A SPECTRAL/hp NONLINEAR FINITE ELEMENT ANALYSIS OF HIGHER-ORDER BEAM THEORY WITH VISCOELASTICITY

Abstract

In this paper, a finite element model for efficient nonlinear analysis of the mechanical response of viscoelastic beams is presented. The principle of virtual work is utilized in conjunction with the third-order beam theory to develop displacement-based, weak-form Galerkin finite element model for both quasi-static and fully-transient analysis. The displacement field is assumed such that the third-order beam theory admits C 0 Lagrange interpolation of all dependent variables and the constitutive equation can be that of an isotropic material. Also, higher-order interpolation functions of spectral/hp type are employed to efficiently eliminate numerical locking. The mechanical properties are considered to be linear viscoelastic while the beam may undergo von Kármán nonlinear geometric deformations. The constitutive equations are modeled using Prony exponential series with general n-parameter Kelvin chain as its mechanical analogy for quasi-static cases and a simple two-element Maxwell model for dynamic cases. The fully discretized finite element equations are obtained by approximating the convolution integrals from the viscous part of the constitutive relations using a trapezoidal rule. A two-point recurrence scheme is developed that uses the approximation of relaxation moduli with Prony series. This necessitates the data storage for only the last time step and not for the entire deformation history.which makes it more important to study these materials. The theory of viscoelastic behavior is long been established, the reader can refer to Lakes [2009] for a materials perspective and Reddy [2008], Flügge [1975], Christensen [1982], Findley et al. [1976] and Lockett [1975] for a continuum perspective. Within the continuum purview there are analytical methods like integral transforms, Laplace transforms and correspondence principle, etc., to study the mechanical response of structures made of viscoelastic materials. But as with many analytical methods, they are limited to simple cases of geometry and loading. In such scenarios numerical techniques like the finite element method becomes very useful in testing and predicting the properties of a material without actually fabricating them. Numerical methods can be used to obtain approximate solution with desired accuracy.Many researchers [Taylor et al., 1970;Oden and Armstrong, 1971;Holzapfel and Reiter, 1995;Henriksen, 1984;Hartmann, 2002;Roy and Reddy, 1988] have used the finite element method to study viscoelastic materials. The main difficulty with the viscoelastic finite element models is the approximation of convolution integrals that come from viscoelastic constitutive equations. Most of these finite element models try to circumvent the problem with the time-dependent convolution integrals by transforming them to a set of discrete algebraic equations in space. Taylor et al. [1970] and Oden and Armstrong [1971] used recurrence relations such that only the deformation history from last few iterations is needed to be stored instead of entire deformation from beginn...

Dynamic Finite Element Analysis of Flexible Double Wishbone Suspension Systems with Different Damping Mechanisms

“…where, σ xx (x, t), τ xy (x, t) and σ v xx (x, t), τ v xy (x, t) are the elastic and viscous components of normal and shear stress; respectively. Based on the hereditary integral, Findley et al [39], Vallala et al [40], both elastic and viscous components of normal and shear stress can be expressed according to the Wiechert viscoelastic mechanical model as, Abdelrahman and Mahmoud [33]…”

“…The viscoelastic material parameters for a single repeated model are, the fully relaxed modulus, E 8 = 1.96 × 10 7 Pa, the transient modulus E 1 = 7.84 × 10 7 Pa, and the material relaxation time ρ v = 2.24 s while the instantaneous creep compliance D 0 = 1.2041 × 10 −8 Pa −1 , the transient creep compliance D 1 = 4.0816 × 10 −8 Pa −1 , and the viscoelastic retardation time τ = 11.2 s. Beams are discretized into beam elements. The elastodynamic and viscoelastodynamic transient responses are detected by applying the Newmark's numerical integration technique.The analytical quasi-static viscoelastic solution based on the elastic viscoelastic correspondence principle can be expressed for both TBT and EBBT as, Vallala et al[40]…”