Showing papers in "Acta Mechanica in 1992"

A transverse shear deformation theory for homogeneous monoclinic plates

Abstract: A general two-dimensional theory suitable for the static and/or dynamic analysis of a transverse shear deformable plate, constructed of a homogeneous, monoclinic, linearly elastic material and subjected to any type of shear tractions at its lateral planes, is presented. Developed on the basis of Hamilton's principle, in conjunction with the method of Lagrange multipliers, this new theory accounts for an unlimited number of choices of through-thickness displacement distributions, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. For the particular case of a theory operating with five degrees of freedom, special attention is given to displacement expansions producing symmetric, through thicknes, distributions of transverse shear strain. For the cylindrical bending problem of a specially orthotropic plate, the governing equations of that five-degrees-of-freedom theory are solved and for three different choices of symmetric, through tickness, transverse shear deformation, numerical results are obtained and compared with corresponding results based on the exact three-dimensional solution existing in the literature. The comparisons made show clearly, that the multiple options offered by the new theory, by either suitably altering the displacement expansions or gradually increasing the degrees of freedom involved, will be found useful in future studies dealing with the static and/or dynamic analysis of homogeneous plates.

MHD flow of a viscoelastic fluid past a stretching surface

Abstract: The flow of a viscoelastic fluid past a stretching sheet in the presence of a transverse magnetic field is considered. An exact analytical solution of the governing non-linear boundary layer equation is obtained, showing that an external magnetic field has the same effect on the flow as the viscoelasticity.

On the gradient-dependent theory of plasticity and shear banding

Abstract: After a brief review of a recently developed gradient-dependent theory of plasticity various questions related to the yield function and the loading-unloading condition in the presence of higher order strain gradients and the determination of the corresponding phenomenological coefficients are addressed. For rate-independent materials, we construct as before an analytical solution for the strain profile in the postlocalization regime providing the shear band thickness and strain within it but we now compare these results to recently obtained experimental data by assigning appropriate values to the gradient coefficients. We also address some questions recently raised in the literature regarding our nonlinear shear band analysis. For rate-dependent materials, the resulting spatio-temporal differential equation for the strain is solved numerically using the finite difference method. It is shown that the band width does not depend on the grid size, as long as the the grid size is smaller than a certain characteristic length. Various initial imperfections of different amplitudes and sizes are examined, and the possibility of simultaneous development of two shear bands and their interaction is investigated.

Pseudomomentum and material forces in nonlinear elasticity : variational formulations and application to brittle fracture

Abstract: This work examines critically the various formulations of the balance of linear momentum innonlinear inhomogeneous elasticity. The corresponding variational formulations are presented. From the point of view of the theory of elastic inhomogeneities, the most interesting formulations are those which, being either completely material or mixed-Eulerian, exhibit explicitly the inhomogeneities in the form ofmaterial forces. They correspond to the balance ofpseudomomentum, a material covector which is seldom used but which we show to play a fundamental role in the Hamiltonian canonical formulation of nonlinear elasticity. The flux associated with pseudomomentum is none other than theEshelby material tensor. Applying this formulation to the case of an elastic body containing a crack of finite extent, the notion of suction force acting at the tip of the crack follows while afracture criterion a la Griffith can be deduced from a variational inequality. Possible extensions to higher-grade elastic materials and inelastic materials are indicated as well as the role played by pseudomomentum in the quantization of elastic vibrations.

New types of variational principles based on the notion of quasidifferentiability

Abstract: In this paper several new types of variational principles are derived by using the new mathematical notion of quasidifferentiability. As a model problem we consider the static analysis problem of deformable bodies subjected to nonmonotone boundary and interface conditions. The nonmonotone laws are produced from appropriately defined nonsmooth and, nonconvex, quasidifferentiable superpotentials by means of the quasidifferential operator in the sense of V. F. Dem'yanov. The static analysis problem is formulated as a system of variational inequalities which is equivalent, minmax formulation, or equivalently as a multivalued quasidifferential inclusion problem which describes the positions(s) of static equilibrium of the body. The theory is illustrated by numerical examples concerning the calculation of adhesive joints.

Wave propagation in fluid filled nonlinear viscoelastic tubes

Abstract: The present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic tube filled with an incompressible, inviscid fluid. In order to include the geometric dispersion in the analysis, the tube wall inertia effects are added to the pressure-area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the long-wave approximation is examined. In the long-wave approximation, a general equation is obtained, and it is shown that by a proper scaling this equation reduces to the well-known nonlinear evolution equations. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained. In the absence of nonlinear viscoelastic effects all the equations reduce to those of the linear viscoelastic tube.

The Blasius problem formulated as a free boundary value problem

Abstract: In the present paper we point out that the correct way to solve the Blasius problem by numerical means is to reformulate it as free boundary value problem. In the new formulation the truncated boundary (instead of infinity) is the unknown free boundary and it has to be determined as part of the numerical solution. Taking into account the “partial” inavariance of the mathematical model at hand with respect to a stretching group we define a non-iterative transformation method. Further, we compare the improved numerical results, obtained by the method in point, with analytical and numerical ones. Moreover, the numerical results confirm that the question of accuracy depends on the value of the free boundary. Therefore, this indicates that boundary value problems with a boundary condition at infinity should be numerically reformulated as free boundary value problems.

Certain solutions of the equations of the planar motion of a second grade fluid for steady and unsteady cases

Abstract: Solutions for the equations of motion of an incompressible second grade fluid are derived by assuming certain conditions on the stream function. Exact solutions are obtained for a planar motion for both steady and unsteady cases.

Heat transfer in the flow of a viscoelastic fluid over a stretching sheet

Abstract: The problem of heat transfer in the viscoelastic fluid flow over a stretching sheet is examined. The important physical quantities such as the skin-friction coefficient and the heat transfer coefficient, are determined. It is found that the heat transfer coefficient decreases with the non-Newtonian parameter.

Flow of electro-rheological materials

Abstract: Summary. An electro-rheological fluid is a material in which a particulate solid is suspended in an electrically non-conducting fluid such as oil. On the application of an electric field, the viscosity and other material properties undergo dramatic and significant changes. In this paper, the particulate imbedded fluid is considered as a homogeneous continuum. It is assumed that the Cauchy stress depends on the velocity gradient and the electric field vector. A representation for the constitutive equation is developed using standard methods of continuum mechanics. The stress components are calculated for a shear flow in which the electric field vector is normal to the velocity vector. The model predicts (i) a viscosity which depends on the shear rate and electric field and (ii) normal stresses due to the interaction between the shear flow and the electric field. These expressions are used to study several fundamental shear flows: the flow between parallel plates, Couette flow, and flow in an eccentric rotating disc device. Detailed solutions are presented when the shear response is that of a Bingham fluid whose yield stress and viscosity depends on the electric field. During the past few years, there has been a great deal of interest in the manufacture and use of a class of materials which can be classified as field dependent theological materials. These materials are essentially fluids which are imbedded with particulate solids which react to an electrical field in that on the application of a field the viscosity and other material properties undergo dramatic and significant changes. Such materials are being touted as agents for enhancing the performance and efficiency of a variety of engineering devices in very diverse fields. Much of the activity in this area is devoted to producing this material and performing experiments in order to understand the scientific basis for their behavior. Little, if any effort has been devoted to mathematically modeling these materials. The need for understanding the mechanics of such materials and mathematically modeling their behavior is made all the more important as these materials are already finding day-to-day applications in the design of ubiquitous devices like clutches and brakes in cars, vibration dampers and absorbers, lubricating fluids in bearings to name some. In this paper we shall present a mathematical model for field dependent materials which is consistent with the phenomena which have been observed. We shall solve a series of boundary value problems the results of which can be compared with future experiments, as these boundary value problems are in domains which are amenable to experimentation. Unlike the field of magnetohydrodynamies, we do not have an equation like Maxwell's equation which governs the applied field, as the fluid which forms the base for the particulate media is non-conducting. The presence of the field alters the basic material properties of the particulate imbedded fluid, which is considered as a homogeneous continuum. Thus, for instance, the Cauchy stress is dependent on the gradient of the

Dynamics of a nonlinear periodic structure with cyclic symmetry

Abstract: The free and forced motions of a nonlinear periodic structure with cyclic symmetry are studied. The structure consists of a number of identical linear flexural members coupled by means of nonlinear stiffnesses of the third degree. It is found that this system can only possessn “similar” nonlinear modes of free oscillation, and that no other modes are possible. Moreover, there exist pairs of nonlinear modes with mutually orthogonal nodal diameters having, in general, distinct “backbone” curves. A multiple-scales averaging analysis is used to study the nonlinear interaction between a pair of modes with orthogonal nodal diameters. As a result of this analysis, it is found that all pairs of nonliner modes along with all their linear combinations are orbitally unstable, and the only possible orbitally stable periodic motions are free travelling waves, that propagate through the structure in the clockwise and anti-clockwise directions. Under harmonic forcing, a bifuraction of a stable branch of forced travelling waves from a branch of forced normal mode motions is detected, and “jump” phenomena between branches of periodic solutions are observed. The analytical results are in agreement with experimental observations of an earlier work, and, in addition, are verified by numerical simulations.

On vibration and buckling of symmetric laminated plates according to shear deformation theories

Abstract: The frequency and buckling equations of rectangular plates with various boundary conditions are developed within the third-order and the first-order shear deformation plate theories. The third-order theories account for a quadratic distribution of the transverse shear strains through the thickness of the plate. In the first part of this paper, Levinson's third-order theory, derived as a special case from Reddy's third-order theory, is used to study a plate laminated of transversely isotropic layers. The relationship between the original form of the governing equations and the interior and the edge-zone equations of the plate is closely examined and the physical insights from the latter equations are established. In the second part of the paper, the first-order shear deformation theory and the third-order theory of Reddy are studied for vibration and buckling.

Static and dynamic analysis of transversely isotropic, moderately thick sandwich beams by analogy

Abstract: A flexural theory of elastic sandwich beams is derived which renders quite precise results within a wide range of ratios of dimensions, mass densities, and elastic constants of the core and faces. The assumptions of the Timoshenko theory of shear-deformable beams are applied to each of the homogeneous, linear elastic, transversely isotropic layers individually. Core and faces are perfectly bonded. The principle of virtual work is applied to derive the equations of motion of a symmetrically designed three-layer beam and its boundary conditions. By definition of an effective cross-sectional rotation the complex problem is reduced to a problem of a homogeneous beam with effective stiffnesses and with corresponding boundary conditions. Thus, methods of classical mechanics become directly applicable to the higher-order problem. Excellent agreement of the results of illustrative examples is observed when compared to solutions of other higher-order laminate theories as well as to exact solutions of the theory of elasticity.

The effect of inhomogeneity and anisotropy on stoneley waves

Abstract: In this paper we study the dispersion equation of Stoneley waves that are travelling in an inhomogeneous elastic half-space over an anisotropic homogeneous elastic half-space. The phase velocity is calculated as a function of the wave number. The results indicate that the effect of anisotropy on such waves is small and can be neglected, while the effect of inhomogeneity is very pronounced. The results show that Stoneley waves do not exist after some cut-off value of the wave number.

Alternative approach to shakedown as a solution of a min-max problem

Abstract: Melan's classical shakedown theorem for continuous media considered as a problem of methematical programming with constraints is reformulated and reduced to a solution of a certain min-max problem. A similar approach is presented for the structural theory described in terms of generalized variables. A distinction is made between alternating plasticity and incremental collapse modes in the analysis of structures with nonsandwich cross-sections.

Flow equations of particle fluid mixtures

Abstract: A set of flow equations are derived for the velocity components of the solid and liquid phases of a particle fluid mixture. The equations have a limited validity to the case of uniaxial flow accompanied by radial expansion.

Dynamic behavior of saturated poroviscoelastic media

Abstract: The Vardoulakis-Beskos model for the dynamics of nearly and fully saturated poroelastic soils and the Aifantis-Beskos model for the dynamics of fully saturated, fissured, poroelastic rocks are here extended to include viscoelastic material behavior. Linear hereditary isotropic viscoelasticity of the relaxation type for the solid phase is considered. Correspondence principles are established for both models in the Laplace transformed domain. The one-dimensional dynamic column problem associated with both of the above proroviscoelastic soil and rock models is solved by an analytical-numerical procedure to illustrate the theory and assess the effect of viscoelasticity on the response.

Natural damped frequencies and axial response of a rotating finite viscous liquid column

Abstract: For a finite solidly rotating cylindrical liquid column the damped natural axisymmetric frequencies have been determined. The liquid was considered incompressible and viscous. The cases of freely slipping edges and that of anchored edges have been treated. It was found that instability appears in a purely aperiodic root for the spinning liquid bridge. This is in contrast to the instability appearing in the damped oscillatory natural frequency of a nonspinning liquid column at $$\frac{h}{a} \geqq 2\pi$$ . The spinning viscous liquid column exhibits the same instability as the frictionless liquid. It appears at $$\frac{h}{a} \geqq 2\pi /\sqrt {1 + We}$$ for axisymmetric oscillations.

Evolution of weakly non-linear shallow water waves generated by a moving boundary

Abstract: In this study we describe the evolution of weakly non-linear shallow water waves in a rectangular channel of 16 m length which aregenerated by a moving boundary. We present a detailed comparison of computaticnal and observational waveheight-time series and thus verify the theoretical model as presented by Villeneuve and Savage [27].

Continuum damage mechanics for softening of brittle materials

Abstract: Continuum damage theory is used to model the failure behaviour of brittle materials. In the constitutive equations a damage parameter is incorporated. A damage criterion is postulated such that large differences between tension and compression strength can be described. A damage growth law is quantified based on experimental data for concrete. For the elaboration of the mathematical formulation the finite element method is applied. Numerical results obtained for a plane strain example show the merits of the procedure.

Axisymmetric thermoelastic interactions in an unbounded body with cylindrical cavity

Abstract: Thermoelastic interactions in a linear, homogeneous and transversely isotropic unbounded body containing a cylindrical cavity due to a uniform step in stress or temperature applied to the boundary of the cavity are studied. A unified system of governing equations that includes among its particular cases the governing equations of the conventional and generalized thermoelasticity theories is employed. By the use of the Laplace transform technique, short time solutions for the temperature, displacement radial stress and hoop stress are constructed. The discontinuities suffered by these fields at the wavefronts are computed. Comparison with the corresponding results obtained in earlier works is made. Numerical results for a single crystal of zinc are presented.

Unsteady forced convection laminar boundary layer flow over a moving longitudinal cylinder

Abstract: The unsteady nonsimilar forced convection flow over a longitudinal cylinder, which is moving in the same or in the opposite direction to the free stream, has been investigated. The unsteadiness is due to the free stream velocity, cylinder velocity, surface temperature of the cylinder and the mass transfer, and nonsimilarity is due to the transverse curvature. The partial differential equations, governing the flow, have been solved numerically, using an implicit finite-difference scheme in combination with a quasilinearization technique. The results show that both, skin friction and heat-transfer, are appreciably affected by the free stream velocity distributions and by the cylinder velocity. Also, skin friction as well as heat-transfer are found to increase as the transverse curvature or the suction increases, but the effect of injection is just the opposite. The heat-transfer is significantly affected by the viscous dissipation and variation of surface temperature with time. It is observed that results of this problem are crucially dependent on the parameter α, which is the ratio of the velocity of the cylinder to the velocity of the free stream. In particular, it is found that solutions for the upstream moving cylinder exist only for a certain range of this parameter (α), and they are nonunique in a small range of α too.

Free convection about a wedge and a cone subjected to mixed thermal boundary conditions

Abstract: The paper deals with a similarity analysis of free convection about a wedge and a cone which are subjected to mixed thermal boundary conditions. The governing equations and the boundary conditions are reduced to a boundary value problem involving a non-negative parameterm which assumes the values 0,1 and ∞ for the cases of prescribed temperature, prescribed heat flux and radiation boundary condition. A numerical solution has been computed for the case of radiation boundary condition. The results for constant temperature and constant heat flux available in literature are deduced with the aid of a simple transformation. Critical cases have been found for which the solution is same for all values ofm.

Dynamics of oscillating viscous droplets immersed in viscous media

Abstract: Damped oscillations of a viscous droplet immersed in a viscous medium are considered in detail. The characteristic equation is solved numerically for arbitrary, finite fluid properties. The cylinder functions in the characteristic equation are solved using an accurate continued fraction algorithm, and the complex decay factor is searched using a minimization scheme. Oscillation frequency and damping rate results are presented for the fundamental mode, for various cases of practical interest (liquid-gas, and liquid-liquid systems), and the effect of the external medium properties are discussed. Results are compared to exact solutions for limiting cases, and to existing experimental data for both the fundamental and higher order modes. It is shown that the theoretical frequency prediction matches well with the experimental observation. Damping rate predictions, however, underestimate experimental observation in some cases, and this is thought to be due to surface impurities. The application of these results to the measurement of surface tension and viscosity of liquid droplets from single-droplet levitation experiments is also discussed.

Elastic-plastic annular disk with variable thickness subjected to external pressure

Abstract: The plane state of stress in an elastic-plastic annular disk with variable thickness under external pressure is studied. The thicknessh is assumed to vary along the radius in the formh=h 0(r/b)−n . The analysis is based on Tresca's yeld condition, its associated flow rule and linear strain hardening.

Failure criteria for weak-axis quasi-orthotropic woven fabric composites

Abstract: The failure behavior of woven fabric composites in the form of plain weave fiber unidirectional laminae is studied in this paper as defined by their failure stresses in simple tension and compression along the three principal stress directions. Since the transverse weave plane is the strong and isotropic plane of the composite, while the normal to it direction the weak one, the material is approximated as a weak-axis transversely isotropic composite. The elliptic paraboloid failure surface (EPFS) criterion, as introduced by the author [1], was shown to describe satisfactorily this type of interesting modern materials. It was shown that such weak-axis transversely isotropic composites correspond to tension strong composites and their failure surfaces consist of a single-sheet convex surface open to the tension-tension-tension octant of the principal stress space. The main characteristic of such surfaces is that they are oblate along the normal direction to the isotropic plane, in contrast with the typical (EPFS)-criterion for fiber composites, which, all of them, are prolate along the same direction. While the intersection of this (EPFS)-criterion by the (σ1,σ3) stress plane (σ3 is the weak axis) resembles closely the respective intersection for the unidirectional fiber composites the (σ1,σ2)-isotopic plane intersection, which coincides with the weaving strong plane approaches very closely a circle thus indicating that along this isotropic plane the failure stress is hydrostatic and independent of its orientation inside this plane. This property constitutes a significant and most promising property which makes this type of woven composites very attractive in applications. Experimental evidence of failure of such materials, which is very sparse, as derived from tests in a woven T-300 Carbon-epoxy composite corroborated excellently with the theory based on the (EPFS)-criterion.

On transverse variation of velocity and bed shear stress in hydraulic jumps in a rectangular open channel

Abstract: This paper is concerned with an experimental investigation on the transverse variation of the velocity and bed shear stress in hydraulic jumps in a rectangular open channel. The velocity is measured by means of a Pitot tube and small propeller-type flowmeter, while the bed shear stress is determined by the Preston's method. The experimental parameter is the supercritical Froude number $$Fr = U_0 /\sqrt {gz_0 }$$ , whereU 0 is the supercritical velocity,g the acceleration due to the gravity andz 0 the supercritical water depth. It is found that hydraulic jumps can be divided into two groups broadly at the supercritical Froude number Fr≊2.5. In case of the first group, no surface roller is formed and only a little amount of air is entrained into the jump. A series of swelling-depression sequences are formed on the surface along the channel axis, and thus the bed shear stress experiences a sinusoidal variation. On the other hand, in case of the second group a definite roller is formed in the jump region and a considerable amount of air is entrained into the roller. The surface height increases rapidly from the toe line to end line. As the result, the bed shear stress decreases rapidly with increasing the distance from the toe line, but the decay rate is decreased with increasing the supercritical Froude number. It is concluded that in hydraulic jumps the surface height, velocity and bed shear stress are dependent on all of the three spatial coordinates. For example, the bed shear stress is greatly decreased with increasing the transverse distance from the channel axis.

Continuum slip foundations of elasto-viscoplasticity

Abstract: A continuum slip theory for dislocation glide is employed to derive macroscopic, phenomenological models for incompressible viscoplasticity. The microstructural origins of kinematic/isotropic hardening and rate-dependence are examined within the framework of the continuum slip theory of Rice and the single slip scale invariance theory of Aifantis for dislocation glide. In the process, the limitations on primitive assumptions for existing forms of state variable viscoplasticity become apparent. A form for rate-dependent evolution of backstress is derived which supports recent phenomenological approaches. Extensions to finite strain and compressible viscoplasticity are discussed.

Elastodynamics of thin plates with internal dissipative processes Part I. Theoretical foundations

Abstract: In the first part of this paper an integral representation of the dynamic behaviour of thin plates is developed where two distinct types of internal defects, namely viscoplastic flow and ductile microfracture are considered. Nucleation and propagation of such defects constitute a dissipative process. Thus, the total dynamic response of the structure is composed of the reaction due to external loading and of that due to the development of defects, which are considered as an internal excitation. In plates the unity source of this internal excitation assumes the form of a curvature singularity. The evolution equations for the internal variables describing the dissipative processes are derived from thermodynamic considerations. A numerical treatment of the problem is contained in part two.

Nonlinear energy stability and convection near the density maximum

Abstract: A nonlinear energy stability analysis is developed for penetrative convection with the cubic and fifth order equations of state proposed by Merker et al. [1]. A conditional stability limit is obtained by using the method of coupling parameters and an unconditional limit is also obtained by constructing a suitable weighted energy.