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Development and Implementation of Robust Large Deformation and Contact Mechanics Capabilities in Process Modelling of Composites

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Development and Implementation of Robust Large Deformation and Contact Mechanics Capabilities in Process Modelling of Composites DEVELOPMENT AND IMPLEMENTATION OF ROBUST LARGE DEFORMATION AND CONTACT MECHANICS CAPABILITIES IN PROCESS MODELLING OF COMPOSITES by AMIR OSOOLY B.Sc., Sharif University of Technology, 1989 M.Sc., Sharif University of Technology, 1993 A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies (Civil Engineering) The University of British Columbia (Vancouver) April 2008 © Amir Osooly, 2008 Abstract Autoclave processing of large scale, one-piece structural parts made of carbon fiber- reinforced polymer composite materials is the key to decreasing manufacturing costs while at the same time increasing quality. Nonetheless, even in manufacturing simple flat parts, residual strains and stresses are unavoidable. For structural design purposes and to aid in the assembly procedures, it is desirable to have proven numerical tools that can be used to predict these residual geometrical and material properties in advance, thus avoid the costly experimental trial and error methods. A 2-D finite element-based code, COMPRO, has previously been developed in-house for predicting autoclave process-induced deformations and residual stresses in composite parts undergoing an entire cure cycle. To simulate the tool-part interaction, an important source of residual deformations/stresses, COMPRO used a non-zero thickness elastic shear layer as its only interface option. Moreover, the code did not account for the large deformations and strains and the resulting nonlinear effects that can arise during the early stages of the cure cycle when the material is rather compliant. In the present work, a contact surface employing a penalty method formulation is introduced at the tool-part interface. Its material-dependent parameters are a function of temperature, degree of cure, pressure and so forth. This makes the stick-slip condition plus separation between the part and the tool possible. The large displacements/rotations and large shear strains that develop at the early stages of the cure cycle when the resin has a very low elastic modulus provided the impetus to include a large strain/deformation option in COMPRO. A new “co-rotational stress formulation” was developed and found to provide a robust method for numerical treatment of very large deformation/strain problems involving anisotropic materials of interest here. Several verification and validation examples are used to calibrate the contact interface parameters and to demonstrate the correctness of implementation and the accuracy of the proposed method. A number of comparisons are made with exact solutions, other -ii- methods, other experiments and the same models in other commercial codes. Finally, several interesting cases are examined to explore the results of COMPRO predictions with the added options. -iii- Table of Contents Abstract........................................................................................................................ ii Table of Contents........................................................................................................ iv List of Tables ............................................................................................................. vii List of Figures............................................................................................................. ix List of Symbols.......................................................................................................xviii Abbreviations.......................................................................................................... xxiv Acknowledgements.................................................................................................. xxv Chapter 1 - INTRODUCTION ................................................................................ 1 1.1 Autoclave Processing Overview.......................................................................... 2 1.2 Process Modeling................................................................................................. 4 1.2.1 Liquid Moulding......................................................................................... 5 1.2.2 Sheet-Forming............................................................................................. 8 1.2.3 Autoclave Processing................................................................................ 10 1.3 Residual Stresses in the Autoclave Process....................................................... 14 1.3.1 Tool-Part Interaction................................................................................. 15 1.3.2 Sliding Approach...................................................................................... 17 1.4 Motivation and Research Objectives ................................................................. 20 Chapter 2 - LARGE DEFORMATIONS............................................................... 25 2.1 Background........................................................................................................ 26 2.1.1 Stress and Strain Measures ....................................................................... 26 2.1.2 Conjugate Stress and Strain Definition..................................................... 29 2.1.3 Polar Decomposition of the Deformation Gradient Tensor...................... 32 2.1.4 Objectivity of the Stress Rate ................................................................... 33 2.1.5 Examples of Objective Stress Rates ......................................................... 35 2.1.5.1 Truesdell Stress Rate.......................................................................... 35 2.1.5.2 Jaumann Stress Rate .......................................................................... 37 2.1.5.3 Green-Naghdi Stress Rate.................................................................. 38 2.1.5.4 Logarithmic Stress Rate..................................................................... 39 2.1.6 General Form of Co-rotational Stress Rate............................................... 39 2.1.7 Role of Material Stiffness Tensor in Objective Stress Rates.................... 41 2.2 Co-rotational Stress Formulation for Isotropic Materials.................................. 42 2.2.1 Linear Hyperelastic Materials................................................................... 47 2.2.2 Non-linear Hyperelastic Materials............................................................ 48 2.3 Co-rotational Stress Formulation for Anisotropic Materials ............................. 49 2.3.1 Rotation of the Material Stiffness Tensor................................................. 50 2.3.2 Hypoelastic Constitutive Equation........................................................... 52 2.4 Summary and Discussion................................................................................... 53 Chapter 3 - NUMERICAL IMPLEMENTATION AND VERIFICATION OF LARGE DEFORMATION FORMULATION................................... 57 -iv- 3.1 Logarithmic Strain Rate..................................................................................... 57 3.2 Co-rotational Constitutive Equation .................................................................. 60 3.3 Determination of Total True Stress.................................................................... 61 3.4 Determination of the Element Tangent Stiffness Matrix................................... 62 3.5 Determination of the Internal Forces ................................................................. 67 3.6 Determination of the External Loads................................................................. 68 3.7 Iteration Scheme ................................................................................................ 70 3.8 Verification ........................................................................................................ 71 3.8.1 Linear Elastic Isotropic Materials............................................................. 72 3.8.1.1 Simple Shear...................................................................................... 73 3.8.1.2 Uniaxial Loading............................................................................... 78 3.8.1.3 Closed Loop Stress Path .................................................................... 79 3.8.2 Non-linear Elastic Isotropic Materials...................................................... 79 3.8.3 Linear Elastic Transversely Isotropic Materials ....................................... 80 3.8.3.1 Simple Shear...................................................................................... 81 3.8.4 Thermal Loading....................................................................................... 81 3.8.4.1 Isotropic Element............................................................................... 82 3.8.4.2 Bi-material Bar...................................................................................82 3.8.5 Response of a Cantilever Beam Undergoing Large Rotations ................. 83 3.8.6 Applicable Range of Strains in Finite Element Codes.............................. 84 3.9 Summary and Discussion................................................................................... 86 Chapter 4 - CONTACT FORMULATION.......................................................... 111 4.1 Background...................................................................................................... 111 4.2 Development of Constitutive Equation...........................................................
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