intro to adv finite element analysis...!!!!

Lecture Notes

Advanced Finite Element Methods

Dr.-Ing. habil. D. KuhlUniv. Prof. Dr. techn. G. Meschke

May 2005

Ruhr University BochumInstitute for Structural Mechanics

Lecture Notes

Advanced Finite Element Methods

Dr.-Ing. habil. D. KuhlUniv. Prof. Dr. techn. G. Meschke

May 2005

Ruhr University BochumInstitute for Structural MechanicsUniversitatsstraße 150 IA6D-44780 BochumTelefon: +49 (0) 234 / 32 29055Telefax: +49 (0) 234 / 32 14149E-Mail: [email protected]: http://www.sd.ruhr-uni-bochum.de

Contents

1 Fundamentals of Linear Structural Mechanics 1

1.1 Continuum Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Definition of a Non-Linear Strain Measure . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Definition of a Linear Strain Measure . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Continuum Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Balance of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Initial Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Classification of Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . 12

1.3.2 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.3 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.4 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Hyperelastic Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.1 Fundamental Assumptions and Classification . . . . . . . . . . . . . . . . . . . . . 16

1.4.2 Elastic Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.3 Isotropic, Elastic Material Relation of Continuum . . . . . . . . . . . . . . . . . . 17

1.4.4 Plane Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.5 Plane Strain State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.6 The Classical Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Initial Boundary Value Problem of Elastomechanics . . . . . . . . . . . . . . . . . . . . . 24

1.5.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5.2 Geometrically and Materially Linear Elastodynamics . . . . . . . . . . . . . . . . . 25

1.5.3 Geometrically and Materially Linear Elastostatics . . . . . . . . . . . . . . . . . . 26

1.6 Weak Form of The Initial Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . 26

1.6.1 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6.2 Properties of The Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . 29

2 Spatial Isoparametric Truss Elements 31

2.1 Fundamental Equations of One-dimensional Continua . . . . . . . . . . . . . . . . . . . . 32

2.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

i

ii Kuhl & Meschke, Finite Element Methods in Linear Structural Mechanics

2.1.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.4 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


2.1.6 Euler Differential Equation and Neumann Boundary Conditions . . . . . . . . . . 38

2.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.1 Partitioning of The Structure into Elements . . . . . . . . . . . . . . . . . . . . . . 39

2.2.2 Approximation of Variables of One-dimensional Continua . . . . . . . . . . . . . . 40

2.2.3 Truss Element with Linear Shape Functions . . . . . . . . . . . . . . . . . . . . . . 45

2.2.4 Truss Element with Quadratic Shape Functions . . . . . . . . . . . . . . . . . . . . 52

2.2.5 Truss Element with Cubic Shape Functions . . . . . . . . . . . . . . . . . . . . . . 55

2.2.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.3 Assembly of the Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.3.1 Transformation of the Element Matrices and Vectors . . . . . . . . . . . . . . . . . 62

2.3.2 Assembly of the Elements to the System . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4 Solution of the System Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.4.1 Linear Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.4.2 Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.4.3 Solution of the Linear System of Equations . . . . . . . . . . . . . . . . . . . . . . 78

2.5 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.5.1 Separation and Transformation of the Element Degrees of Freedom . . . . . . . . . 79

2.5.2 Computation of Strains, Stresses and Section Loads . . . . . . . . . . . . . . . . . 79

2.5.3 Aspects of Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Plane Finite Elements 81

3.1 Basic Equations of Planar Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.1.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.4 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


3.1.6 Euler Differential Equation and Neumann Boundary Conditions . . . . . . . . . . 86

3.2 Finite Elemente Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.2.1 Partitioning into Elements and Discretization . . . . . . . . . . . . . . . . . . . . . 88

3.2.2 Classification of Plane Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2.3 Shape Functions of Plane Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.3 Bilinear Lagrange element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3.1 Ansatz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3.3 Jacobi transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.3.4 Approximation of element quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Institute for Structural Mechanics, Ruhr University Bochum, May 2005 iii

3.3.5 Strain vector approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.3.6 Appproximation of internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . 101

3.3.7 Approximation of dynamic virtual work . . . . . . . . . . . . . . . . . . . . . . . . 102

3.3.8 Approximation of virtual work of external loads . . . . . . . . . . . . . . . . . . . . 103

3.3.9 Rectangular Bilinear Lagrange Element . . . . . . . . . . . . . . . . . . . . . . . . 106

3.4 Rectangular biquadratic Lagrange element . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


3.4.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.4.3 Jacoby transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124


3.4.5 Approximation of the strain vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.4.6 Element matrices and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.5 Biquadratic serendipity element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125


3.5.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127


3.6 Triangular plane finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3.6.1 Natural coordinates of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129


3.6.3 Isoparametric approximation of continuous quantities . . . . . . . . . . . . . . . . 133

3.6.4 Element matrices and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.6.5 Constant Strain Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.7 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.7.1 Quadrangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3.7.2 Triangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4 Finite volume elements 143

4.1 Fundamental equations of three-dimensional continua . . . . . . . . . . . . . . . . . . . . 144

4.2 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.2.1 Natural coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.2.2 Ansatz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.2.4 Jacobi transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.2.5 Differential Operator B(ξ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2.6 Element Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.2.7 Element Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5 Basics of non-linear structural mechanics 151

5.1 Non-linearities of structural mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.2 Material non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2.1 Mathematical formulation of material non-linearity . . . . . . . . . . . . . . . . . . 153

iv Kuhl & Meschke, Finite Element Methods in Linear Structural Mechanics

5.3 Geometrical non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.3.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.3.3 Constitutive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.3.4 Principle of virtual displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.3.5 Internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.3.6 Elastic internal potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.3.7 Remarks regarding combined material and geometric non-linearity . . . . . . . . . 163

5.4 Consistent linearization of internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . 163

5.4.1 Linearization background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.4.2 Gateaux derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.4.3 Gateaux derivative of internal virtual work . . . . . . . . . . . . . . . . . . . . . . 164

5.4.4 Linearization of Green Lagrange strains . . . . . . . . . . . . . . . . . . . . . . . . 166

5.4.5 Linearization of variation of Green Lagrange strains . . . . . . . . . . . . . . . . . 167

6 Finite element discretization of geometrically non-linear continua 171

6.1 Finite volume elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.1.1 Discretization of internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.1.2 Non-linear semi-discrete initial value problem . . . . . . . . . . . . . . . . . . . . . 177

6.1.3 Non-linear discrete static equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.1.4 Discretization of linearized internal virtual work . . . . . . . . . . . . . . . . . . . 178

6.1.5 Linearization of internal forces vector . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.2 Finite truss elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.2.1 Non-linear continuum-mechanical formulation . . . . . . . . . . . . . . . . . . . . . 182

6.2.2 Truss elements of arbitrary polynomial degree . . . . . . . . . . . . . . . . . . . . . 183

6.2.3 Linear truss element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7 Solution of non-linear static structural equations 189

7.1 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.2 Iteration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.2.1 Single step method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.2.2 Pure Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.2.3 Modified Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.3 Control of iteration procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.3.1 Load-incrementing and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.3.2 Arc-length controlling method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

References 202

Preface

These lecture notes, which in actual fact are an English translation of the German lecturenotes ’Finite Elemente Methoden I’ of the diploma study course, were created in the contextof the lecture ’Finite Element Methods I’ which was first held in this form during the winterterm 1998/1999. ’Finite Element Methods in Linear Structural Mechanics’ thus represents theteachings of finite element methods in the area of linear structural mechanics with the focuson showing of possibilities and limits of the numerical method as well as the development ofisoparametric finite elements. These notes are to support the students in following up the lectureand to prepare them for the exam. They cannot possibly substitute the lecture or the exerciseentities. In addition to the lecture and the notes, mathematical programmes for deepeningthe lecture contents are available at the homepage of the Institute for Structural Mechanicshttp://www.sd.ruhr-uni-bochum.de/.

Here, the authors would like to thank Mr. Jorn Mosler and Mr. Stefan Jox for the excellentconduction of the theoretical and practical exercise entities accompanying the lecture ’FiniteElement Methods I’. Moreover, the authors give their thanks to Ms. Barbara Kalkhoff, graphicaldesigner, for the high quality drawings as well as to Ms. Monika Rotthaus, Ms. Wiebke Breil,Ms. Sandra Krimpmann, Ms. Julia Mergenheim, Mr. Christian Becker, Mr. Alexander Beer,Mr. Sonke Carstens and Mr. Janosch Stascheit for their indispensable efforts in creating theselecture notes.

Last but not least the authors would like to thank Mr. Ivaylo Vladimirov, Mr. Hrvoje Vucemilovicand Ms. Amelie Gray who helped to translate the notes into the English language. At the sametime we would like to excuse the fact that the description of the drawings are in German.Nevertheless, we believe that the meaning becomes clear. The authors are continually workingon improving the lecture notes. Therefore, please feel free to communicate your comments, ideasand corrections.

For all students who intend to continue with the lecture ’Finite Element Methods II’ withthe emphasis on non-linear structural mechanics, the lecture notes are complemented by thecorresponding chapters 5 to 7 as well as by the indication of further literature. The chaptersconcerning the non-linear finite element methods are also available in the form of lecture notes(’Finite Elemente Methoden II’, 3. edition, October 2002, in German language) at the Institutefor Structural Mechanics, IA 6/127.

Bochum, May 2005 Gunther Meschke and Detlef Kuhl

v

Chapter 5

Basics of non-linear structuralmechanics

In the scope of Finite Element discretization in linear structural mechanics presented in theprevious chapters, two major very important simplifying assumptions were made which did notalways result in adequate modelling of real structural behaviour. In these assumptions, it waspostulated that the material behaviour is linear elastic (materially linear) and that deforma-tions are small (geometrically linear). The former assumption was prescribing the validity ofthe Hooke law and a priori excluding the modelling of irreversible material behaviour suchas plastification or damaging. From the latter assumption, it follows that equilibrium can beestablished on an undeformed structure and that non-linear terms of Green Lagrange straintensor E can be neglected. The so far discussed linear structural mechanics theory makes upthe classic fundamentals of static and dynamic analysis in civil engineering. Inspite of funda-mental restrictive assumptions just mentioned, this theory will find application in calculationsof deformations and stresses in engineering structures in the future as well, as long as the nec-essary conditions are met (small deformations and stresses which justify the assumptions oflinear material behaviour), since complexity of FE formulation as well as numerical effort forsolving non-linear problems are notably rising compared to linear problems. What makes a dif-ference in engineering praxis is that superposition principle is not valid in non-linear cases andconsequently every analysed load case requires a complete computation.

Apart from the classic linear analysis established in engineering, the demands on models ofstructural engineering will notably increase due to growing replacement of development andverification experiments by cost-reducing, transparent and faster computer simulations. For ex-ample, structures which are slender and light for technical or aesthetic reasons can only beadequately simulated and examined regarding stability with the help of a geometrically non-linear calculus (see e.g. Kramer et al. [110]). On the other hand, concrete or reinforcedconcrete is a material characterized by distinct non-linear behaviour due to inevitable cracks(see e.g. Kratzig, Mancevski &Polling [113]), which has to be taken into account in thestructural analysis calculus. Cupping and shaping processes in the field of industrial produc-tion (see e.g. Glaser [104]) or car crashes (see e.g. Moller [67]) are application examplesfor simulations dependent on modelling of metal plastification with large deformations. On theone hand it is the lasting deformations and on the other hand the dissipated plastification en-ergy that is of crucial significance for the product quality, that is, the safety of the passenger.That shaping processes are impossible without large deformations is self-evident; also the after-math of a crash seldomly justifies the assumption of small deformations. Other structures areso intensively loaded that they are impossible with structural exclusion of non-linear material

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behaviour. Plastic deformation during exploitation has to be accepted and simulated accord-ingly in the design and development process (see e.g. simulation of incineration chambers inKuhl [114], Kuhl, Woschnak &Haidn [117]). The previous list of necessities of non-linearsimulation techniques can be extended almost at will, but we nevertheless want to concentrate onelaboration of geometrically or/and materially non-linear problems, the modelling thereof, andon numeric-algorithmic conversion. If the body deformations are large, the carrying pattern canchange significantly with the deformation what can reflect on the change of structural stiffness,or in view of dynamics, on the change of eigenfrequencies. In the extreme case, this can lead toloss of stability where significant deformations occur without the load increase and the structureeventually collapses. Especially the slender, thin-walled structures or structures optimized withrespect to linear properties tend to behave in a pragmatic, non-linear way with endangered sta-bility. Besides the geometrical non-linearity, modelling and discretization of non-linear problemsought to be investigated, too. Metal materials for example display a linear behaviour until theyreach a certain stress level, the so called flow limit, above which their plastic deformations occurin connection with a notably reduced material stiffness. The consequence is that the afflictedstructures undergo a load redistribution which still leaves them serviceable even though plas-tic deformations were already localized. A similar phenomenon can be observed at concrete orother quasi-brittle materials, particularly ceramics. In these materials microcracks develop dueto loading and degrade the material strength and stiffness.

5.1 Non-linearities of structural mechanics

Structural mechanical simulations can be classified according to the modelled non-linearities andtheir combinations in the order of increased complexity, and according to the numerical analysiseffort, as follows (see figure 1.13):

• materially non-linear and geometrically linear

• materially linear and geometrically non-linear

• materially and geometrically non-linear with assumed moderate strains

• materially and geometrically non-linear with finite strains

Compared to figure 1.13, the differentiation of combined material and geometrical non-linearitywith finite strains is of crucial importance for the formulation and numerical conversion of struc-tural mechanical problematics, but should however not be studied in more detail within the scopeof these lectures. The lectures will limit themselves for most part to pure geometric non-linearlocal impulse equilibrium and its formulation, discretization, linearization, numerical solutionwith diverse algorithms, as well as to stability observations. The material non-linearity will bepresented only schematically and the corresponding FE discretization will be elaborated briefly.For material formulations on the material point level, which is equivalent to a Gauß’schenintegration point in the numerical realization of FE methods, and their algorithmic conver-sion, refer to lectures of Computational Plasticity and to technical literature (e.g. Groß [54],Hill [108], Krajcinovic [111], Krajcinovic &Lemaitre [112], Lemaitre [118], Lemaitre

&Chaboche [61], Lubliner [119] and Simo &Hughes [133]).

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5.2 Material non-linearity

5.2.1 Mathematical formulation of material non-linearity

The rate-independent material non-linearity is, contrary to materially linear formulations (seechapter 1.4 and equation (1.50)), characterized by the fact that the stress state

σ = C : ε (5.1)

cannot be obtained by linear mapping of the strain state with the help of material tensor C.The stress tensor or vector σ is rather an arbitrary function of the strain tensor or vector ε,and other values α, described as internal variables or as time history variables that characterizenon-elastic deformations or damage

σ = σ(ε,α) (5.2)

where the partial derivative of the stress tensor/vector with respect to the strain tensor definesthe tangential material tensor or the tangential material matrix.

Ctan(ε,α) =∂σ(ε,α)

∂ε(5.3)

In order to make the solution of mechanical boundary problems or initial value problems possible,the stress function ((5.2) has to be supplemented with the so called evolution equations of internalvariables in the form

α = α(ε,α) (5.4)

In the special case of non-linear elastic material laws, the stress state is only a function of thestrain state

σ = σ(ε) Ctan(ε) =∂σ(ε)

∂ε(5.5)

and the evolution equations are dropped. For the simulation of the non-linear material lawmostly

• non-linear elastic,

• elasto-plastic

• and elasto-damaged

material models and presented ground type combinations are in use. As shown in figure 5.1, thecurves of three materially non-linear phenomena can basically be one and the same for loadingsequence, whereas the differences in material formulations are decisive in the unloading sequence.In the non-linear elastic case, the stress-strain diagram for unloading runs along the loading pathand after full unloading a strain-free state is reached and the new cycle is identical to the first one.In case of the elasto-plastic material model, the unloading sequence runs parallel to the initialrate E and after full unloading the structure is not strain-free because plastic strains remain.The next cycle is therefore different from the first one. As opposed to that, upon unloading in thecase of elasto-damage model, no permanent strains remain. Unlike the linear-elastic model, the


(non-linear elastic elasto-plastic elasto-damaged

no change plastic strain elasticity modulus

-

6

ε

σ

E

::

99

-

6

ε

σ

E

::

- -εp εp-

6

ε

σ

E (1−d)E

::

Figure 5.1: Cyclic loading and unloading of elastic, elasto-plastic, and elasto-damaged bilinearmaterial models

unloading sequence does not follow the loading path but it runs linearly to the diagram origin.A new load introduction is influenced by degradation of stiffness with the damage parameter d,due to which the repeated load cycles are not identical in their effects on the material and thestructure.

5.3 Geometrical non-linearity

Alterations regarding geometrically linear observation:

• Consideration of non-linear terms of the strain tensor E (kinematics)

• Establishing the forces equilibrium or application of the impulse theorem in the deformedconfiguration (kinetics)

We use:

• (total) Lagrange point of view which is also described as the material point of view

• Stress and strain quantities in the undeformed configuration (second Piola Kirchhoff

stress tensor and Green Lagrange strain tensor)

Literature:

Altenbach&Altenbach [38], Antman [84], Basar [88], Betten [43], de Boer [44], Bonet

&Wood [46], Malvern [63], Marsden &Hughes [65], Smith [73], Stein &Barthold [74]and Truesdell &Noll [78]

5.3.1 Kinematics

The fundamental of the geometrically non-linear formulation of structural mechanics is based onthe material deformation gradient F , which was already used within the scope of deriving the


e 1

e 2

e 3

Y

XP

Qd X q

d xpyx

R e f e r e n z k o n f i g u r a t i o nu n d e f o r m i e r t e L a g e

d e f o r m i e r t e L a g eM o m e n t a n k o n f i g u r a t i o n

A b b i l d u n g j ( X , t )

Figure 5.2: Current and reference configuration of a deformable material body

linear strain tensor ε with the help of a non-linear deformation analysis of a material body, andwithin the scope of subsequent linearization in chapter 1.1.2, equation (1.5), but not elaboratedbecause in linear observations it possesses no further significance. Non-linear observations area different story where the material deformation gradient defines the transformation fromreference to current configuration or from undeformed to deformed state and vice versa. Thesetransformations are referred to in technical literature as push forward and pull back. Thematerial deformation gradient is defined by a transformation of a line element dX of thereference configuration to the current configuration dx (see figure 5.2).

dx = F · dX F =∂x

∂X= ∇x (5.6)

The Green Lagrange strain tensor E was also already derived and is given in equation(1.12) as function of the displacement gradient ∇u and its transpose ∇Tu. If we describe themotion of the material point from the reference to the current configuration with help of thedisplacement vector x = X + u, the Green Lagrange strain tensor

E =1

2

[F T · F − 1

]= ∇symu +

1

2∇Tu · ∇u =

1

2

[∇Tu + ∇u + ∇Tu · ∇u

](5.7)

can be represented as a function of the material deformation gradient.

F =∂

∂X(X + u) = 1 + ∇u (5.8)


According to this relation, the components of the term

∇symu =

u1,112(u1,2 + u2,1)

12(u1,3 + u3,1)

12(u1,2 + u2,1) u2,2

12(u2,3 + u3,2)

12(u1,3 + u3,1)

12(u2,3 + u3,2) u3,3

(5.9)

of the linear part of the strain tensor E, given explicitly in equations (1.13) and (1.14), aresupplemented by the term 1/2∇Tu · ∇u, in the scope of geometrically non-linear theory, whichcan again be calculated with the displacement vector gradient according to eq. (1.7)

∇u =

u1,1 u1,2 u1,3

u2,1 u2,2 u2,3

u3,1 u3,2 u3,3

(5.10)

by matrix multiplication.

1

2∇Tu · ∇u =

1

2

uk,1 uk,1 uk,1 uk,2 uk,1 uk,3



(5.11)

It should be noted that the summation is performed over k = 1, 2, 3, respectively. The compo-nent presentation of the Green Lagrange strain tensor finally yields the following:

Eij =1

2(ui,j + uj,i + uk,i uk,j) E = Eij Ei ⊗ Ej (5.12)

In order to formulate the non-linear Finite Element methods, it remains to convert the calcula-tion rule of the strain tensor, given in eq. (5.7) or (5.12), into the calculation rule of the strainvector in a suitable way . The linear part of the strain tensor can be expressed as a vector byapplication of the differential operator Dε to the displacement vector u, as described in eq.(1.16). However, for the non-linear part of the strain tensor, no suitable operator presentationcan be found.

E(u) =

E11

E22

E33

2E12

2E23

2E13

=

u1,1

u2,2

u3,3

u1,2 + u2,1

u2,3 + u3,2

u1,3 + u3,1

+

1/2 (u1,1 u1,1 + u2,1 u2,1 + u3,1 u3,1)

1/2 (u1,2 u1,2 + u2,2 u2,2 + u3,2 u3,2)

1/2 (u1,3 u1,3 + u2,3 u2,3 + u3,3 u3,3)

u1,1 u1,2 + u2,1 u2,2 + u3,1 u3,2

u1,2 u1,3 + u2,2 u2,3 + u3,2 u3,3

u1,1 u1,3 + u2,1 u2,3 + u3,1 u3,3

(5.13)

According to eq (5.13), the Green Lagrange strain tensor of geometrically non-lineardeformations is obtained by addition of the well-known linear part Dεu and the non-linear partEnl(u),

E(u) = Dε u + Enl(u) (5.14)


where Dεu and Enl(u) are defined as follows (summation over k = 1, 2, 3).

Dε u =

u1,1

u2,2

u3,3

u1,2 + u2,1

u2,3 + u3,2

u1,3 + u3,1

Enl(u) =

1/2 uk,1 uk,1

1/2 uk,2 uk,2

1/2 uk,3 uk,3

uk,1 uk,2

uk,2 uk,3

uk,1 uk,3

(5.15)

Formulation of Dirichlet boundary conditions (eq. (1.33))

u(X) = u?(X) ∀ X ∈ Γu (5.16)

and of initial conditions remains unaltered compared to linear structural mechanics (eq. (1.45)).

u(X , t = 0) = u?(X)

oder

u(X , t = 0) = u?(X)

∀ X ∈ Ω (5.17)

5.3.2 Kinetics

Unlike the linear structural analysis, its non-linear counterpart requires that the dynamic orthe static forces equilibrium be observed in the deformed configuration. This firstly calls forthe evaluation of mass distribution which gives the relation between density in the currentconfiguration ρc and the one in the reference configuration ρ, with the help of determinant |F |of the material deformation gradient (see e.g. Marsden &Hughes [65]).

ρ = |F | ρc (5.18)

The forces equilibrium of a geometrically linear approximation, presented in chapter 1.2, eq.(1.25), was obtained by pure kinetic analysis of a differential volume element. Analoguous anal-ysis of a volume element in a deformed configuration gives us the Cauchy motion equation inthe so called spatial or Euler formulation.

ρc u = divσ + ρc b ∀ x ∈ Ω (5.19)

divσ symbolises the tensor divergence of the real stresses or the Cauchy stress tensor, relatedto the current configuration. As the first description of this stress tensor might lead us to antic-ipate, this stress quantity is defined by a differential load in the current configuration, effectinga deformed surface element da, arbitrarily oriented with a normal vector n inside the body,leading to the consequence that the actual stresses occurring in the material can be described.

It is of advantage to numerical conversion in structural mechanics to utilize the motion equationin the materialor Lagrange formulation. To perform this it is necessary to relate the Cauchy

motion equation to the undeformed configuration. Multiplication of motion equation (5.19)by determinant of the material deformation gradient already transforms the density in thereference configuration, according to (5.18).


|F | ρc u = |F | divσ + |F | ρc b ρ u = |F | divσ + ρ b (5.20)

It remains to transform the middle term of the equation. Since this transformation requires theprofound familiarity with non-linear continuum mechanics, we shall leave out the derivationprocedure and resort to the related literature for the result (e.g. Malvern [63], Marsden

& Hughes [65], Smith [73]). To clarify the outcome,we shall give the resulting identity insymbolic presentation and it component notation.

|F | divσ = DIV(F · S) |F |∂σij

∂xj

=∂(Fik Skj)

∂Xj

(5.21)

DIV symbolises the divergence operator with respect to the reference configuration. Tensor 1

S, which is here used for the first time, is the second Piola Kirchhoff stress tensor defined withrespect to the reference configuration. It should be noted that the Piola Kirchhoff stresstensor S, unlike the Cauchy stress tensor σ, does not refer to actual stresses but to ’pseudostresses’. They are defined with respect to the reference configuration by a differential loadeffecting a surface element of the reference configuration dA, which is oriented with a normalvector N . Introduction of equation (5.21) into equation (5.20) eventually yields the material orthe Lagrange formulation of impulse rule, that is, the Cauchy motion equation.

ρ u = DIV(F · S) + ρ b ∀ X ∈ Ω (5.22)

In order to formulate a well-defined problem, it is necessary to supplement the impulse rulewith the static or Neumann boundary condition. The latter is given in spatial formulationanalogous to equation (1.40). It is possible to apply the procedure for derivation of equilibriumat the boundary, demonstrated in the geometrically linear case, to the deformed configuration

σ · n = t? ∀ x ∈ Γσ (5.23)

and thereupon to transform it. Material formulation of the Neumann boundary conditions isin the geometrically non-linear case defined by

F · S · N = T ? ∀ X ∈ Γσ (5.24)

(see e. g. Marsden & Hughes [65]), where N presents the normal vector of referenceconfiguration. Stress vector T ? (first Piola Kirchhoff stress tensor) is defined by a differentialload vector of current configuration acting on a surface element of the reference configurationdA parallel to t?.

T ? dA = t? da (5.25)

5.3.3 Constitutive Law

For moderate strains, to which we shall limit ourselves here, we can introduce the Saint

Venant Kirchhoff material model to project the Green Lagrange strains on to second

1The Product P = F · S presents the first Piola Kirchhoff stress tensor, which is not used or discussed in the

lectures


Piola Kirchhoff stresses in analogy with equation (1.56) of linear structural mechanics.

S = C : E C = 2µ I + λ 1 ⊗ 1 (5.26)

5.3.4 Principle of virtual displacements

5.3.4.1 Weak formulation of initial value problem

Weak formulation of the initial value problem of geometrically non-linear structural mechanicscan be obtained in a way analogous with linear formulation in chapter 1.6.1, by choice of dis-placement vector transformation δu as special test function, and by applying the calculationrule (1.87) to δu · DIV(F · S),= DIV(δu · (F · S)) −∇δu : (F · S) , where the volume integralof the first right-hand side term DIV(δu · (F · S)) can be transformed to upper surface integralvia δu · F · S · N, by applying the Gauß integral law (see chapter 1.6.1). With the help ofmentioned transformations we get the virtual displacements principle.

∫

Ω

δu · u ρ dV +

∫

Ω

∇δu : (F · S) dV =

∫

Ω

δu · b ρ dV +

∫

Γσ

δu · T ? dΓσ (5.27)

dV and dΓ describe the volume element, that is, the line element in the reference configuration.First term in equation (5.30) is the virtual work of inertial forces δWdyn, the second term isthe virtual work δWint. The sum of the third and the fourth term makes up the virtual workof external loads δWext. Internal virtual work in equation (5.27) should be transformed in sucha way that one obtains the form of virtual displacements equivalent to equation (1.94). To getthis form, we first vary the term δE : S according to definition of E in equation (5.7), where inorder to achieve further transformation we use the symmetry of the stress tensor.

δE : S =1

2δ(F T · F − 1

): S =

1

2

(δF T · F + F T · δF

): S = F T δF : S = δF : F · S (5.28)

With the definition of the material deformation gradient in equation (5.6) and the unchangingcoordinates of the reference configuration (δX = 0),

δF =∂

∂X(δX + δu) =

∂δu

∂X= ∇δu (5.29)

we can write the virtual displacement principle with equations (5.27), (5.28) and (5.29) inthe form preferred for further derivations (in analogy with equation (1.94) of geometricallynon-linear structural mechanics).

∫

Ω

δu · u ρ dV +

∫

Ω

δE : S dV =

∫

Ω

δu · b ρ dV +

∫

Γσ

δu · T ? dΓσ (5.30)

It may be noticed once again that the Lagrange formulation of the impulse rule is used, andas a consequence, the integration over the volume, that is, over the boundary of a material bodyhas to be performed in the reference configuration. From equation (5.30),it can be concluded


that virtual work of inertial forces δWdyn, and of external loads δWext, did not undergoany formal change in comparison with linear observations in chapter 1.6.1. This practicallymeans that for isoparametric one-, two- and three-dimensional finite elements elaborated inchapters 2 to 4, calculation of the mass matrix me, and of the kinematically equivalent loadsor the consistent element load vectors re

p and ren, can be inferred from the linear formulation.

This claim is valid for structural elements only in special cases due to rotational degrees offreedom (see e.g. Argyris [85] or Betsch, Menzel &Stein [91]), used to describe kinematicsand deformation (displacement-based description of rotation parameters, isoparametric shear-susceptible elements - Timoshenko, Kirchhoff-Love and Nagdi), and needs to be discussednext. Within the scope of a generalised approach to geometrically non-linear formulation offinite element methods we shall limit ourselves to finite elements which allow the assumptionsmade. The necessary changes are occasionally considered, discussed, and in some cases replacedby approximations.

As opposed to the terms of virtual work of inertial forces and of external loads, the internalvirtual work δWint for geometrically non-linear observations is crucially different. Instead ofvariation of Green strain tensor ε that is strain vector ε, comes the Green Lagrange straintensor E that is strain vector E, given in equation (5.7) or equation (5.14). The Cauchy stresstensor/vector σ has to be replaced with the second Piola Kirchhoff stress tensor/vector S

which is linked with the Green Lagrange strain tensor/vector through equation (5.26) in thematerially linear case.

δWint =

∫

Ω

δE : S dV =

∫

Ω

δE : C : E dV =

∫

Ω

δE · C E dV (5.31)

5.3.4.2 Variation of Green Lagrange strains

Equation (5.31) contains the variation of Green Lagrange strain tensor δE. Before we de-rive this variation, we first review the geometrically non-linear theory. In case of geometricnon-linearity, the Green strain vector ε could be computed with the help of the deformation-independent differential operator Dε and the displacement vector u , ε = Dεu (see equation(1.16)). Accordingly, the variation of Green strain vector was obtained by variation of the linearmapping just mentioned δε = Dεδu. The corresponding relation should be derived as prepara-tion for the variation of Green Lagrange strain vector, needed to discretize and formulatefinite elements in chapter 6.

δE =

[

δE11 δE22 δE33 2δE12 2δE23 2δE13

]T

(5.32)

Variation of strain components Eij with i, j = 1, 2, 3 can be calculated as follows with equation(5.12)

Eij =1

2

(∂ui

∂Xj

+∂uj

∂Xi

+∂uk

∂Xi

∂uk

∂Xj

)

(5.33)

and with the exchangeability of sequence of variation and partial derivatives when applying theproduct rule.

δEij =1

2

(∂δui

∂Xj

+∂δuj

∂Xi

+∂uk

∂Xj

∂δuk

∂Xi

+∂uk

∂Xi

∂δuk

∂Xj

)

(5.34)


Here it is summated over k = 1, 2, 3. Setting i, j = 1, 2, 3 yields the components of the strainvector variation.

δE11 =∂δu1

∂X1

+∂uk

∂X1

∂δuk

∂X1

δE22 =∂δu2

∂X2

+∂uk

∂X2

∂δuk

∂X2

δE33 =∂δu3

∂X3

+∂uk

∂X3

∂δuk

∂X3

2δE12 =∂δu1

∂X2

+∂δu2

∂X1

+∂uk

∂X2

∂δuk

∂X1

+∂uk

∂X1

∂δuk

∂X2

2δE23 =∂δu2

∂X3

+∂δu3

∂X2

+∂uk

∂X3

∂δuk

∂X2

+∂uk

∂X2

∂δuk

∂X3

2δE13 =∂δu1

∂X3

+∂δu3

∂X1

+∂uk

∂X3

∂δuk

∂X1

+∂uk

∂X1

∂δuk

∂X3

(5.35)

Having precised the components of the variation of strain vector δEij in matrix form takinginto account the summation convention of k and l, yields the following presentation which isexaminable by multiplication:

δE11

δE22

δE33

2δE12

2δE23

2δE13

=

∂∂X1

+u1,1

∂

∂X1

u2,1

∂

∂X1

u3,1

∂

∂X1

u1,2

∂

∂X2

∂

∂X2

+u2,2

∂

∂X2

u3,2

∂

∂X2

u1,3

∂

∂X3

u2,3

∂

∂X3

∂

∂X3

+u3,3

∂

∂X3

∂

∂X2

+u1,2

∂

∂X1

+ u1,1

∂

∂X2

∂

∂X1

+u2,2

∂

∂X1

+ u2,1

∂

∂X2

u3,2

∂

∂X1

+ u3,1

∂

∂X2

u1,3

∂

∂X2

+ u1,2

∂

∂X3

∂

∂X3

+u2,3

∂

∂X2

+ u2,2

∂

∂X3

∂

∂X2

+u3,3

∂

∂X2

+ u3,2

∂

∂X3

∂

∂X3

+u1,3∂

∂X1

+ u1,1∂

∂X3

u2,3

∂

∂X1

+ u2,1

∂

∂X3

∂

∂X1

+u3,3

∂

∂X1

+ u3,1

∂

∂X3

δu1

δu2

δu3

(5.36)

If we have a closer look at the equation, it becomes evident that the differential operator whichmaps the displacement vector variation to the Green Lagrange strain vector is composed byaddition of the constant part already defined in equation (1.16)

Dε =

∂

∂X1

0 0

0∂

∂X2

0

0 0∂

∂X3∂

∂X2

∂

∂X1

0

0∂

∂X3

∂

∂X2∂

∂X3

0∂

∂X1

(5.37)


and of the deformation-dependent part.

Dnlε (u) =

u1,1

∂

∂X1

u2,1

∂

∂X1

u3,1

∂

∂X1

u1,2∂

∂X2

u2,2∂

∂X2

u3,2∂

∂X2

u1,3∂

∂X3

u2,3∂

∂X3

u3,3∂

∂X3

u1,2

∂

∂X1

+ u1,1

∂

∂X2

u2,2

∂

∂X1

+ u2,1

∂

∂X2

u3,2

∂

∂X1

+ u3,1

∂

∂X2

u1,3

∂

∂X2

+ u1,2

∂

∂X3

u2,3

∂

∂X2

+ u2,2

∂

∂X3

u3,3

∂

∂X2

+ u3,2

∂

∂X3

u1,3∂

∂X1

+ u1,1∂

∂X3

u2,3∂

∂X1

+ u2,1∂

∂X3

u3,3∂

∂X1

+ u3,1∂

∂X3

(5.38)

The connection between variation of continuous displacements and variation of the Green

Lagrange strain tensor can be demonstrated with assistance of the differential operator Dε

and Dnlε (u), where the product Dnl

ε δu stands for the variation of the non-linear strain vectorterm δEnl(u).

δE(u) = Dε δu + δEnl(u) =(

Dε + Dnlε (u)

)

δu (5.39)

5.3.5 Internal virtual work

With developments of strains and their variation in linear and non-linear parts, we can nowtransform the internal virtual work according to equation (5.31). Parts of Green Lagrange

strain vector E that arise and their variation δE are described in equations (5.13) and (5.39).We can also infer the material matrix of a three-dimensional continuum from equation (1.62)or equation (1.63).

δWint =

∫

Ω

δE ·C E dV =

∫

Ω

[(

Dε + Dnlε (u)

)

δu]

· C[

Dε u + Enl(u)]

dV (5.40)

5.3.6 Elastic internal potential

Because of its importance for derivation of solution algorithms in geometrically non-linear struc-tural mechanics and for development of direct methods which determine singular points (sta-bility), we should write down the internal potential Πint shall be written as function of elasticpotential function W (E) or of Green Lagrange strain tensor E in connection with the ma-


terial tensor C.

Πint =

∫

Ω

W (E) dV =1

2

∫

Ω

E : S dV =1

2

∫

Ω

E : C : E dV (5.41)

The variation of internal potential is identical with the internal virtual work, with assumptionof a material law having a potential character (see equation (5.31)).

δΠint =1

2

∫

Ω

(δE : C : E + E : C : δE) dV =

∫

Ω

δE : C : E dV =

∫

Ω

δE : S dV = δWint (5.42)

5.3.7 Remarks regarding combined material and geometric non-linearity

If one should combine the non-linearities dealt with in this and the previous chapter 5.2, namelymaterial and geometric non-linearity, one should note that non-linear material models are for-mulated in true, that is in Cauchy stresses but also in strains (Euler strain tensor) relatedto current configuration. For this reason, the actual Green Lagrange strain tensor generallyhas to be related to the momentary configuration with a push forward. There, the Cauchy

stress tensor is computed with the help of the used material model and thereafter related to thereference configuration with a pull back. The second Piola Kirchhoff stress tensor, which isthe result of this procedure, is thereby a function of the Green Lagrange strain tensor, of thematerial deformation gradient (push forward and pull back) and of internal variables dependenton the material model.

S = S(E,F ,α) (5.43)

Compared to a pure material non-linearity, the reference configuration related stress and strainquantities have to be utilized here instead of σ and ε as well as the material deformationgradient.

5.4 Consistent linearization of internal virtual work

The basis for the solution of geometrically non-linear finite element systems is the linearizationof internal virtual work. Before we perform the linearization in the next chapter 6, we mustdefine the directional or the Gateaux derivative of a scalar or a vector.

5.4.1 Linearization background

5.4.2 Gateaux derivative

Gateaux derivative of a scalar, vector, matrix or a tensor will henceforth be designated witha ∆ symbol. The definition is given with the help of an arbitrary scalar function f(u).

∆f(u) =d

dη[f(u + η∆u)]|η=0 =

∂f(u)

∂u· ∆u (5.44)

In this definition, u stands for the actual displacement vector and ∆u stands for an incremental


change of u. ddη

[f(u + η∆u)] is the increase or the derivative of function f along a direction

or a straight line set by ∆u. If this derivative is evaluated at η = 0, we get the function fderivative of the displacement state u in direction ∆u. On the other hand, the scalar product ofthe normal vector defined by ∂f(u)/∂u gives, along with the vector of incremental displacement∆u , the change ∆f of function f for the change ∆u in a tangential plane at f(u). The secondidentity can be shown if f(u + η∆u) is replaced by f(u(η)) with u(η) = u + η∆u, and if weapply the chain rule.

d

dηf(u + η∆u) =

d

dηf(u(η)) =

∂f(u(η))

∂u·∂u(η)

∂η=∂f(u + η∆u)

∂u· ∆u (5.45)

Equation (5.45) evaluated at η = 0 finally gives the sought second identity of equation (5.44).

5.4.3 Gateaux derivative of internal virtual work

Application of Gateaux derivative definition (5.44) to the internal virtual work δWint, accordingto equation (5.40) with variation of the strain tensor according to equation (5.39), determinesthe linearization of internal virtual work.

∆δWint =∂δWint

∂u∆u =

d

dη

[ ∫

Ω

[

Dε δ(u + η∆u) + δEnl(u + η∆u)]

· C

[

Dε (u + η∆u) + Enl(u + η∆u)]

dV

]∣∣∣∣η=0

(5.46)

Gateaux derivative is assembled by differentiation with respect to scalar η and by applyingthe chain rule and evaluating at η = 0.

∆δWint =

∫

Ω

[Dε δ∆u +d

dη(δEnl(u + η∆u))] · C

[Dε (u + η∆u) + Enl(u + η∆u)] dV

∣∣∣∣η=0

+

∫

Ω

[Dε δ(u + η∆u) + δEnl(u + η∆u)] ·C

[Dε ∆u +d

dη(Enl(u + η∆u))] dV

∣∣∣∣η=0

=

∫

Ω

[Dε δ∆u +d

dη(δEnl(u + η∆u))|η=0] ·C [Dε u + Enl(u)] dV

+

∫

Ω

[Dε δu + δEnl(u)] ·C [Dε ∆u +d

dηEnl(u + η∆u)|η=0] dV

(5.47)

If we observe the Gateaux derivative definition in equation (5.44) applied to the derivativesyet to be performed in equation (5.47), and if we take into consideration that, according to itsproperties formulated in chapter 1.6.1, the virtual displacement δu is arbitrary that is indepen-dent of the displacement state u, making ∆δu = δ∆u = 0, we get the Gateaux derivative of


internal virtual work,

∆δWint =

∫

Ω

∆δEnl(u) ·C [Dε u + Enl(u)] dV

+

∫

Ω

[Dε δu + δEnl(u)] ·C [Dε ∆u + ∆Enl(u)] dV

(5.48)

which can further be transformed after introduction of equation (5.39) applied to the termδEnl(u).

∆δWint =

∫

Ω

∆δEnl(u) ·C [Dε u + Enl(u)] dV

+

∫

Ω

[(Dε + Dnlε (u)) δu] ·C [Dε ∆u + ∆Enl(u)] dV

(5.49)

This equation represents the Gateaux derivative or the directional derivative of internal virtualwork related to the incremental change of displacement state by ∆u. The Gateaux derivativeis also called the consistent linearization of internal virtual work due to its strict mathematicalderivation. This consistent linearized virtual work presents a milestone of non-linear structuralmechanics since it is the basis of all iterative, incremental solution strategies (see chapter 7).

We should notice that Dε∆u + ∆Enl(u) presents the linearization of the Green Lagrange

strain vector ∆E(u) and that the Gateaux derivative of the second Piola Kirchhoff stressvector for materially non-linear models is given by

∆S(u) = C [Dε ∆u + ∆Enl(u)] = C ∆E(u) (5.50)

We should further notice the identity of linearization of the Green Lagrange strain vectorvariation and its non-linear part

∆δE(u) = Dε ∆δu + ∆δEnl(u) = ∆δEnl(u) (5.51)

due to the vanishing linearization of the linear part. Consequently we can write the linearizedinternal virtual work (5.49) in a compact form with equations (5.14), (5.50) and (5.51).

∆δWint =

∫

Ω

[∆δE(u) · S(u) + δE(u) · ∆S(u)] dV (5.52)

In order to discretisize the directional derivative or linearize the internal virtual work, accordingto equation (5.49) or (5.52), the linearization of the non-linear part of the Green Lagrange

strain vector ∆Enl(u) and its variation ∆δEnl(u) needs to be determined. The non-linear partof the Green Lagrange strain vector Enl(u) and differential operators Dε and Dnl

ε (u) arealready given in equations (5.14), (5.38) and (5.39).


5.4.4 Linearization of Green Lagrange strains

When using a linear material law, the linearization of the Green Lagrange strain vector,according to equation (5.50), multiplied by the material matrix is equivalent to the linearizationof the second Piola Kirchhoff stress vector. In order to determine the linearization of theGreen Lagrange strain vector, it suffices to investigate the non-linear part since the linearpart is already known to be Dε.∆u.

∆E(u) = Dε ∆u + ∆Enl(u) ∆Enl(u) =d

dη(Enl(u + η∆u))

∣∣∣∣η=0

(5.53)

In order to generate ∆Enl, the components of the non-linear part of the strain vector should beobserved in index notation and linearized according to equation (5.12) that is equation (5.33).

Enlij =

1

2

∂

∂Xi

uk

∂

∂Xj

uk (5.54)

The application of formalism of the Gateaux derivative yields:

∆Enlij =

1

2

d

dη

[∂

∂Xi

(uk + η∆uk)∂

∂Xj

(uk + η∆uk)

]∣∣∣∣η=0

=1

2

[∂

∂Xi

∆uk

∂

∂Xj

(uk + η∆uk) +∂

∂Xi

(uk + η∆uk)∂

∂Xj

∆uk

]∣∣∣∣η=0

=1

2

[∂

∂Xi

∆uk

∂

∂Xj

uk +∂

∂Xi

uk

∂

∂Xj

∆uk

]

=1

2

[

uk,j

∂

∂Xi

∆uk + uk,i

∂

∂Xj

∆uk

]

(5.55)

By comparing the last row of the above equation with the variation of the corresponding straincomponent δEnl

ij in equation (5.34), we showed the equivalence of variation and linearizationof strains. As a consequence, we conclude that linearization of the Green Lagrange strainvector is given directly with equation (5.39), where the variation symbol δ needs to be replacedby the linearization symbol ∆.

∆E(u) = Dε ∆u + ∆Enl(u) =(

Dε + Dnlε (u)

)

∆u (5.56)


5.4.5 Linearization of variation of Green Lagrange strains

The linearization of variation of Green Lagrange strains follows from considering equation (5.51)based on equation (5.34) with the help of the Gateaux derivative in components.

∆δEij = ∆δEnlij

=1

2

d

dη

[∂

∂Xi

δ(uk + η∆uk)∂

∂Xj

(uk + η∆uk)

+∂

∂Xi

(uk + η∆uk)∂

∂Xj

δ(uk + η∆uk)

]∣∣∣∣η=0

=1

2

[∂

∂Xi

δ∆uk

∂

∂Xj

(uk + η∆uk) +∂

∂Xi

δ(uk + η∆uk)∂

∂Xj

∆uk

+∂

∂Xi

∆uk

∂

∂Xj

δ(uk + η∆uk) +∂

∂Xi

(uk + η∆uk)∂

∂Xj

δ∆uk

]∣∣∣∣η=0

=1

2

[∂

∂Xi

δ∆uk

∂

∂Xj

uk +∂

∂Xi

δuk

∂

∂Xj

∆uk

+∂

∂Xi

∆uk

∂

∂Xj

δuk +∂

∂Xi

uk

∂

∂Xj

δ∆uk

]

(5.57)

As already elaborated in 5.4.3, the linearization of the virtual displacement δ∆uk = 0 vanishesand therefore also all terms in equation (5.57) which contain this term.

∆δEij = ∆δEnlij =

1

2

[∂

∂Xi

∆uk

∂

∂Xj

δuk +∂

∂Xj

∆uk

∂

∂Xi

δuk

]

(5.58)

Executing equation (5.58) for all permutations of i, j = 1, 2, 3 and summation over k = 1, 2, 3,we get components of linearized variation of the Green Lagrange strain vector.

∆δE11 =∂

∂X1

∆uk

∂

∂X1

δuk

∆δE22 =∂

∂X2

∆uk

∂

∂X2

δuk

∆δE33 =∂

∂X3

∆uk

∂

∂X3

δuk

∆δE12 =1

2

[∂

∂X1

∆uk

∂

∂X2

δuk +∂

∂X2

∆uk

∂

∂X1

δuk

]

∆δE23 =1

2

[∂

∂X2

∆uk

∂

∂X3

δuk +∂

∂X3

∆uk

∂

∂X2

δuk

]

∆δE13 =1

2

[∂

∂X1

∆uk

∂

∂X3

δuk +∂

∂X3

∆uk

∂

∂X1

δuk

]

(5.59)

In order to obtain greater clarity on linearization of variation of the strain vector we should, asalternative to the above derivations, vary Green Lagrange strains in tensor notation accordingto equation (5.7), with exchangeability of variation and linearization sequence,

δE =1

2

[∇Tδu + ∇δu + ∇Tδu · ∇u + ∇Tu · ∇δu

](5.60)


and thereupon linearize them with the help of Gateaux derivative definition according to (5.44),where the identity ∆δu = 0 must be heeded.

∆δE =1

2

d

dη

[

∇Tδ(u + η∆u) + ∇δ(u + η∆u)

+∇Tδ(u + η∆u) · ∇(u + η∆u) + ∇T(u + η∆u) · ∇δ(u + η∆u)

]∣∣∣∣η=0

=1

2

[

∇Tδu · ∇∆u + ∇T∆u · ∇δu

]

(5.61)

With that ∆δE is determined in a written form. It is evident that this expression does not dependon the given state of displacement u. For the following discretization of internal virtual workin a pure and linearized form in chapter 6, some additional considerations and transformationsshould by all means be done beforehand. The linearized variation of Green Lagrange strainsappears in the linearized internal virtual work (equation (5.52)) as a scalar valued product withthe second Piola Kirchhoff stresses. In tensor notation, this can be written as follows andtransformed and summarized due to the symmetry of the stress tensor.

∆δE : S =1

2

[∇Tδu · ∇∆u

]: S +

1

2

[∇T∆u · ∇δu

]: S

=[∇Tδu · ∇∆u

]: S =

[∇T∆u · ∇δu

]: S = ∆δE · S

(5.62)

The proof of equation (5.62) comes in components, where symmetry of the stress tensor Sij = Sji

and exchangeability of dummy (summation) indices enable the particular steps of the proof.

∆δEij =1

2[δuk,j∆uk,i + δuk,i∆uk,j]

∆δEijSij =1

2[δuk,j∆uk,iSij + δuk,i∆uk,jSij] =

1

2[δuk,j∆uk,iSij + δuk,i∆uk,jSji]

=1

2[δuk,j∆uk,iSij + δuk,j∆uk,iSij] = δuk,j∆uk,iSij

(5.63)

After this, it remains to specify the expression (analogy with calculation of ∇Tu · ∇u, given inequation (5.11))

[∇Tδu · ∇∆u] : S =

δuk,1 ∆uk,1 δuk,1 ∆uk,2 δuk,1 ∆uk,3



:

S11 S12 S13

S12 S12 S23

S13 S23 S33

(5.64)

and to transfer it into a suitable form in matrix notation.

[∇Tδu · ∇∆u] : S = δu · S ∆u (5.65)

Here, u defines a vector component of the displacement vector gradient ∇u.

δu =

[

δu1,1 δu1,2 δu1,3 δu2,1 δu2,2 δu2,3 δu3,1 δu3,2 δu3,3

]T

∆u =

[

∆u1,1 ∆u1,2 ∆u1,3 ∆u2,1 ∆u2,2 ∆u2,3 ∆u3,1 ∆u3,2 ∆u3,3

]T (5.66)


or

δu =

∂δu1

∂X∂δu2

∂X∂δu3

∂X

=

∇δu1

∇δu2

∇δu3

∆u =

∂∆u1

∂X∂∆u2

∂X∂∆u3

∂X

=

∇∆u1

∇∆u2

∇∆u3

(5.67)

and S defines the hyper-diagonal matrix of the second Piola Kirchhoff stress components.

S =

S11 S12 S13

S12 S22 S23

S13 S23 S33

S11 S12 S13

S12 S22 S23

S13 S23 S33

S11 S12 S13

S12 S22 S23

S13 S23 S33

, S =

S

S

S

(5.68)

The validity of identity (5.65) can be examined by calculation of the corresponding scalaraccording to both left and right side of this equation, and also by applying definitions (5.66)and (5.68). Intotal the identity relevant to discretization of linearized internal work is obtainedwith equations (5.62) and (5.65).

∆δE · S = δu · S ∆u (5.69)

Chapter 6

Finite element discretization ofgeometrically non-linear continua

The finite element discretization of the weak form of impulse balance, or of the principle ofvirtual work (5.30), gives the static or dynamic equilibrium in the form of a non-linear vectorequation or vector differential equation. For the finite elements dealt with in this chapter, thelatter differs from the discrete formulation of the principle of virtual work for small deformations(geometrically linear observation) only in the term of discretisized internal virtual work. Theresult of discretization of this different term is the deformation-dependent vector of internal

forces.

Besides the discretization of principle of virtual work, when considering definite deformations(geometrically non-linear), the discretization of linearized internal virtual work (5.49) or (5.52)is of crucial significance for numerical solution of geometrically non-linear elasto-mechanics.Discretization of the linearized internal virtual work defines the so called tangential stiffness

matrix. This tangential stiffness, identical to linearization of vector of internal forces, forms thebasis of iterative Newton procedures on the one hand, and is of importance for characterizationof stability properties of the structure on the other hand.

Based on the fundamental understanding of finite element development in linear structuralmechanics, it is effective to first derive the discretization of the three-dimensional non-linearcontinuum with isoparametric finite volume elements, and thereafter develop the correspondingone- and two-dimensional finite elements analogously.

6.1 Finite volume elements

The discretization of the geometrically non-linear three-dimensional continuum can be doneanalogously to the linear case (chapter 4). In comparison to geometrically linear observation,the following developing steps remain unaltered:

• Approximation of continuous values with the ansatz function matrix N(ξ) and of elementvectors Xe, ue, δue and ue of a NN -noded hexagon element according to equations (4.10)and (4.11), where the generation of ansatz functions is performed according to table ??.

• The generation of Jacobi matrix J(ξ) and Jacobi determinant |J(ξ)| according to equa-tions (4.12) and (4.14) remains the same. Also, the inversion for calculation of Jacobi

matrix inverse J−1(ξ), which is defined in equation (4.13).

171


• The spatial Gauß-Legendre integration of element matrices and vectors.

• The mass matrix me is identical to linear observation in equation (4.19).

• The calculation of consistent equivalent loads of volume loads rep (4.20) remains unmodified

as well, and one obtains the consistent equivalent load of boundary loads ren by replacing

boundary stresses t? by boundary stresses T ?, see equations (5.23) and (5.24).

In other words, the internal virtual work and its linearization remain to be discretisized, wherein order to discretisize δWint only the virtual displacement by ansatz functions and variation ofelement free values are approximated. Contrarily, to discretisize ∆δWint, the approximation toδu and ∆u is used. All further displacement-dependent terms remain undiscretisized.

6.1.1 Discretization of internal virtual work

After the preceding derivations, it remains to discuss the discretization if internal virtual workof a finite element e according to equation (5.31). In order to formulate the isoparametric finitevolume element, the equation (5.40) should be applied to the element volume, and the differentialvolume dΩ should be transformed into natural coordinates dΩ = |J(ξ)| dξ1dξ2dξ3 according toequation (4.14), with the help of the Jacobi determinant |J(ξ)|. In the differential operatorsDε, Dnl

ε (u) as well as in the non-linear part of the Green Lagrange strain vector Enl(u),the derivatives of physical coordinates must be substituted by derivatives of natural coordinateswith the help of the Jacobi transformation. Formally this is designated by index Jacobi. Thedifferential operator Dεξ was already given in equation (4.15). The explicit generation of furtherterms follows in the next two sections. For the further development of internal virtual work,equation (5.40) is resolved.

δW eint =

1∫

−1

1∫

−1

1∫

−1

δEξ(u, ξ) ·C Eξ(u, ξ) |J(ξ)| dξ1 dξ2 dξ3

=

1∫

−1

1∫

−1

1∫

−1

(Dεξ(ξ) δu(ξ)) ·C Dεξ(ξ) u(ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

(Dεξ(ξ) δu(ξ)) ·C Enlξ (u, ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

(

Dnlεξ(u, ξ) δu(ξ)

)

·C Dεξ(ξ) u(ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

(

Dnlεξ(u, ξ) δu(ξ)

)

·C Enlξ (u, ξ) |J(ξ)| dξ1 dξ2 dξ3

(6.1)

The first summand corresponds to the internal virtual work of geometrically linear elastome-

chanics δW e linint = δue · ke

eue according to (1.96), which can again be discretisized with the

stiffness matrix (equation (4.18)), which is marked as elastic stiffness matrix kee for distinction

purposes, and with the element displacement vector ue as well as its variation δue (equation(4.10)). Independent of this realization, the simplification of the scientific notation for purposesof a uniform presentation of all terms, and most of all, in order to effectively compute thetangential stiffness matrix (should at first not be considered). Since the non-linear part of the


strain vector Enl(u) cannot be obtained by a linear mapping of the displacement vector (seeequation (5.13)), a strategy alternative to linear theory must be applied for discretization ofinternal virtual work. For that reason, the strain variation δE ξ(u, ξ) is discretisized with thehelp of the approximation of the displacement variation based on the product of matrix ofansatz functions N(ξ) (equation (4.11)) and variation of the element displacement vector δue

according to equation (4.10).

δu(ξ) ≈ δu(ξ) = N(ξ) δue δue =[

δue11 δue1

2 δue13 · · · δueNN

3

]T(6.2)

Dεξ(ξ) δu ≈ Dεξ(ξ) N(ξ) δue = B(ξ) δue

δEnlξ (u, ξ) ≈ Dnl

εξ(u, ξ) N(ξ) δue = Bnl(u, ξ)δue

δEξ(u, ξ) ≈(

Dεξ(ξ) + Dnlεξ(u, ξ)

)

N(ξ) δue = B(u, ξ) δue

(6.3)

The remaining linear and non-linear terms of the strain vector Dεξ(ξ)u and Enlξ (u, ξ) are not

subject to discretization. The reason for that is evident in chapter 7 in the scope of iterativesolution procedures of non-linear structural mechanics. Here, the result of investigations thatare yet to come should only be sketched: In order to realize the sought algorithms, the strainsare calculated just for one estimated or approximated displacement state uek or continuousdisplacement state uk(ξ), where the estimated continuous displacements are calculable fromthe displacement vector with the help of approximation (4.10). 1

uk(ξ) ≈ uk(ξ) = N(ξ) uek uek =[

uek11 uek1

2 uek13 · · · uekNN

3

]T(6.4)

Accordingly, also the Green Lagrange strains Enlξ (uk, ξ) can be determined with equation

(5.13). These statements are analogously valid for deformation-dependent terms Dnlεξ(u

k, ξ) that

is Bnl(uk, ξ) which are contained in equations (6.3). They are determined with the displacementstate uk and equations (5.38) or (6.3). After this elaboration, from now on the so called iterationindex k will be omitted; all undiscretisized displacements should be understood as iterativelyapproximated displacements. If we apply the approximation (6.3) to the element-specific

1Already in advance, in the scope of non-linear algorithms, uek presents the element displacement vector of

the last iteration step


formulation of internal virtual work (6.1), we get its discrete approximation.

δW eint ≈ δue ·

1∫

−1

1∫

−1

1∫

−1

BT (ξ) C Dεξ(ξ) u(ξ) |J(ξ)| dξ1 dξ2 dξ3

+ δue ·

1∫

−1

1∫

−1

1∫

−1

BT (ξ) C Enlξ (u, ξ) |J(ξ)| dξ1 dξ2 dξ3

+ δue ·

1∫

−1

1∫

−1

1∫

−1

Bnl T (u, ξ) C Dεξ(ξ) u(ξ) |J(ξ)| dξ1 dξ2 dξ3

+ δue ·

1∫

−1

1∫

−1

1∫

−1

Bnl T (u, ξ) C Enlξ (u, ξ) |J(ξ)| dξ1 dξ2 dξ3

(6.5)

The internal virtual work can also be calculated with definition of the deformation-dependentelement vector of internal forces re

i ,

rei (u

e) =

1∫

−1

1∫

−1

1∫

−1

BT (ξ) C Dεξ(ξ) u(ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

BT (ξ) C Enlξ (u, ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

Bnl T (u, ξ) C Dεξ(ξ) u(ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

Bnl T (u, ξ) C Enlξ (u, ξ) |J(ξ)| dξ1 dξ2 dξ3

(6.6)

and with variation of the displacement vector δue.

δW eint ≈ δue · re

i (ue) (6.7)

Alternatively, the vector of internal forces can be presented in a more compact form if thepartitioning of the strain vector and the differential operator is taken back and the equation(5.14) is utilized.

rei (u

e) =

1∫

−1

1∫

−1

1∫

−1

[

BT (ξ) + Bnl T (u, ξ)]

C Eξ(u, ξ)|J(ξ)|dξ1dξ2dξ3 (6.8)

Generally, the internal forces are calculable by integration of the product of the non-lineardifferential operator B(u, ξ) = B(ξ) + Bnl(u, ξ) and the stress vector Sξ(u, ξ) = CEξ(u, ξ),where of course the Jacobi transformation needs to be included.

rei (u

e) =

1∫

−1

1∫

−1

1∫

−1

BT (u, ξ) Sξ(u, ξ) |J(ξ)| dξ1 dξ2 dξ3 (6.9)


The deformational dependence of internal forces was characterised by the functional dependenceof the displacement vector ue. This implies that for calculation of internal forces in general, andfor calculation of Bnl, Dεξu as well as Enl

ξ , especially the element displacement vector has tobe either known or assessable.

6.1.1.1 Jacoby transformation of δE

When previously discretisizing the internal virtual work, the variation of the strain vectorδEξ(u, ξ) formulated in natural coordinates was already used, but not determined. By means ofthe Jacobi transformation of equation (5.39), δE ξ(u, ξ) can be expressed with the assistanceof Dεξ(ξ) and Dnl

εξ(u, ξ).

δEξ(u) = Dεξ δu + δEnlξ (u) =

(

Dεξ + Dnlεξ(u)

)

δu (6.10)

The differential operator Dεξ and consequently also the B-operator B(ξ) are already knownfrom the development of a finite volume element of linear structural mechanics (equations (4.15)to (4.17)). It now only remains to proceed with the development of the operator Dnl

εξ(u, ξ) and

of the result of its application to the matrix of ansatz functions Dnlεξ(u, ξ)N(ξ)=Bnl(u, ξ). The

differential operator Dnlε (u, ξ), defined in equation (5.38), contains derivatives of displacement

components with respect to physical coordinates as well as differentiation rules with respect tophysical coordinates of a generalised form:

uj,k

∂

∂Xl

=∂uj

∂Xk

∂

∂Xl

(6.11)

With the help of components of the inverse Jacobi matrix according to equation (4.13), thetransformation can be performed in natural coordinates

∂

∂Xl

=∂ξm∂Xl

∂

∂ξmuj,k =

∂uj

∂Xk

=∂ξn∂Xk

∂uj

∂ξn(6.12)

where the summation is over m = 1, 2, 3, that is over n = 1, 2, 3. Derivatives of the displacementvector with respect to natural coordinates are set by the derivative of ansatz functions.

∂u(ξ)

∂ξn= N;n(ξ) ue ∂uj(ξ)

∂ξn=

NN∑

i=1

N i;n(ξ) uei

j (6.13)

For generation of the differential operator Dnlεξ, we write only the derivative ∂/∂Xl. The deriva-

tives of displacement components with respect to natural coordinates are implicitly included


according to equations (6.12)2 and (6.13).

Dnlεξ(u, ξ) =

u1,1

∂ξm

∂X1

∂

∂ξm

u2,1

∂ξm

∂X1

∂

∂ξm

u3,1

∂ξm

∂X1

∂

∂ξm

u1,2

∂ξm

∂X2

∂

∂ξm

u2,2

∂ξm

∂X2

∂

∂ξm

u3,2

∂ξm

∂X2

∂

∂ξm

u1,3

∂ξm

∂X3

∂

∂ξm

u2,3

∂ξm

∂X3

∂

∂ξm

u3,3

∂ξm

∂X3

∂

∂ξm

u1,2

∂ξm

∂X1

∂

∂ξm

+ u1,1

∂ξm

∂X2

∂

∂ξm

u2,2

∂ξm

∂X1

∂

∂ξm

+ u2,1

∂ξm

∂X2

∂

∂ξm

u3,2

∂ξm

∂X1

∂

∂ξm

+ u3,1

∂ξm

∂X2

∂

∂ξm

u1,3

∂ξm

∂X2

∂

∂ξm

+ u1,2

∂ξm

∂X3

∂

∂ξm

u2,3

∂ξm

∂X2

∂

∂ξm

+ u2,2

∂ξm

∂X3

∂

∂ξm

u3,3

∂ξm

∂X2

∂

∂ξm

+ u3,2

∂ξm

∂X3

∂

∂ξm

u1,3

∂ξm

∂X1

∂

∂ξm

+ u1,1

∂ξm

∂X3

∂

∂ξm

u2,3

∂ξm

∂X1

∂

∂ξm

+ u2,1

∂ξm

∂X3

∂

∂ξm

u3,3

∂ξm

∂X1

∂

∂ξm

+ u3,1

∂ξm

∂X3

∂

∂ξm

(6.14)

Application of the differential operator Dnlεξ to the matrix of ansatz functions Ni(ξ) belonging

to node i, yields the non-linear part of B-operator Bnli (u, ξ) = Dnl

εξ(u, ξ)Ni(ξ),

Bnlij (u, ξ) =

uj,1∂ξm∂X1

N i;m

uj,2∂ξm∂X2

N i;m

uj,3

∂ξm∂X3

N i;m

uj,2

∂ξm∂X1

N i;m + uj,1

∂ξm∂X2

N i;m

uj,3∂ξm∂X2

N i;m + uj,2

∂ξm∂X3

N i;m

uj,3∂ξm∂X1

N i;m + uj,1

∂ξm∂X3

N i;m

Bnli (u, ξ) =

[

Bnli1 Bnl

i2 Bnli3

]

(6.15)

where the non-linear operator of the finite volume element is assembled analogously to equation(4.17).

Bnl(u, ξ) =

[

Bnl1 (u, ξ) Bnl

2 (u, ξ) · · · BnlNN(u, ξ)

]

(6.16)

6.1.1.2 Jacoby transformation of E

The derivatives of physical coordinates of the non-linear part of the strains Enl can be replacedby derivatives of natural coordinates, too. If the observed strains are composed from the linearand the non-linear part of equation (5.14), one gets the strains in natural coordinates after theJacobi transformation,

Eξ(u) = Dεξ u + Enlξ (u) (6.17)

in case of which the linear part Dεξu is already determined (equation (4.15)). The vectorEnl(u) contains, according to equation (5.12) that is (5.13), terms of the form uj,k that can be


transformed with the help of components of the Jacobi matrix according to equation (6.12).

Enlij =

1

2

∂

∂Xi

uk

∂

∂Xj

uk Enlξij =

1

2

∂ξm∂Xi

∂

∂ξmuk

∂ξn∂Xj

∂

∂ξnuk (6.18)

As an example of generation of Enlξ , the normal component Enl

ξ11 is derived.

Enl11(u, ξ) =

1

2(u1,1 u1,1 + u2,1 u2,1 + u3,1 u3,1)

Enlξ11(u, ξ) =

1

2

(∂ξm∂X1

∂u1

∂ξm

∂ξn∂X1

∂u1

∂ξn+∂ξm∂X1

∂u2

∂ξm

∂ξn∂X1

∂u2

∂ξn+∂ξm∂X1

∂u3

∂ξm

∂ξn∂X1

∂u3

∂ξn

) (6.19)

Further components of the strain tensor can be derived in the same manner.

6.1.2 Non-linear semi-discrete initial value problem

Due to the spatial discretization presented in preceding sections, the principle of virtual workcan be approximated in the element plane.

δue ·me ue + δue · rei (u

e) = δue · (rep + re

n) (6.20)

By summing equation (6.20) or explicitly, the vector of internal loads,

ri(u) =NE⋃

e = 1

rei (u

e) (6.21)

with the vector of external loads rep + re

n and the mass matrix me in analogy with equation(2.146), we obtain the system-related spatially discrete formulation of the principle of virtualwork,

δu ·M u + δu · ri(u) = δu · r (6.22)

which can be transferred to the semi-discrete initial value problem of non-linear elasto-dynamicsby application of the fundamental lemma of variation calculus (see chapter 2.2 and 2.3). Theproblem is defined by the semi-discrete differential equation of motion of the second order

M u + ri(u) = r (6.23)

and by the discretisized initial conditions.

u(0) = u?

ri(u(0)) = [r(0) −M u(0)]

u(0) = u?

u(0) = M−1 [r(0) − ri(u(0))](6.24)

The substantial difference to linear elasto-dynamics is that due to geometric non-linearity, thereis a non-linear equation of motion with the deformation-dependent vector of internal loadsri(u), instead of the product of constant stiffness matrix, and with the displacement vectorKu. As a further characteristic, we should bring up the non-linearity of the initial conditionwhen prescribing accelerations u(0) = u?.


6.1.3 Non-linear discrete static equilibrium

For static or quasi-static problems of structural mechanics we can formulate the discreteequation of non-linear static equilibrium by neglecting the inertial forces Mu = 0.

ri(u) = r (6.25)

This equation presents a non-linear algebraic vector equation. The solution of the non-linearcoupled vector equation (6.25) requires the discretization of linearized internal virtual work(chapter 5.4) and the solution of incremental equilibrium relations with special numericalprocedures, elaborated in chapter 7.

6.1.4 Discretization of linearized internal virtual work

For finite element discretization of the linearized internal virtual work, formed in equation (5.49)and (5.56) that is equation (5.52), the integrands first have to be applied to the element volumeand thereafter transformed to natural coordinates ξ1 ξ2 and ξ3.

∆δW e

int =

1∫

−1

1∫

−1

1∫

−1

[∆δEξ(u, ξ) · Sξ(u, ξ) + δEξ(u, ξ) · ∆Sξ(u, ξ)] |J(ξ)| dξ1 dξ2 dξ3

=

1∫

−1

1∫

−1

1∫

−1

∆δEnlξ (u, ξ) · C Eξ(u, ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

[(Dεξ(ξ) + Dnlεξ(u, ξ)) δu] ·C

[Dεξ(ξ) + Dnlεξ(u, ξ)] ∆u |J(ξ)| dξ1 dξ2 dξ3

(6.26)

After the derivations of the previous chapter 6 and 5.4.3, only the discretization of thisexpression requires the discretization of the non-linear part of strain vector linearization∆Eξ(u, ξ), that is of stress vector ∆Sξ(u, ξ) and of term ∆δEξ(u, ξ) ·Sξ(u, ξ). Due to analogywith the variation and linearization, the linearization of the Green Lagrange strain vector∆Eξ(u, ξ) can be written directly in natural coordinates, in analogy with equation (6.10),

∆Eξ(u) = Dεξ ∆u + ∆Enlξ (u) =

(

Dεξ + Dnlεξ(u)

)

∆u (6.27)

and be discretisized.

Dεξ(ξ) ∆u ≈ B(ξ) ∆ue ∆Enlξ (u, ξ) ≈ Bnl(u, ξ) ∆ue (6.28)

Thereby, the second summand of ∆δW eint can be written according to the operators, their

variation, linearization, linearized variation and discretization, in its spatially discretisized formalready derived in equation (6.26). This should however be postponed until the first summandof ∆δW e

int is discretisized, too. Therefore, the transformation is used which was derived in the

preceding section (equation (5.69)) in which the stress matrix S and the vectors δu that is


∆u, must be given in natural coordinates (Sξ, δuξ and ∆uξ). The components of the stress

matrix Sξ can be computed based on strains Eξ, according to equation (6.17) that is (6.18)with the help of the constitutive law (5.26). The Jacobi transformation from δu is analoguousto those with ∆u, which is why this vector δu is shown as an example. The derivatives ofphysical coordinates δui,j are replaced by derivatives of natural coordinates with the Jacobi

transformation according to equation (6.12) (summation over k = 1, 2, 3).

δuξ =

∂ξk∂X1

∂

∂ξkδu1

∂ξk∂X2

∂

∂ξkδu1

∂ξk∂X3

∂

∂ξkδu1

∂ξk∂X1

∂

∂ξkδu2

∂ξk∂X2

∂

∂ξkδu2

∂ξk∂X3

∂

∂ξkδu2

∂ξk∂X1

∂

∂ξkδu3

∂ξk∂X2

∂

∂ξkδu3

∂ξk∂X3

∂

∂ξkδu3

=

∂ξk∂X1

∂

∂ξk∂ξk∂X2

∂

∂ξk∂ξk∂X3

∂

∂ξk∂ξk∂X1

∂

∂ξk∂ξk∂X2

∂

∂ξk∂ξk∂X3

∂

∂ξk∂ξk∂X1

∂

∂ξk∂ξk∂X2

∂

∂ξk∂ξk∂X3

∂

∂ξk

δu1

δu2

δu3

(6.29)

With the discretization of vector δu, which was defined in equation (4.10) with the help of thematrix of ansatz functions N(ξ) (equation (4.11)), and variation of the element displacementvector δue (δu = N(ξ)δue), we obtain the approximation of δuξ. The same can be done for∆uξ analogous to δuξ.

δuξ ≈ Bg(ξ) δue ∆uξ ≈ Bg(ξ) ∆ue (6.30)

The differential operator Bg(ξ), which is dependent solely on natural coordinates, can becomposed with the differential operator

Bgi (ξ) =

∂N i(ξ)

∂X∂N i(ξ)

∂X∂N i(ξ)

∂X

∂N i(ξ)

∂X=

∂ξk∂X1

N i;k(ξ)

∂ξk∂X2

N i;k(ξ)

∂ξk∂X3

N i;k(ξ)

(6.31)

associated with the element node i.

Bg(ξ) =

[

Bg1(ξ) B

g2(ξ) · · · B

gNN (ξ)

]

(6.32)

Therefore, it is possible to approximate the scalar product of the linearized variation of Green

Lagrange strains and the second Piola Kirchhoff stresses with the help of differential


operator Bg(ξ), and with the hyper-matrix of stresses Sξ(u, ξ) (equations (5.69) and (6.30)):

∆δEξ(u, ξ) · Sξ(u, ξ) = δu · Sξ(u, ξ) ∆u ≈ δue · Bg T (ξ) Sξ(u, ξ) Bg(ξ) ∆ue (6.33)

The linear internal virtual work can finally be entirely discretisized. Furthermore, we takeadvantage of the independence of element displacement vector ue from natural coordinates ξ

in order to extract δue and ∆ue from the integral expressions.

∆δW e

int =

1∫

−1

1∫

−1

1∫

−1

∆δEξ(u, ξ) · Sξ(u, ξ) |J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

[(Dεξ(ξ) + Dnlεξ(u, ξ)) δu] ·C

[Dεξ(ξ) ∆u + ∆Enlξ (u, ξ)] |J(ξ)| dξ1 dξ2 dξ3

(6.34)

∆δW eint ≈ δue ·

1∫

−1

1∫

−1

1∫

−1

Bg T (ξ) Sξ(u, ξ) Bg(ξ) |J(ξ)| dξ1 dξ2 dξ3 ∆ue

+ δue ·

1∫

−1

1∫

−1

1∫

−1

(BT (ξ) + Bnl T (u, ξ)) C

(B(ξ) + Bnl(u, ξ)) |J(ξ)| dξ1 dξ2 dξ3 ∆ue

(6.34)

The compact form of linearized internal virtual work is obtained with definitions of thedeformation-dependent geometric element stiffness matrix

keg(u

e) =

1∫

−1

1∫

−1

1∫

−1

Bg T (ξ) Sξ(u, ξ) Bg(ξ) |J(ξ)| dξ1 dξ2 dξ3 (6.35)

and the deformation-dependent material element stiffness matrix

kem(ue) =

1∫

−1

1∫

−1

1∫

−1

[

BT (ξ) + Bnl T (u, ξ)]

C[

B(ξ) + Bnl(u, ξ)]

|J(ξ)| dξ1 dξ2 dξ3 (6.36)

with the sum of the two yielding the tangential element stiffness matrix:

∆δW eint ≈ δue ·

(ke

g(ue) + ke

m(ue))

∆ue = δue · ket (u

e) ∆ue (6.37)

If we use the additive composition of the B-operator of the material stiffness matrix, we


can break up the elastic element stiffness matrix kee, identical to the linear theory, and the

deformation-dependent matrix of initial displacements keu(ue).

kem(ue) = ke

e + keu(ue)

kee =

1∫

−1

1∫

−1

1∫

−1

BT (ξ) C B(ξ) |J(ξ)| dξ1 dξ2 dξ3

keu(ue) =

1∫

−1

1∫

−1

1∫

−1

[

BT (ξ) C Bnl(u, ξ) + Bnl T (u, ξ) C B(ξ)]

|J(ξ)| dξ1 dξ2 dξ3

+

1∫

−1

1∫

−1

1∫

−1

Bnl T (u, ξ) C Bnl(u, ξ) |J(ξ)| dξ1 dξ2 dξ3

(6.38)

It is obvious that all parts of the tangential stiffness matrix are symmetric. Symmetry of elementstiffness matrices ke

g(ue), ke

m(ue) and kee can be inferred directly from respective definitions. The

symmetry of keu(ue) follows from the symmetry of ke

m(ue) and kee and equation (6.38).

6.1.5 Linearization of internal forces vector

In the scope of the iterative solution of non-linear dynamic differential vector equations (6.23)or the non-linear static vector equation (6.25), the linearization of internal forces will be ofgreat importance. In the element plane, the linearization of internal forces r i(u) is defined bylinearization of the internal virtual work in equation (6.7), what, by comparison to equation(6.37), can be put down to the tangential element stiffness and the increment of the elementdisplacement vector.

∆δW eint ≈ δue · ∆re

i (ue) = δue ·

∂rei (u

e)

∂ue∆ue = δue ·

(ke

g(ue) + ke

m(ue))

∆ue (6.39)

By summing the linearized internal forces and parts of the tangential stiffness matrix

∆ri(u) =NE⋃

e = 1

∆rei (u

e)

Kt(u) =NE⋃

e = 1

ket (u

e)

Kg(u) =NE⋃

e = 1

keg(u

e)

Km(u) =NE⋃

e = 1

kem(ue)

(6.40)

we eventually get the linearization of the system vector of internal forces ∆r i(u) as function ofthe material system stiffness matrix Km(u), the geometric system stiffness matrix Kg(u), thetangential system stiffness matrix Kt(u) and the increment of the system displacement vector∆u.

∆ri(u) = (Kg(u) + Km(u)) ∆u = Kt(u) ∆u (6.41)


6.2 Finite truss elements

6.2.1 Non-linear continuum-mechanical formulation

6.2.1.1 Kinematics

The Green Lagrange strain in the longitudinal direction of the truss E11 is determined basedon equation (5.12) that is (5.13) with u2,1 = u3,1 = 0.

E11 =1

2(u1,1 + u1,1 + u1,1 u1,1) = u1,1 +

1

2u1,1 u1,1 (6.42)

The presentation of the Green Lagrange truss strain, analoguous with the presentation of thedifferential operator (5.14) can be acquired by the following transformation.

E11 =∂

∂X1

u1 +1

2u1,1 u1,1 = Dε u1 +Enl

11(u1) (6.43)

6.2.1.2 Kinetics

The Lagrange formulation of the forces equilibrium of a truss element is obtained with equation(5.22) for S22 = S33 = S12 = S23 = S13 = 0, and the vanishing derivatives in e2 and e3 directions(divergence DIV)

ρ u1 =∂

∂X1

(∂x1

∂X1

S11

)

+ ρ b1 (6.44)

6.2.1.3 Constitutive law

The one-dimensional special case of the constitutive equation (5.26) can be described only withthe assistance of the elasticity modulus.

S11 = E E11 (6.45)

6.2.1.4 Principle of virtual work

The principle of virtual displacements of a one-dimensional continuum is obtained by reductionof the three-dimensional formulation (5.30), or by adjusting the linear formulation ((2.24). Incase of the latter procedure, the Green strain ε11 and the Cauchy stress σ11 have to be replacedby the Green Lagrange strain E11 and the second Piola Kirchhoff stress S11, respectively,and the non-specified length l by the reference length L.

L2∫

−

L2

δu1 u1 ρ A dX1 +

L2∫

−

L2

δE11 S11 A dX1 = δue11 N e1

1 + δue21 N e2

1 +

L2∫

−

L2

δu1 p1 dX1 (6.46)

The inertial term as well as terms of internal and external loads are the same as in the linearobservation. For the internal virtual work, the integrand is already known, with the exception of


the variation of the Green Lagrange strain. δE11 is obtained by reduction of equation (5.34)for the one-dimensional continuum or by variation of equation (6.42).

δE11 =∂

∂X1

δu1 +1

2

∂

∂X1

δu1

∂

∂X1

u1 +1

2

∂

∂X1

u1

∂

∂X1

δu1 =∂

∂X1

δu1 + u1,1∂

∂X1

δu1 (6.47)

This can be presented by means of analogy to equation (5.39) with the help of the differentialoperator Dε and Dnl

ε (u1).

δE11 = Dε δu1 + δEnl11(u1) =

(

Dε +Dnlε (u1)

)

δu1 (6.48)

6.2.1.5 Linearization of internal virtual work

The linearized internal virtual work can be extracted from the internal virtual work of the three-dimensional continuum (5.49), or much simpler, by linearization of internal virtual work givenin equation (6.46).

∆δWint =

L2∫

−

L2

A (∆δE11 S11 + δE11 ∆S11) dX1 (6.49)

In this expression, we need to generate the linearization of variation of the Green Lagrange

strain and of the second Piola Kirchhoff stress. The first linearization yields:

∆δE11 = Dε ∆δu1 + ∆δEnl11(u1) = ∆δEnl

11(u1) = ∆u1,1 δu1,1 + u1,1 ∆δu1,1 = ∆u1,1 δu1,1 (6.50)

The linearization of the second Piola Kirchhoff stress can be put down to the linearization ofthe Green Lagrange strain with the help of the constitutive equation (6.45), which is definedby the equivalence of variation and linearization in analogy to equation (6.48).

∆S11 = E ∆E11 = E(

Dε +Dnlε (u1)

)

∆u1 (6.51)

For the discretization of linearized internal virtual work of the three-dimensional continuum,it has turned out to be favourable to properly transform the expression ∆δE · S equivalentto ∆δE11S11. For a deeper understanding of this transformation, we should do it for the one-dimensional continuum. Since ∆δE11 and S11 are scalar expressions, this transformation is ofno further significance.

∆δE11 S11 = δu1,1 ∆u1,1 S11 = δu1,1 S11 ∆u1,1 =∂

∂X1

δu1 S11

∂

∂X1

∆u1 = δu1 S11 ∆u1 (6.52)

6.2.2 Truss elements of arbitrary polynomial degree

In order to develop a family of truss elements for the modelling of geometrically non-linearstructural behaviour, we should assume equidistant element nodes. The development is separatedinto determination of the element vector of internal forces by discretization of the internal virtualwork, and generation of the tangential element stiffness matrix based on linearized internalvirtual work. Afterwards, one derives the special case of the linear truss element.


6.2.2.1 Element vector of internal forces

As shown within the scope of finite volume element development, the element vector of internalforces is, developed with the discretization of virtual displacement in the internal virtual workexpression (6.46). The basis of standardized discretization is the formulation of internal virtualwork in natural coordinates, where the Jacobi matrix and the Jacobi determinant are identicalto formulations of geometrically linear truss elements (chapter 2.2.2), as long as they are formedwith the reference truss length L.

δWint =

1∫

−1

A δEξ11 Sξ11

L

2dξ1 =

1∫

−1

A(

Dεξ +Dnlεξ(u1)

)

δu1 Sξ11

L

2dξ1 (6.53)

The differential operators Dεξ and Dnlεξ(u1) as well as the stress Sξ11 are formed with the Jacoby

transformation of corresponding values in physical coordinates (see equation (2.38)).

Dεξ =∂ξ1∂X1

∂

∂ξ1=

2

L

∂

∂ξ1

Dnlεξ(u1) =

∂ξ1∂X1

∂u1

∂ξ1

∂ξ1∂X1

∂

∂ξ1=

4

L2

∂u1

∂ξ1

∂

∂ξ1

Sξ11(u1) =E

[∂ξ1∂X1

∂u1

∂ξ1+

1

2

∂ξ1∂X1

∂u1

∂ξ1

∂ξ1∂X1

∂u1

∂ξ1

]

=2 E

L

[∂u1

∂ξ1+

1

L

∂u1

∂ξ1

∂u1

∂ξ1

]

(6.54)

The discretization of virtual displacement δu1 is done by applying equation (2.42), by meansof which also the derivation of displacement u1 with respect to natural coordinate ξ1 can becomputed for an approximate displacement state ue, with equation (2.46).

δu1(ξ1) ≈

NN∑

i=1

δuei1 N i(ξ1) =

[

N1(ξ1) · · · NNN (ξ1)]

δue11...

δueNN1

= N(ξ1) δue

∂u1(ξ1)

∂ξ1≈

NN∑

i=1

uei1 N i

;1(ξ1) =[

N1;1(ξ1) · · · NNN

;1 (ξ1)]

δue11...

δueNN1

= N;1(ξ1) ue

(6.55)

Insertion of equation (6.55) into equation (6.53) yields the approximation of internal virtualwork.

δWint ≈

1∫

−1

A(

Dεξ +Dnlεξ(u

e))

N(ξ1) δue Sξ11(u

e)L

2dξ1

=

1∫

−1

A(

B(ξ1) + Bnl(ue, ξ1))

δue Sξ11(ue)L

2dξ1 = δue · re

i (ue)

(6.56)

Here, the approximation of the Green Lagrange strain was accomplished with the help of theB-operator B(ξ1) and Bnl(ue, ξ1) and the functional dependence on the displacement vector ue

was specified as consequence of equation (6.55).

δEξ11 ≈(

Dεξ +Dnlεξ(u

e))

N(ξ1) δue =

(


δue (6.57)


By transformation of equation (6.56), the internal forces vector

rei (u

e) =

1∫

−1

A(

BT (ξ1) + Bnl T (ue, ξ1))

Sξ11(ue)L

2dξ1 (6.58)

can be developed with the differential operator B(ξ1) of geometrically linear truss elements(generalisation of equation (2.60)) and the non-linear deformation-dependent differentialoperator Bnl(u1, ξ1)

B(ξ1) =2

LN;1(ξ1) Bnl(ue, ξ1) =

4

L2

∂u1

∂ξ1N;1(ξ1) (6.59)

with the second Piola Kirchhoff stress in natural coordinates Sξ11 being computable withthe help of equations (6.54) and (6.55) for a prescribed displacement state ue.

6.2.2.2 Tangential element stiffness matrix

By discretization of linearized internal virtual work (6.49) in natural coordinates,

∆δWint =

1∫

−1

A (∆δEξ11 Sξ11 + δEξ11 ∆Sξ11)L

2dξ1 (6.60)

the tangential stiffness matrix of hierarchically generated truss elements is derived. From theterms in equation (6.60), Sξ11 is already known (equations (6.54) und (6.55)) and δEξ11 is alreadyderived in equations (6.53) and (6.54) and discretisized in equation (6.57). The approximationof ∆Sξ11 can be done with the help of the constitutive law (6.45) and with the equivalence ofvariation and linearization based on equation (6.57).

∆Sξ11 = E∆Eξ11 ≈ E(

Dεξ +Dnlεξ(u

e))

N(ξ1) ∆ue = E(


∆ue (6.61)

∆δEξ11 is derived from (6.50), by applying the Jacobi transformation (see (6.54)).

∆δEξ11 =2

L

∂

∂ξ1∆u1

2

L

∂

∂ξ1δu1 =

4

L2

∂

∂ξ1∆u1

∂

∂ξ1δu1 = Dεξ∆u1 Dεξδu1 (6.62)

The approximation of term ∆δEξ11 is performed with the help of differential operator Bg(ξ1),which in the special case of one-dimensional elements corresponds to the differential operatorof geometrically linear truss elements. This can be shown with discretization of incrementaldisplacement ∆u1 and the virtual displacement δu1 in equation (6.62).

∆δEξ11 ≈ δue ·Bg T (ξ1) Bg(ξ1) ∆ue = δue ·BT (ξ1) B(ξ1) ∆ue (6.63)

Insertion of equations (6.57), (6.61) and (6.63) into equation (6.60) yields the approximation ofthe linearized internal virtual work

∆δWint ≈ δue ·(ke

g(ue) + ke

m(ue))

∆ue (6.64)


with the deformation-dependent geometric element stiffness matrix (Bg(ξ1) = B(ξ1))

keg(u

e) =

1∫

−1

A Bg T (ξ1) Sξ11(ue) Bg(ξ1)

L

2dξ1 (6.65)

and the deformation-dependent material element stiffness matrix.

kem(ue) =

1∫

−1

EA(

BT (ξ1) + Bnl T (u1, ξ1)) (

B(ξ1) + Bnl(u1, ξ1)) L

2dξ1 (6.66)

Linearization of the element vector of internal forces is also determined with equation (6.64).

∆rei (u

e) =(ke

g(ue) + ke

m(ue))

∆ue (6.67)

It should be noted once more that the utilization of the B-operator B(ξ1), instead of Bg(ξ1)operator, in equation (6.65) is limited to the special case of one-dimensional elements.

6.2.3 Linear truss element

The detailed elaboration of the two-noded linear finite truss element, as part of the familyof hierarchically generated finite truss elements, presented in chapter 6.2.2, is performed byreduction of element vectors, differential operators, matrix of ansatz functions and elementmatrices. The definition of the matrix of ansatz functions N(ξ1), the element displacement vectorue, as well as the differential operator B(ξ1) = B can be inferred from derivations dealing withthe linear theory in chapter 2.2.2 (equations (2.52), (2.53) and (2.60)). The non-linear part of theB-operator Bnl(u1, ξ1) = Bnl(u1) is calculated according to (6.59) by introducing the derivativesof ansatz functions N 1

;1 = −1/2 and N 2;1 = 1/2 according to (2.51),

B(ξ1) = B = Bg =1

L

[

−1 1

]

Bnl(u1) =2

L2

∂u1

∂ξ1

[

−1 1

]

(6.68)

with the approximation

∂u1

∂ξ1≈ N;1(ξ1) ue = N;1 ue =

1

2

[

−1 1

]

ue (6.69)

being constant in ξ1. When generating the element vector of internal forces according to (6.58),especially when integrating, it turned out to be advantageous that, due to the linear displacementansatz, the B operators (6.68) and the normal stress ((6.54) be independent of the naturalcoordinate ξ1.

rei (u

e) = A L Sξ11(ue)

(

BT + Bnl T (ue))

(6.70)


The use of differential operators according to equation (6.68) in the end yields the elementvector of internal forces of the two-noded truss element.

rei (u

e) = A Sξ11(ue)

(

1 +2

L

∂u1

∂ξ1

) [

−1

1

]

(6.71)

By executing the products of matrices in equations (6.65) und (6.66) with differential operatorsof the two-noded truss element (6.68) we get the geometric

keg(u

e) =A

LSξ11(u

e)

[

1 −1

−1 1

]

(6.72)

and the material element stiffness matrix.

kem(ue) =

A E

L

(

1 +2

L

∂u1

∂ξ1

)2[

1 −1

−1 1

]

(6.73)

For ue = 0, the material element stiffness matrix equals the elastic element stiffness matrixke

m(ue = 0) = kee = ke given in (2.67).

Chapter 7

Solution of non-linear staticstructural equations

The revolving point of this chapter is the numerical solution of non-linear vector equations(6.25)

ri(u) = r (7.1)

of time-independent structural mechanics as well as the assessment of deformation statesregarding the stability of the structure involved, as part of a geometrically non-linear finiteelement analysis.

7.1 Strategies

The procedures for solving equation (7.1) consist of two components:

• - The control methods or control procedures describe the successive load application.

• - The iteration procedures serve the purpose of finding the equilibrium states for an incre-mental load change.

Only the combination of both, however, enables the solution of high degree non-linear problemsin civil engineering.

As shown in figure 7.1, the load application control is realized by controlling the load, thedisplacement or the so called arc length along the equilibrium path which characterises theequality of internal forces and external loads. The control of load application is accomplishedby parametrization of equation (7.1) with the load factor λ,

ri(u) = λr [0, λ] =

NT⋃

n=1

λn+1 − λn (7.2)

where λ is applied step-wise for the numerical transformation of the solution. The load stepsinduced in such a way are numbered in analogy with the time steps of transient problems,

189


u0 u1 u2 un un+1u

λ0

λ1

λ2

λn

λn+1

λ

Iteration

Control parameter

LoadDisplacement

Arc-length

un = u0n+1 u1

n+1 ukn+1u

k+1n+1 un+1

u

λn

λn+1λ

Figure 7.1: Load parameter control (λn → rin) and local iteration (Newton-Raphson)

with n = 1, . . . , NT . The equilibrium state corresponding to the load factor λn, is describedwith displacements un. This explains why with the numerical solution of the non-linearequation (7.1), only discrete supporting points of the path can be acquired, not the continuousequilibrium path. The number, the separation, and the arrangement of supporting pointsis determined by the control procedure which shall be derived in detail in chapter 7.3. Thestep-wise numerical solution from one supporting point to the other is accomplished for exampleby means of the pure Newton-Raphson method, the modified Newton-Raphson methodor the Quasi-Newton methods. Starting from an equilibrium state characterised by thedisplacement state un and the internal forces ri(un) calculable from that state, we numericallycompute the equilibrium point for the load factor λn+1, by means of solution (un+1, ri(un+1)).In figure 7.1, we see an example of the schematic presentation of equilibrium iteration of theload-controlled pure Newton-Raphson procedure of a one-dimensional non-linear equationri(u) = r. The derivation and the numerical properties of this and the other methods arediscussed in chapter 7.2.

7.2 Iteration methods

All methods presented here are based on the multi-dimensional Taylor row expansion of theinternal forces vector at a deformation state determined by the structural displacement vectoruk

n+1.

ri(uk+1n+1) = ri(u

kn+1) +

∂ri(ukn+1)

∂ukn+1

(

uk+1n+1 − uk

n+1

)

+1

2

∂2ri(ukn+1)

∂uk 2n+1

(

uk+1n+1 − uk

n+1

)2

+ · · · (7.3)

The result of a Taylor row is approximated by the vector of internal forces of the displacementstate uk+1

n+1 in the neighbourhood of point ukn+1. It should be noted that the index k, used to des-

ignate both neighbouring displacement states, stands for the iteration index in dealing with theiteration procedure. To generate the iteration procedure, the Taylor row expansion is abortedafter the linear term and the difference of both displacement states is termed displacement


increment ∆u = uk+1n+1 − uk

n+1.

ri(uk+1n+1) = ri(u

kn+1) +

∂ri(ukn+1)

∂ukn+1

∆u (7.4)

The partial derivative of the system vector of internal forces with respect to the actualdisplacement state uk

n+1 contained in this equation is the tangential stiffness matrix of thisstate (see chapter 6.1.5).

ri(uk+1n+1) = ri(u

kn+1) + Kt(u

kn+1) ∆u ∆u = uk+1

n+1 − ukn+1 (7.5)

According to equation (6.41), Kt(ukn+1)∆u is identical to the linearization of the vector of

internal forces ∆ri(ukn+1), so that the equation (7.5) can be interpreted as incremental change

of internal forces for an incremental change of the displacement state.

7.2.1 Single step method

Single step methods are no iteration methods of course. For reasons of completeness of discussionand possibility of comparison to true iteration methods, these procedures should neverthelessbe shortly elaborated. The quality of the numerical solution of single step methods is poor indirect comparison to iteration procedures. These procedures were however frequently used in thepast and implemented in commercial FEM programs as well. The reasons for this were smallercomputation times compared to iteration procedures and the programming structure of FEMsoftware which is transferable nearly without any alterations. Today, these procedures are nolonger in use thanks to advanced development of computers and FEM software.

The fundamental guideline of the single step methods rests on the incremental load application,where within one step the displacement state un goes over to the displacement state un+1 bymeans of a quasi-linear calculation. For that we use the Taylor row expansion (7.5) at the stateun and we seek the fulfilment of the non-linear equation (7.2).

ri(un+1) = ri(un) + Kt(un) (un+1 − un) = λn+1 r (7.6)

If we further postulate the equality of internal forces r i(un) and the external loads λnr - whichby the way, is far from being true - we can obtain un+1.

un+1 = K−1t (un) (λn+1 − λn) r + un (7.7)

Since the internal forces are approximated only linearly, the solution vector un+1 does notsatisfy the equilibrium condition (7.2) resulting in the fact that for a successive load increasethe equilibrium condition is generally not fulfilled in the beginning of the load step. Thereby,the solution path gets more and more remote from the equilibrium path.

7.2.2 Pure Newton-Raphson method

The development of the vector of internal forces into a Taylor row according to equation (7.5),and the requirement for fulfilment of the equilibrium relation (7.2), comprise the basis of theNewton-Raphson method which was first published by Cauchy in 1847.

ri(uk+1n+1) = ri(u

kn+1) + Kt(u

kn+1) ∆u = λn+1 r (7.8)


In this context, the displacement increment ∆u is often denoted as the Newton correction. Asit will soon become clear, in the beginning of load step n+1 we can set out from an equilibriumstate.

ri(un) = λn r (7.9)

In order to locate the equilibrium for load factor λn+1, we now demand that the approxima-tion of internal forces of the deformation state u1

n+1, based on Taylor row expansion at thedevelopment point un = u0

n+1, fulfils the equilibrium relation at the end of the load step.

ri(u1n+1) = ri(u

0n+1) + Kt(u

0n+1) ∆u = λn+1 r (7.10)

The solution of this equation yields the displacement increment, which gives the approximationof solution (u1

n+1) with equation (7.5).

∆u = K−1t (u0

n+1)(λn+1 r − ri(u

0n+1)

)u1

n+1 = u0n+1 + ∆u (7.11)

Generally, the first calculation step of an iteration method is termed predictor. The quality ofsolution u1

n+1 can be evaluated by means of the so called residuum λn+1 r − ri(u1n+1). In the

case of equilibrium which is attainable only hypothetically, the residuum vanishes. In order toclose in on this hypothetical ideal case, we can repeat the procedure defined by equations (7.10)and (7.11) for calculating the displacement vector u2

n+1 based on u1n+1. The displacement

vector uk+1n+1 is generally calculated based on the approximated solution uk

n+1. The methodfor iterative solution of the non-linear vector equation (7.2) generated by this is the pureNewton-Raphson procedure, or shortly Newton-Raphson method. The updating relation,described as corrector, and the resulting improvement of the solution vector uk+1

n+1 follow fromtransformation of equation (7.8), having in mind the definition of the displacement increment∆u.

∆u = K−1t (uk

n+1)(

λn+1 r − ri(ukn+1)

)

uk+1n+1 = uk

n+1 + ∆u (7.12)

The procedure of solution improvement according to (7.12) is repeated until the equilibrium isreached approximately (see figure 7.2). Here it should be stressed that the exact equilibriumcannot be reached. Since this is a vector-value problem, we generate a scalar-valued criterion, aso called convergence criterion in order to judge whether a sufficient accuracy of the equilibriumapproximation is reached. Due to the versatility of possibilities and the general application ofthis criterion to arbitrary iteration procedures, the derivation of suitable convergence criteriais left to inquiries of chapter ??. If the criterion is fulfilled, (here, because of completeness ofderivations, as example the criterion

ηk+1u =

‖∆u‖

‖uk+1n+1 − un‖

≤ ηu (7.13)

is given as an example (ηu is given by the user) the solution uk+1n+1 is approximated accurately

enough and the next load step can be solved analogously.

The pure Newton-Raphson method has the following properties (see Deuflhard &

Hohmann [51]):


un = u0n+1 u1

n+1|ukn+1 uk+1

n+1 un+1u

∆u ∆u

ri(u0n+1) = λn r

ri(u1n+1)|ri(u

kn+1)

ri(uk+1n+1)

ri(un+1) = λn+1 rri

λn+1r − ri(u0n+1)

λn+1r − ri(ukn+1)

λn+1r − ri(uk+1n+1)

Kt(u0n+1)

Kt(ukn+1)

uk+1n+1 = uk

n+1 + ∆u

Figure 7.2: Pure Newton-Raphson Iteration

• - The pure Newton-Raphson method converges to a solution if the start vector u0n+1 is

close enough to the solution.

• - The pure Newton-Raphson method converges locally quadratic.

The condition for convergence of the Newton-Raphson method can be fulfilled by succes-sive loading of the structure. If convergence is not reached, the load step must be smaller. Wecan, for example, adapt the size of the load step based on the convergence behaviour (see e.g.Reitinger [127]), which guaranties the convergent pursuit of the equilibrium path within thescope of a non-linear FEM analysis. It remains to clarify the notion of local quadratic conver-gence. Local means that the quadratic convergence is reached in the vicinity of solution, whichpractically means in the last one to two steps. The convergence rate describes the number ofnecessary iterations, the higher the convergence rate, the less iteration steps are needed to solvethe non-linear equilibrium. It can be determined by means of evolution of the displacementincrement. The convergence rate ... is defined by the following equation (see equation (7.13)).

If the displacement icrement decreases by the square, we talk about a quadratic convergence.

With the pure Newton-Raphson method we have presented the standard tool of modern non-linear FEM analysis. We can grade it as robust in connection with suitable procedures for loadparameter control. For solution of larger systems with a correspondingly higher number of DOF,this method however has proven to be very computation-time-consuming, since in each iterationstep a linear equation system (7.12) must be solved. In order to circumvent this equation solution,which often needs to be performed and on top of that is costly, methods were developed whichapproximate the deformation-dependent tangential stiffness matrix or its inverse. One of thesemethods is the modified Newton-Raphson method; an entire group of such methods comprisethe Quasi-Newton procedures.

7.2.3 Modified Newton-Raphson method

The development of the modified Newton-Raphson method requires that we replace, inequation (7.12), the deformation-dependent tangential stiffness matrix, which therefore must


un = u0n+1 u1

n+1|ukn+1 u

k+1n+1 un+1

u

∆u ∆u

ri(u0n+1) = λn r

ri(u1n+1)|ri(u

kn+1)

ri(uk+1n+1)

ri(un+1) = λn+1 rri

λn+1r − ri(u0n+1)

λn+1r − ri(ukn+1)

λn+1r − ri(uk+1n+1)

Kt(un)

Kt(un)

uk+1n+1 = uk

n+1 + ∆u

Figure 7.3: Modified Newton-Raphson Iteration

be generated anew in each step, with the tangential stiffness matrix Kt(ukn+1) of the last known

equilibrium state Kt(un) = Kt(u0n+1) which is constant throughout the iteration.

∆u = K−1t (u0

n+1)(

λn+1 r − ri(ukn+1)

)

uk+1n+1 = uk

n+1 + ∆u (7.14)

The tangential stiffness has to be generated in the first iteration step and then inverted. Inthe next load steps it suffices to multiply the inverse tangent with the iteratively changeableright side. Here, we have a practically convenient operation of triangle projection and then theback-substitution. As a consequence, the number of required operations per iteration step isnotably reduced in comparison with the pure Newton-Raphson method. It is unfavourable,on the other hand, that with this modification the property of quadratic convergence is lost,which calls for more iteration steps than originally, within a single load step. To go on, themodified method is not as robust as the pure Newton-Raphson method, which practicallymeans that with this reduced procedure a convergent solution is impossible, unlike the originalprocedure which converges to the equilibrium state.

7.3 Control of iteration procedures

After the derivations in chapter 7.1, we now must develop strategies for controlling the loadparameter λ. The latter globally controls the successive load increase, and, within the scope ofarc-length procedures and the displacement-controlled calculus, locally the applied external load(λn+1 is altering with k).

7.3.1 Load-incrementing and control

The simplest possibility to control the loads acting on the system is the step-wise increase of theload with each load step n. This external load is constant during the equilibrium iteration which


Load step loop n = 0, NT − 1

n+ 1 −→ n

Iteration step loop k = 0,konv

k + 1 −→ k

Generation of external loads λn+1r, Pre-described displacements u0n+1 = un (7.2)

Calculation of internal forces ri(ukn+1) (6.8)

Calculation of tangential stiffness matrix Kt(ukn+1) (6.35-6.37)

Calculation of increment ∆u = K−1t (uk

n+1) (λn+1 r − ri(ukn+1)) (7.12)

Updating of displacements uk+1n+1 = uk

n+1 + ∆u (7.12)

Convergence criterion check, e.g. ηk+1u ≤ ηu (??)

Figure 7.4: Algorithm of load-controlled pure Newton-Raphson iteration

ensures the attainment of equilibrium state correlated to the corresponding load λn+1, aftersuccessful iteration. The equilibrium path and the equilibrium states obtained in the scope ofload-controlled pure Newton-Raphson iterations were already sketched in figure 7.1, showing aone-dimensional NEQ = 1) example. By combining the load control and the iteration procedurespresented in chapter 7.2, the following algorithms for numerical solution of non-linear vectorequation (7.1) are generated:

• In connection with the pure Newton-Raphson method the algorithm sketched in thefigure 7.4 should be used.

• In the iteration step k = 0, the vector of internal forces is given according to equation (7.2)with ri(u

0n+1) = λnr, assumming that the system is in equilibrium at the beginning of the

load step. Further we can drop the check of the convergence criterion since the solutionafter one iteration step is the correct one only in the linear case. This makes the pureNewton-Raphson method easier to arrange with less numerical effort, if the algorithmis executed as a predictor-corrector method.

• The algorithm of the modified Newton-Raphson method differs from the predictor-corrector formulation of the original method as follows: inversion of tangent stiffness inthe predictor is accomplished by means of triangle projection, in the corrector the tangentdoes not need to be calculated, and instead of inversion of up-to-date tangential stiffness,the back-substitution with the triangle-projected tangent Kt(u

0n+1) comes into play

• In order to realize the Broyden-Fletcher-Goldfarb-Shanno-procedure, in compar-ison to the modified Newton-Raphson method, the inverse tangential stiffness matrix inthe predictor should be stored and the approximated inverse of tangential stiffness matrixin the corrector should be updated. The back-substitution is replaced by multiplication ofthe right equation side with the approximated inverses of the tangential stiffness matrix.

Figures 7.5 and 7.6 show the results of the pure Newton-Raphson and the modified Newton-

Raphson method within the scope of a load controlled analysis of a one-dimensional (NEQ = 1)example1.

1Three-joint-frame in exercises


u0 u1 u2 un un+1u

λ0

λ1

λ2

λn

λn+1

λ

un = u0n+1 u1

n+1 ukn+1u

k+1n+1 un+1

u

λn

λn+1λ

Figure 7.5: Combination of load control and pure Newton-Raphson method

u0 u1 u2 un un+1u

λ0

λ1

λ2

λn

λn+1

λ

un = u0n+1 u1

n+1 ukn+1u

k+1n+1 un+1

u

λn

λn+1λ

Figure 7.6: Combination of load control and modified Newton-Raphson iteration

7.3.2 Arc-length controlling method

Literature: Monographies Wessels [137], Wagner [134], Crisfield [19] und Reitinger [127].

Essays: Batoz &Dhatt [90], Crisfield [96], Crisfield [99], Crisfield [97], Crisfield [98],Kouhia & Mikkola [109], Ramm [124], Ramm [125], Riks [128], Riks [129], Riks [130],Schweizerho &Wriggers [131], Wempner [136]

In order to control the load application in the arc-length method, the load parameter canvary during the iteration. This evolution of the load parameter is formulated with a generallynon-linear scalar additional condition of the form f(u, λ) = 0, dependent on the up-to-datedisplacement state u, and the load parameter λ. As illustrated in figure 7.7, this additional


un un+1u

λn

λn+1

λ

Additional condition

f(un+1, λn+1) = 0

Equilibrium path

ri(u) − λr = 0

unu1

n+1 un + ∆uλu

λn

λn+1λ

∆uλ

∆u

∆λ

1

s

s0

Kt(un)

Figure 7.7: Extended system (7.16) and construction of predictor (7.29)-(7.32)

condition supplements to the non-linear vector equation (7.2) both forming the so-calledextended system.

ri(u) − λr = 0

f(u, λ) = 0[0, λ] =

NT⋃

n=1

λn+1 − λn (7.15)

If we want to work out alternative extended systems, we need to specify additional conditionsin form of equation (7.15). Firstly, we should discuss the general solution of extended systemsdefined by equations (7.15). Starting from an equilibrium state of load step n, which becomesan abstract numerical solution step, we should calculate the equilibrium state n + 1, whichought to satisfy both equilibrium relation and the additional condition.

ri(un+1) − λn+1r = 0

f(un+1, λn+1) = 0

(7.16)

The methods for solving the extended system (7.16) can be classified into the exact or thelinearized form, regarding the inclusion of the additional condition into the procedure. The realexecution of the arc-length method of both categories, which is elaborated next, requires thespecification of the function of additional condition f(u, λ). There is a multitude of possibilitiesin the specialized literature for this (see Wagner [134], Reitinger [127] und Crisfield [97]).

As examples, the development of two methods will be demonstrated in the remainder of thismanuscript. All other extended systems can be developed in analoguous procedures. The firstmethod is the iteration in the tangential plane. In this method, the additional condition whichdescribes the tangential plane of the previous iteration step is introduced exactly. The iter-ation in the tangential planes presents a modification of the iteration in the initial normalplane (Riks [128, 129] und Wempner [136]) and of the iteration in the updated normal plane(Ramm [124, 125]).


The second method discussed here was suggested by Crisfield [96]. In this method, the ad-ditional condition is the equation of a general sphere, which is characterized by radius s andthe mid-point designated by un and λn. The inclusion of the additional condition is commonin the exact as well as in the linearized form. Crisfield [96] and Ramm [124] concluded thatfor multiple DOF systems (In the order of FEM simulations), the load parameter has smallsignificance in the iteration on the sphere surface, and suggested that this term be neglected.This way the method of iteration on the cylinder surface (ψ2 = 0) was created.

7.3.2.1 Iterative solution of extended equation system

As in the case of solution of the non-linear equilibrium, the numerical solution of non-linearextended system (7.16) is based on the Taylor row expansion of the equilibrium relation, towhich the Taylor row expansion of additional condition is added. The development point ofTaylor rows regarding parameters u and λ is defined by the prescribed state uk

n+1, λkn+1. Here

it should be noted that due to the functional dependence of equilibrium on the load parameter,additional terms appear in the linearization, compared to equation (7.5). The row expansion ofequilibrium, halted after the linear term, is given by the following equation.

ri(uk+1n+1) − λk+1

n+1r = ri(ukn+1) − λk

n+1r

+∂ri(u

kn+1)

∂ukn+1

︸︷︷︸

Kt(ukn+1)

(uk+1n+1 − uk

n+1)

︸︷︷︸

∆u

−∂(λk

n+1r)

∂ukn+1

︸︷︷︸

0

(uk+1n+1 − uk

n+1)

+∂ri(u

kn+1)

∂λkn+1

︸︷︷︸

0

(λk+1n+1 − λk

n+1) −∂(λk

n+1r)

∂λkn+1

︸︷︷︸

r

(λk+1n+1 − λk

n+1)

︸︷︷︸

∆λ

(7.17)

In this equation, we can recognize the partial derivatives of internal forces with respect todisplacements, the tangential stiffness matrix as well as the increment of displacements ∆u andthe analogously defined increment of load parameter ∆λ.

∆u = uk+1n+1 − uk

n+1 ∆λ = λk+1n+1 − λk

n+1 (7.18)

The internal forces are independent of the load parameter and the external loads are deformation-dependent, what makes the corresponding derivatives vanish. The generation of derivatives ofexternal loads λk

n+1r according to the load parameter can be written directly. With that, weeventually obtain the Taylor expansion of equilibrium relation, the right side of which equalszero.

ri(uk+1n+1) − λk+1

n+1r = ri(ukn+1) − λk

n+1r + Kt(ukn+1)∆u − r∆λ = 0 (7.19)

The Taylor row expansion of additional condition can be done purely formally without detailedspecification.

f(uk+1n+1, λ

k+1n+1) = f(uk

n+1, λkn+1) +

∂f(ukn+1, λ

kn+1)

∂ukn+1

︸︷︷︸

f,u(ukn+1, λ

kn+1)

·∆u +∂f(uk

n+1, λkn+1)

∂λkn+1

︸︷︷︸

f,λ(ukn+1, λ

kn+1)

∆λ (7.20)


With definitions

f,u(ukn+1, λ

kn+1) =

∂f(ukn+1, λ

kn+1)

∂ukn+1

f,λ(ukn+1, λ

kn+1) =

∂f(ukn+1, λ

kn+1)

∂λkn+1

(7.21)

equation (7.20) can be formed compactly. Here, the validity of (7.16) is reguired.

f(uk+1n+1, λ

k+1n+1) = f(uk

n+1, λkn+1) + f,u(uk

n+1, λkn+1) · ∆u + f,λ(uk

n+1, λkn+1) ∆λ = 0 (7.22)

Equations (7.19) and (7.22) together yield, along with conditions r i(uk+1n+1) − λk+1

n+1r = 0 and

f(uk+1n+1, λ

k+1n+1) = 0, a linear equation system with the solution vector of displacement and load

parameter increments.

Kt(ukn+1) −r

fT,u(uk

n+1, λkn+1) f,λ(uk

n+1, λkn+1)

∆u

∆λ

=

λkn+1r − ri(u

kn+1)

−f(ukn+1, λ

kn+1)

(7.23)

Since this equation system is non-symmetric, and unlike the tangential stiffness matrix has abanded configuration, the direct solution calls for a substantial numerical effort. To circumventthis problem, the so-called partitioning techniques or block elimination procedures were firstsuggested by Batoz &Dhatt [90] to obtain the solution. To develop this procedure, the firstrow of (7.23) is solved for the displacement increment.

∆u = K−1t (uk

n+1)(λkn+1r − ri(u

kn+1)) + K−1

t (ukn+1)r∆λ (7.24)

Since ∆λ is not known, the displacement increment cannot be calculated. Instead, it is possibleto generate block solutions,

∆ur = K−1t (uk

n+1)(λkn+1r − ri(u

kn+1)) ∆uλ = K−1

t (ukn+1)r (7.25)

which give the displacement increment ∆u for the known ∆λ.

∆u = ∆ur + ∆uλ∆λ (7.26)

To determine the load factor increment, the second row of (7.23) is solved and the displacementincrement ∆u is replaced by block solutions ∆ur and ∆uλ according to (7.26).

f,u(ukn+1, λ

kn+1) · ∆u + f,λ(uk

n+1, λkn+1) ∆λ= −f(uk

n+1, λkn+1)

f,u(ukn+1, λ

kn+1) · (∆ur + ∆uλ∆λ) + f,λ(uk

n+1, λkn+1) ∆λ= −f(uk

n+1, λkn+1)

(7.27)


This equation contains ∆λ as the only unknown, which makes it the appropriate equation fordetermination of ∆λ.

∆λ = −f(uk

n+1, λkn+1) + f,u(uk

n+1, λkn+1) · ∆ur

f,u(ukn+1, λ

kn+1) · ∆uλ + f,λ(uk

n+1, λkn+1)

(7.28)

Thus it possible to replace the solution of the non-symmetrical linear equation system (7.23)by a series of evaluations of equations (7.25), (7.28), and finally (7.26). At the end of thissolution process, ∆u and ∆λ are determined, so that the displacements and the load parametercan be updated by means of (7.18). The iterative repetition of the procedure, generated withincrement calculation and updating of system variables, gives us, after the fulfilment of theconvergence criterion (e.g. equation (??)), an equilibrium solution characterized by un+1, whichsimultaneously fulfils the additional condition f(un+1, λn+1) = 0.

7.3.2.2 Predictor

If we observe the additional conditions of extended systems, we find out that the additionalconditions are not determined in the predictor step. That means the application of arc-lengthmethods, which are next elaborated, is limited to corrector iteration. That implies thatprocedures other than those already shown must be derived to calculate the predictor. Theprocedure elaborated here was developed in Wagner [134], based on another extended system.Still, we want to rely on a transparent argumentation to develop this procedure (see figure 7.7).Starting from the last equilibrium state (un, λn), the load parameter is increased by one andthe corresponding displacement increment

∆uλ = K−1t (un) [(λn + 1) − λn] r = K−1

t (un) 1 r (7.29)

is calculated as well as the distance to the previous equilibrium point.

s0 =√

∆uλ · ∆uλ + 1 (7.30)

It should be noted that ∆uλ of the predictor is identical to displacement increment ∆u, sinceri(un) = λnr describes an equilibrium state. This statement can be verified by means ofequation (7.25) with ∆λ = 1.

∆ur = K−1t (un)(λnr − ri(un)) = 0 ∆u = ∆ur + ∆uλ∆λ+ ∆uλ1 (7.31)

By comparing the distance s0 and the prescribed arc-length, we can scale load parameterincrements and displacements.

∆λ =s

s01 =

s√

∆uλ · ∆uλ + 1∆u =

s

s0∆uλ =

s√

∆uλ · ∆uλ + 1∆uλ (7.32)

The updating of displacements and load parameters follows in a standard way.

u1n+1 = un + ∆u λ1

n+1 = λn + ∆λ (7.33)


7.3.2.3 Iteration on a general sphere

The additional condition introduced by Crisfield [96] describes a general sphere defined byNEQ system displacements and the load parameter, the centre point of which is determined bythe ultimately reached equilibrium state and the radius of which s (arc-length) is prescribed.

f(uk+1n+1, λ

k+1n+1) =

√

(uk+1n+1 − un) · (uk+1

n+1 − un) + ψ2 (λk+1n+1 − λn)2 − s = sk+1 − s (7.34)

In equation (7.34), the scaling factor ψ was introduced in the formulation according to Re-

itinger [127]. This factor enables the adjustment of physical units as well as weighting of thedisplacement term and the load parameter term. For ψ = 0 equation (7.34), in the additionalcondition of iteration, turns into a cylinder surface. For ψ = 1, we get the formulations as theywere utilized by Wagner [134] and for ψ2 = Ψ2 r ·r, we get the formulation by Crisfield [19].The limit value ψ → ∞ defines a procedure equivalent to the load control analysis. In orderto perform the block elimination procedure, we must generate the linearization of additionalconditions with respect to the displacements and the load parameter. While the gradient withrespect to up-to-date displacements is determined by means of Gateaux derivative,

∂f(ukn+1, λ

kn+1)

∂ukn+1

· ∆u =d

dη

[√

‖ukn+1 + η∆u − un‖2 + ψ2 (λk

n+1 − λn)2 − s

]∣∣∣∣η=0

=∆u · (uk

n+1 − un) + (ukn+1 − un) · ∆u

2√

(ukn+1 − un) · (uk

n+1 − un) + ψ2 (λkn+1 − λn)2

=uk

n+1 − un

sk · ∆u

(7.35)

for the derivative with respect to λ the ordinary differentiation with respect to the scalar-valuedload factor is sufficient.

∂f(ukn+1, λ

kn+1)

∂ukn+1

=uk

n+1 − un

sk

∂f(ukn+1, λ

kn+1)

∂λkn+1

=λk

n+1 − λn

sk(7.36)

Based on equation (7.28) and the just generated derivatives, we can calculate the load factorincrement in order to iterate on a general sphere surface according to (7.28).

∆λ = −f(uk

n+1, λkn+1) s

k + (ukn+1 − un) · ∆ur

(ukn+1 − un) · ∆uλ + ψ2 (λk

n+1 − λn)(7.37)

By introducing sk = f(ukn+1, λ

kn+1) + s according to (7.34), we obtain the following form for

calculating the load increment ∆λ.

∆λ = −f(uk

n+1, λkn+1) (f(uk

n+1, λkn+1) + s) + (uk

n+1 − un) · ∆ur

(ukn+1 − un) · ∆uλ + ψ2 (λk

n+1 − λn)(7.38)

The displacement increment ∆u can be calculated upon evaluation of the above equation bymeans of block solutions ∆ur and ∆uλ according to (7.26). After this, the displacements andthe load parameter must be updated according to (7.18) and convergence of the solution mustbe checked.

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intro to adv finite element analysis...!!!!

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