How To Tackle Non-Linear Finite Element Analysis

Crocombe, A D   

First Published - March 2002
Softback, 90 Pages

There has been a proliferation in accessible finite element codes in recent years. This has led to the situation where most engineering undergraduates have had exposure to this important subset of engineering software. This exposure may have taken the form of hands on experience as part of design or project modules, or even as a more fundamental module in finite element principles. There is a lot of truth behind many old sayings and certainly two spring to mind in this context: A little knowledge is a dangerous thing and Rubbish in equals rubbish out . To counter this, it is important a) to impart both the limitations as well as the benefits of FEA and b) to encourage a thorough review of results before presenting them to others.

Nowhere is this more true than in the field of non-linear analysis. This area, which not so long ago was considered a domain for experts, now has a rapidly expanding user base of engineers and designers who have a commendable, yet sometimes misplaced, desire to model reality more closely. With modern analysis software, it is often possible to obtain solutions to non-linear problems. However these can easily be inappropriate and skill is required to determine their validity. Care should be taken to specify appropriate model and solution parameters. Understanding the problem, the role played by these parameters and a planned and logical approach will do much to ensure a successful solution is obtained where one exists.

As the title (and size) suggests, this “How-To” book is a practical guide, which is seen as the starting point for undertaking non-linear analyses. It is not a reference book, and is primarily targeted at a person who has experience with linear finite element analyses (FEA) and now wants to undertake non-linear analyses. A typical reader could be a recent engineering undergraduate whose degree is likely to include linear finite element analysis, or a more experienced stress analyst who has currently only undertaken linear analyses. Such people will be familiar with the concepts of finite element analyses and need to be made aware of how this can be extended to include non-linearity.

Many texts have been written on non-linear FEA. In the main these provide a detailed treatise of non-linear FEA, giving extensive mathematical derivations. The other commonly available sources of information about non-linear FEA are the user manuals of various FE codes. These generally tell you how to undertake a specific non-linear solution, but generally assume a working knowledge of non-linear FEA. What is missing is a practical guide to non-linear FEA that would enable a user to become familiar enough with the concepts to enable them to carry out meaningful analyses. That defines the role of this booklet, which seeks to present most of the salient aspects of non-linear FEA whilst minimising the mathematical derivations.

Contents

Preface

Contents

1. Introduction

• 1.1 Purpose of this booklet
• 1.2 Other sources of information
• 1.3 Structure of the booklet
• 1.4 What is non-linearity?
  • 1.4.1 Some sources of non-linearity
  • 1.4.2 Outline of a non-linear solution 

2. Common Causes of Non-Linearity 

• 2.1 Measures of stress and strain
  • 2.1.1 Stress measures 
  • 2.1.2 Strain measures 
• 2.2 Material non-linearity 
• • 2.2.1 Time independent elasto-plasticity models
  • 2.2.2 Time dependent non-linear material models 
  • 2.2.3 Non-linear elastic behaviour
• 2.3 Geometric non-linearity
• 2.4 Boundary condition non-linearity
• 2.5 Explicit dynamics

3. Flowcharts for Planning Non-Linear Analyses

• 3.1 Geometric non-linearity
  • 3.1.1 Compression of a car door seal
  • 3.1.2 Thin elastic cantilever beam
  • 3.1.3 Clamped square plate under uniform pressure 
  • 3.1.4 Thin cylinder with axial compressive loading 
• 3.2 Material non-linearity
  • 3.2.1 Compression of a car door seal
  • 3.2.2 Thick walled steel cylinder with internal pressure. 
  • 3.2.3 Beam bending at high temperature
  • 3.2.4 Pressing of sheet metal parts 
• 3.3 Boundary non-linearity
  • 3.3.1 Compression of a car door seal 
  • 3.3.2 Two contacting cylinders 
  • 3.3.3 Crushing of a box section column 

4. A Checklist for Achieving and Validating Non-Linear Results 

• 4.1 Benchmarking simple models 
• 4.2 The use of solution tools 
  • 4.2.1 Evaluating the stiffness matrix
  • 4.2.2 Updating the nodal unknowns 
  • 4.2.3 Assessing convergence 
  • 4.2.4 The size of the load incrementation
  • 4.2.5 Softening behaviour 
• 4.3 Observation of deformation and stress fields
• 4.4 Assessment of element size for the non-linear zone 
• 4.5 Observation of constitutive data at a sampling point throughout loading
• 4.6 Use of load displacement plots to assess global non-linearity 54
• 4.7 Output
• 4.8 Some differences between linear and non-linear finite element analyses

5. Glossary

• Arc-length method
• Buckling analysis
• Consistent tangent modulus matrix
• Continuum tangent modulus matrix
• Creep strain tolerance
• Deformation gradient [F] 
• Deviatoric stress
• Drucker-Prager
• Engineering (or nominal) strain
• Explicit creep integration 
• Explicit dynamics
• Flow rule 
• Gauss points 
• Green’s strain 
• Hardening curve 
• Hydrostatic stress 
• Hyperbolic-sine law creep 
• Implicit creep integration
• Isotropic hardening 
• Kinematic hardening
• Large displacement small strain
• Large strain 
• Line search technique
• Logarithmic strain
• Mooney-Rivlin
• NAFEMS benchmark tests 
• Newton-Raphson
• ORNL creep model
• Plastic collapse 
• Post buckling analysis
• Power law creep 
• Predictor technique 
• Principal stress space
• Proportional loading 
• Secant modulus 
• Secant stiffness
• Stability 
• Strain energy function
• Stress stiffening
• Tangent stiffness matrix 
• Tresca plasticity
• Visco-elasticity
• Visco-plasticity
• Von Mises equivalent stress
• Von Mises plasticity
• Yield stress
• Yield surface

6. Conclusion

7. References

8. Index