Showing papers in "International Journal of Non-linear Mechanics in 1977"
TL;DR: In this paper, the second law of thermodynamics and the consistency equation in the strain space formulation are expressed by two inequalities, and explicit necessary and sufficient conditions on the elastic moduli and their change with plastic deformation are derived for the two inequalities to be satisfied.
Abstract: Elasto-plastic coupling is studied within a general thermodynamic train space formulation of rate-independent plasticity by means of plastic internal variables. The strain space formulation offers a unified approach in treating both stable and unstable material behavior simultaneously. The conditions on elasto-plastic coupling are imposed by the second law of thermodynamics and the consistency equation in the strain space formulation, and are expressed by two inequalities. This is further illustrated by specific examples where explicit necessary and sufficient conditions on the elastic moduli and their change with plastic deformation are derived for the two inequalities to be satisfied.
39 citations
TL;DR: In this paper, the Fourier series is used to solve the Foppl large deflection equations for laterally loaded membranes for uniform load and for edges which are fixed normal to the edge but are free to move parallel to an edge.
Abstract: The Foppl large deflection equations for laterally loaded membranes are solved for uniform load and for edges which are fixed normal to the edge but are free to move parallel to the edge. Both the deflection function and the Airy stress function are expanded in Fourier series. The resulting coupled non-linear cubic equations for the deflection function coefficients are truncated and solved by means of an iterative procedure. Results for the center normal deflection and the stress resultants at selected points are calculated with the use of 100 or more equations and are found to differ significantly from the previously accepted approximate results.
39 citations
TL;DR: In this paper, the problem of finite-amplitude, axisymmetric free and forced vibration of a circular plate is examined with various boundary conditions, and the non-linear boundary value problem is converted into the corresponding eigenvalue problem by elimination of the time variable.
Abstract: The problem of finite-amplitude, axisymmetric free and forced vibration of a circular plate is examined with various boundary conditions. The non-linear boundary-value problem is converted into the corresponding eigenvalue problem by elimination of the time variable. Then by a Newton-Raphson iteration scheme, and the concept of analytical continuation, the solution to the non-linear eigenvalue problem for the vibrations is obtained in a discrete form. It is seen that the removal of radial restraint causes drastic changes in the plate responses and the patterns of membrane stresses. Comparison with solutions based on the Berger assumption reveals the unsuitability of the assumption when the plate is not radially restrained.
37 citations
Abstract: While studies of post-buckling behavior and load-carrying capacities of thin plates subjected to uniaxial compression have been limited to stable conditions, further post-buckling loading generates an unstable condition. The secondary buckling which occurs with snap-through to higher-order deflections under such unstable conditions has not been analyzed in detail as yet. In the first part of this paper, a thin square plate under uniaxial compression, which is simply supported along four edges, is considered. A method based on the second variation of the total potential energy is then proposed for evaluating the stability of the post-buckling equilibrium state and inevitable secondary buckling is derived analytically. The effects of various factors, such as initial imperfections, assumed virtual displacement pattern, post-buckling deflection pattern and in-plane boundary conditions, on the secondary buckling values are discussed. In part 2, secondary buckling of clamped plates is analyzed by use of the finite element method and the resultant numerical results are compared with experimental results.
35 citations
TL;DR: In this article, the effect of the nonlinearity of the governing equations on the propagation of waves in fluid filled elastic tubes is investigated by the method of characteristics for a particular form of pressure pulse applied at the end of a semi-infinite initially uniform tube.
Abstract: The effect of the non-linearity of the governing equations on the propagation of waves in fluid filled elastic tubes is investigated. Results are obtained by the method of characteristics for a particular form of pressure pulse applied at the end of a semi-infinite initially uniform tube. An expression is obtained for the distance along the tube at which shock formation is predicted. Two different hyperelastic materials whose elastic properties model those of biological tissue are considered for the tube walls. Numerical results are presented in graphical form.
24 citations
TL;DR: In this article, the authors deal with the phenomenon of reflection of plane elastic waves in a free surface when the medium is initially stressed and show that the reflected P and SV waves depend on initial stresses present in the medium.
Abstract: The paper deals with the phenomenon of reflection of plane elastic waves in a free surface when the medium is initially stressed. It has been shown analytically that the reflected P and SV waves depend on initial stresses present in the medium. The numerical values of reflection coefficients for different initial stresses and the angle of incidence have been calculated by the Computer I.C.L. 1901-A and the results are given in the form of graphs. Many interesting results are found in the paper which are not seen in an initially stress-free medium.
24 citations
TL;DR: In this article, the non-linear behavior of a thick walled sphere under internal and external pressure is considered, where the material is assumed to be incrementally elasto-plastic in the classical sense.
Abstract: The non-linear behaviour of a thick walled sphere under internal and external pressure is considered. The material is assumed to be incrementally elasto-plastic in the classical sense. No restriction on the magnitude of the deformation is imposed. For internal pressure a solution is obtained in terms of closed integrals. It is shown that the three dimensional solution reduces exactly to the membrane solution when the thickness of the shell becomes very small. Numerical examples are given for some practical materials used in the aerospace industry. It is shown that at a certain critical pressure instability of the second kind (inflation) is obtained.
21 citations
TL;DR: In this paper, a two-degree-of-freedom lumped-mass model was used to gain understanding of the equilibrium and stability of a circularly towed cable, in particular for no drag, viscous drag, and viscous charge with a cross wind.
Abstract: A two-degree-of-freedom lumped-mass model is used to gain understanding of the equilibrium and stability of a circularly towed cable. Particular cases considered are those of no drag, viscous drag, and viscous drag with a crosswind.
20 citations
TL;DR: In this article, the propagation of non-linear deformation waves in a dissipativc medium is described by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves.
Abstract: The propagation of non-linear deformation waves in a dissipativc medium is described by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves. The mathematical models are derived from the theories of thermoclasticity or viscoclasticity taking into account the geometric and physical non-linearities and dispersion. On the basis of eikonal equations for the associated linear problem the transport equations of the nth order are obtained. In the multidimensional case the method of matched separation of initial equations is proposed. The interaction problems which occur in head-on collisions and in reflection from boundaries or interfaces are analyzed. Conditions are also studied when the interaction of non-linear waves does not take place. The inverse problem of determining materials properties according to pulse shape changes is discussed.
18 citations
TL;DR: In this paper, the extensional equations of motion for a cantilever bar rotating about an axis fixed in space are derived, and it is shown that the form of the nonlinear strain-displacement relation is important in determining the nature of the relationship between the frequency of extensional oscillations and the rotational speed.
Abstract: The extensional equations of motion for a cantilever bar rotating about an axis fixed in space are derived. It is shown that the form of the non-linear strain-displacement relation is important in determining the nature of the relationship between the frequency of extensional oscillations and the rotational speed. In particular, the frequency may or may not increase monotonically with rotational speed, depending on the degree of hardening in the effective extensional spring. The determination whether an instability occurs as the rotational speed increases is beyond the limits of engineering beam theory.
12 citations
TL;DR: In this article, a modified version of the Linstedt-Poincare perturbation procedure is employed to facilitate the solution of the governing shell equations, and several numerical results are presented.
Abstract: The problem of circumferentially traveling radial loads on rings and infinitely long cylindrical shells will be considered in this paper. Since strong transitional excitations are considered, the response involves moderately large rotations. Hence the governing shell equations are non-linear. To facilitate their solution, a modified version of the Linstedt-Poincare perturbation procedure is employed. Based on this solution, several numerical results are presented. In addition to considering the effects of displacement induced non-linearity, special emphasis is given to the response behavior in load speed zones which mark transitions from sub- to supercritical waveforms.
TL;DR: In this paper, the variational and modified forms of the von Karman-type nonlinear plate equations are considered in the context of the Rayleigh-Ritz and Galerkin methods.
Abstract: The variational and modified forms of the von Karman-type non-linear plate equations are considered in the context of the Rayleigh-Ritz and Galerkin methods. An approximate analysis of the non-linear vibrations of thin elastic plates including inplane inertia is presented. The quantitative study confirms that the inplane inertia effects are negligible for thin plates provided the non-linearity is not too large. It is observed that the non-linear inertia terms in the transverse equation of motion should be retained in any such study. The analysis is simplified by neglecting the inplane inertia and applied to constrained and unconstrained plates. A different type of inplane boundary condition termed ‘the partially constrained’ is studied, and the inadequacy of replacing the unconstrained condition by means of an average-zero stress condition is clearly demonstrated. It is observed that in most of the cases considered the Galerkin method yields lower bounds for the non-linear coefficient of the modal equation. In all cases the Galerkin results yield less stiff models than the Rayleigh-Ritz method. The general significance of the convergence of the two methods beyond the scope of the title problem is highlighted.
TL;DR: In this paper, the authors considered two-degree-of-freedom Hamiltonian systems and proved that the periodic solutions are isoenergetically stable for sufficiently small e. The proof is an application of the Twist Theorem of Kolmogorov-Arnol'd-Moser.
Abstract: In this note we consider certain two-degree-of-freedom Hamiltonian systems which may be regarded as perturbations of integrable systems governed by a real parameter e. We wish to study the stability, at fixed energy, of certain periodic solutions. Two constants are defined, computable in terms of the original Hamiltonian function and the energy. The main theorem then states that if these constants are not zero, the periodic solutions are isoenergetically stable for sufficiently small e. The proof is an application of the Twist Theorem of Kolmogorov-Arnol'd-Moser. By way of illustration, we apply the theorem to a mechanical system consisting of coupled non-linear oscillators. The periodic solutions are the “normal modes” and e governs the non-linearity of the system. One obtains stability criteria for arbitrary energies and small e, or, alternatively, for arbitrary e and small energies.
TL;DR: In this article, exact periodic solutions for a dynamic system with several degrees of freedom consisting of a series of Reid springs with piecewise-linear, non-linear characteristis are derived.
Abstract: Exact periodic solutions are derived for a dynamic system with several degrees of freedom consisting of a series of ‘Reid springs’ with piecewise-linear, non-linear characteristis; however, the solutions are restricted to a class of harmonic excitation in the ‘modal form’ described subsequently in the paper. Conditions are derived for the asymptotic stability of the periodic solution and an example has been worked out in detail on the response of a dynamic system with two degrees of freedom.
TL;DR: In this paper, a concise survey of the known stability results for non-linear time invariant and time varying feedback systems is given, and some comments are made on the stability analysis of systems governed by partial differential equations.
Abstract: After formulating the problem of stability and instability of linear and non-linear time varying feedback systems, we give a concise survey of a few known stability results for non-linear time invariant and time varying systems. This serves as motivation for the establishment of general frequency domain stability and instability criteria for time varying linear and non-linear feedback systems. All the details are omitted in an attempt to give an overall view. Finally some comments are made on the stability analysis of systems governed by partial differential equations.
TL;DR: In this article, an interpretation geometrique de la methode du premier harmonique de Haag-Krylov-Bogolioubov-Mitropolski, appliquee a un oscillateur a hysteresis, conduit a nouvelle methode d'etude dynamique d'un tel systeme which consiste a remplacer l'equation de mouvement par une autre, approchee, conservant la non linearite de l'sequation originale.
Abstract: Resumen Une interpretation geometrique de la methode du premier harmonique de Haag-Krylov-Bogolioubov-Mitropolski, appliquee a un oscillateur a hysteresis, conduit a une nouvelle methode d'etude dynamique d'un tel systeme qui consiste a remplacer l'equation de mouvement par une autre, approchee, conservant la non linearite de l'equation originale. Cette methode est valable si l'excitation est aleatoire. En particulier, si cette excitation est un bruit blanc, on peut, moyennant certaines hypotheses physiquement realisables, ecrire l'equation de Fokker-Planck du systeme et, par consequent, avoir tout renseignement utile concernant la reponse.
TL;DR: In this paper, general expressions for both amplitude-dependent and speed-dependent damping were derived as a function of nonlinear restoring forces and arbitrary nonlinear damping forces, and the results of the analysis suggest how experimental data can be utilized to identify and evaluate the damping parameters for a given non-linear oscillator.
Abstract: The logarithmic damping decrement is obtained as a function of arbitrary non-linear restoring forces and arbitrary, but small, non-linear damping forces. General expressions are obtained for both amplitude-dependent and speed-dependent damping. The special case of a cubic restoring force with quadratic amplitude-dependent damping and the special case of a cubic restoring force with quadratic speed-dependent damping are considered in detail. The results of the analysis suggest how experimental data can be utilized to identify and evaluate the damping parameters for a given non-linear oscillator.
TL;DR: In this paper, the problem of first strain gradient solid stability under non-cons rvative loadings is studied and an energy Liapunov functional is constructed and the sufficiency criteria for the stability of a loaded equilibrium configuration are derived.
Abstract: The problem of thermoelastic stability of a first strain gradient solid under noncons rvative loadings is studied. The Liapunov-Movchan method of elastic stability analysis is reviewed. An energy Liapunov functional is constructed and the sufficiency criteria for the stability of a loaded equilibrium configuration are derived. The effects of motion dependent surface tractions and couples are discussed. The special case of isothermal elastic stability of solids with couple stress is also considered.
TL;DR: In this article, the Lagrangian formulation for continuous media is derived from variational principles for perfectly general mechanical systems and the basic conservation laws for such systems are generated from first principles and these are all referred to general curvilinear coordinates.
Abstract: The Lagrangian formulation for continuous media is derived from variational principles for perfectly general mechanical systems. The basic conservation laws for such systems are generated from first principles and these are all referred to general curvilinear coordinates. Specific results are catalogued for spherical and cylindrical coordinate systems. The equilibrium equations for isotropic, homogeneous, and incompressible elastic media are recovered as are some previous results of Bland.
TL;DR: In this article, a general technique is developed to obtain a relationship between the magnitude of the small parameter and the permissible error in the approximate solution, which is then demonstrated by its application to two examples.
Abstract: The Method of Averaging is an asymptotic method that can be used to obtain approximate solutions for many parameter dependent non-linear systems. The resulting approximate solutions are as accurate as desired provided the system parameter is sufficiently small. In this paper, a general technique is developed to obtain a relationship between the magnitude of the small parameter and the permissible error in the approximate solution. The technique is then demonstrated by its application to two examples.
TL;DR: In this paper, the relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied, and it is shown that the energy criterion for neutral equilibria is equivalent to the one for the non-homogeneous equilibrium.
Abstract: The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V2[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.
TL;DR: In this article, two theorems on the behaviour of a single degree of freedom oscillator with linear stiffness and even non-linear damping terms were proved for the case of linear damping.
Abstract: In this short note we prove two theorems on the behaviour of a single degree of freedom oscillator with linear stiffness and even non-linear damping terms.
TL;DR: In this article, a random van der Pol oscillator is assumed to be subjected to two different kinds of perturbations: standard Wiener process and homogeneous process with independent increments, finite second order moments, mean zero and no continuous sample functions.
Abstract: This paper deals with a random van der Pol oscillator. It is assumed that the oscillator is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the oscillator, two functionals are defined over its phase plane sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations, is suggested, and its efficiency and applicability are demonstrated with examples.
TL;DR: In this article, an approximate analytical approach for determining the steady-state response of a class of systems with spatially localized nonlinearity is presented, and fundamental properties of the response are identified.
Abstract: This paper presents an approximate analytical approach for determining the steady-state response of a class of systems with spatially localized non-linearity. Fundamental properties of the response are identified. An example illustrates the nature and accuracy of the results of the approximate analysis.
TL;DR: The motion of a small gas bubble, assumed to retain its geometrical shape and contained in a rotating liquid, has been investigated in this article, showing that under certain conditions (spin axis and direction of gravity are perpendicular to each other) the bubble travels on a circular path about the axis of rotation, as seen from an observer moving with the bulk of the liquid.
Abstract: The motion of a small gas bubble, presumed to retain its geometrical shape and contained in a rotating liquid, has been investigated. The fluid system liquid-gas is subject to the influence of a reduced gravitational field. It is demonstrated that under certain conditions (spin axis and direction of gravity are perpendicular to each other) the bubble travels on a circular path about the axis of rotation, as seen from an observer moving with the bulk of the liquid.
TL;DR: In this paper, an admissible form of rate-type constitutive equation of inelastic materials is given from the viewpoint of irreversible thermodynamics, where the displacement gradient tensor F referred to the temporarily fixed reference frame which coincides with the Euler frame at the instant of the reference time is decomposed linearly into elastic and inelastically parts so that the procedure of formulation is simplified.
Abstract: From the viewpoint of irreversible thermodynamics an admissible form of rate-type constitutive equation of inelastic materials is given. The displacement gradient tensor F referred to the temporarily fixed reference frame which coincides with the Euler frame at the instant of the reference time is decomposed linearly into elastic and inelastic parts so that the procedure of formulation is simplified and clarified. The inelastic deformation rate is directly related to the internal production rate of entropy. The existence of an inelastic potential of the usual sense is not assumed, though the result can be understood to include the conventional flow theory based on an inelastic potential. An example of an elastoviscoplastic constitutive equation is given and some properties of yield surfaces are discussed.
TL;DR: In this paper, the velocity at an oblique shock in a compressible fluid is derived in dyadic form similar to that for refraction of light rays at an interface, where the assumption of conservation of mass and equality of tangential velocity components is made.
Abstract: A refraction law for the velocity at an oblique shock in a compressible fluid is derived in dyadic form similar to that for refraction of light rays at an interface. The shock tensor embodies only the assumptions of conservation of mass and equality of tangential velocity components. Given the shock inclination and density ratio, a quadratic equation in the ratio of the flow speeds can be found with flow turning angle as a parameter. Analysis of the two solutions shows that they lie on a circle in the polar plane, a result independent of the equation of state or other conservation laws. If the density ratio is allowed to vary, a pencil of circles is generated in the hodograph plane ; or, equivalently a right, elliptic cone with two nappes appears in the three-space formed when the density ratio coordinate is added at right angles to the hodograph plane. The further requirements that momentum and energy be conserved taken together with weak restrictions on the functional form of the equation of state are sufficient to permit the development of a general theory of shock polars. The allowed shock states are seen to lie on the space curve formed by intersection of a surface called the Hugoniot cylinder with the elliptic cone. The projection of this space curve on the hodograph plane is the shock polar. The theory is applied to the special case of a polytropic gas by way of illustration.
TL;DR: In this article, a class of complete integrals of the plane eikonal equation | grad u| 2 = f(x,y) for harmonic u(x andy) is determined by using complex variables.
Abstract: A class of complete integrals of the plane eikonal equation | grad u| 2 = f(x,y) for harmonic u(x,y) is determined by using complex variables. The case in which z = 0 is a singular point of the analytic function whose real part is log f is also treated. Illustrative examples are given.
TL;DR: In this paper, some examples of elastoplastic constitutive equation are presented using the general theory reported in the preceding paper (Part I), some examinations of them are given to show that the theory is self-consistent and useful especially for anisotropic materials or materials with anisotropy resulting from plastic deformation.
Abstract: Some examples of elastoplastic constitutive equation are presented using the general theory reported in the preceding paper (Part I). Some examinations of them are given to show that the theory is self-consistent and useful especially for anisotropic materials or materials with anisotropy resulting from plastic deformation. Mises' and Yoshimura's yield functions and a kind of quadratic function are adopted as the yield function. Formulae of r-value after arbitrary pre-straining are given which are of paramount importance in the field of press-forming of sheet metals. Several examples of stress-strain curves for various loading paths are also given.
TL;DR: In this paper, Salvadori's method of a one parameter family of Liapunov functions is applied to the two-body problem in the presence of some friction forces and when the reference frame is non-inertial.
Abstract: Considering a closed set M of some x-space and a solution x(t), y(t) of a differential system x = X(x, y, t), y = Y(x, y, t), we give sufficient conditions in order that x(t) approaches M. We use several auxiliary functions and employ Salvadori's method of a one parameter family of Liapunov functions. An application is given to the two-body problem in the presence of some friction forces and when the reference frame is non-inertial.