Neil McCartney - Properties for Design of Composite Structures

(4.177)

(4.178)

(4.179)

(4.180)

(4.181)

(4.182)

These results ( 4.173)–( 4.182) can also be written in the form of a mixtures estimate plus a correction term as follows

(4.183)

(4.184)

(4.185)

(4.186)

(4.187)

(4.188)

(4.189)

(4.190)

(4.191)

where the parameter λ is defined by

(4.192)

Bounds for the effective properties of unidirectional fibre-reinforced composites have been derived using variational techniques by Hashin and Shtrikman [7], and in the review by Hashin [3]. In this section, these bounds are now associated with the results in this chapter derived using Maxwell’s methodology. The results of Section 4.8, derived using Maxwell’s methodology or using the concentric cylinder model of a composite, correspond exactly to one of the bounds. Furthermore, if one bound is known, then the other is obtained simply by interchanging the properties and volume fractions. It is of interest to determine the condition that specifies whether the extreme value is an upper or lower bound.

The thermal conductivity bounds are very easily determined from the result ( 4.183), which is such that the denominator of the third term on the right-hand side is the only term to change value when the properties and volume fractions are interchanged. It then follows that the bounds may be written as

(4.193)

whenever κm≤κTf. Whenever κm≥κTf, the bounds defined by ( 4.193) should be reversed.

The bounds for the axial Young’s modulus are given by

(4.194)

which are valid only if μTf≥μm, and the bounds are reversed if μTf≤μm.

The bounds for the axial Poisson’s ratio are given by

(4.195)

which are valid only if (νAf−νm)(kTf−kTm)(μtf−μm)≥0, and the bounds are reversed if (νAf−νm)(kTf−kTm)(μtf−μm)≤0.

The bounds for the transverse bulk modulus are given by

(4.196)

which are valid only if μtf≤μm, and the bounds are reversed if μtf≥μm.

The bounds for transverse shear modulus are given by

(4.197)

which are valid only if

The bounds are reversed if

The bounds for the axial shear modulus are given by

(4.198)

which are valid only if μm≤μAf, and the bounds are reversed if μm≥μAf.

The bounds for the axial thermal expansion are given by

(4.199)

where

(4.200)

These bounds are valid only if (νAf−νm)(α^Tf−α^Tm)(μtf−μm)≥0, and the bounds are reversed if (νAf−νm)(α^Tf−α^Tm)(μtf−μm)≤​​0.

The bounds for the transverse thermal expansion are given by

(4.201)