Showing papers in "International Journal of Non-linear Mechanics in 1976"

An historical perspective of yield surface investigations for metals

Abstract: This survey reviews work dealing either directly or indirectly with initial and subsequent yield surfaces. Emphasis is placed primarily on experimental investigations. Features of the work surveyed which are discussed include motivation, discrepancies, attempts to overcome discrepancies, and the nature of further work needed.

Short-wave asymptotics for weak shock waves and solitons in mechanics

Abstract: Some recent applications of the theory of non-linear waves in smoothly inhomogeneous and weakly dissipative media are discussed in the paper. The possibilities of “linear-ray” approximation when the non-linear self-refraction effects may be neglected in comparison with the non-linear wave distortion along the rays are demonstrated for weak acoustic shocks in stratified atmospheres and ocean, the solitary waves in shallow water of variable depth and the solitons in elastic rods.

Stability of an elastic cube under dead loading: Two equal forces

Abstract: A cube of incompressible neo-Hookean material undergoes a pure homogeneous deformation and is held in equilibrium by three specified pairs of equal and opposite forces, two of which are the same, applied normally to its faces and uniformly distributed over them. The possible equilibrium states are determined and the stability of each is studied with respect to arbitrary superposed infinitesimal deformations. The stability limits are found to be different from those obtained when only infinitesimal deformations having the same principal directions as those of the basic equilibrium state are considered. The differences arise from rotational and shearing types of instabilities that may occur in the general case. A critical inference is drawn concerning the nature of the dead loading conditions employed.

Large amplitude free vibrations of irregular plates placed on an elastic foundation

Abstract: A unified method for determining the lowest natural frequency of large amplitude free vibrations of thin elastic plates of any shape and placed on elastic foundation is given. The conformal mapping technique is introduced and Galerkin's method is used to calculate approximate values of the lowest natural frequency. Time periods for circular, square and cornered plates placed on elastic foundation have been determined for simply supported and clamped edge boundary conditions. Practical values have also been determined experimentally. The results are presented in the form of graphs and they are compared with other known results.

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Abstract: Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σt = φ(e, σ)et + ψ(e, σ), e = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x − c)/(t − d)e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.

Constitutive equations for elastic dielectrics

Abstract: Assuming the free energy function to depend on the deformation gradient tensor, the polarization vector, the polarization gradient tensor and the temperature, material and spatial forms of the constitutive equations are obtained which involve the invariants constituting the minimal isotropy integrity basis for the free energy function. A special form of stored energy function, involving material descriptors, is used for anisotropic dielectrics. Upon suitable linearization. the results obtained here reduce to the constitutive equations for Voigt's piezoelectricity theory and Mindlin's theory for elastic dielectrics with polarization gradient.

On equilibrium instability for conservative and partially dissipative systems

Abstract: The investigation brings some contributions to the classical problem of inverting the Lagrange-Dirichlet stability theorem. First, an example is given of a conservative holonomic mechanical system with a stable equilibrium at the origin, although the potential function is strictly negative along some rays issuing from the origin. Then, one establishes a new instability result in the conservative case. Last, by means of a vector auxiliary function, one proves an instability theorem for holonomic systems with partial dissipation.

A perturbation solution to a plate postbuckling problem

Abstract: The modal coupling behavior of a compressed, imperfect plate is examined as a six degree of freedom structural system. A direct variational solution procedure is developed by the use of the functional for the total potential energy of the system. The non-linear algebraic equations generated are solved by use of a continued perturbation technique. The segmented deflection history is then developed for a variety of imperfection compositions and aspect ratios.

Effect of time dependent compressibility on non-linear viscoelastic wave propagation

Abstract: The work presented consists essentially of two parts: the first deals with the development of a non-linear constitutive equation for a three-dimensional viscoelastic material with instantaneous and time dependent compressibility; the second deals with the solution of some specific wave propagation problems for three simple three-dimensional geometries. The constitutive equation is based on the existence of elastic and creep potentials and is expressed in terms of single memory integrals with non-linear kernels. The wave propagation problems are solved by numerical integration along the characteristics of the governing equations. The primary conclusion drawn deals with the effect of time dependent compressibility on the dynamic stress, strain and velocity fields. Results indicate that the dynamic response of even slightly time dependent compressible materials varies dramatically from those assumed to have only an instantaneous elastic compressibility.

Postbuckling behavior of circular cylindrical shells under compression

Abstract: Applying the Galerkin procedure to the Donnell basic equations, reasonably accurate solutions are obtained for the postbuckling behavior of clamped circular cylindrical shells under axial compression. To make a distinct comparison with the previous experimental results, calculations are carried out for shells with the same elastic and geometric parameters and the relations between the waveform, axial shortening and maximum deflections with applied loads are clarified. The results here obtained are found to be in reasonable agreement with experimental ones throughout the regions with fairly large deformations.

Response of a non-linearly damped oscillator to combined periodic parametric and random external excitation

Abstract: An analytical solution, using the Fokker-Planck-Kolmogorov equation, is obtained for the problem of response of a non-linearly damped oscillator to combined periodic parametric and random external excitation. The solution yields first-order probability densities of amplitude and phase. These expressions are employed to distinguish between oscillations excited by external and parametric periodic forces in the presence of additional broadband random external excitation. Through decoupling of fast and slow motions an approximate expression is obtained for expected value of time to phase “switch”.

On bounds and limit theorems for secondary creep in symmetric pressure vessels

Abstract: The field equations governing creep in spherical and incompressible cylindrical pressure vessels subject to a nondecreasing internal pressure are reduced to a single equation in the effective stress. Using this equation, bounds are obtained for the effective stress and the displacement at any point in the body at any time. Also, in the case where the pressure tends to a limit as t → ∞. limit theorems are obtained which describe the long term behavior of the effective stress and the displacement.

Methode de centrage-estimation de l'erreur et comportement des trajectoires dans l'espace des phases

Abstract: Resume Nous etudions des estimations de l'erreur dans la methode de centrage pour des equations differentielles dans des espaces de Banach dans l'intervalle [0, ∞ [ de la variable t . Si les equations exacte et centree ont une meme position d'equilibre, nous etudions, sous certaines hypotheses de stabilite, l'allure des solutions de l'equation exacte en fonction du type de la position d'equilibre de l'equation centree (noeud, foyer, …).

Nonholonomic systems with non-linear constraint equations

Abstract: The development of a form of Lagrange's equations applicable with nonholonomic systems with non-linear constraint equations is presented and discussed. The analysis is based upon, and is an extension of. a method developed by the authors for nonholonomic systems with linear constraint equations in the generalized coordinate derivatives. The method is illustrated with the problem of the “balancing pole”.

Determination of mechanical response of non-linear viscoelastic solids based on frechet expansion

Abstract: This paper deals with the determination of mechanical response of a non-linear viscoelastic solid based on a Frechet expansion In the first portion, a modified superposition principle, suggested earlier by various authors, is arrived at from a slightly different point of view The effect of such a principle on the kernel functions and a geometric interpretation are discussed In the later portion, the tests required for the experimental determination of the kernel functions are discussed It is seen that besides the tests suggested by Onaran and Findley for the two-dimensional case, three additional tests are needed for the three-dimensional case to compute the response under constant combined stress

On the non-linear dynamic stability problem for thin elastic plates

Abstract: In this paper the non-linear response of thin elastic plates under parametric excitation is investigated. A new analytical method is proposed. It gives the possibility to obtain all the characteristic features of the phenomenon considered, which are known from experiments—the existing of beats, their dependence on the excitation parameter, the influence of the initial conditions, the typical character of the vibrations in the different regions. Analog computer studies are carried out, and they show clearly the influence of different parameters on the output of the problem considered.

Generalized non-linear free vibrations of prestressed plates and shells

Abstract: This paper investigates the non-linear free vibration of prestressed plates and shells in a general form. The analysis includes the effects of in-plane inertia. The analysis is based on the non-linear equations of motion and uses a perturbation procedure. No assumption is made a priori for the form of the time or space mode. The boundary conditions are treated in a general manner including boundary conditions where non-linear stress resultants are specified. The method is illustrated by three examples.

Inflation of a bonded non-linear viscoelastic toroidal membrane

Abstract: The inflation of a bonded viscoelastic toroidal membrane under finite deformations is considered. Three new variables, viz. the two principal stretch ratios and the angle between the normal vector of a deformed membrane and the axis of symmetry are introduced as dependent variables. The governing equations are reduced thereafter to a set of three first-order partial differential integral equations. The constitutive equation developed by Pipkin and Rogers for the non-linear response of a viscoelastic material is used. The creep phenomenon for an inflated viscoelastic toroidal membrane under a constant pressure is presented.

Stability analysis of the periodic solution of a piecewise-linear non-linear dynamic system

Abstract: The stability of the periodic solution under harmonic excitation of a non-linear dynamic system with “linear hysteretic damping” is examined proceeding from first principles. The method can be extended to the case of multi-degree of freedom systems unlike regular perturbation procedure.

Dynamical bending of rigid-plastic annular plates

Abstract: Dynamical bending of circular rigid-plastic annular plates, fixed along the exterior boundary and free on the interior boundary, when subjected to instantaneously applied transverse uniformly distributed blast-type load[1], is investigated in this paper. It is shown that annular plates are preferable to plates without holes, since their load capacity increases while residual deflections decrease. A so-called boundary parameter is introduced to estimate the effect of boundary conditions on the radial bending moment. A procedure for determining the residual deflections at every point of a plate is developed for use on an electronic computer. Numerical examples are given. In the end of the paper, the particularities of solution of our problem for annular plates, corresponding to the setting of Wang[2], Wang and Hopkins[3] for plates without holes are discussed.

Non-linear vibrating systems in resonance governed by hyperbolic differential equations

Abstract: The asymptotic solutions of second order hyperbolic differential equations with weak non-linearities in the case of internal and external resonance are found. The method used is an extension of the Krylov-Bogoliubov-Mitropolskii method. An application is made to the longitudinal vibrations of a rod in which non-linear elastic behaviour and linear viscoelastic damping occur.

Second harmonic generation in elastic surface waves on an isotropic solid

Abstract: A procedure for solving the problem of non-linear propagation of elastic surface waves is given. An expression for the particle displacement of the second harmonic is obtained. It is shown that the amplitude of the harmonic increases linearly with distance and time and is proportional to the square of the amplitude of the fundamental frequency wave.

On non-linear periodic differential equations and associated point mappings. Application to stability problems

Abstract: In this paper a method is given allowing the determination of the lower degree terms of a point mapping (i.e. recurrent relationship) associated to a non-linear differential equation with time periodic coefficients. This permits the study of some resonance phenomena. An example of application is given.

Stationary small noise problems

Abstract: An approximation technique is demonstrated for estimating stationary averages for a mildly non-linear dynamical system perturbed by additive white noise with a small coefficient. Numerical examples illustrating the effectiveness of this technique are given.

Stability of an elastic thick walled tube under end thrust and external pressure

Abstract: The stability of a thick-walled Neo-Hookean tube under end thrust and external pressure is examined by the use of the theory of an infinitesimal deformation superimposed on a finite deformation. Loss of stability occurs when a non-trivial solution for the infinitesimal deformation is available. The resulting equations are solved numerically and the critical conditions presented graphically for axisymmetric and non-axisymmetric buckling. The effects of external pressure are shown. The effect of placing a simple end condition on the tube is also demonstrated.

Damage induced instability in beam bending

Abstract: The effect of internal damage creation on the load carrying capacity of a beam is studied. Time independent relations are postulated for the development of strain and damage with increasing net stress. The resulting relations between load and deformation for a Bernoulli-Navier beam lamina with rectangular cross-section are then derived. A state of instability is shown to exist, characterized by unlimited rate of increase in deformation and damage with load. An instability locus in the plane of bending moment and normal force is defined. The shape of this locus is studied for varying parameters in the deformation and damage laws.

Second-order torsion problem of a homogeneous isotropic compressible multiply-connected elastic cylinder

Abstract: For simply-connected regions, some solutions are available for the second-order torsion problem of homogeneous isotropic compressible elastic cylinders based on the theory given by Green and others. In the present paper, these theories are extended to cover the second-order torsion problem for multiply-connected regions. As an example, results for torsion of a confocal elliptical ring are given.

A hyperbolic theory of plasticity

Abstract: The finite deformation model which was adopted for investigation was restricted so that it would exhibit non-analytic behaviour. Specifically, discontinuities in the first order partial derivatives of stress and strain were permitted. Mathematical difficulties limited the investigation to the solution of the problems of generalized plane strain and axial symmetry. For both of these problems, it was determined that the analysis yielded stress-strain laws of the flow type. An example was included to illustrate the application of the theory to both of these problems.

Non-linear flexural oscillations of a partially tapered annular plate

Abstract: The free finite amplitude axisymmetric oscillations of an isotropic annular plate with partially tapered thickness are investigated. The time variable is eliminated by a Ritz-Kantorovich averaging method. The von Karman plate equations are then reduced to two non-linear ordinary differential equations, which form a non-linear eigenvalue problem. Solutions to the problem are obtained by utilizing a direct computational method. The results reveal the effects of large amplitude upon the dynamic responses. Also, an annulus of constant thickness, which has the same boundary conditions and the same volume as the partially tapered one, is investigated. Their results, which may shed light on the optimal design of annular plates, are compared.

An experimental investigation of wave propagation in a rubber string impacted by a projectile

Abstract: Transverse perpendicular impact on long rubber strings of 0.140 and 0.277 in. dia was studied using a special apparatus consisting of a system of oscilloscopes, stroboscopes, camera and air pressure gun. The rubbers used in tests came from two manufacturers and were classified as pure gum rubbers. Three different initial stretches λ0 were chosen for each string, and the impact velocities ranged from 1000 to 3000 in. sec. Maximum stretch, kink angles and impact velocities were measured and the following relations recorded : nominal stress vs stretch, kink angle vs impact velocity, maximum stretch λmax vs impact velocity and the difference λmax − λ0 vs non-dimensional impact velocity. Some of the results are compared with the theoretical data for the Mooney-Rivlin and the Isihara-Hashitsume-Tatibana-Zahorski materials, and others with the results of theoretical equations upon inserting experimental data.