The hybrid Laplace transform/finite element method applied to the quasi‐static and dynamic analysis of viscoelastic Timoshenko beams

Abstract

SUMMARYThe quasi-static and dynamic responses of a linear viscoelastic beam are solved numerically by using the hybrid Laplace transform/finite element method. In the analysis, the Timoshenko beam theory, which includes the transverse shear and rotatory inertia effect and conventional beam theory, are used to solve this problem. The temperature field is assumed to be constant and homogeneous and that the relaxation modulus has the form of the Prony series. In the hybrid method, the Laplace transform with respect to time is applied to the coupled equations and the finite element model is developed by applying Hamilton's variational principle without any integral transformation. The numerical results of quasi-static and dynamic responses for the models of Maxwell fluid and three parameter solid types are presented and discussed.

The elusive and fickle viscoelastic Poisson’s ratio and its relation to the elastic-viscoelastic correspondence principle

“…Therefore, the requirements on the F(t) and F c (t) functions of (57) are unrealistic means to simply achieve the desired time-independent PRs. These severe restrictions necessary for the existence of separable variable solutions are discussed in [Hilton 1996;Alfrey 1944;1948;Christensen 1982]. Each and every one of the following conditions must be enforced for separation of variable formulations to exist:…”

“…Viscoelastic Timoshenko beams. Although the definition of the elastic Timoshenko shear coefficient is somewhat arbitrary, in that it is based on equalities of strain energies [Bert 1973] or deformations [Cowper 1966] between exact and approximate solutions to mention a few examples, the concept leads to relatively simple expressions depending only on beam cross sectional geometry and its elastic material properties. However, under either definitions the shear coefficient is dependent on the elastic PR, thus making it impossible to construct an elastic-viscoelastic analogy for this problem [Hilton 2009].…”

“…In a number of instances [Gottenberg and Christensen 1963;Olesiak 1966;Paulino and Jin 2001a;2001b;Jin and Paulino 2002;Jin 2006;Ko et al 2003;Freudenthal and Henry 1960;Bieniek et al 1981;Librescu and Chandiramani 1989b;1989a;O'Brien et al 2001;Zhu 2000;Shrotriya and Sottos 1998;Zhu et al 2003;Andrianov et al 2004;di Bernedetto et al 2007;Noh and Whitcomb 2003;Klasztorny 2004;Bert 1973;Cowper 1966;Hilton 2009;Therriault 2003], time-independent PR assumptions lead to overdetermined ill-posed problems and cause use of the elasticviscoelastic correspondence principle to become unjustified. In other analyses [Jin and Paulino 2002;Jin 2006;Ko et al 2003;Freudenthal and Henry 1960;Bieniek et al 1981;Librescu and Chandiramani 1989b;1989a;O'Brien et al 2001;Zhu 2000;Shrotriya and Sottos 1998;Zhu et al 2003;Andrianov et al 2004;di Bernedetto et al 2007;Noh and Whitcomb 2003;Klasztorny 2004;Bert 1973;Cowper 1966;Hilton 2009;Therriault 2003;…”

“…In other analyses [Jin and Paulino 2002;Jin 2006;Ko et al 2003;Freudenthal and Henry 1960;Bieniek et al 1981;Librescu and Chandiramani 1989b;1989a;O'Brien et al 2001;Zhu 2000;Shrotriya and Sottos 1998;Zhu et al 2003;Andrianov et al 2004;di Bernedetto et al 2007;Noh and Whitcomb 2003;Klasztorny 2004;Bert 1973;Cowper 1966;Hilton 2009;Therriault 2003;Nakao et al 1985;Singh and Abdelnaser 1993;Chen 1995;Hilton and Vail 1993], the elastic-viscoelastic correspondence principle (analogy) has been applied improperly by extending it to viscoelastic time-dependent PRs. In this paper the applicability and predominent inapplicability of this analogy as it relates to Poisson's ratio in elastic and viscoelastic expressions involving bulk, shear and Young's moduli is examined.…”

Viscoelastic Plate Analysis Based on Gâteaux Differential

“…The application of the numerical methods to viscoelastic problems has been presented by a number of authors [3][4]. However, to the best of the present authors' knowledge, there are very few published studies on the viscoelastic analysis of plates [5][6][7][8][9].…”