Showing papers in "International Journal of Non-linear Mechanics in 1979"
TL;DR: In this article, a non-linear differential equation for uniaxial viscoplasticity is transformed into an equivalent integral equation, which employs total strain only and is symmetric with respect to the origin and applies for tension and compression.
Abstract: A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.
78 citations
TL;DR: In this article, the authors presented results for eigenfunction-expansion solutions to the forward Fokker-Planck equation associated with a specific, non-linear, first-order system subject to white noise excitation.
Abstract: In an earlier paper the authors presented results for eigenfunction-expansion solutions to the forward Fokker-Planck equation associated with a specific, non-linear, first-order system subject to white noise excitation. This work is concerned with eigenfunction-expansion solutions to the forward and backward Fokker-Planck equations associated with a specific, non-linear, second-order system subject to white noise excitation. Expansion terms through the fourth-order have been generated using a digital computer. Using this new information, inverted Domb-Sykes plots revealed a pattern in the coefficients for certain values of the parameters. Through this pattern, Dingle's theory of terminants was used to recast the series into a more favorable computational form.
31 citations
TL;DR: In this paper, the authors deal with the phenomena of reflection and refraction of plane elastic waves at a plane interface between two semi-infinite elastic solid media in contact, when both the media are initially stressed.
Abstract: The paper deals with the phenomena of reflection and refraction of plane elastic waves at a plane interface between two semi-infinite elastic solid media in contact, when both the media are initially stressed. It has been shown analytically that both reflected and refracted P and SV waves depend on initial stresses present in the media. The numerical values of reflection and refraction coefficients for different initial stresses and the angle of incidence have been calculated by computer and the results are given in the form of graphs. Many results are found in the paper which are not seen in initially stress-free media.
27 citations
TL;DR: Minimal normal modes (MNMs) as discussed by the authors are defined as non-linear normal modes which give a true minimum to Jacobi's Principle of Least Action, and for a certain class of two degree of freedom nonlinear conservative systems, MNMs generically occur in pairs.
Abstract: Minimal normal modes (MNMs) are defined as non-linear normal modes which give a true minimum to Jacobi's Principle of Least Action. It is shown that for a certain class of two degree of freedom non-linear conservative systems, MNMs generically occur in pairs. The nature of both generic and non-generic bifurcations of MNMs is derived and illustrative examples are given.
22 citations
TL;DR: In this article, the non-linear partial differential equations obtained from von Karman's large deflection plate theory have been solved by using the Chebyshev series in the space domain and the Houbolt numerical integration scheme in the time domain.
Abstract: In the present paper, Chebyshev series are employed to obtain the non-linear static and dynamic response of isotropic and orthotropic annular plates. The non-linear partial differential equations obtained from von Karman's large deflection plate theory have been solved by using the Chebyshev series in the space domain and the Houbolt numerical integration scheme in the time domain. Two different sets of boundary conditions of the annulus are investigated and detailed numerical results have been obtained for different cases of orthotropy and geometry.
21 citations
TL;DR: In this paper, a non-linear bending theory for beams is constructed which accommodates shear and longitudinal deformations, and an analytical solution for the cantilever beam subject to a compressive load is derived.
Abstract: A non-linear bending theory for beams is constructed which accommodates shear and longitudinal deformations. Using the theory, an analytical solution for the cantilever beam subject to a compressive load (the elastica ) and a series solution for the horizontal cantilever under weight loading are derived. The effects of including shear and longitudinal deformations are found to be negligible for configurations in which the two deformations tend to offset one another—such as at the onset of buckling under a compressive load—but are shown to be significant for certain configurations in which the two deformations are additive—as in some instances of weight loading.
17 citations
TL;DR: In this paper, the motion of a simple undamped pendulum that is confined to a single vertical plane is modified by shortening the string at a constant speed U 0 through a fixed point.
Abstract: The motion of a simple undamped pendulum that is confined to a single vertical plane is modified by shortening the string at a constant speed U0 through a fixed point
13 citations
TL;DR: In this article, it was shown that the fundamental equations in discrete mechanics are consistency relations for a parametric spline function approximation, and that the formula which relates displacement and acceleration is of O((Δt)2)2 and unconditionally stable in the sense of Dahlquist.
Abstract: It is shown that the fundamental equations in discrete mechanics are consistency relations for a parametric spline function approximation. The formula which relates displacement and acceleration is of O((Δt)2 and unconditionally stable in the sense of Dahlquist. This method is equivalent to two first order difference equations which relate velocity and acceleration, and displacement and velocity.
13 citations
TL;DR: In this article, the axial stiffness of a twisted m-wire spring in which contact is maintained can be made by treating the spring as m untwisted helical wires acting independently, provided the strand twist is not too severe.
Abstract: The large static deflection of an axially loaded helical spring formed of a twisted strand of smooth circular wires is considered. Contact between the wires in the strand may or may not be maintained upon loading, depending upon the type of construction and the type of loading. It is found that the making or breaking of wire contact within the strand has a drastic ettect upon the extension and twist of a wire and upon the extension of the strand, but has practically no effect upon the twist of the strand, and only a moderate effect upon the overall response of the spring. Limited experimental data tend to verify the theory. It is found that a good engineering approximation for the axial stiffness of a twisted m -wire spring in which contact is maintained can be made by treating the spring as m untwisted helical wires acting independently, provided the strand twist is not too severe.
13 citations
TL;DR: In this article, boundary value and eigenvalue problems of the Emden-Fowler equation (t α u′)′ + λt β ⨍(u) = 0, √ u = u γ and eu = u ǫ) were studied using the simple one parameter group properties.
Abstract: Two pragmatic boundary value and eigenvalue problems of the Emden-Fowler equation (t α u′)′ + λt β ⨍(u) = 0,⨍(u) = u γ and eu are studied using the simple one parameter group properties In all cases boundary value problems are converted into initial value problems using the property of the invariance group With ⨍(u) = u γ an eigenvalue problem is detailed and calculations presented
12 citations
TL;DR: In this paper, an analysis of the postbuckling behavior of isotropic cylindrical and conical shells under axial compression is presented, based on Lagrange's variational principle.
Abstract: This paper presents an analysis of the post-buckling behaviour of isotropic cylindrical and conical shells under axial compression. The starting point of the paper is Lagrange's variational principle, the application of which consists in assuming a kinematically admissible strain and displacement field. The field is determined considering the geometry of quasi-isometric deformations of the shell after buckling. That permits solving the problem with no limitation on the magnitude of the displacements.
TL;DR: In this paper, an expression for the traction vector t on a solid surface which is adjacent to an incompressible fluid of grade three which is compatible with thermodynamics was obtained. But this expression assumes that the velocity field of the fluid is uniform.
Abstract: An expression is obtained for the traction vector t on a solid surface which is adjacent to an incompressible fluid of grade three which is compatible with thermodynamics. It is found that unlike fluids of grade two wherein there is no additional drag due to the non-Newtonian nature of the fluid for bodies with certain geometric symmetries (e.g. sphere), fluids of grade three provide an additional drag which is of the same sign as that provided by the viscous terms, provided certain symmetry conditions are met by the velocity field.
TL;DR: In this article, a method for obtaining periodic solutions to forced oscillations of non-linear systems governed by equations of the form u ss − u yy −e f ( u, u, y, u y y, s ) = 0.
Abstract: A method is presented for obtaining periodic solutions to forced oscillations of non-linear systems governed by equations of the form u ss − u yy −e f ( u , u , y , u yy …, s ) = 0. The method is presented by application to the equation u ss − u yy −e u 2 y u yy = 0 which governs the vibrations of a soil layer that is free on the top surface and is forced harmonically at the bedrock. It is shown that unlike the ODE case (Duffing equation), the PDE requires an infinite number of periodicity conditions to correctly characterize the resonant region and these conditions lead to an infinite number of branches in the dispersion spectrum. Calculations indicate that these branches tend to an envelope curve. The uniform approach presented by Millmann and Keller is discussed in order to determine in what sense it can be viewed as an effective approximation for the fundamental mode.
TL;DR: In this paper, the mechanics of granular and porous media are considered in the light of the modern theories of structured continuum and the basic laws of motion are presented and several constitutive relations are derived.
Abstract: The mechanics of granular and porous media is considered in the light of the modern theories of structured continuum. The basic laws of motion are presented and several constitutive relations are derived. The special case of elastic porous media is considered in detail and the basic field equations are derived and the possible application of the results to soil dynamics is pointed out. The theory of the flow of granular media is also considered and basic equations of motion are derived where the results of Goodman and Cowin are recovered. The viscoplastic flow of porous media is studied and the possible application to soil mechanics is also discussed.
TL;DR: In this article, the authors extended the method presented in Part I to cover the damped and transient behavior of nonlinear systems described by equations of the form uss−uyy−ef(u,uy, uyys,…,s) = 0.25.
Abstract: The method presented in Part I is extended to cover the damped and transient behavior of nonlinear systems described by equations of the form uss−uyy−ef(u,uy, uyys,…,s) = 0. The method is presented by application to the equation uss−uyy−euyys−eu2yuyy= 0. Similar to the undamped case it is again shown that the PDE requires an infinite number of periodicity conditions to correctly characterize the resonant region. However, damping eliminates some of the branches of the amplitude-frequency spectrum of the undamped case. In fact, for e = 0.25 all but the outermost branch disappear. A method of multiple time scales is presented for the study of the transient behavior and the stability of the branches for steady vibrations. The stability analysis yields an interior stable point in the amplitude-frequency spectrum which has no analog in the Duffing equation. Finally via the multiple scale procedure in the spirit of the early work of Zabusky and Kruskal one obtains forced Burgers and Korteweg-de Vries equations on a finite interval.
TL;DR: In this paper, a method of multiple time scales is used to study the non-linear oscillations of impulsively forced systems under conditions of internal resonance, and a partial analytical solution is obtained.
Abstract: The method of multiple time scales is used to study the non-linear oscillations of impulsively forced systems under conditions of internal resonance. A partial analytical solution is obtained. The method is illustrated by an example in which the internal resonance effects are shown to be significant.
TL;DR: In this paper, the range of validity of the Poincare method is studied by comparison with the exact solution for the anharmonic and Morse oscillators in terms of elliptic and inverse trigonometric functions.
Abstract: The range of validity of the Poincare method is studied by comparison with the exact solution for the anharmonic and Morse oscillators. The exact solutions for these cases are expressible respectively in terms of elliptic and inverse trigonometric functions. The oscillation frequency is taken as the basis for the comparison. It is seen that the Poincare perturbation gives fairly accurate results up to amplitudes that are 20–30 per cent of the maximum value allowed for periodic solutions depending on the form of the potential energy. A new method is presented as a slight variation over the standard Poincare method. This method differs from the former only by a rearrangement of the differential equation through a collocation approximation for the potential. In spite of its simplicity, the method proves to be a better approximation than the standard Poincare method and gives remarkably accurate results for amplitudes up to 60–70 per cent of the maximum value allowed for periodic solutions.
TL;DR: In this article, the existence of a global relation between the solutions of the two systems is established, and it is easy to deduce the difference between two correspondent solutions, and the theorems of existence for particular solutions (periodic, quasiperiodic,…) or integral manifolds, of the initial system.
Abstract: One considers a perturbed ordinary differential system which is then reduced to the non-perturbed corresponding system; i.e. the existence of a global relation between the solutions of the two systems is established. From this it is easy to deduce the difference between two correspondent solutions, and the theorems of existence for particular solutions (periodic, quasiperiodic,…) or integral manifolds, of the initial system.
TL;DR: In this paper, it is shown that the system is integrable even if it is only holonomic and scleronomic and has one quasi-cyclic coordinate, provided the kinetic energy satisfies a certain additional condition.
Abstract: It is well known that conservative holonomic and scleromic systems with two degrees of freedom which have one cyclic coordinate are ‘integrable’. This means that the solution to the equations of motion can be given analytically in terms of quadratures, due to the existence of the two first integrals: the energy integral and the integral corresponding to the cyclic coordinate. In the present paper it is shown that the system is ‘integrable’ even if it is only holonomic and scleronomic and has one ‘quasi-cyclic’ coordinate, and even if the generalized forces are non-conservative provided the kinetic energy satisfies a certain additional condition.
TL;DR: In this paper, the second-order solutions of the torsion problem for simply-connected regions are available based on the theory given by Green and others both for compressible and incompressible materials.
Abstract: Some second-order solutions of the torsion problem for simply-connected regions are available based on the theory given by Green and others both for compressible and incompressible materials. Bhargava and Gupta [1] have recently extended the theory for torsion problem of multiply-connected regions. In the present paper these theories are extended further to account for the composite regions. The complex variable formulation is employed. As an illustration, results for the torsion problem of a composite cylinder of concentric circular cross-section are given.
TL;DR: In this article, the authors present an analysis for the case when the machine characteristic can be expressed by a continuous unique curve as well as for that when the characteristic is neither a unique nor even a smooth curve.
Abstract: Investigations concerned with the stability of stationary states and the possibility of self-excited oscillation (surge) occurring in systems with a centrifugal compressor (or a centrifugal pump) lead, for a simplified model, to an analysis of a set of two first-order differential equations. The paper presents such an analysis for the case when the machine characteristic can be expressed by a continuous unique curve as well as for that when the characteristic is neither a unique nor even a smooth curve. It is shown which of the singular points is the saddle point and in the case of the latter type of characteristic, which point can be taken for the saddle; this approach is believed to make practical analyses more straightforward.
TL;DR: In this article, the authors examined the following conjecture concerning equilibrium instability for conservative systems: if the potential is analytic and has no minimum at the origin, the vanishing solution is unstable, and they proved that the conjecture is still true if the Hessian matrix is zero and the third-order terms of the potential do not constitute a perfect cube.
Abstract: The author examines the following conjecture concerning equilibrium instability for conservative systems: if the potential is analytic and has no minimum at the origin, the vanishing solution is unstable. It is well known that for a system with two degrees of freedom, this conjecture is true if the Hessian matrix of the potential is not zero. It is proved that the conjecture is still true if the Hessian matrix is zero and the third-order terms of the potential do not constitute a perfect cube. Then, for an /b n/-dimensional system, a result of Koiter (1965) is extended to the case of a potential for which the Hessian matrix has (/b n/-2) strictly positive eigenvalues and two zero eigenvalues.
TL;DR: In this paper, a mechanical-optical constitutive relation is derived that can model rate dependent behavior without itself having any explicit dependence on the rates, and the use of the birefringent effect for solving non-linear time dependent problems is discussed and the inapplicability of some existing photoelastic procedures pointed out.
Abstract: Many transparent materials when deformed and viewed in polarized light exhibit a birfringent effect. The present study is an attempt to consider, within a consistent framework, many of the non-linear aspects of the phenomenon of birefringence in polymers and how they may be used in the development of a method for the stress analysis of non-linear problems. The kinematics in a rate form for the large deformations of a medium exhibiting permanent as well as recoverable deformations is first given. The constitutive problem for rate dependent materials is then treated. The proposed differential model can exhibit a very wide range of material behavior and is in a form that satisfies the requirements for the proper formulation of constitutive relations. A mechanical-optical constitutive relation is derived that can model rate dependent behavior without itself having any explicit dependence on the rates. The use of the birefringent effect for solving non-linear time dependent problems is discussed and the inapplicability of some of the existing photoelastic procedures pointed out. The theory and techniques of analysis are checked by first performing the quasistatic and dynamic characterization of a polyester blend to about 25% strain, and then solving some stress analysis problems.
TL;DR: In this article, the Granato-Lucke theory of internal friction is used as the basis of a model of shock-wave formation and propagation in elasto-plastic solids below the general yield point.
Abstract: The Granato-Lucke theory of internal friction is used as the basis of a model of shock-wave formation and propagation in elasto-plastic solids below the general yield point. The structure of shock layers in such a model is shown to be in general asymmetric and, at sufficiently large jumps in strain, to exhibit oscillations in the strain at its trailing edge.
TL;DR: In this paper, the optimal control of a random sine wave oscillator is studied and sufficient conditions on the optimal controls are derived by a set of two coupled non-linear partial integro-differential equations.
Abstract: This paper deals with the optimal control of a random non-linear sine wave oscillator. It is assumed that the oscillator is subjected to two different kinds of perturbation. The first kind of perturbation is represented by a vector of independent standard Wiener processes and the second kind by a generalized type of a Poisson process. Sufficient conditions on the optimal controls are derived. These conditions are given by a set of two coupled non-linear partial integro-differential equations. A numerical procedure for the solution of these equations is suggested and its efficiency and applicability are demonstrated with examples.
TL;DR: In this paper, an asymptotic method for the determination of the region of self-excitation and the amplitudes and phase angles for both stationary and non-stationary responses is outlined.
Abstract: The parametric response of a metallic column at elevated temperature is investigated, taking into account its non-linear viscous characteristics. An asymptotic method for the determination of the region of self-excitation and the amplitudes and phase angles for both stationary and non-stationary responses is outlined briefly. This method is applied to predict the parametric response of a 2618-T61 Al alloy column at 200°C. It is shown that the region of selfexcitation shifts away from the elastic frequency axis, that the amplitude of the stationary response increases sharply as the excitation parameters move from the stability region into the selfexcitation region and that the amplitude of the non-stationary response can be approximated by the stationary response solution for slow varying excitation frequency.
TL;DR: In this paper, the authors studied periodic solutions of autonomous quasiharmonic systems in the resonant case if the branching equation has multiple roots and provided sufficient conditions of the solutions depend upon the defining equation.
Abstract: Periodic solutions of autonomous quasiharmonic systems are studied in the resonant case if the branching equation has multiple roots. In order to find all the real solutions of this equation, we use Newton's diagram. This problem may have no real periodic solution. This depends upon the configuration of the descending section of Newton's diagram and upon the roots of the appropriate defining equations. The stability of these periodic solutions is also considered. Sufficient conditions of the solutions depend upon the defining equation.
TL;DR: In this paper, the problem of determining the control force that will bring, by radial deformation, a circular cylinder and a thin spherical shell from one configuration to the other, in a minimum time was solved.
Abstract: The problem of determining the control force that will bring, by radial deformation, (a) a circular cylinder and (b) a thin spherical shell from one configuration to the other, in a minimum time, is solved. The material is assumed to be incompressible and elastic of the Mooney-Rivlin type. Two numerical examples are presented.
TL;DR: In this article, a semi-implicit difference scheme was proposed to solve the equations of the unsteady incompressible laminar boundary layers about circular and elliptic cylinders.
Abstract: The equations of the unsteady incompressible laminar boundary layers about circular and elliptic cylinders started impulsively from rest are solved, after an original coordinate transformation, with the help of a semi-implicit difference scheme linearly stable without conditions. The knowledge of the phenomenon results only from that of the flow characteristics at the initial time; this makes mandatory the use of an analytical method of the Blasius type. When the ratio of the semi-axes of the elliptic cylinder becomes infinite, the present method gives the solution of the stagnation flow. When the flow presents a separation point in the steady state (circular cylinder), a criterion for stopping computation is proposed. It may be noted that to obtain the stagnation flow, we can continue the computations as far as possible and a steady solution is found with excellent accuracy.
TL;DR: In this paper, a multiaxial stress-strain damage relations, describing both time-independent and time-dependent strain and damage creation, were investigated for a beam under tension and bending.
Abstract: Multiaxial stress-strain damage relations, describing both time-independent and time-dependent strain and damage creation are postulated. The influence of damage creation on the load carrying capacity of simple structures is discussed. The time-independent deformation, and loss of stability, of a thick-walled cylinder under torque and internal pressure are analysed. The results are shown to be similar to previously found results for a beam under tension and bending.