Abstract
Representative volume element (RVE) has commonly been used to predict the stiffness of undamaged composite materials using finite element analysis (FEA). However, never has been an independently measured true microstructural damage quantity used in FEA to predict composite stiffness. Hence, in this work, measured fiber crack density in unidirectional fiber composite (generated using controlled fatigue loading) was used to predict reduction in stiffness using a RVE. It was found that the stiffness changes with change in depth of the volume element along the fiber direction and asymptotically reaches a constant value beyond a critical length called representative depth. It was argued that this representative depth should be more than the minimum of two characteristic length scales, twice of ineffective length and average length of broken fibers. Effective stiffness obtained from FEA of the optimum-sized RVE was in excellent agreement with the experimental results for given microstructural damage state.
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Notes
The terms modulus and stiffness are used interchangeably.
The term VE was used instead of RVE since the change in size of a VE changes the stiffness and hence it is not a truly representative of the material under consideration. However, once the representative depth was obtained, it can be termed as RVE.
Interface was modelled with elastic behaviour, when the matrix was elastic. It (interface) was modelled with elastic-perfectly plastic behaviour, when matrix too was elastic–plastic. Elastic–plastic behaviour of interface was not considered when the matrix was elastic or vice versa.
This volume was chosen so that integer values of fiber cracks in the VE lead to volumetric densities, N v , of fiber crack density that were similar to that obtained in the experiments (N v was of the order of 10−5 µm−3).
It should be noted that the three VEs do not represent exactly the same spatial arrangements of the fibers. However, they have the similar local spatial arrangements of the fibers and the exact same volume fraction.
In FEA, displacement (strain) was applied until the average stress in the VE was equal to the applied stress from the experiment.
Applied load corresponds to the load, along the longitudinal direction or fiber axis, used in the quasi-static analysis (FEA) for effective stiffness prediction.
Stress used in the experiment was 0.85 times the static strength of CFRP composite.
Single fiber VE could not be used for determining stiffness degradation as one fiber crack can cause the whole VE to fail (upon load application beyond matrix yield strength).
Normalized applied load corresponds to the applied load that was normalized with respect to the static strength of the CFRP composite.
Ineffective length is the length over which load transfer from the matrix to the fiber takes place until the load in fiber reaches to the peak.
Stress–strain curve was reported until the normalized applied load of 0.85 as used in the experiments (Ref 1).
No failure model (damage initiation or damage progression) was considered either for fiber or matrix or interface during quasi-static tension analysis through FE simulation. In the experiments (Ref 1), fatigue testing was carried out for the given peak stress under load control and the microstructural damage was measured independently. Hence, even in FE analysis, damage progression was not simulated as the CFRP specimen (in experiments) could take the given load with the known microstructural damage state.
This is strictly true only for low fiber crack density, where there is no interaction between the cracks.
λ/2 = (σ r)/(2τ); where, σ is the axial stress in the fiber, r is the fiber radius and τ is the interface shear strength.
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Acknowledgment
The authors gratefully acknowledge the support provided for this work by the “National Centre for Aerospace Innovation and Research, IIT-Bombay, a Department of Science and Technology, Government of India, The Boeing Company and IIT Bombay Collaboration”; and the “Ministry of Textiles, Government of India”.
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Hiremath, C.P., Senthilnathan, K., Naik, N.K. et al. Numerical Study and Experimental Validation of Effect of Varying Fiber Crack Density on Stiffness Reduction in CFRP Composites. J. of Materi Eng and Perform 27, 1685–1693 (2018). https://doi.org/10.1007/s11665-018-3275-0
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DOI: https://doi.org/10.1007/s11665-018-3275-0