Showing papers in "Acta Mechanica in 1993"
TL;DR: In this article, a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity is presented. But the method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.
Abstract: We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.
401 citations
TL;DR: In this paper, a numerical simulation of patterns of shear bands in biaxial compression tests using an elasto-plastic Cosserat constitutive equation is presented.
Abstract: Numerical simulation of patterns of shear bands in biaxial compression tests using an elasto-plastic Cosserat constitutive equation is presented. Random distribution of the material properties acts as a trigger for the localized deformation. Two types of stress-strain curves, namely strain softening and strain softening followed by strain hardening, are investigated. It is shown that the characteristic of the stress-strain curve is crucial for the patterning of shear bands. While calculations with the stress-strain curve with solely softening yield only one single shear band, a flock of shear bands can be obtained with the stress-strain curve with softening followed by hardening. Benefited from the characteristic length provided by the Cosserat elasto-plastic constitutive equation, the dependence of the calculation on the mesh-size is avoided.
143 citations
TL;DR: In this article, the motion of an unconfined finite mass of a granular material released from rest on an inclined plane is treated as a frictional Coulomb-like continuum with a Coulomblike basal friction law.
Abstract: This paper is concerned with the motion of an unconfined finite mass of a granular material released from rest on an inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth averaged equations are deduced from the three-dimensional dynamical equations by scaling the equations and imposing the shallowness assumption that the moving piles are long and wide but not deep. Several distinguished limits for small depth to length and depth to width ratios can be analysed. We develop an approximate theory based upon the full dynamical equations parallel to the inclined plane and imposed hydrostatic pressure conditions perpendicular to it. The resulting model equations are then applied to construct either yet simpler model equations or else solutions for particular cases. In a first application the transverse distributions of the velocity fields and of the depth profile are prescribed, while representative values of these functions (such as the cross sectional averages or maxima) as functions of time and the downhill coordinate are left unspecified. For these quantities evolution equations are obtained from a lateral averaging of the vertically averaged equations. In a second application approximate similarity solutions of the spatially two-dimensional equations are derived. The depth and velocity profiles for the moving mass are determined in analytical form, and the evolution equation for the total length and the total width of the pile is integrated numerically. A parameter study illustrates the performance of the model.
135 citations
TL;DR: In this paper, the authors approximate the non-convex free energy by a train of two intersecting parabolae, which can be used to carry out all calculation explicitly and it seems to describe most phenomena observed in pseudo-elasticity.
Abstract: The phenomenon of pseudo-elasticity is connected with a phase transition. Its description requires a non-convex free energy and a non-monotone load-deformation curve. Realistic functions like that are too complicated to permit the analytic calculation of phase equilibria and the evaluation of, stability properties of a phase mixture. One has to resort to graphical methods, see [1], or to numerical calculations. However, here we approximate the non-convex free energy by a train of two intersecting parabolae. This simplification enables us to carry out all calculation explicitly and it seems to describe — at least qualitatively — most phenomena observed in pseudoelasticity.
85 citations
TL;DR: In this article, the effects of free convection and the presence of heat generation/absorption on the flow and heat transfer characteristics are considered, and the equations of conservation of momentum, mass, and energy are solved numerically by using a variable order, variable step size finite-difference method.
Abstract: Analysis of convection flow and heat transfer of a viscous heat-generating fluid near an infinite vertical stretching surface is carried out. The effects of free convection and the presence of heat generation/absorption on the flow and heat transfer characteristics are considered. The equations of conservation of momentum, mass, and energy, which govern the flow and heat transfer problem, are solved numerically by using a variable order, variable step size finite-difference method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. These behaviors warrant further study of the effects of free convection on the flow and heat transfer characteristics.
79 citations
TL;DR: In this paper, a complete development for the first two terms of the crack tip fields for both Mode I and Mode II loading of a hardening material in either plane stress or plane strain is performed, including the elastic deformation in the analysis.
Abstract: A complete development for the first two terms of the crack tip fields for both Mode I and Mode II loading of a hardening material in either plane stress or plane strain is performed, including the elastic deformation in the analysis. It is shown that the determination of the order of the second term depends on bothn and whether plane stress or plane strain is considered. In addition, regions of HRR dominance at a crack tip for the field variables are estimated. Comparison of the analytic predictions with finite element results indicates that the analytic results for the zone of HRR dominance are in agreement with numerical predictions.
78 citations
TL;DR: In this article, it was shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticities for slow deformation processes.
Abstract: The classical theories of continuum mechanics — linear elasticity, viscoelasticity, plasticity and hydrodynamics — are defined by special constitutive equations. These can be understood to be asymptotic approximations of a quite general constitutive model, valid under restrictive assumptions for the stress functional or the input processes. The general theory of material behavior develops systematic methods to represent material properties in a context of physical evidence and mathematical consistency. According to experimental observations material behavior may be rate independent or rate dependent with or without equilibrium hysteresis. This motivates four different constitutive theories, namely elasticity, plasticity, viscoelasticity and viscoplasticity. Constitutive equations can be formulated explicitly as functionals. Then, the particular constitutive models correspond to continuity properties of these functionals, related to convenient function spaces. On the other hand, a system of differential equations may lead to an implicit definition of a stress functional. In this case additional variables are introduced, which are called internal variables. For these variables additional evolution equations must be formulated, specifying the rate of change of the internal variables in dependence on their present values and the strain (or stress) input. In the context of different models of inelastic material behavior the evolution equations have different mathematical characteristics. These concern the existence of equilibrium solutions and their stability properties. Rate independent material behavior is modelled by means of evolution equations, which are related to an arclength instead of the time as independent variable. It can be shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticity for slow deformation processes.
67 citations
TL;DR: In this paper, the authors examined the fully developed flow of a generalized fluid of second grade between heated parallel plates, due to a pressure gradient along the plate, and obtained the solution for the case when the temperature changes only in the direction normal to the plates for the two most commonly used viscosity models.
Abstract: We examine the fully developed flow of a generalized fluid of second grade between heated parallel plates, due to a pressure gradient along the plate. The constant coefficient of shear viscosity of a fluid of second grade is replaced by a shear dependent viscosity with an exponentm. If the normal stress coefficients are set equal to zero, this model reduces to the standard power-law model. We obtain the solution for the case when the temperature changes only in the direction normal to the plates for the two most commonly used viscosity models, i.e. (i) when the viscosity does not depend on temperature, and (ii) when the viscosity is an exponentially decaying function of temperature.
65 citations
TL;DR: In this paper, a numerical study for magnetohydrodynamic free convection of an electrically conducting fluid in a two-dimensional rectangular enclosure in which two side walls are maintained at uniform heat flux condition.
Abstract: A numerical study is presented for magnetohydrodynamic free convection of an electrically conducting fluid in a two-dimensional rectangular enclosure in which two side walls are maintained at uniform heat flux condition. The horizontal top and bottom walls are thermally insulated. A finite difference scheme comprising of modified ADI (Alternating Direction Implicit) method and SOR (Successive-Over-Relaxation) method is used to solve the governing equations. Computations are carried out over a wide range of Grashof number, Gr and Hartmann number, Ha for an enclosure of aspect ratio 1 and 2. The influences of these parameters on the flow pattern and the associated heat transfer characteristics are discussed. Numerical results show that with the application of an external magnetic field, the temperature and velocity fields are significantly modified. When the Grashof number is low and Hartmann number is high, the central streamlines are elongated and the isotherms are almost parallel representing a conduction state. For sufficiently large magnetic field strength the convection is suppressed for all values of Gr. The average Nusselt number decreases with an increase of Hartmann number and hence a magnetic field can be used as an effective mechanism to control the convection in an enclosure.
59 citations
TL;DR: In this paper, the effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied.
Abstract: Effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied. The surface absorbs the fluid with a constant suction and the free stream velocity oscillates about a constant mean value. Analytical expressions for the velocity and the angular velocity fields have been obtained, using the regular perturbation technique. The effects of Grashof numberG; material parameters α and β; Prandtl numberP; permeability parameterK and frequency parametern on the velocity and the angular velocity are discussed. The effects of cooling and heating of a polar fluid compared to a Newtonian fluid have also been discussed. The velocity of a polar fluid is found to decrease as compared to the Newtonian fluid.
57 citations
TL;DR: In this paper, the authors examined the complexities associated with the kinematics of finite elastoplastic deformations and other issues related to the development of constitutive equations, and derived the elastic and plastic strain rate tensors by considering both a phenomenological energy approach and a physically motivatedmesomechanical approach based on the double-slip idealization.
Abstract: In this paper we examine the complexities associated with the kinematics of finite elastoplastic deformations and other issues related to the development of constitutive equations. The decomposition of the total strain and strain rate tensors into elastic and plastic constituents is investigated by considering both a multiplicative decomposition of the deformation gradient and an additive decomposition of the deformation vector field. Physically based definitions for the elastic and plastic strain rate tensors are given and compared with other values found in the literature. Constitutive equations for the plastic flow are derived by considering both a phenomenological-energy approach and a physically motivatedmesomechanical approach based on the double-slip idealization. It is shown that by resorting to the mechanics of the double slip, specific relations for the plastic stretching and plastic spin can be rigorously derived, taking into account the effect of noncoaxiality and material rotation. Finally, the implication of such effects to large deformations is examined in connection with the localization phenomenon.
TL;DR: The vibrational behavior of geometrically imperfect single and multilayered composite double-curved shallow panels subjected to a system of tangential compressive/tensile edge loads in the pre- and postbuckling ranges is investigated in this paper.
Abstract: The vibrational behavior of geometrically imperfect single and multilayered composite double-curved shallow panels subjected to a system of tangential compressive/tensile edge loads in the pre- and postbuckling ranges is investigated. The effects of transverse shear deformations, lamination, the character of in-plane boundary conditions, and of transverse normal stress are incorporated and their influence is emphasized. Numerical illustrations enabling one to compare the obtained results based on higher order and first order shell theories with their classical counterparts, based on the Love-Kirchhoff model are presented and conclusions related to their range of applicability are outlined.
TL;DR: In this paper, a general constitutive formulation for large inelastic deformations is presented, employing multiple constitutive and plastic spins in the rate equations of evolution for the tensorial internal variables.
Abstract: A general constitutive formulation for large inelastic deformations is presented, employing multiple constitutive and plastic spins in the rate equations of evolution for the tensorial internal variables. The multiplicity of the spins enables the macroscopic description of complex texture development, by allowing differential orientations of the internal variables in the course of deformation. The general case is illustrated by the study of a combined kinematic/orthotropic hardening model. The study is focused on the orientational evolution of the back-stress and, in particular, the orthotropic directions, in what can be called orthotropic texture development. Use of macroscopic and continuum micromechanical models provides interesting results on the texture development in simple shear, expressed by the variation of the plastic spin. The results are compared with available simulations by a Taylor-type polycrystalline model.
TL;DR: In this paper, a mathematical regularization approach is proposed to remove the nearlysingular and singular integrals occurring in the boundary integral formulations for the solution of the boundary value problems with a "pathologica" integration boundary due to very near or coinciding parts of the total boundary.
Abstract: This paper presents a unique approach, named “mathematical regularization”, to remove the nearly-singular and singular integrals occurring in the boundary integral formulations for the solution of the boundary value problems with a “pathologica” integration boundary due to the very near or coinciding parts of the total boundary. Nonsingular boundary integral equations are derived for thin-walled structure problems. In the case of crack-like problems, we present two kinds of the nonsingular integral representations of the secondary fields and the derivative boundary integral equations.
TL;DR: In this paper, a set of balance equations and entropy production inequality are formulated in spatial, referential and intermediate description for an elastoplastic body at finite strain within the framework of classical mechanics and field theories.
Abstract: Within the framework of classical mechanics and field theories, equivalent sets of balance equations and entropy production inequality are formulated in spatial, referential and intermediate description for an elastoplastic body at finite strain. The entropy production inequality formulated with respect to the intermediate configuration together with an assumption about the functional form of the free energy are used to substantiate new constitutive equations. Special attention is focused to the cases of small elastic/large plastic strain deformation (metal plasticity) and rigid-plastic deformation.
TL;DR: A cubic spline collocation numerical method and a simple transposition theorem have been used to study the free convection in the flow of a micropolar fluid along irregular vertical surfaces.
Abstract: A cubic spline collocation numerical method and a simple transposition theorem have been used to study the free convection in the flow of a micropolar fluid along irregular vertical surfaces. A sinusoidal surface is used to elucidate the amplitude wavelength ratio effects on the free convection in a micropolar boundary layer. The effects of micropolar parameterR and geometries on the velocity and temperature fields have been graphically studied. The skin friction stress on the wall has also been studied and discussed. It is observed that the frequency of the local heat transfer rate is twice that of the wavy surface, irrespective of whether the fluid is a Newtonian fluid or micropolar fluid; the same result is also obtained for the skin friction on the wall.
TL;DR: In this paper, the effect of foundation inertia on the response of an infinitely long beam resting on a foundation of finite depth is studied by modeling the foundation as a series of closely spaced axially vibrating rods, fixed at bottom and connected to the beam at the top.
Abstract: In this paper, dynamic response of an infinitely long beam resting on a foundation of finite depth, under a moving force is studied. The effect of foundation inertia is included in the analysis by modelling the foundation as a series of closely spaced axially vibrating rods of finite depth, fixed at the bottom and connected to the beam at the top. Viscous damping in the beam and foundation is included in the analysis. Steady state response of the beam-foundation system is obtained. Detailed numerical results are presented to study the effect of various parameters such as foundation mass, velocity of the moving load, damping and axial force on the beam. It is shown that foundation inertia can considerably reduce the critical velocity and can also amplify the beam response.
TL;DR: In this article, for the analysis of composite laminates various finite-rotation theories are presented in a single formulation, which allows a quadratic shear deformation distribution across the thickness.
Abstract: For the finite-element analysis of arbitrary composite laminates various finite-rotation theories are presented in a single formulation. First a refined theory is derived which allows a quadratic shear deformation distribution across the thickness. The so-called difference vector appearing in the kinematic relations is expressed in terms of rotational degrees of freedom permitting a clear determination of the deformed normal vector in every nonlinear range. The constitutive relations derived are applicable to orthotropic material properties varying arbitrarily across the thickness and to curvilinear laminate coordinates as well. This refined theory is then transformed into simplified theories of Kirchhoff-Love and Mindlin-Reissner types. Finally, the last formulation is used to formulate a layer-wise theory able to grasp the shear deformations very accurately. Kinematic relations and the corresponding constraints are presented in two alternative forms suitable for the application of isoparametric and classical finite-element formulations.
TL;DR: In this paper, a general series solution to the problem of interacting circular inhomogeneities in plane elastostatics was proposed. But the complexity of the complex stress potentials was not considered.
Abstract: This paper provides a general series solution to the problem of interacting circular inhomogeneities in plane elastostatics. The analysis is based upon the use of the complex stress potentials of Muskhelishvili and the Laurent series expansion method. The general forms of the complex potentials are derived explicitly for the circular inhomogeneity problem under arbitrary plane loading. Using the superposition principle, these general expressions were subsequently employed to treat the problem of an infinitely extended matrix containing any number ofarbitrarily located inhomogeneities. The above procedure reduces the problem to a set of linear algebraic equations which are solved with the aid of a perturbation technique. The current method is shown to be capable of yielding approximate closed-form solutions for multiple inhomogeneities, thus providing the explicit dependence of the solution upon the partinent parameters.
TL;DR: In this article, the authors investigated nonlinear axisymmetric waves in compressible hyperelastic circular cylindrical rods and derived the corresponding simplified model equations, which gave the framework for studying problems like wave-interactions arising through collision or reflection.
Abstract: This paper investigates nonlinear axisymmetric waves in compressible hyperelastic circular cylindrical rods. We consider first a compressible Mooney-Rivlin material to obtain exact governing equations. To further study the problem, we introduce the notion of long finite amplitude waves and derive the corresponding simplified model equations, which gives the framework for studying problems like wave-interactions arising through collision or reflection. The asymptotically valid far-field equation is consequently deduced from the simplified model equations. Then, using a strained-coordinate method, we obtain the second-order solitary wave solution. The result is not only of interest itself, but also provides a suitable initial condition for wave interaction problems. Finally, the results for a general hyperelastic rod are presented.
TL;DR: In this paper, a boundary element method formulation and numerical implementation of elasticity problems in nonhomogeneous media is presented and the fundamental solutions for elasticity in homogeneous media are employed and the nonsingular formulation is derived.
Abstract: This paper presents a new boundary element method formulation and numerical implementation of elasticity problems in nonhomogeneous media. The fundamental solutions for elasticity in homogeneous media are employed and the nonsingular formulation is derived. A physically and mathematically meaningful elimination of internal degrees of freedom is proposed. The solution at an arbitrary point is expressed in terms of boundary displacements and tractions. The rank of the system matrix (for computation of relevant unknowns) is dependent only on the discretization of the boundary.
TL;DR: In this paper, the ferrofluid lubrication of cylindrical rollers under combined rolling and normal motion was analyzed for general cases where the magnetization vectors need not be parallel to the applied magnetic field.
Abstract: This paper analyses the ferrofluid lubrication of cylindrical rollers under combined rolling and normal motion. The analysis, which takes into account the rotation of magnetic particles, has been made for general cases where the magnetization vectors need not be parallel to the applied magnetic field. Cavitation boundary conditions are used and the applied magnetic field is assumed to be imposed in a direction transverse to the fluid motion. A perturbation scheme in terms of non-dimensional Brownian time relaxation parameter has been used and the effects of various parameters on bearing characteristics have been studied.
TL;DR: In this article, it is shown that the Zaremba-Jaumann derivative necessarily reduces to theLie one, and the tangent modulus relating the objective rate of the stress tensor to the strain velocity tensor is no more constant but assumed to depend on the stress state itself.
Abstract: Demanding that rate-type constitutive equations in solid mechanics has to be in accord with the thermodynamical requirement of the existence of a free energy function, the restrictions to be imposed on stress evolution equations in terms of theLie (Oldroyd) as well as theZaremba-Jaumann objective rate of theCauchy stress tensor are discussed. Explicit forms of generalized constitutive equations are given according to which the tangent modulus relating the objective rate of the stress tensor to the strain velocity tensor is no more constant but assumed to depend on the stress state itself. The use of theKirchhoff stress tensor is handled as well. It is shown that theZaremba-Jaumann derivative necessarily reduces to theLie one. As an application the case of simple shear is discussed where a monotonously increasing stress-strain relation is obtained. The paper closes with remarks concerning elasto-plasticity.
TL;DR: In this article, the linear stability of axial parallel Poiseuille-Couette flow in an annulus between concentric circular cylinders is considered using a long-wave version of the axisymmetric Orr-Sommerfeld equation.
Abstract: The linear stability of axial parallel Poiseuille-Couette flow in an annulus between concentric circular cylinders is considered. Using a long-wave version of the axisymmetric Orr-Sommerfeld equation the stability chart of this flow in the velocity ratio-radius ratio plane is derived. It is shown that pure sliding Couette flow can become unstable if the radius ratio is below a specific threshold value. Finally, applying the results to other flow geometries, it is shown that the boundary layer along a slender cylinder can become unstable in a confined region downstream the leading edge only.
TL;DR: In this paper, a general solution technique for transient thermoelastic problems of transversely isotropic solids in Cartesian coordinates is proposed, which consists of five fundamental solutions.
Abstract: In the present paper we propose a new general solution technique for transient thermoelastic problems of transversely isotropic solids in Cartesian coordinates. The solution technique consists of five fundamental solutions. By considering the relations among the material constants of transverse isotropy, the solution technique is classified into five groups. One among those corresponds to Goodier's thermoelastic potential function as well as the generalized Boussinesq solutions and the Michell function. For an application of the solution technique, an inverse problem of transient thermoelasticity in a transversely isotropic semi-infinite solid is analyzed.
TL;DR: In this paper, the thermal post buckling behavior of simply-supported, antisymmetric cross-ply plates with immovable edges is investigated and conditions under which the thermal buckling in the classical sense occurs are derived.
Abstract: The thermal post buckling behavior of simply-supported, antisymmetric cross-ply plates with immovable edges is investigated in this paper. For this purpose, a one term approximation for the inplane and transverse displacements is assumed and Rayleigh-Ritz method is used to obtain the equations of equilibrium. The question, whether the true thermal buckling (bifurcation buckling under thermal loads) exists for such plates is also addressed to. The conditions under which the thermal buckling in the classical sense occurs are derived. It is shown that nonlinear bending stiffness for such plates is direction dependent, because of an extra quadratic nonlinear term in addition to the usual cubic nonlinear term in the governing equilibrium equation. Results are presented for various plate configurations.
TL;DR: In this paper, the solution of high amplitude cantilever spatial beam vibration problems is described, where the standard linear damping model has been taken into consideration, and the rigid finite element method has been introduced to discretize the beam.
Abstract: This paper describes the solution of high amplitude cantilever spatial beam vibration problems. The standard linear damping model has been taken into consideration. In order to allow the large displacement analysis the rigid finite element method has been introduced to discretize the beam. Denavit-Hartenberg's matrices and Lagrange's equations have been used to formulate the equations of motion. Both, numerical calculations and conclusions on the accuracy of the above mentioned method are shown.
TL;DR: The contact line of a liquid with a solid does in many cases, depending on the smoothness of the solid, the viscosity, the surface tension and the excitation force, flow along the solid during oscillations.
Abstract: The contact line of a liquid with a solid does in many cases—depending on the smoothness of the solid, the viscosity, the surface tension and the excitation force—apparently flow along the solid during oscillations. The influence of this effect upon the natural frequencies, the stability and the response of the system has been investigated at an oscillating and spinning cylindrical liquid column.
TL;DR: In this article, the mixed convection in an axisymmetric stagnation flow over a vertical cylinder with arbitrary temperature variations is studied and numerical solutions are given for the governing momentum and energy equations.
Abstract: An analysis is presented to study the mixed convection in an axisymmetric stagnation flow over a vertical cylinder with arbitrary temperature variations. Numerical solutions are given for the governing momentum and energy equations Two flow regions, namely, the buoyancy-assisted and buoyancy-opposed cases are analyzed. It is observed that the friction factor and Nusselt number increase or decrease with the buoyancy force parameter depending upon which flow regime is being considered. The effect of the Prandtl number on the flow field in both the flow regimes is discussed.
TL;DR: In this paper, a new stress rate, based on the spin of a material triad that coincides momentarily with the principal stress axes, is applied to the analysis of constrained and unconstrained shear with and without superposed normal stresses of hypoelastic and rigid plastic, kinematically hardening solids.
Abstract: A new stress rate, based on the spin of a material triad that coincides momentarily with the principal stress axes, is applied to the analysis of constrained and unconstrained shear with and without superposed normal stresses of hypoelastic and rigid plastic, kinematically hardening solids. A comparative study of various alternative stress rates that have been proposed, including the Jaumann rate, favours the new formulation from the viewpoints of qualitative behaviour, general applicability and ease of implementation. In particular, the present rate is the only one which furnishes a limiting value for tan β (β being inclination of a vertical fall of a material cube) in unconstrained shear with and without normal stresses.