Gabriel Cederbaum

Dynamic stability of nonlinear viscoelastic plates

Abstract The dynamic stability analysis of isotropic plates made of a nonlinear viscoelastic material is performed within the concept of the Lyapunov exponents. The material behavior is modeled according to the Leaderman representation of nonlinear viscoelasticity. The influence of the various parameters involved on the possibility of instability to occur is investigated. It is also shown that in some cases the system is chaotic.

Dynamic instability of shear-deformable viscoelastic laminated plates by lyapunov exponents

Abstract The dynamic stability of viscoelastic laminated plates, subjected to a harmonic in-plane excitation, is analyzed. The viscoelastic behavior is caused by the polymeric matrix of the fiberreinforced material, and a micromechanical analysis provides the time-dependent relaxation functions of the unidirectional lamina. The Boltzmann representation involved in the stress-strain relations of the laminated plate leads to an integro-differential equation of motion, obtained within the first-order shear deformation theory. For this case, a dynamic stability analysis which employs the concept of Lyapunov exponents is performed, and is shown to be very efficient.

Analysis of parametrically excited laminated shells

The dynamic stability of a shear-deformable circular cylindrical shell subjected to a periodic axial loading P(t) = Ps + Pd cos ωt is investigated. The simply-supported laminated shell of finite length is analyzed within Loves first-approximation theory, with the addition of transverse shear deformation and rotary inertia. Using the method of multiple scales, analytical expressions for the instability regions are obtained at ω = Ωj ± Ωi, where Ωi are the natural frequencies of the shell. Yet, it is shown that instability cannot occur for the case ω = Ωj − Ωi due to the symmetric properties of the problem. It is also shown that, besides the principal instability region at ω = 2Ω1 (Ω1 is the fundamental frequency), other cases of ω = Ωi + Ωj can be of major importance and yield a significantly enlarged instability region.

Theory of poroelastic beams with axial diffusion

The governing equations for a transversely isotropic poroelastic rod subjected to axial or transverse loads are formulated for the case of microgeometries which permit fluid motion in the axial direction only. The quasi-static problem of beam bending is analysed and series solutions are found for normal loadings with various mechanical and diffusion boundary conditions. The unique features of the time-dependent behavior of such beams are pointed out.

Periodic and chaotic behavior of viscoelastic nonlinear (elastica) bars under harmonic excitations

The analysis of viscoelastic homogeneous bars subjected to harmonic distributed loadings is presented. The material behavior is given in terms of the Boltzmann superposition principle. The equation of motion derived for the elastica, and by including changes in the bars length, is an integro-differential variation of the Duffing equation. The classical tools of nonlinear dynamics, such as the phase plane portrait, the Poincare map, the Fourier spectrum and the Lyapunov exponents analysis, are applied in order to investigate the different kinds of behaviors observed

Postbuckling of non-linear viscoelastic imperfect laminated plates Part I: material considerations

A method of predicting the non-linear relaxation behavior from creep experiments done on non-linear viscoelastic orthotropic materials is presented. Such a method is desirable since creep experiments are those mainly performed for these materials. It is shown that for given non-linear creep properties, and creep compliance represented by the Prony series, the Schapery creep model for non-linear viscoelastic orthotropic material can be transformed into a set of first-order non-linear equations. The solution of these equations makes it possible to obtain the non-linear stress relaxation curves, from which the strain-dependent constitutive equation of any non-linear viscoelastic model can be constructed, as needed for engineering applications.

Stress Relaxation of Nonlinear Thermoviscoelastic Materials Predicted from Known Creep

A method to predict the stress relaxation response of nonlinear thermoviscoelastic materials from known creep data is presented. For given nonlinear creep properties, and creep compliance represented by the Prony series, it is shown that the Schapery creep model can be transformed into a set of first order nonlinear differential equations. By solving these equations the nonlinear stress relaxation curves for different strain and temperature levels are established. The strain/temperature-dependent constitutive equation can then be constructed for any nonlinear thermoviscoelastic model, as needed for engineering applications.

Dynamic instability of antisymmetric laminated plates

Abstract The dynamic instability of antisymmetric angle-ply and cross-ply laminated plates, subjected to periodic in-plane loads P(t) = P0 + P1 cos θt, is investigated. Within the classical lamination theory, the motion is governed by three partial differential equations. By using the method of multiple scales, analytical expressions for the instability regions are obtained at θ = Ω i + Ω j , where Ω i are the natural frequences of the system. It is shown that in some cases, beside the principal instability region at θ = 2Ω 1 , where Ω 1 is the fundamental frequency, other cases of θ = Ω i + Ω j can be of major importance, and the final instability regions are significantly enlarged.

Postbuckling of non-linear viscoelastic imperfect laminated plates Part II: structural analysis

The postbuckling behavior of geometrically imperfect laminated plates with non-linear viscoelastic materials is investigated. The relaxation properties for the various Schapery single-integrals of a unidirectional layer are derived from the stress-relaxation curves obtained in part I. The non-linear plate (von Karman) equations are derived symbolically using Mathematica in the form of a system of first-order non-linear differential equations. A numerical example of graphite/epoxy cross-ply laminates is presented and discussed.

Post buckling analysis of imperfect nonlinear viscoelastic columns

The post buckling behavior of imperfect columns made of a nonlinear viscoelastic material is investigated. The material is modeled according to the Leaderman representation of nonlinear viscoelasticity. Solutions are developed to calculate the growth of the initial imperfection in time and within the elastica. The numerical results show that unlike the case of the post-buckling analysis of linear elastic or viscoelastic materials, here the ratio of h/l plays an important part in the structure response. It is shown that the post-buckling behavior of columns made of nonlinear viscoelastic materials is qualitatively and quantitatively better than in the linear viscoelastic case. Conclusions concerning creep buckling within the small deflection theory are also presented.