Showing papers in "International Journal of Non-linear Mechanics in 1975"
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TL;DR: In this paper, large amplitude whirling motions of a simply supported beam constrained to have a fixed length are investigated, taking into account bending in two planes and longitudinal deformations, using the method of harmonic balance, response curves for certain planar and nonplanar steady state, forced motions are obtained.
Abstract: Large amplitude whirling motions of a simply supported beam constrained to have a fixed length are investigated. Equations of motion taking into account bending in two planes and longitudinal deformations are developed. Using the method of harmonic balance, response curves for certain planar and non-planar steady state, forced motions are obtained. Another approximate scheme is used to study the stability of these motions. Stable regions corresponding to non-planar motions are found, thus confirming the existence of whirling motions. Numerical results are presented and discussed for several specific cases.
52 citations
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TL;DR: In this paper, the non-linear equations of motion of a slender bar rotating at constant angular velocity about a transverse axis are formulated under the assumption that a small perturbed motion occurs about an initially stressed equilibrium configuration.
Abstract: The non-linear equations of motion of a slender bar rotating at constant angular velocity about a transverse axis are formulated. Under the assumption that a small perturbed motion occurs about an initially stressed equilibrium configuration, linearized equations of motion for the longitudinal and flexural deformations of a rotating bar carrying a tip mass are derived. Numerical computations for the natural frequencies of the lowest three modes of free vibration reveal that the values of the extensional frequencies increase monotonically, contrary to previously published results, as the angular velocity of rotation increases.
47 citations
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TL;DR: In this article, the authors derived non-linear modal equations for thin elastic shells of arbitrary geometry by utilizing the strain-displacement relations of the Sanders-Koiter nonlinear shell theory.
Abstract: In this paper we derive non-linear modal equations for thin elastic shells of arbitrary geometry. Geometric non-linearities are accounted for by utilizing the strain-displacement relations of the Sanders-Koiter non-linear shell theory. Arbitrary initial imperfections are accounted for and the shell thickness is free to vary within the limits of thin shell theory. The derivation gives the coefficients of the modal equations as integral expressions over the surface of the shell. The resulting equations are well-suited for practical applications. Weighting factors are introduced to allow for reduction of our results to the Love shell theory and to the Donnell approximation. The equations are specialized for a finite simply supported circular cylinder and numerical results are compared to those previously published in the literature.
43 citations
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TL;DR: In this article, the theory of non-local viscoelastic media is studied and the constitutive equation of strain and strain rate dependent, stress and stress rate dependent as well as continuous memory dependent are derived and discussed.
Abstract: The theory of non-local viscoelastic media is studied. The constitutive equation of strain and strain rate dependent, stress and stress rate dependent as well as continuous memory dependent are derived and discussed. The thermodynamical restrictions are also studied.
25 citations
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TL;DR: Berger's field equations are generalized to dynamic phenomena in anisotropic plates and shallow shells the anisotropy being either of a cylindrically orthotropic or rectilinearly orthotropic type as discussed by the authors.
Abstract: Berger's field equations are generalized to dynamic phenomena in anisotropic plates and shallow shells the anisotropy being either of a cylindrically orthotropic or rectilinearly orthotropic type. Under the assumption that the rim of the structure is prevented from inplane motions explicit equations for the coupling parameter are given. Method of solution in the dynamic case is illustrated by an example involving isotropy and original axial symmetry of the structure. It is shown that for the non-linear oscillations of built-in circular plates a close agreement is reached with the results obtained by means of the Von Karman field equations and a different coordinate function. The procedure suggested for solution of dynamic problems associated with the discontinuities of the boundary conditions is discussed and illustrated on an isotropic case involving a circular plate partially simply supported and partially clamped at the periphery. A numerical example is given concerning static behavior of an infinite isotropic strip uniformly loaded and simply supported along the edges except for two symmetrically situated built-in segments. The dependence of the average moment of clamping on the width of the built-in segments as well as on the load intensity for a fixed width of the segments is displayed on graphs.
19 citations
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TL;DR: In this paper, the eigenfunction expansion method is used to obtain local solutions to some non-Newtonian slow viscous flows, including power-law fluids, and the critical corner angle for eddy formation is obtained.
Abstract: The eigenfunction expansion method is used to obtain local solutions to some non-Newtonian slow viscous flows. The forms of viscosity variation amenable to such analysis are restricted but do include power-law fluids. Power-law flow near a sharp corner between plane boundaries is analysed and results are obtained for the critical corner angle for eddy formation. Flows near a 90° corner with either a moving boundary or a finite flow rate at the corner are also considered. The “stick-slip” behaviour of a power-law fluid at a plane solid boundary is shown to obey a simple law.
16 citations
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TL;DR: In this paper, the problem of the brachystochrone with dry friction is governed by an autonomous set of highly non-linear ordinary differential equations, which contain a small parameter μ (the coefficient of friction), and using perturbation technique an approximate solution is constructed.
Abstract: The problem of the brachystochrone with dry friction is governed by an autonomous set of highly non-linear ordinary differential equations, which contain a small parameter μ (the coefficient of friction). Using perturbation technique an approximate solution is constructed.
16 citations
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TL;DR: In this article, a direct computational method for the post-buckling of an orthotropic annular plate is presented, where the von Karman plate equations are transformed to a non-linear eigenvalue problem.
Abstract: In the present paper, a direct computational method for the post-buckling of an orthotropic annular plate is presented. Two different boundary cases of an annular are investigated. The von Karman plate equations are transformed to a non-linear eigenvalue problem. Solutions are obtained by employing the related initial value problem in conjunction with a numerical method of integration, and by using the method of continuation. The solution yields a complete description of post-buckling loads and stresses, and it reveals the non-linear behavior of the annulus beyond the linear buckling load.
15 citations
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TL;DR: In this article, a uniform study of all types of resonances that can occur in non-linear, dissipative multi-degree-of-freedom systems subject to sinusoidal excitation is presented.
Abstract: A uniform study of all types of resonances that can occur in non-linear, dissipative multi-degree-of-freedom systems subject to sinusoidal excitation is presented. The theoretical investigation is based on a harmonic or multi-harmonic solution and the Ritz method. The new approach suggests that non-linear normal mode shape or the so-called “coupled” non-linear mode shapes are those which are retained in resonance conditions, no matter what type of resonancemain, or secondary, periodic or almost-periodic. By introducing the concept of non-linear normal coordinates the response of an n -degree-of-freedom system is described, to a satisfactory degree of accuracy, by a single coordinate in the case of main or secondary-periodic resonance, or by p coordinates in the case of almost-periodic (combination) resonance with p harmonic components. Numerical examples indicate good agreement of theoretical and analog computer results and illustrate considerable discrepancies between resonance curves calculated by the commonly used “single linear mode approach” and the suggested “single non-linear mode approach”.
11 citations
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TL;DR: In this article, general solutions for axial velocity of the string and the velocity of wave propagation, based on a transformation in terms of the particle function, are presented and a set of particular solutions are then sought to investigate physical aspects of the problem.
Abstract: Motion of elliptic ballooning for a traveling string at equilibrium is treated in detail. Some invariants of motion are derived. General solutions for axial velocity of the string and the velocity of wave propagation, based on a transformation in terms of the particle function, are presented. A set of particular solutions is then sought to investigate physical aspects of the problem. Properties of solutions for the transverse displacement in the characteristic plane are analyzed.
11 citations
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TL;DR: In this paper, the non-linear transverse vibrations of a uniform beam with ends restrained to remain a fixed distance apart and forced by a two mode function which is harmonic in time, are studied by a corresponding two mode approach.
Abstract: The non-linear transverse vibrations of a uniform beam with ends restrained to remain a fixed distance apart and forced by a two mode function which is harmonic in time, are studied by a corresponding two mode approach. The existence of sub-harmonic response of order 1 3 and harmonic response in the sub-harmonic resonance region of the forcing frequency is proved. Approximate solutions are found by Urabe's numerical method applied to Galerkin's procedure and by an analytical harmonic balance-perturbation method. Error bounds of the Galerkin approximations are given.
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TL;DR: In this article, the dynamic compression of a fluid-filled isotropic Hookean elasto-porous medium with non-constant permeability is considered and the governing non-linear partial differential equations for two simple models are set up with the aid of an appropriate approximation and solved numerically for the medium displacements and fluid pressure.
Abstract: The dynamic compression of a fluid-filled isotropic Hookean elasto-porous medium with non-constant permeability is considered. The governing non-linear partial differential equations for two simple models are set up with the aid of an appropriate approximation and solved numerically for the medium displacements and fluid pressure. The results are compared with a series solution for the case of constant permeability. They indicate that the neglect of variable permeability could result in errors of order one hundred per cent.
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TL;DR: In this paper, a rheological model to describe non-linear deformations and failure is proposed for the case of constant stress expressions of instantaneous and time-dependent deformations.
Abstract: A rheological model to describe non-linear deformations and failure is proposed. For the case of constant stress expressions of instantaneous and time-dependent deformations are developed. There is an acceptable agreement between theoretical predictions and experimental data.
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TL;DR: In this paper, free axisymmetric vibrations of a stretched circular membrane are studied using a membrane theory consisting of a pair of non-linear partial differential equations coupled between the transverse and radial displacements of the membrane.
Abstract: Free axisymmetric vibrations of a stretched circular membrane are studied using a membrane theory consisting of a pair of non-linear partial differential equations coupled between the transverse and radial displacements of the membrane. A systematic perturbation method, in which the amplitude of the transverse displacement is taken as the perturbation parameter, is used to obtain periodic solutions of the non-linear equations. The initial membrane strain enters the problem as a parameter which is allowed to vary over a range of values. A case of self-resonance is encountered when the initial membrane strain approaches some critical values. This self-resonance case is also treated through an appropriate modification of the perturbation method.
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TL;DR: In this article, the problem of a rubber-like hollow circular cylinder subjected to a finite uniform initial stress in the axial direction is considered and an infinitesimal axial-symmetric deformation is then superposed on the prestressed hollow cylinder.
Abstract: In this paper, the problem of a rubber-like hollow circular cylinder subjected to a finite uniform initial stress in the axial direction is considered. An infinitesimal axial-symmetric deformation is then superposed on the prestressed hollow cylinder and the axial-symmetric instability of the cylinder caused by the initial stress is investigated. The analysis is based on the general theory of mechanics of incremental deformations established by Biot[ll]. It is shown that introducing a displacement potential function, φ, the field equation previously derived by Sun[15] is obtained. The solution of this field equation is shown to be given in terms of modified Bessel functions of the first and the second kind in r and trigonometric functions of z . By satisfying the homogeneous boundary conditions, the characteristic equation governing stability of the hollow cylinder is obtained. Numerical results examining the critical initial stress as a function of the slenderness ratio and the geometry of the cylinder are presented and discussed.
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TL;DR: In this article, the possibility of holding certain non-uniform temperature fields in finitely deformed spherical sectors is considered and an exact solution in spherical coordinates to the coupled equations of thermoelasticity for Fourier-like materials is given.
Abstract: The possibility of holding certain non-uniform temperature fields in finitely deformed spherical sectors is considered. An exact solution in spherical coordinates to the coupled equations of thermoelasticity for Fourier-like materials is given.
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TL;DR: In this paper, a variational principle is derived for the mixed initial-boundary value problem of non-linear elastodynamics, which involves stress quantities only, and an incremental procedure for the numerical solution is described.
Abstract: A variational principle is derived for the mixed initial-boundary value problem of non-linear elastodynamics. This principle involves stress quantities only. It is an extension of a similar one derived by Gurtin[1] for linear elastodynamics. The principle is then specialized to the class of semilinear materials, and generalized for the use in the hybrid stress model of finite element analysis. An incremental procedure for the numerical solution is described.
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TL;DR: In this article, the authors studied perturbed systems of differential equations which are close to Hamiltonian systems of the type for which small divisor problems arise and provided a partial justification for truncating the Fourier series occurring in these problems.
Abstract: Perturbed systems of differential equations are studied which are close to Hamiltonian systems of the type for which small divisor problems arise. If the non-Hamiltonian terms involve both damping and self-excitation, behavior characteristic of Hamiltonian systems can occur but is confined to small regions of space in which these opposite effects are in balance. The results provide a partial justification for truncating the Fourier series occurring in these problems. The principle method is local averaging, which is used to obtain global results by compactness arguments.
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TL;DR: In this article, the orthogonality principle was used to prove that systems treated in classical thermodynamics are nongyroscopic, and the dissipative forces must be determined by the dissipation function.
Abstract: The systems treated in classical thermodynamics are nongyroscopic. In such systems, the dissipative forces must be determined by the dissipation function. For elementary processes, this statement leads to the orthogonality principle proposed by the author in his earlier work.
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TL;DR: In this article, the exponential stability of a non-linear feedback system (consisting of a time-invariant block G in the forward path and a nonlinear time varying gain φ.k(t) in the feedback path) is presented.
Abstract: Explicit geometric criteria for the exponential stability of a non-linear feedback system (consisting of a time-invariant block G in the forward path and a non-linear time varying gain φ.k(t) in the feedback path) are presented when φ(.) belongs to certain classes of non-linear functions. The resulting bound on [ ( dk dt ) k ] is less stringent than those found in the literature.
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TL;DR: In this article, a method for constructing Lyapunov functionals for dynamical systems governed by partial differential equations is presented, where the functionals are obtained as path integrals in a suitably chosen state space of a generalized gradient operator.
Abstract: A method is given for constructing Lyapunov Functionals for dynamical systems governed by partial differential equations. The functionals are obtained as path integrals in a suitably chosen state space of a generalized gradient operator, and the method may be viewed as an extension to infinite dimensional systems of the variable gradient technique. Some of the fundamental concepts underlying the formalism are reviewed, and examples of applications to some linear, non-linear and hybrid systems are given.
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TL;DR: In this article, a general propagation condition is derived which permits the calculation of the speed of propagation of a second-order acceleration wave passing through a particular non-linear incompressible viscoelastic fluid.
Abstract: A general propagation condition is derived which permits the calculation of the speed of propagation of a second-order acceleration wave passing through a particular non-linear incompressible viscoelastic fluid. The viscoelastic fluid is taken to obey the Bernstein, Kearsley and Zapas single integral constitutive model. The analysis is valid for arbitrary finite amplitude waves propagating through a medium undergoing an arbitrary large deformation. Three examples, rest history, steady simple shearing flow and steady simple extensional flow, are given to demonstrate the utility of the propagation condition.
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TL;DR: In this paper, the authors define a cycle de conditions initiales approprie servant de plateforme for un tir dont les diverses etapes sont reglees a l'aide d'une periode d'echantillonnage convenablement choisie.
Abstract: Resume Etant donne un systeme continu x = X(x,y), y = Y(x,y) dont le portrait topologique des courbes integrales est caracterise par au moins un point singulier asymptotiquement instable, entoure d'un cycle limite stable, dans un domaine donne, on determine la situation du cycle limite par une methode appelee balisto-stroboscopique. A partir d'un systeme discret associe au modele continu, on definit d'une part un cycle de conditions initiales approprie servant de plateforme pour un tir dont les diverses etapes sont reglees a l'aide d'une periode d'echantillonnage convenablement choisie ; a chaque periode, le cycle initial se deforme par stroboscopique et converge vers le cycle limite de la solution periodique. Une application numerique detaillee est donnee pour un modele de Van der Pol dans le cas ϵ = 1 ; on examine aussi les cas H = 0,1 et H = 10.
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TL;DR: An extension of Prager-Ziegler's work-hardening rule for infinitesimal elastic-plastic deformation to a workhardening rules for finite elastic plastic deformation of polycrystalline metals at elevated temperatures is developed in this paper.
Abstract: An extension of Prager-Ziegler's work-hardening rule for infinitesimal elastic-plastic deformation to a work-hardening rule for finite elastic-plastic deformation of polycrystalline metals at elevated temperatures is developed. Attention is given to the development to satisfy certain invariant, continuity and thermodynamic requirements.
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TL;DR: The existence and asymptotic behavior as ϵ → 0+ of periodic, almost periodic, and bounded solutions of the differential system x = f(t, x, y, ϵ), Ωy′ = g(t. x, f; are n-vectors.
Abstract: The existence and asymptotic behavior as ϵ → 0+ of periodic, almost periodic, and bounded solutions of the differential system x = f(t, x, y, ϵ), Ωy′ = g(t, x, y, ϵ), are considered where x, f; are n-vectors, y, g are m-vectors and Ω = diag{ϵh1}…, ϵhm for integral hi, h1 ≦ h2 ≦ …, ≦ hm The principal tools are a lemma of Nagumo which allows the construction of appropriate upper and lower solutions and the asymptotic theory of singularly perturbed linear differential systems
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TL;DR: In this paper, a generalization of the effective stiffness theory of Sun, Achenbach and Herrmann is used to derive a set of approximate equations describing the propagation of a stress wave in a non-linear, geometrically dispersive composite material.
Abstract: A generalization of the effective stiffness theory of Sun, Achenbach and Herrmann is used to derive a set of approximate equations describing the propagation of a stress wave in a non-linear, geometrically dispersive composite material. The equations which are derived are singular perturbation equations having two kinds of singularities, one caused by dispersion and one caused by the non-linearity. A non-linear dispersion equation which describes the propagation of a periodic wave is derived using an averaged Lagrangian technique developed by Whitham. It is shown that the resulting solution is a simple wave system. With this technique, the singularity associated with the non-linearity may be eliminated and a solution which is convergent beyond the time of shock formation found. Application of this solution to a stress pulse and the group velocity for this system are discussed.
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TL;DR: In this article, non-linear stress-strain equations for incompressible, transversely isotropic elastic materials are developed and the expressions for a strain energy function are derived.
Abstract: Non-linear stress-strain equations for incompressible, transversely isotropic elastic materials are developed. In order to obtain these equations, the expressions for a strain energy function is found. The derivation of the strain energy function follows a geometrical approach and a method suggested by Mooney. These stress-strain relations are expressed in terms of three principal stretches to the sixth order.
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TL;DR: In this paper, the general propagation condition for acceleration waves is derived for a class of elastic heat conductors with two distinct temperatures, and the results show that the wave velocity is always greater than in the single-temperature theory and that every wave is homentropic.
Abstract: In this paper, the general propagation condition for acceleration waves is derived for a class of elastic heat conductors with two distinct temperatures. The results show that the wave velocity is always greater than in the single-temperature theory and that every wave is homentropic.
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TL;DR: In this paper, the instability of von Mises truss and shallow sinusoidal arch is investigated either in symmetric or asymmetric buckling forms and the constitutive equation of the material is assumed to be the general linear hereditary integral equation or the power series presentation of hereditary integrals in which the first considered equation forms the term of the first order.
Abstract: The instability of von Mises truss and shallow sinusoidal arch is investigated either in symmetric or asymmetric buckling forms The constitutive equation of the material is assumed to be the general linear hereditary integral equation or the power series presentation of hereditary integrals in which the first considered equation forms the term of the first order In the linear material equation case the limit load value for asymptotically constant, stable state of equilibrium is analytically established and the relationship between load and critical time is numerically established In the case of non-linear material equation, only numerical treatment of the problem is outlined and demonstrated by an example
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TL;DR: In this paper, the authors employed the known universal, controllable solution of finite deformations of incompressible, isotropic elastic dielectric material, a perturbation scheme helps in extending the solution to the slightly compressible case.
Abstract: Employing the known universal, controllable solution of finite deformations of incompressible, isotropic elastic dielectric material, a perturbation scheme helps in extending the solution to the slightly compressible case. This technique is used to investigate the stress and displacement fields around a finite screw-dislocation in a slightly compressible dielectric. With polarization, the strength of the dislocation is increased.