E-Book, Englisch, 872 Seiten
Reihe: Woodhead Publishing Series in Composites Science and Engineering
Innovative Modelling Methods and Intelligent Design
E-Book, Englisch, 872 Seiten
Reihe: Woodhead Publishing Series in Composites Science and Engineering
ISBN: 978-0-08-100138-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark
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1 Composite micromechanics
From carbon fibres to graphene
R.J. Young University of Manchester, Manchester, UK Abstract
Composite materials are being used increasingly in high-performance engineering applications. The development of the subject of composite micromechanics to understand stress transfer to a high-performance fibre in a low-modulus matrix in a composite is described. It is shown how Raman spectroscopy can now be used to measure stress and strain of individual fibres and so validate the predictions even though reinforcement was considered originally only from a theoretical viewpoint. Recent developments in the field of polymer nanocomposites are then described and it is demonstrated how it is possible to adapt and modify the theories of fibre reinforcement to matrices reinforced by nanoplatelets such as graphene. It is shown that Raman spectroscopy can again be used to map stress and strain in the graphene and that continuum micromechanics is still applicable at the nanolevel. Future trends and challenges in the field are then discussed and a number of unsolved problems are highlighted. Keywords
Carbon fibres; Composite micromechanics; Graphene; Interfacial stress transfer; Nanocomposites 1.1. Introduction
In the quest to understand fibre reinforcement, this chapter traces the development of the subject of composite micromechanics from its earliest roots to the most recent analysis of the deformation of graphene-reinforced nanocomposites. It is shown first how, employing concepts introduced by Kelly, it is possible through the use of shear-lag theory to predict the distribution of stress and strain in a single discontinuous fibre in a low-modulus matrix. For a number of years the shear-lag approach could only be used theoretically as there were no techniques available to monitor the stresses within a fibre in a resin. It will be shown that Raman spectroscopy and the discovery of stress-induced Raman bands shifts in reinforcing fibres have enabled us to map out the stresses in individual fibres in a transparent resin matrix, and thereby both test and develop Kelly's pioneering analytical approach. In view of the growing interest in the study of polymer-based nanocomposites, it is shown how the shear-lag methodology can be modified to predict the distribution of stress and strain in nanoplatelets reinforcing a polymer matrix and an analogous set of relationships is obtained for nanoplatelet reinforcement to those obtained for fibre reinforcement. Because of the very strong resonance Raman scattering, it is shown that well-defined Raman spectra can be obtained from graphene monolayers. Large stress-induced band shifts can be obtained from these sheets when embedded in a polymer matrix, which has enabled the prediction of the shear-lag model to be validated. It is shown further how the shear-lag model can be used to model reinforcement by bilayer graphene. 1.2. Fibre reinforcement – theory
1.2.1. Composite micromechanics
Interest in the mechanics of fibre reinforcement can be traced back to the first uses of high-modulus fibres to reinforce a low-modulus matrix. A useful relationship developed to describe this reinforcement is the so-called ‘rule of mixtures’ in which, for stress parallel to the fibre direction, the Young's modulus of a composite Ec consisting of infinitely long aligned fibres is given by an equation of the form (Young & Lovell, 2011). c=EfVf+EmVm (1.1) where Ef and Em are the Young's modulus of the fibre and matrix and Vf and Vm are the volume fraction of the fibre and matrix, respectively. This equation captures the essence of fibre reinforcement and is found to work well in the specific conditions outlined above when high-modulus fibres are incorporated into low-modulus matrix materials. Since the strain in the fibre and matrix are the same, the stress in the fibres is much higher than that in the matrix, hence the fibres take most of the load and so reinforce the polymer matrix. In reality, however, composites do not consist of infinitely long aligned fibres and are not always stressed parallel to the fibre direction. The full analysis of the situation in reality is the subject of many composites textbooks. The deformation of composites containing fibres of finite length deformed axially has been considered by a number of authors including Krenchel (1964). In addition, he also analysed the situation with fibres aligned randomly in plane and also randomly in three dimensions (Krenchel, 1964). The problem of transfer of stress from the matrix to a fibre and the subsequent variation of stress along a fibre of finite length in a matrix was first tackled properly by Kelly (1966) in his classical text, Strong Solids. This ground-breaking work involved both the revival of the shear-lag concept of Cox (1952) and considerable intuition on his part. Indeed, in the introductory text to Chapter 5 of Strong Solids (Kelly, 1966) he makes the following statement: ‘In this chapter we will discuss firstly how stress can be transferred between the matrix and fibre. This will be done in a semi- intuitive fashion since it is a difficult problem to solve exactly’. Kelly's analysis became the foundation of a new research field known as ‘composite micromechanics’. It will be shown how this field gave us the framework for the study of fibre reinforcement at both a theoretical and practical level, also enabling us to use the approach to tailor the properties of fibre–matrix interfaces in composites. 1.2.2. Discontinuous fibres
In the case of discontinuous fibres reinforcing a composite matrix, stress transfer from the matrix to the fibre takes place through a shear stress at the fibre–matrix interface as shown in Figure 1.1. It is envisaged that parallel lines perpendicular to the fibre can be drawn from the matrix through the fibre before deformation. When the system is subjected to an axial stress s1 parallel to the fibre axis, the lines become distorted since the Young's modulus of the matrix is much lower than that of the fibre. This induces a shear stress at the fibre/matrix interface, and the axial stress in the fibre builds up from zero at the fibre ends to a maximum value in the middle of the fibre. The assumption of uniform strain means that in the middle of the fibre the strain in the fibre equals that in the matrix, if the fibre is long enough. Since the fibres generally have a much higher Young's modulus than the matrix, the fibres then carry most of the stress (and hence load) in the composite – this is essentially how composites work. It is now necessary to introduce the concept of interfacial shear stress (Kelly, 1966). The relationship between the interfacial shear stress ti near the fibre ends and the fibre stress sf can be determined by using a balance of the shear forces at the interface and the tensile forces in a fibre element, as shown in Figure 1.2. The main assumption is that the force due to the shear stress ti at the interface is balanced by the force due to the variation of axial stress dsf in the fibre such that
Figure 1.1 Deformation patterns for a discontinuous high-modulus fibre in a low-modulus polymer matrix. The top diagram shows the situation before deformation, and the bottom diagram shows the effect of the application of a tensile stress, s1, parallel to the fibre. prti?x=-pr2?sf (1.2) so?sf?x=-2tir (1.3)
Figure 1.2 Balance of stresses acting on an element of the fibre of thickness dx in the composite.
Figure 1.3 Model of a fibre undergoing deformation within a resin used in shear-lag theory. The shear stress t acts at a radius ? from the fibre centre. 1.2.2. Elastic stress transfer
The behaviour of a discontinuous fibre in a matrix can be modelled using shear-lag theory, developed initially by Cox (1952) to model the mechanical properties of paper. It is assumed in the theory that the fibre is surrounded by a cylinder of resin extending to a radius ? from the fibre centre, as shown in Figure 1.3. In this model it is assumed that both the fibre and matrix deform elastically and that the fibre–matrix interface remains intact. If u is the displacement of the matrix in the fibre axial direction at a radius ?, then the shear strain ? at that position is given by =?u?? (1.4) The shear modulus of the matrix is defined as Gm = t/?, hence it follows that u??=tGm (1.5) The shear force per unit length carried by the matrix cylinder surface is 2p?t and is transmitted to the fibre surface though the layers of resin, and...
