# Introduction to Non-Linear Finite Element Analysis

This book is intended for a reader with a background in linear finite element analysis who wish to gain some insight in non-linear FE stress analysis. Before attempting this text, prospective readers should have first studied an introductory text on linear finite element analysis such as the NAFEMS Finite Element Primer.
Although it is recognised that many different types of nonlinear problems may be solved using the Finite Element Method,  our attention is here focused on nonlinear finite element stress analysis and in particular static or quasi-static problems (i.e. situation in which inertia forces may be neglected).

This book in not intended to be deeply theoretical text. Consequently, some of the explanations rely heavily on simple presentations of complex ideas. Rigour and comprehensiveness have been sacrified for the sake of clarity and ease of explanation. Boxes are used throughout the text and usually include algorithmic details which the more inquisitive reader may require - this part of the text may be skipped by the more casual reader.

Nonlinear Finite Element Stress analysis has its own particular language or jargon. Such jargon is explained as it is introduced in this text, but a glossary of terms used is also provided at the end of the book as a convenient, easily accessible aide-mémoire. An attempt has been made to standardise on notation- this inevitably differs to some degree from the notation used in the NAFEMS Finite Element Primer as our present task is a little bit more arduous. Various aspects concerning the notation used in this book are discussed in Chapter 1 and a local notation list is placed at the end of each chapter.

The NAFEMS Working groups have over the years commissioned a number of benchmark tests for nonlinear Finite Element stress analysis. This text will make use of some of these tests as they have been shown to be an excellent means of training novices.

## Contents

• Chapter 1 provides a general introduction to the text.

• Geometric nonlinearities are considered in Chapter 2.

• Material nonlinearities which time independent and time dependent are dealt with in Chapter 3 and 4 respectively.

• In Chapter 5, incremental-iterative solutions are described.

• Further complexities (including  transient dynamics, coupled problems, boundary nonlinearities and non linear fracture mechanics) are reviewed in Chapter 6.

• The practical use of FE software in nonlinear analysis is examined in Chapter 7.

• Finally, the glossary of terms provides a brief definition of terms used throughout the text.

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