Effective mechanical properties of fiber-reinforced composites strongly depend on the microstructure, including the fibers' orientation. Studying this dependency, we identify the variety of fiber orientation tensors up to fourth-order using irreducible tensors and material symmetry. The case of planar fiber orientation tensors, relevant for sheet molding compound, is presented completely. Consequences for the reconstruction of fiber distributions and mean field homogenization are presented.
Author(s): Julian Karl Bauer
Series: Schriftenreihe Kontinuumsmechanik im Maschinenbau
Publisher: KIT Scientific Publishing
Year: 2023
Language: English
Pages: 250
City: Karlsruhe
Zusammenfassung
Summary
Acknowledgments
Contents
1 Introduction
1.1 Motivation
1.2 Research objectives
1.3 Originality
1.4 State of the art
1.4.1 Fiber orientation tensors
1.4.2 Fiber orientation distribution reconstruction
1.4.3 Orientation-averaging mean field homogenization
1.5 Outline
1.6 Notation, frequently used acronyms, symbols, and operators
2 Continuum Mechanics
2.1 Motivation
2.2 Kinematics and deformation
2.3 Time derivatives
2.4 Stress
2.5 Balance equations
2.5.1 Divergence theorem
2.5.2 Transport theorem
2.5.3 General balance equation
2.5.4 Specific balance equations at regular points
2.6 Material modeling and closure of balance equations
2.7 Linear elasticity
3 Tensoralgebra
3.1 Motivation
3.2 Linear invariant decomposition
3.3 Irreducible tensors
3.4 Material symmetry
3.5 Irreducible tensors constrained by material symmetry
3.5.1 Second-order tensors
3.5.2 Fourth-order tensors
4 Variety of Fiber Orientation Tensors
4.1 Introduction
4.2 Fiber orientation
4.3 Orientation tensors of first kind
4.4 Orientation tensors of third kind
4.5 Variety of second-order orientation tensors
4.5.1 Parameterizations of the orientation triangle
4.6 Variety of fourth-order orientation tensors
4.6.1 Harmonic decomposition
4.6.2 Parameterizations and admissible parameter ranges
4.6.3 Transversely isotropic case
4.6.4 Orthotropic case
4.6.5 Planar case
4.7 A note on closure approximations
4.8 Summary and conclusions
5 Fiber Orientation Distributions Based on Planar Fiber Orientation Tensors of Fourth Order
5.1 Introduction
5.2 Directional measures as microstructure descriptors
5.2.1 Fiber orientation distribution function
5.2.2 Fiber orientation tensors
5.3 Admissible and distinct planar fiber orientation tensors
5.4 Reconstructed fiber orientation distribution functions
5.4.1 Transition from 3D into 2D
5.4.2 Truncated fiber orientation distribution function in a 2D-framework
5.4.3 Maximum entropy reconstruction
5.4.4 Visualization of reconstructed planar fiber orientation distribution functions
5.4.5 Reconstruction solely based on second-order fiber orientation tensors
5.5 Summary and conclusion
6 On the Dependence of Orientation Averaging Mean Field Homogenization on Planar Fourth-Order Fiber Orientation Tensors
6.1 Introduction
6.2 Sheet molding compound and planar microstructures
6.3 Planar fourth-order fiber orientation tensors
6.4 Orientation averages
6.4.1 Reformulation of the explicit Advani-Tucker orientation average
6.4.2 Direct numerical integration and the adaptive scheme based on angular central Gaussian distributions
6.4.3 Orientation average by reconstructed planar FODF based on a maximum entropy method
6.5 Orientation-averaging mean field homogenization
6.5.1 Two-step Hashin-Shtrikman
6.5.2 Orientation-averaging Mori-Tanaka following Benveniste1986
6.5.3 Direct orientation average of a transversely isotropic stiffness
6.5.4 Direct orientation average of a transversely isotropic compliance
6.6 Graphical representation of elasticity tensors
6.7 Effective stiffnesses: Polar plots and the dependence on planar fourth-order fiber orientations
6.7.1 Visualization setup
6.7.2 Observations on bounds
6.7.3 Observations on the shape of the Young's modulus
6.7.4 Observations on the shape of the generalized bulk modulus
6.7.5 Implications of closure approximations based on second-order fiber orientation tensors
6.8 Summary and conclusions
7 Conclusions and Outlook
7.1 Conclusions
7.2 Outlook
A Appendices to Chapter 4
A.1 Material symmetries of second-order tensors
A.2 Parameter sets of specific second-order orientation tensors
A.3 Coefficient-wise extrema of moment tensors
A.4 Parameterization of admissible N^ortho with Isotropic N
B Appendices to Chapter 5
B.1 Kelvin-Mandel notation and completely symmetric tensors of fourth order
B.2 Parameter sets in polar plots
C Appendices to Chapter 6
C.1 Kelvin-Mandel notation and completely symmetric tensors of fourth order
C.2 Harmonic decomposition of transversely isotropic elasticity tensors
C.3 Reformulation of the Advani-Tucker orientation average
C.4 Advani-Tucker Orientation Average of Minor Symmetric Tensors
C.5 Connection to Notation in Kehrer2020
C.6 Component representations of tensor inversions
C.7 Parameter sets in polar plots
C.8 Effective complementary elastic energy density
Bibliography
