Non-linear Modeling and Analysis of Solids and Structures

Abstract

This book presents a theoretical treatment of nonlinear behaviour of solids and structures in such a way that it is suitable for numerical computation, typically using the Finite Element Method. Starting out from elementary concepts, the author systematically uses the principle of virtual work, initially illustrated by truss structures, to give a self-contained and rigorous account of the basic methods. The author illustrates the combination of translations and rotations by finite deformation beam theories in absolute and co-rotation format, and describes the deformation of a three-dimensional continuum in material form. A concise introduction to finite elasticity is followed by an extension to elasto-plastic materials via internal variables and the maximum dissipation principle. Finally, the author presents numerical techniques for solution of the nonlinear global equations and summarises recent results on momentum and energy conserving integration of time-dependent problems. Exercises, examples and algorithms are included throughout.

Hybrid state-space time integration of rotating beams

“…However, due to the applied small displacement beam formulation the results lack in capturing the apparent axial shortening during the acceleration phase, whereas the small elongation due to the centrifugal force in the final near stationary phase is present. The effect of shortening due to bending may be reproduced by a simple post-processing of the results using the beam column theory from [7] as illustrated by a dashed line in Fig. 2 (b).…”

Direct calculation of critical points in parameter sensitive systems

“…The simple truss-spring system (Figure 4) was investigated by several authors [28,46,47]. The perfect structure consists of two truss bars in the x − y plane, laterally supported by a horizontal spring in the z direction.…”

Variational formulations of the nonlinear equilibrium and stability problems of elastic rods

“…Traditional approach of variational formulation of the nonlinear rod deformation problems is to use the variational equation in the form of principle of virtual displacements (Goloskokov, Zhilin), (Elisseev, 1994), (Zhilin, Sergeev, 1994), (Zhilin, et al 1997), (Zhilin, 2007), (Eliseev, Zinov'eva, 2008), (Jelenic, Crisfield, 1999), (Shabana,Yakoub, 2001), (REDDY, 2004), (Antman, 2005), (Gerstmayr, Shabana, 2006), (Shabana, 2008), (Wriggers, 2008), (Krenk, 2009), (Ibrahimbegovic, 2009). In this work, considering the example of plane problem, it is shown that using of the conjugate pairs of strain and stress vectors (Lalin, 2004), the variational problem can be defined as a problem of search for the stationary point of the Lagrangian functional.…”