Aligned fibre composites systems

For the following fibre composite systems, find the fibre mass firaction at which the axial coefficient of linear thermal expansion becomes zero (assume fibres to be aligned and continuous) ... 

There is an extensive literature on composite materials stemming from the seminal papers of Eshelby [8], who considered the elastic field in and around an elliptic inclusion in an infinite matrix. His theory assumed a single particle in an infinite matrix and therefore was valid only for low-volume fractions ( 1%). The extension to more concentrated systems was undertaken by Mori and Tanaka [9], whose method was used by Tandon and Weng [10] to derive the elastic constants of an aligned fibre composite. The composite moduli for this model are given by... 

Similar considerations of symmetry apply in other systems, for example nematic liquid crystals and aligned short fibre composites have symmetry D h, smectic A liquid crystals D , while in copolymers and certain fibre composites examples of hexagonal symmetry may be found and translational symmetry may also be present, which is not found in petrology. 

It has been demonstrated that polydiacetylene single crystal fibres are relatively perfect and have excellent molecular alignment. In consequence they display high values of stiffness and strength and are very resistant to creep. It has been shown that such fibres have considerable promise as reinforcing fibres in an epoxy resin matrix and the study of such composite systems has enabled considerable fundamental information to be obtained concerning the mechanisms of fibre reinforcement. 

The fibres in short fibre composites are not usually fiilly aligned. An early attempt to deal with such systems was made by Brody and Ward [12], who applied the aggregate model of Section 7.5 above assuming that the elastic constants of a representative unit of structure could be predicted by Equations (8.7a)-(8.7e). It was found that this simple theory fitted the results for composites reinforced with short fibres of carbon or glass reasonably well, with the moduli lying close to the lower Reuss bounds. Recent studies by Ward and co-workers [13] have shown that this approach is viable provided that the elastic constants of the representative unit are calculated more exactly. 

The combined effect of orientation (aligned and random arrays) and length on the overall efficiency, as calculated by Laws [4] is shown in Figure 4.6. She demonstrated that the efficiency is dependent on the frictional bond before slip (static) and the frictional bond after fibre slip (dynamic). The higher efficiency values for the systems where dynamic slip is present (dashed line in Figure 4.5) are the result of the contribution of this type of frictional resistance to the strength of the short fibre composite. 

The data included in this text on several specialized types of composite are rather limited. These (mainly thermosetting matrix) systems include those based on aligned short fibres, hybrids consisting of two or more types of fibre in a common matrix and sandwich structures consisting of a light-weight core and composite skins. These systems either have relatively low properties or such a range of possible constitutions as to make it difficult to find representative structures. Nevertheless, they are widely used in many areas, often because of their ease of fabrication. 

In the continuous form, the fibres can be aligned in a preferred orientation, which is controlled by the production process (orientation of winding, or lay-up direction of the mat) and the structure of the mat. This type of fibre reinforcement bears some resemblance to ferrocement applications it is less common in FRC composites which are usually reinforced by discrete, short fibres, but has recently been the focus of intense development efforts (see Chapter 13 for details). In the case of dispersed fibres the dispersion in the matrix is more uniform, and the short fibres tend to assume a more random orientation. Flowever, even in these systems the fibre distribution is rarely completely uniform, and their orientation is not... 

---