5.1 Elastic stress transfer, in Chapter 5. Composite mechanics: Short fibers, in Load Bearing Fiber Composites, by Michael R. Piggott, 2002
This time reading notes on a section of a well-known mechanics book are presented, due to its extreme clarity and reader-friendliness in explaining the stress transfer in fiber-reinforced composites. Personal understanding for the micromechanical model and derivations are included accordingly without specific indications.
The reinforcing mechanism between fiber and matrix was initially dealt with by Cox in 1952, with considerable intuition/ingenuity, although this is only a small part of his paper and the derivation is not easy to digest. This has been referred to as shear-lag theory (the matrix transfers load to the fibers through shear deformation). An enormous number of researchers dedicate to this field (composite micromechanics) using and improving Cox’s theory, although whether the refinements have really provided significantly improved predictions are not certain]. But there are indeed several classical works greatly illuminated the Cox’s equations for fiber axial stress and fiber-matrix interface shear stress distribution, e.g., the introductory text in Chapter V of Strong Solids by Kelly in 1966 elaborates the details, Nairn’s strict examination and re-derivation in 1997. I have to admit that I still have not completely understood/digested these old papers, but Prof. Piggott’s text in Chapter 5.1 in his book has enlightened me greatly in understanding the basics in shear lag theory. Once understood these, then the vast number of papers on this topic in the literature could be hopefully understood fluently.
This is why I provide reading notes on this here; I wish that researchers who share similar situations to mine (struggling in understanding old, classical papers in mechanics in composites) can get some true helpful clues/insights. By the way, the stress distributions predicted by both classical and following shear-lag theories have been experimentally verified, such as Raman spectra and photoelasticity [Young 2016].
Goal: determine the elastic stress transfer from fiber to matrix, including (1) the axial fiber stress distribution, (2) the matrix-fiber interface (interface) shear stress distribution, (3) the average Young’s modulus and (4) the average strength of the composite;
Model/the physical process: a fiber with finite length is embedded in matrix, with each fiber is surrounded by other fibers packed in an ordered fashion; the unit (one fiber in matrix) is loaded axially by external stress σ;
Upon loading, the matrix is stretched, but the fiber constrains the matrix deformation, resulting in successive shear strains varying from the interface radially in the matrix; thus, the stress is transferred to the fiber at the interface (loads are supported by both, but the fiber carries a major part); the displacements in the fiber and the matrix are different due to different elastic moduli;
The matrix tensile strain is assumed to be constant at the ring of nearest neighbors, at diatance R from the fiber, and this matrix strain, εm, is equal to the composite tensile strain, ε1. The matrix displacement, w, and shear stress, τ, are functions of r from the interface.
Assumptions: fiber and matrix behave elastically; the fiber-matrix interface bonding is perfect; the stress transfer across fiber ends and stress concentrations there are neglected; the axial stress in the fiber is uniform on fiber cross section;
Route:
(1) by matrix equilibrium find the relationship between matrix shear stress and interfacial shear stress, then use integration radially and shear modulus-elastic modulus relationship, thus obtain the axial displacements relationship; (2) by differentiation and Hooke’s law, obtain axial stress v.s. interfacial shear stress relationship; (3) by fiber segment equilibrium find the interfacial shear stress v.s. axial stress relationship.
Then by combining (2) and (3) can solve for the axial fiber stress expression. Then the interfacial shear stress equation can be obtained by (3), the matrix shear stress distribution can be found by (1), and the average elastic modulus and strength can be determined by using rule of averages.
Detailed derivations can be found in the book, and are not repeated here (please let me know if you want a pdf copy of the book, I would be happy to provide it OR there will be a link for it later in this post).
Key steps/thoughts:
(1) Force equilibriums for the matrix and the fiber, each provides useful equations/relationships;
(2) Integration and differentiation are powerful operations to construct relations between equations and reduce variables;
(3) After constructing the mechanical model/physical procedure, then (re-)consider boundary conditions, continuity, and assumptions to enable/allow the realization of the derivation.
Several notes: (1) it is strongly suggested to read the original books and papers, since they not only provide classical knowledge that stands the test of time but also teach you silently how to be a man when reading between the lines;
(2) for researchers in materials & mechanics (not pure mechanics), (a) there are subtle differences among the theoretical derivations between Cox, Kelly, Nairn, and Piggott, e.g., assuming only one fiber or fibers in certain packing manner, and assuming the composite strain to be the matrix strain at certain places/conditions to reduce variables; we should bear this in mind when we use the respective theories; (b) the classical shear-lag transfer theories are the basic, but may be not directly applicable to common engineering or biological composites, because the former ones did not consider the significant interactions between the reinforcing fibers/platelets. Using these theoretical equations directly would lead to predictions that deviate from/underestimate experimental measurements a lot, e.g., Facca et al. 2006, Kotha et al. 2000.
A free online version of Piggott’s classical book in pdf is available at: http://individual.utoronto.ca/michael_piggott/
References:
Cox, H. L. (1952). The elasticity and strength of paper and other fibrous materials. British journal of applied physics, 3(3), 72.
Piggott, M. (2002). Load Bearing Fibre Composites. Springer Science & Business Media.
Kelly A. (1966). Strong Solids. Clarendon Press Oxford.
Nairn, J. A. (1997). On the use of shear-lag methods for analysis of stress transfer in unidirectional composites. Mechanics of Materials, 26(2), 63-80.
Young, R. J. (2016). Carbon Fibre Composites: Deformation Micromechanics Analysed using Raman Spectroscopy. In Structure and Multiscale Mechanics of Carbon Nanomaterials (pp. 29-50). Springer, Vienna.
Facca, A. G., Kortschot, M. T., & Yan, N. (2006). Predicting the elastic modulus of natural fibre reinforced thermoplastics. Composites Part A: Applied Science and Manufacturing, 37(10), 1660-1671.
Kotha, S. P., Kotha, S., & Guzelsu, N. (2000). A shear-lag model to account for interaction effects between inclusions in composites reinforced with rectangular platelets. Composites Science and Technology, 60(11), 2147-2158.