is shown that, different from the results obtained by previous researchers, the liquid-like surface/interface tension, as a residual stress-type term, also influences the effective property of the composite.

Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes

Abstract: Vibration and buckling of in-plane loaded simply supported double-walled carbon nanotubes were investigated using the nonlocal Timoshenko-beam theory. The influence of in-plane loads on the natural frequencies was determined. The results show that while the natural frequencies decrease with increasing compressive in-plane loads, an increase in frequencies is observed for tension type of in-plane loads. Effects of in-plane loads are more pronounced for lower modes, and some mode changes are observed at critical in-plane loads. A comparison of nonlocal elasticity solutions with local elasticity solutions indicates that the nonlocal effects should be considered for higher modes of vibration of double-walled carbon nanotubes.

Love wave propagation in layered magneto-electro-elastic structures with initial stress

Abstract: An analytical approach is taken to investigate Love wave propagation in layered magneto-electro-elastic structures with initial stress, where a piezomagnetic (piezoelectric) material thin layer is bonded to a semi-infinite piezoelectric (piezomagnetic) substrate. The magneto-electrically open and short conditions are applied to solve the problem. The phase velocity of the Love wave is numerically calculated for the magneto-electrically open and short cases, respectively. The effect of the initial stress on the phase velocity and the magneto-electromechanical coupling factor are studied in detail for piezomagnetic ceramics CoFe2O4 and piezoelectric ceramics BaTiO3. We find that the initial stress has an important effect on the Love wave propagation in layered piezomagnetic/piezoelectric structures.

On some helical flows of Oldroyd-B fluids

Abstract: In this study the velocity fields and the associated tangential stresses corresponding to some helical flows of Oldroyd-B fluids between two infinite coaxial circular cylinders and within an infinite circular cylinder are determined in forms of series in terms of Bessel functions. At time t = 0 the fluid is at rest and the motion is produced by the combined action of rotating and sliding cylinders. The solutions that have been obtained satisfy the governing differential equations and all imposed initial and boundary conditions. For λr = 0, λ = 0 or λr = λ = 0 they reduce to the similar solutions for a Maxwell, second grade or Newtonian fluid, respectively. Finally, for comparison, the velocity profiles corresponding to the four models are plotted for different values of t.

Vibration of functionally graded cylindrical shells based on higher order shear deformation plate theory with ring support

Abstract: In this paper, a study on the vibration of thin cylindrical shells with ring supports made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which impose a zero lateral deflection. The study is carried out based on third order shear deformation shell theory (T.S.D.T). The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies. The properties are graded in the thickness direction according to the volume fraction power-law distribution. The analysis is carried out with strains-displacement relations from Love's shell theory. The governing equations are obtained using an energy functional with the Rayleigh-Ritz method. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.

An analysis of peristaltic transport for flow of a Jeffrey fluid

Abstract: The peristaltic mechanism of a Jeffrey fluid in a circular tube is investigated. The rheological effects and compressibility of the fluid are taken into account. The modeled equations are solved using perturbation technique when the ratio of the wave amplitude to the radius of the pore is small. In the second order approximation, a net flow due to a travelling wave is obtained and effects of Reynolds number, relaxation and retardation times, compressibility of the fluid and tube radius are studied. It is noticed that for the Jeffrey fluid the back flow only occurs for large values of the relaxation time and small values of the retardation time (less than 10 in the present analysis). Another interesting observation is that oscillatory behavior of the net flow rate in the Jeffrey fluid is less than that of a Maxwell fluid. Several results of other fluid models can be deduced as the limiting cases of our situation.

Newtonian flow with nonlinear Navier boundary condition

Abstract: The generalized nonlinear Navier boundary condition advocated by Thompson and Troian in the journal Nature, and motivated from molecular dynamical simulations, is applied to the conventional continuum mechanical description of fluid flow for three simple pressure-driven flows through a pipe, a channel and an annulus, with a view to examining possible non-uniqueness arising from the nonlinear nature of the boundary condition. For the pipe and the channel it is shown that the results with the nonlinear Navier boundary condition may be obtained from a pseudo linear Navier boundary condition but with a modified slip length. For the annulus, two sets of physically acceptable solutions are obtained corresponding to the chosen sign of the normal derivative of the velocity at each solid boundary. Closer examination reveals that although the generalized Navier boundary condition is highly nonlinear, in terms of the assumed form of solution the integration constants obtained are still unique for the three simple pressure-driven flows presented here, provided that care is taken in its application and noting that the multiplicity of solutions obtained for the annulus arise as a consequence of adopting different signs for the normal derivatives of velocity at the boundaries.

Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid

Abstract: The present work examines the flow of a third grade fluid and heat transfer analysis between two stationary porous plates. The governing non-linear flow problem is solved analytically using homotopy analysis method (HAM). After combining the solution for the velocity, the temperature profile is determined for the constant surface temperature case. Graphs for the velocity and temperature profiles are presented and discussed for various values of parameters entering the problem.

Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet

Abstract: The problem of axisymmetric flow of a third grade fluid over a radially stretching sheet is studied. By means of similarity transformation, the governing non-linear partial differential equations are reduced to a non-linear ordinary differential equation. The ordinary differential equation is analytically solved using homotopy analysis method (HAM). The solution for the velocity is obtained. The series solution is developed and the convergence of the results is discussed. Finally, the results are discussed with various graphs.

SH wave propagation in a cylindrically layered piezoelectric structure with initial stress

Abstract: SH wave propagation in a cylindrically layered piezoelectric structure with initial stress is investigated analytically. By means of transformation, the governing equations of the coupled waves are reduced to Bessel and Laplace equations. The boundary conditions imply that the displacements, shear stresses, electric potential, and electric displacements are continuous across the interface between the layer and the substrate. The electrically open and short conditions at the cylindrical surface are applied to solve the problem. The phase velocity is numerically calculated for the electrically open and short cases, respectively, for different wavenumber and thickness of the layer. The effect of the initial stress on the phase velocity and the electromechanical coupling factor are discussed in detail for piezoelectric ceramics PZT-5H. We find that the initial stress has an important effect on the SH wave propagation in the cylindrically layered piezoelectric structures. The results also show that the ratio of the layer thickness to the wavelength has a remarkable effect on the SH wave phase velocity and electromechanical coupling factor.

SH-waves at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space

Abstract: A problem of reflection and transmission of a plane SH-wave incident at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space is investigated. Rayleigh's method of approximation is applied to derive the reflection and transmission coefficients for the first and second order approximation of the corrugation. The expressions for reflection and transmission coefficients for the first order approximation of the corrugation are obtained in closed form for a special type of interface, and the energy partition relation is derived. It is found that these coefficients are proportional to the amplitude of corrugation and are functions of elastic properties of the half-spaces and also of the angle of incidence. Numerical examples illustrating the effects of the sandiness, the anisotropy, the corrugation of the interface, the frequency, and the angle of the incidence on the coefficients are presented.

Axisymmetric stagnation-point flow over a lubricated surface

Abstract: Axisymmetric stagnation-point flow is considered. A Newtonian fluid impinges orthogonally on a plane surface lubricated by a thin non-Newtonian liquid film of variable thickness. A slip-flow boundary condition is deduced, which allows for partial slip at the surface. The amount of slip, from full slip to no-slip, is controlled by a dimensionless slip coefficient. Similarity solutions are generally prohibited by the slip-flow boundary condition, except for one particular value of the power-law index of the lubricant. Solutions are presented for this case in order to demonstrate the influence of partial slip on the stagnation point flow. With increasing slip and reduced surface stress, a thinning of the viscous boundary layer is observed. The classical Homann flow is recovered in the no-slip limit.

Thermodynamic format and heat generation of isotropic hardening plasticity

Abstract: Heat generation due to plastic deformation of metals and steel is studied. Whereas in many investigations it is assumed that the fraction η of the plastic work transformed into heat is constant throughout the deformation process, the fraction η is here derived from thermodynamic considerations in a large-strain setting. It is shown that for elasto-plasticity the fraction η follows as a result of the choice of free energy, potential function and yield function. Taking the stress-strain response and the dissipative properties of the material as basis for calibration, it is shown that the thermodynamic framework of a thermoplastic material is non-unique for the general situation of non-associated plasticity. In the investigation conducted here, the mechanical response and the portion of the plastic work converted into heat (or into stored energy) during plastic deformations is predicted by means of isotropic hardening von Mises plasticity. It is shown that for a situation in which the internal variable is taken as the effective plastic, close fitting to experimental data requires a non-associated format of the evolution law for the internal variable.

Complete solution of elastica for a clamped-hinged beam, and its applications to a carbon nanotube

Abstract: This paper treats an exact elastica solution for a clamped-hinged beam, and its applications to a carbon nanotube. Although the elastica has a long history, and the exact post-buckling solution for the Euler buckling problem has been known for at least 150 years, it seems that the elastica solution for a post-buckled clamped-hinged beam has never been obtained. Therefore, the exact solution obtained in this paper constitutes an addition to the existing family of elastica solutions. As an application of the results, a post-buckling analysis of a single wall carbon nanotube is studied. Also, a potential use of the post-buckling analysis of the carbon nanotube for the determination of its Young's modulus has been indicated.

Evaluation of the Micropolar elasticity constants for honeycombs

Abstract: The Micropolar and Lame constants for a circular cell polycarbonate honeycomb are calculated using a finite element representation of the honeycomb microstructure. A hexagonally packed, circular cell, honeycomb sheet with a rigid circular inclusion was numerically analyzed under uniaxial tension. Micropolar Elasticity was found to be the best continuum representation of the discrete honeycomb. This conclusion was arrived at by matching the strain field in the discrete honeycomb with that predicted by a micropolar elastic continuum representation of the honeycomb. The minimum ratio of inclusion radius a to cell diameter d, for the honeycomb to be approximated accurately as a continuum was between 16 and 20. A radius of 20d and 32d showed a near perfect approximation to a continuum.

Effect of fabric anisotropy on shear localization in sand during plane strain compression

Abstract: The paper focuses on the effect of fabric anisotropy on shear localization in cohesionless granular materials. For the numerical simulation, a hypoplastic constitutive model was used. In order to take into account a characteristic length of the micro-structure, the constitutive model was extended to include the second gradient of the Euclidian norm of the deformation rate. The hypoplastic model captures the salient features of granular bodies in a wide range of density and pressure with a single set of parameters. Transversal isotropy is described by the dyadic product of the normal vector of the space orientation of the plane of symmetry. FE-simulations of plane strain compression under constant lateral pressure were carried out with a medium dense specimen for both uniform and stochastic distribution of the initial void ratio. The effect of the direction of the bedding plane and the initial void ratio distribution on the load-deformation behavior was investigated. Moreover, the location, thickness and inclination of the shear zone were also analyzed.

Scattering of antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface

Abstract: We present analytical solutions for the scattering of an antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface. We first consider the simple case in which the imperfection is homogeneous along the interface. Two typical imperfect interfaces are addressed: 1) mechanically compliant and dielectrically weakly conducting interface, and 2) mechanically compliant and dielectrically highly conducting interface. The expressions for the directivity pattern and scattering cross-section of the scattered shear waves are derived. We then investigate the more difficult problem in which the imperfection is circumferentially inhomogeneous along the interface. A concise expression for an inhomogeneously compliant and weakly conducting interface is derived by means of matrix notation. Numerical examples are presented to demonstrate the effect of the imperfection and the circumferential inhomogeneity of the interface on the directivity patterns and scattering cross-sections of the scattered shear wave. The circumferentially inhomogeneous interface is also utilized to model the interface where an arbitrary number of cracks exist. Results show that when every part of the interface is rather compliant, large low-frequency peaks of the scattered cross-sections, which correspond to the resonance scattering, can be observed no matter if the interface is homogeneous or inhomogeneous. The appearance of large low-frequency peaks can be well explained by estimating the natural frequency of the corresponding reduced mass-spring system where the cylinder is assumed as a rigid body. Peaks of the scattered cross-sections spanning from low frequencies to high frequencies can be observed for a cylinder with a partially debonded interface.

Compound matrix block diagonalization for efficient solution of eigenproblems in structural mechanics

Abstract: Recently, methods were developed for the decomposition of special matrices, leading to canonical forms I-IV. In this paper, the conditions required for the decomposability of such matrices are studied. It is shown that these canonical forms are obtainable by special block diagonalization of matrices, having certain properties. Here, tri- to penta-block diagonal matrices are studied and methods are developed for their decomposition. These methods are incorporated in the efficient solution of numerous eigenproblems of structural mechanics.

On the applicability of tension field theory to a wrinkling instability problem

Abstract: The wrinkling instability of a pre-stressed annular membrane loaded uniformly along its inner boundary is investigated with the help of a tension field theory. The theoretical solution is shown to capture the essential qualitative features involved in axisymmetric wrinkling, but displays several limitations regarding the quantitative aspects of this issue. The relationship with some related experimental work reported recently in the literature is also discussed.

Gravity-driven film flow down an inclined wall with three-dimensional corrugations

Abstract: The gravity-driven flow of a liquid film down an inclined wall with three-dimensional doubly periodic corrugations is investigated in the limit of vanishing Reynolds number. The film surface may exhibit constant or variable surface tension due to an insoluble surfactant. A perturbation analysis for small-amplitude corrugations is performed, wherein the wall geometry is expressed as a Fourier series consisting of a linear superposition of two-dimensional oblique waves defined by two base vectors. Each of the constituent perturbation flows over the individual oblique waves is further decomposed into a two-dimensional flow transverse to the oblique waves and a unidirectional flow parallel to the waves. Both the transverse and the parallel flow are calculated by carrying out an analysis in oblique coordinates, similar to that conducted for two-dimensional flow. The particular cases of flow down a wall with oblique two-dimensional, orthogonal three-dimensional, and hexagonal three-dimensional corrugations are considered. The results illustrate the surface velocity field and the distribution of the surfactant. The three-dimensional wall geometry is found to reduce the surface deformation with respect to its two-dimensional counterpart by increasing the effective wave numbers and decreasing the effective capillary number encapsulating the effect of surface tension.

Field equations, equations of motion, and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness from a three-dimensional theory

Abstract: This work uses tensor calculus to derive a complete set of three-dimensional field equations well-suited for determining the behavior of thick shells of revolution having arbitrary curvature and variable thickness. The material is assumed to be homogeneous, isotropic and linearly elastic. The equations are expressed in terms of coordinates tangent and normal to the shell middle surface. The relationships are combined to yield equations of motion in terms of orthogonal displacement components taken in the meridional, normal and circumferential directions. Strain energy and kinetic energy functionals are also presented. The equations of motion and energy functionals may be used to determine the static or dynamic displacements and stresses in shells of revolution, including free and forced vibration and wave propagation.

Mixed convection with viscous dissipation in a vertical channel filled with a porous medium

Abstract: The fully developed laminar mixed convection flow in a vertical plane parallel channel filled with a porous medium and subject to isoflux ÷ isothermal wall conditions is investigated assuming that (i) the Darcy law and the Boussinesq approximation hold, (ii) the effect of viscous dissipation is significant, and (iii) the average flow velocity U m (as an experimentally accessible quantity) is prescribed. It is shown that under these conditions both upward (U m > 0) and downward (U m < 0) laminar flow solutions may exist as long as U m does not exceed a maximum value U m, max. The velocity field can either be unidirectional or bidirectional. Moreover, bidirectional flow configurations are possible also for U m = 0. A remarkable feature of the problem is that for U m < U m, max even two solution branches (dual solutions) exist, which merge when U m approaches its maximum value U m, max. The mechanical and thermal characteristics of the flow configurations associated with the dual solutions are investigated in the paper analytically and numerically in detail.

Finite deformation of a pressurized magnetoelastic membrane in a stationary dipole field

Abstract: A nonlinear boundary-value problem for the equilibrium of a pressurized magnetoelastic membrane acted upon by an applied magnetic field is derived from three-dimensional magnetoelasticity. The model is specified entirely by differential equations in the limit of weak material magnetization, these replacing the integro-differential equations of the general theory. The model is further specialized to axisymmetry, and several problems of technical interest are solved numerically.

Toward an equation of state for solid materials with memory by use of the half-order derivative

Abstract: The time derivative operator does not depend upon the difference between the current time and the past times; however, the fractional time derivative operator does Thus, it is reasonable to expect that the fractional derivative would be useful in describing the mathematical theory of the behavior of materials with memory An equation of state is proposed for solid materials with memory by introducing the half-order fractional calculus derivative in order to relate to the empirical expression used in the fundamental work of Tobolsky and Catsiff This theory replaces the three empirical functions used by Tobolsky and Catsiff in reducing their experimental data for the low temperature glassy region, the transition region and the quasi-static rubbery plateau region The square root differential operator with respect to time, D 1/2, has built in memory since the kernel of this operator depends upon the difference between the current time and the past time D 1/2 is a special case of the Abel operator, which is used in the theory of integral equations The present theory introduces integrals into the standard linear solid resulting in an integral differential equation governing the stress, strain and temperature It is shown that this proposed linear equation of state for a solid material, which undergoes a second order transition, requires only four phenomenological constants to completely determine the behavior of the solid material These four phenomenological constants are two relaxation times and two creep times, both of which are functions of the temperature

Note on an exact solution for the pipe flow of a third-grade fluid

Abstract: We obtain a new exact power law solution for the pipe flow of a third-grade fluid. Moreover, we provide general analytical expressions from which all previous known solutions can be constructed.

Optimal control of a pretwisted shearable smart composite rotating beam

Abstract: The optimal vibration control of a rotating, composite, pretwisted, single-celled box beam, exhibiting transverse shear flexibility and restrained warping, is analyzed. A higher-order shear deformation theory (HSDT) enabling satisfaction of traction-free boundary conditions is employed. An orthotropic host with Circumferentially Uniform Stiffness ply angle configuration and transversely isotropic sensors-actuator pairs that are surface embedded along the span are considered. The total output from sensors is fed to a controller and then uniformly applied to actuators. The extended Galerkin method, along with either instantaneous or classical LQR methods, is used. Instantaneous LQR provides greater response attenuation in case of sustained external forcing. Results are obtained for a linear spanwise variation of pretwist. Compared to the unshearable and first-order shearable (FSDT) models, the HSDT appears most sensitive to pretwist and – when a saturation constraint is considered – it predicts the lowest settling time. The HSDT predicts significant attenuation in response and power required as compared to the FSDT, the differences being especially pronounced for constrained input control. Parametric studies involving the ply angle, pretwist, and patch length are performed. An optimum pretwist that yields lowest response, power, and settling time is obtained. Its value differs when the saturation constraint is used. Tailoring provides greater attenuation at the expense of an increase in settling time. Using constrained input control, an order-of-magnitude reduction in power requirement is possible via tailoring. The results underscore the importance of shear strain variation across the beam wall, and also of synthesizing active control with tailoring, for achieving efficient control.

Transient dynamic stress intensity factors around two rectangular cracks in a nonhomogeneous interfacial layer between two dissimilar elastic half-spaces under impact load

Abstract: Transient dynamic stresses around two rectangular cracks in a nonhomogeneous interfacial layer sandwiched between two dissimilar elastic half-spaces are examined. The material properties vary continuously in the layer within a range from those of the upper half-space to those of the lower half-space. An incoming shock stress wave impinges perpendicular on the crack surfaces. In order to solve the problem, the interfacial layer is divided into several homogeneous layers that have different material properties. Application of Laplace and Fourier transforms reduces the problem to the solution of a pair of dual integral equations. To solve the equations, the differences in the crack surface displacements are expanded into a series of functions that vanish outside the crack. The unknown coefficients in the series are solved using the Schmidt method. The stress intensity factors are defined in the Laplace transform domain and these are inverted numerically in physical space. Numerical calculations are carried out for composite materials made of a ceramic half-space and a steel half-space.

The integrability criterion in finite elastoplasticity and its constitutive implications

Abstract: In traditional Eulerian rate formulation of finite elastoplasticity, an Eulerian rate equation of hypoelastic type is used as one of the basic constituents to relate the elastic part of the stretching to a stress rate. On account of the fact that the elastic-like behavior should be expected prior to yielding, the foregoing basic elastic rate equation should be exactly integrable to deliver a conventional hyperelastic stress-strain relation. Physically, it requires a consistent combination of hypoelasticity and hyperelasticity into a single Eulerian rate equation. Since this criterion is purely a physical consistency requirement and since the basic elastic rate equation involves no strain concept and allows for any stress rate in its own right, from a physical standpoint it may be of both interest and significance to know what consequences it will imply concerning the stress rate and the finite strain measure. By a simple, straightforward procedure we demonstrate in a general sense that both Hencky strain and the logarithmic rate emerge naturally as direct consequences of the foregoing criterion. This result may be regarded to reveal the physical essence behind Hencky strain and the logarithmic rate in connection with a basic physical consistency requirement in finite elastoplasticity. Constitutive implications are discussed in a few relevant respects concerning representative formulations of finite elastoplasticity.

Unconditional nonlinear stability for convection in a porous medium with vertical throughflow

Abstract: Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key terms, a weighted functional is used in the energy analysis. The linear instability and nonlinear stability thresholds show considerable agreement when the vertical throughflow is small, although there is substantial deterioration of this agreement as the vertical throughflow increases.