 | Kreikemeier J. A two scale finite-element-approach to analyse the damage state of composite structures: Diss. … Dr.-Ing. / Deutsches Zentrum für Luft- und Raumfahrt, Institut für Faserverbundleichtbau und Adaptronik, Braunschweig. - Köln: DLR, Bibliotheks- und Informationswesen, 2011. - vi, 132 p.: ill., graph. - (Forschungsbericht; 2011-05). - Bibliogr.: p.119-130. - Ind.: p.131-132. - ISSN 1434-8454
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1 Introduction ................................................. 1
1.1 Objectives of the present work .......................... 4
1.2 List of symbols and sets ................................ 5
1.3 Notation ................................................ 7
2 Fracture and damage modelling, state of the art .............. 9
2.1 Introduction ............................................ 9
2.2 Analytical approaches .................................. 13
2.2.1 Strength assessment ............................. 13
2.2.2 Fracture mechanical approaches .................. 16
2.2.3 Cohesive zone models ............................ 19
2.2.4 Continuum damage mechanics ...................... 19
2.2.5 Compression strength of composite materials ..... 20
2.3 Numerical treatment of fracture and damage ............. 22
2.3.1 Standard continuum mechanics approaches ......... 22
2.4 Multiscale modelling ................................... 24
2.4.1 Multiscale Finite Element Analysis .............. 26
2.4.2 Multiscale modelling by means of Fast Fourier
Transformation .................................. 28
2.4.3 VORONOI Cell Finite Element Analysis ............ 29
2.5 Conclusion ............................................. 30
3 Remarks on continuum mechanics .............................. 31
3.1 Balance equations and conservation laws ................ 33
3.2 Entropy inequality ..................................... 34
3.3 The classical boundary value problem of elasticity ..... 34
3.4 Variational formulation and weak form of equilibrium ... 35
3.5 Strain measures ........................................ 36
3.6 Stress measures ........................................ 37
4 The two scale approach ...................................... 39
4.1 Introduction ........................................... 39
4.2 The constitutive relation .............................. 40
4.3 The damage approach .................................... 41
4.3.1 The damage function ............................. 41
4.3.2 Anisotropic damage propagation .................. 43
4.4 Thermodynamical consistency ............................ 48
4.5 Localisation phenomenon ................................ 49
4.5.1 The viscous regularisation ...................... 55
4.6 Finite Element Formulation ............................. 56
4.6.1 Linearisation of the weak form .................. 56
4.6.2 Discretisation .................................. 58
4.6.3 Convergence of finite element solutions ......... 60
4.6.4 Element stiffness matrix on microscale .......... 62
4.6.5 Realisation of periodic displacement boundary
conditions ...................................... 65
4.6.6 Numerical modelling of phase boundaries ......... 69
4.6.7 Verification of the GAUSS-point method .......... 72
4.7 Homogenisation of micro information .................... 79
4.7.1 Analytical methods .............................. 79
4.7.2 Homogenisation at finite deformation ............ 81
4.8 Coupling with commercial FE tool ....................... 83
4.8.1 Element stiffness matrix on macroscale .......... 83
4.9 Conclusion ............................................. 85
5 Numerical examples .......................................... 87
5.1 Verification of the viscous regularisation ............. 87
5.2 Shell with a hole ...................................... 93
5.3 Crimped end fittings ................................... 99
5.3.1 Effective material parameters ................... 99
5.3.2 Experimental observations ...................... 103
5.3.3 Numerical model ................................ 103
5.3.4 Results ........................................ 107
5.4 Conclusion ............................................ 113
6 Conclusion and Outlook ..................................... 115
6.1 Conclusion ............................................ 115
6.2 Outlook ............................................... 116
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