Finite Element Modelling of the Microstresses ^ Regular Hexagonal

Fibre Array

On the basis that the temperature gradients within the laminate are negligible and the resulting stresses not high enough to initiate failure of the materials it was decided to concentrate on the residual thermal microstresses, created as a result of the different thermal expansion coefficients of the epoxy resin and the carbon fibres. Since the geometry in the longitudinal direction of the fibres is constant the stresses could be adequately modelled by analyzing the stresses in a transverse section. In addition assuming a regular array of fibres, that is a distribution in which a structure regularly

repeats itself throughout the array, the modelling could involve a unit cell approach. The most highly packed structures conform to a regular hexagonal array - see Figure 6, with fibre volume fractions of up to 90.6%. By comparison the highest fibre volume fractions

achievable in regular square arrays are 78.5% see Figure 7. It was decided to model the hexagonal array as being the more probable

distribution at the high volume fractions obtained commercially of 65% [93] and as has been assumed in the majority of the literature.

A triangular unit cell of the transverse section of a regular hexagonal array of fibres in an epoxy resin matrix was modelled

assuming a typical fibre diameter of 7 nm and interfibre distance of

4 /im [93] as shown in Figure 6. This gave a fibre volume fraction of 37%. Material properties were used based on those for Torayca T300 carbon fibres and Ciba-Geigy BSL 914 epoxy resin as given below.

Material Carbon Fibre Epoxy Resin

Young's modulus (GPa) 15.8 3.9

Poisson's ratio 0.42 0.41

Thermal expansion coefficient (°C-1) 8.0 x 10“6 48 x IQ"6

These were obtained from characterization tests undertaken on the epoxy resin at 23°C by Ciba-Geigy for Rolls-Royce pic and transverse properties given for the fibres in the literature [73,83].

The unit cell was assumed to undergo a uniform temperature decrease of 170°C from 190°C to 20°C.

Two cases were considered: that of a "radially restrained" unit cell restrained along the lines OA and OB, in a direction perpendicular to these lines, programs ES01 (plane stress) and ES03 (plane strain), and that of a "totally restrained" unit cell restrained along the outside edges, OA, OB and AB, perpendicular to these lines, program ES02

(plane strain) - See Figure 6. The former "radially restrained" unit cells used absolute values for the thermal expansion coefficients of the fibre and the matrix and the latter "totally restrained" unit cells used thermal expansion coefficients of the fibre and the matrix expressed relative to a mean overall value for the unit cell. The mean value was calculated from those of the fibre and the matrix weighted in proportion to the volume fraction of the material as follows

amean ^carbon fibreacarbon fibre + ^epoxy resinaepoxy resin The restraint applied to the radially restrained unit cell was so as to preserve the symmetry of the array. The use of expansion

coefficients relative to a mean value in the totally restrained unit cell required the condition that the cell as a whole underwent no overall deformation. The former calculated the stresses generated due to differences in displacement from the free displacements of the epoxy resin and the carbon fibre and the latter those due to

differences in displacement from a mean free thermal expansion of the material as a whole. The deformation and stresses were calculated for

conditions of plane strain and plane stress.

The relative thermal expansion coefficients used in the totally restrained model were calculated at

a , _{carbon fibre} = -25 x 10"6

a_{epoxy resin} = 15 x 10-6

from a mean value of 33 x 10“6.

3.5.1 Incorporation of Temperature Dependent Epoxy Resin Properties

The Ciba-Geigy characterization tests showed the properties of Ciba- Geigy BSL 914 epoxy resin to vary appreciably with temperature. This temperature dependence has been incorporated into both the macro- and micro-modelling of the stress distributions. At the microlevel it was decided to concentrate on the radially restrained unit cell under conditions of plane strain. This provided the best approximation to conditions around long fibres surrounded by epoxy resin.

The epoxy resin properties were allowed to vary with temperature in a program, ES04, according to the Ciba-Geigy characterization tests as follows

Temperature (°C) -20 23 85 140 175 200 Thermal Expansion Coeff.( x 10“6°C“i) - 48 53 65 81 104 Young's Modulus (GPa) 4.4 3.9 3.4 2.8 Poisson's Ratio 0.398 0.407 0.413 0.423

3.6 Finite Element Modelling of the Microstresses ^ Other Regular

Fibre Arrays

The micromodelling so far had considered one specific distribution of fibres, that is hexagonal packing of 7 /zm diameter fibres separated by a distance of 4 /zm. To model alternative arrangements of fibres a similar right angled triangle was used for the mesh but allow but the acute angles of the triangle were allowed to vary in a way dependent on the proposed arrangement of fibres.

In addition the modelling employed a fibre volume fraction of 36.7%. Commercially fibre volume fractions of 65% are achievable and fibre diameters can range from 6 /zm to 11 /zm. By altering the dimensions of the micromesh it was proposed to study the effect of varying fibre volume fraction, fibre diameter and consequent variation in interfibre distance on the microstresses.

Triangular meshes with various "co-ordination angles", 6 subtended at the fibre centre were modelled. Angles of 30°, 45°, 60° and 15° were chosen since these represent arrangements in which the fibres have 6, 8, 3 and 12 nearest neighbours, respectively see Figures 5, 6,

7 and 8. The regular square array obtained for 0 = 45° has been