Properties for Design of Composite Structures. Neil McCartney
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Название: Properties for Design of Composite Structures

Автор: Neil McCartney

Издательство: John Wiley & Sons Limited

Жанр: Техническая литература

Серия:

isbn: 9781118789780

isbn:

СКАЧАТЬ from (4.130), (4.133) and (4.134) that

      left-parenthesis StartFraction 1 Over mu Subscript m Baseline EndFraction plus StartFraction 1 Over mu Subscript t Superscript f Baseline EndFraction plus StartFraction 2 Over k Subscript upper T Superscript m Baseline EndFraction right-parenthesis StartFraction upper B Subscript m Baseline Over a squared EndFraction equals minus StartFraction k Subscript upper T Superscript m Baseline plus mu Subscript m Baseline Over k Subscript upper T Superscript m Baseline EndFraction left-parenthesis StartFraction 1 Over mu Subscript m Baseline EndFraction minus StartFraction 1 Over mu Subscript t Superscript f Baseline EndFraction right-parenthesis StartFraction tau Over mu Subscript m Baseline EndFraction comma upper C Subscript f Baseline equals 0 comma(4.135)

      As Cf=0, it follows from (4.115)–(4.117) that the stress and strain fields are uniform within the fibre. From (4.133) and (4.134)

      On substituting (4.136) into (4.137) to eliminate Dm, it can be shown that

      upper A Subscript f Baseline equals zero width space zero width space zero width space zero width space StartFraction mu Subscript upper T Superscript f Baseline k Subscript upper T Superscript m Baseline plus mu Subscript t Superscript f Baseline mu Subscript m Baseline Over mu Subscript t Superscript f Baseline k Subscript upper T Superscript m Baseline plus k Subscript upper T Superscript m Baseline mu Subscript m Baseline plus 2 mu Subscript t Superscript f Baseline mu Subscript m Baseline EndFraction StartFraction tau Over mu Subscript t Superscript f Baseline EndFraction period(4.138)

      It should be noted that the displacement and stress fields in the fibre and the matrix can now be calculated. It is clear from (4.118)–(4.121) that at large distances from the fibre the perturbations of the displacement and stress fields arising from the presence of the fibre are characterised by the values of the parameters Bm and Dm which are related according to relation (4.134) and depend on fibre properties. It is also clear that the far-field is insensitive to the actual location of the fibre. This means that a cluster of weakly interacting fibres can easily be considered, and this is the basis of Maxwell’s method, which is now described.

      4.5.5 Applying Maxwell’s Approach to Multiphase Fibre Composites

      Owing to the use of the far-field in Maxwell’s method for estimating the properties of fibre composites, it is possible to consider multiple fibre reinforcements. Suppose in a cluster of fibres that there are N different types such that for i = 1, …, N, there are nifibres of radius ai. The properties of the fibres of type i are denoted by a superscript i. The cluster is assumed to be homogeneous regarding the distribution of fibres, and leads to transverse isotropic effective properties.

      Consider the following asymptotic form of radial displacement field in the matrix as r→∞ that is derived from (4.118)

      For the case of multiple phases, relation (4.139) is generalised to the following form

      When the result (4.140) is applied to a single fibre of radius b having effective properties corresponding to the multiphase cluster of fibres, it follows that

      The СКАЧАТЬ