main-content

## Über dieses Buch

This book lays the foundation of knowledge that will allow a better understanding of nonlinear phenomena that occur in structural dynamics.

This work is intended for graduate engineering students who want to expand their knowledge on the dynamic behavior of structures, specifically in the nonlinear field, by presenting the basis of dynamic balance in non‐linear behavior structures due to the material and kinematics mechanical effects.

Particularly, this publication shows the solution of the equation of dynamic equilibrium for structure with nonlinear time‐independent materials (plasticity, damage and frequencies evolution), as well as those time dependent non‐linear behavior materials (viscoelasticity and viscoplasticity). The convergence conditions for the non‐linear dynamic structure solution are studied and the theoretical concepts and its programming algorithms are presented.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
Structural dynamics studies the structural equilibrium over time among external forces, elastic forces, mass forces and viscous forces for a discrete structural system with points that are internally linked to each other and all linked to a fixed reference system. These internal links between points describing the structural system may be elastic or not. If they are not elastic, the behavior of the system of points is non-conservative and therefore the structural material has a nonlinear dissipative constitutive behavior. Additionally to this nonlinear behavior, there is also a nonlinear dissipative behavior due to the effects of the material viscosity that leads to viscous forces dependent on the system velocity. In simpler cases, the damping non linearity is due to the development of viscous forces proportional to the velocity; however, in more complex cases the viscosity term may be time-dependent. Also, the system’s non linearity can be observed in systems having large displacements and where the system works beyond its original geometric configuration, leading to a nonlinear kinematic behavior. Such non linearity is even more pronounced when large strain occurs along with large displacements, turning the solution of the structure’s dynamic problem more complex.
Sergio Oller

### 2. Thermodynamic Basic of the Equation of Motion

Abstract
The thermodynamic basis defining the linear or nonlinear behavior of a solid during the mechanical process is introduced in this chapter. The synthesized concepts here help to understand the solid nonlinear behavior and to clearly set equilibrium at every time.
Sergio Oller

### 3. Solution of the Equation of Motion

Abstract
This chapter deals with the solution of the equation of motion in its semi-discrete form in the time domain (see equilibrium equation, section 2.5). Below is the assembly of equation 2.72 (or 2.73 if the equilibrium is achieved in the reference configuration) which defines the equilibrium in the solid at time t + Δt,
$$\underset{\varOmega^e}{\mathrm{A}}{}^i\left[{\displaystyle \underset{{\mathrm{V}}^e}{\int }{\sigma}_{ij}{\nabla}_i^S{N}_{jk} dV}\right]_{\varOmega^e}^{t+\varDelta t}=\underset{\varOmega^e}{\mathrm{A}}{}^i\left[{\displaystyle \underset{S^e}{\oint }{t}_i{N}_{ik}\ dS}+{\displaystyle \underset{V^e}{\int}\uprho\ {b}_i{N}_{ik} dV}\right]_{\varOmega^e}^{t+\varDelta t}-\underset{\varOmega^e}{\mathrm{A}}{}^i\left[{\displaystyle \underset{V^e}{\int}\uprho\;{N}_{ki}\;{N}_{ij} dV}\right]_{\varOmega^e}^{t+\varDelta t}\;{\left.{\ddot{U}}_j\right|}_{\varOmega^e}^{t+\varDelta t}$$
Sergio Oller

### 4. Convergence Analysis of the Dynamic Solution

Abstract
In the first part of this chapter the dynamic equation (3.1) is particularized for linear problems to study the convergence of the solution for different numerical methods in the time domain. Strictly speaking, the concept of convergence cannot be guaranteed in the second-order nonlinear differential equations as studied in the solution shown in chapter B3 because the convergence involves stability in the solution and this cannot be guaranteed. Nevertheless, the “linearized stability” concept will be studied. It is the most commonly used concept. It can only guarantee the minimum stability conditions, although not enough.
Sergio Oller

### 5. Time-independent Models (*)

Abstract
Some basic concepts of the theory of elasticity and its mechanical variables are reviewed in this chapter. Particularly, a brief summary of the classic plasticity theory is presented as well as its modification in order to make it more general. Moreover, a brief presentation of the continuous damage theory will be offered. All this will be carried out within the kinematic system with small displacements that hypothetically introduce small deformations. Basic knowledge of the mechanics of continuous media, , is recommended. Answers will be found to analyze the subject in depth. It is important, however, to set the criteria, hypothesis and notations and also to remember the most important concepts of the subject addressed in this work.
Sergio Oller

### 6. Time-dependent Models

Abstract
As shown in Chapters 3 and 4, the nonlinearity in dynamics is caused by changes in the direction of external forces due to large movements and to the nonlinearity of internal forces, caused by non-time-dependent phenomena studied in Chapter 5 and time-dependent phenomena that will be presented in this chapter.
Sergio Oller

### Backmatter

Weitere Informationen