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## Über dieses Buch

This monograph describes the numerical analysis of non-linearities in structural mechanics, i.e. large rotations, large strain (geometric non-linearities), non-linear material behaviour, in particular elasto-plasticity as well as time-dependent behaviour, and contact. Based on that, the book treats stability problems and limit-load analyses, as well as non-linear equations of a large number of variables. Moreover, the author presents a wide range of problem sets and their solutions. The target audience primarily comprises advanced undergraduate and graduate students of mechanical and civil engineering, but the book may also be beneficial for practising engineers in industry.

## Inhaltsverzeichnis

### Chapter 1. Basic Mathematical Methods

This chapter occurs here because it is of relevance for all following sections. It is possible to skip it until the first applications are formulated.

Wilhelm Rust

### Chapter 2. Geometrically Nonlinear Behaviour

The assumptions of a geometrically linear theory are:

Wilhelm Rust

### Chapter 3. Stability Problems

A beam is loaded in its axial direction by a compressive force. The force is increased. Suddenly the beam moves perpendicular to its axis: it buckles (Fig. 3.1).

Wilhelm Rust

### Chapter 4. Load Incrementation in a Non-linear Analysis

A structure can be loaded by a number of loads of different type. We restrict ourselves to force-type loads such as forces, moments and distributed loads like line and surface loads, e.g. pressure. The reference loads contained in the vector

f

0

ext

can be increased proportionally. The proportionality factor is

λ

, in the following referred to as

:

f

e

x

t

=

λ

f

0

e

x

t

$${\mathbf{f}}^{ext}=\lambda {\mathbf{f}}_0^{ext}$$

f

0

ext

can be for example the planned load in use and

λ

a safety factor for the system while the maximum

λ

should be determined. Here we generalise that the system response depending on

λ

is of interest.

Wilhelm Rust

### Chapter 5. Fundamentals of Material Models

Material models describe the relation between strain and stress. In the following, however, simple components are considered, the force–displacement behaviour of which is commonly known. As base for a material law the displacement must simply be replaced by strain and the force by stress.

Wilhelm Rust

### Chapter 6. Theory and Numerics of the Linear Visco-elasticity

The visco-elastic material model can be interpreted as a linear spring in parallel to

n

Maxwell elements (linear spring and damper in a row, see Sect.

5.2.2

and Fig.

5.17

).

Wilhelm Rust

### Chapter 7. Theory and Numerics of Creep

At first

creep

means creep in the classical sense (Fig. 7.1), i.e. the time-dependent increase of strain under constant stress whereas the other limiting case is called (Fig. 7.2) and means the time-dependent decrease of stress under constant strain. Now the expression

creep

is extended to all processes in between, the occurrence of time-dependent strain becoming permanent after load release.

Wilhelm Rust

### Chapter 8. Theory and Numerics of Elasto-plasticity

In case of ductile materials like steel for which this theory is developed it is assumed that the behaviour is linearly elastic until reaching are certain stress, the yield strength

σ

y

. It is described by Hooke’s law, thus by Young’s modulus

E

and Poisson’s ratio

ν

. This holds strictly speaking for materials with a distinct yield strength like in Fig. 8.1. The peak value in front of the ideally elastic region also shown in Fig. 8.1 is usually not modelled.

Wilhelm Rust

### Chapter 9. Contact Analysis: Introduction, Kinematics

We distinguish the following cases:

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### Chapter 10. Fulfilling the Contact Condition

In this chapter we firstly consider the following simple model problem (Fig. 10.1):

Wilhelm Rust

### Chapter 11. Aspects of Modelling Contact

Some special aspects can be shown here on the example of the penalty method but for the other methods similar effects and problems occur.

Wilhelm Rust

### Chapter 12. Contact Detection

Besides achieving convergence effective contact detection is the most critical point in programming a contact algorithm. Much experience is necessary to cover all possible situations. Not everything is published. Therefore, only some basic ideas can be outlined here.

Wilhelm Rust

### Chapter 13. Contact with Shell- and Beam-Elements

In case of beam elements only one dimension, in case of shell elements the two directions of the reference plane, usually the mid-surface, are discretised. They both represent three-dimensional system. For beams in 2d the height resp. the distance of the outer edges from the axis, for shells the thickness must be taken into account. Neither the contact of two beams in 3d nor the contact of shell edges is considered here (Fig. 13.1).

Wilhelm Rust

### Backmatter

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