Results from some of the earlier programs modelling a typical regular hexagonal array of 3.5 fim diameter fibres at 37% fibre volume fraction
are now summarized (Figure 6) These have been selected as the most
informative. Models were originally developed to cover all the conditions of plane strain, plane stress, "radial restraint" and "total restraint". This was due to uncertainty concerning how best to represent the actual conditions.
ES01 ("radially restrained", plane stress) - This models a unit cell restrained along the lines, OA and OB in a direction perpendicular to these lines. Absolute thermal expansion coefficients under conditions
of plane stress are assumed.
ES03 ("radially restrained", plane strain) - This is as ES01 but with the conditions of plane stress changed to plane strain.
ES04 ("radially restrained", plane strain, temperature dependent, epoxy resin properties) - This is as ES03 but with temperature dependent epoxy resin properties.
ES02 ("totally restrained", plane strain) - This as ES03 but with the unit cell restrained along the lines, OA, OB and AB, in a direction perpendicular to these lines. Thermal expansion coefficients relative to a mean value for the unit cell are assumed.
Results are given in the Figures for the model which was selected as
the most realistic, ES04. This represents a state of plane strain and incorporates "radial restraints" and temperature dependent properties Labels, OXABY, where XY represents the undeformed fibre/matrix
interface and OAB, represents the undeformed lines of symmetry of the unit cell, have been included on the figure representing the
deformation (Figure 76). The presence of these lines permits ease of reference in discussing the results.
4.2.1.1 Deformation
Figure 76. Deformations were produced (having scaled maximum
displacements of 1 cm) for each of the programs, ES01, ES02, ES03 and ES04. These showed the desired restraints to be achieved. Figure 76 gives the result for the model ES04 with the deformed shape,
OX'A'B'Y', superimposed on the undeformed shape, OXABY. Models ES01 and ES03 were similarly "radially restrained" along the lines, OA and
OB, perpendicular to these lines, with the end plane, AB remaining approximately parallel to the z axis (a small deviation from this parallel line led to modification of the restraints in the later programming). This is necessary to retain the hexagonal symmetry of
the array. Maximum displacements occurred at the point B. In addition program ES02, representing the deformation relative to the overall average deformation of the unit cell was "totally restrained" along the outside edges, OA, OB and AB, perpendicular to these lines. The magnitude of the maximum screen displacement, see Table 3, was noted to increase from conditions of plane stress - 35.5 nm (ES01) through conditions of plane strain - 51.8 nm (ES03) to conditions of plane strain with the incorporation of temperature dependent
properties - 66.5 nm (ES04). Under conditions of longitudinal
contraction the transverse displacement would be expected to increase as conditions change from plane stress to plane strain. Similarly with incorporation of the temperature dependent properties where the
Young's modulus decreases with temperature further increases in displacements would be expected. The maximum relative displacement obtained for the "totally restrained" model, ES02 was at the
fibre/matrix interface and was equal to 20.1 n m
b.2.1.2 Stresses
i) "Radially Restrained" Models, ES01, ES03, ES04:
A similar pattern of stresses was apparent in all the calculations where the unit cell was radially restrained. The magnitude of the stresses however was noted to increase from conditions of plane
stress (ES01) through plane strain (ES03) to plane strain with temperature dependent properties (ES04). This can be explained by a similar reasoning as for the deformations.
Since the stress contours obtained for the three programs ES01, ES03 and ES04 follow approximately similar patterns although of slightly differing magnitudes it is simplest to examine the stresses of one program in detail. ES04 representing a state of plane strain as typified by long cylindrical fibres and incorporating temperature dependent properties is chosen as the most realistic of these models. Unaveraged stresses are included as a measure of the accuracy
obtained.
a) Directional stresses:
The elements are meshed in the global y-z plane. Plots were obtained
of the variation in the in-plane a and yy a normal stresses andzz the a shear stresses. These were difficult to interpret since they are based on a Cartesian geometry and bear no relation to the
rotational symmetry of the fibres. They have therefore not been included in the results. It was however noted that the a stresses.zz were found to be more significant, having greater values. These
reached a maximum tensile value of 40 MPa in the epoxy close to the
interface at point X, decreasing to values of around 5.2 MPa at B, and an approximately constant compressive value of around -20.1 MPa in the fibre. By comparison the a stresses had approximately constant
compressive values of around -17.64 MPa in the outer regions of the fibre, whilst in the epoxy resin, values ranged from around -15.45 MPa at point X on the interface decreasing to a maximum tensile value of
3.13 MPa on the line AB.