Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

Properties for Design of Composite Structures - Neil McCartney

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of the form

      Superscript ‘f’ is used to denote anisotropic fibre properties and subscript ‘m’ is used to denote the properties Em, νm, μm and αm of an isotropic matrix such that Em=2μm(1+νm). If the fibres are also isotropic then

      upper E Subscript upper A Superscript f Baseline equals upper E Subscript upper T Superscript f Baseline equals upper E Subscript f Baseline comma mu Subscript upper A Superscript f Baseline equals mu Subscript t Superscript f Baseline equals mu Subscript f Baseline comma nu Subscript upper A Superscript f Baseline equals nu Subscript t Superscript f Baseline equals nu Subscript f Baseline comma alpha Subscript upper A Superscript f Baseline equals alpha Subscript upper T Superscript f Baseline equals alpha Subscript f Baseline period(4.19)

      The equilibrium equations for the fibres and matrix in the absence of body forces are (see (2.125)–(2.127))

      StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript theta theta Baseline Over partial-differential theta EndFraction plus StartFraction partial-differential sigma Subscript theta z Baseline Over partial-differential z EndFraction plus StartFraction 2 sigma Subscript theta r Baseline Over r EndFraction equals 0 comma(4.21)

      4.3.1 Properties Defined from Axisymmetric Distributions

      The following analysis applies to an isolated cylindrical fibre of radius a that is perfectly bonded to an infinite matrix, subject to a uniform temperature change ΔT, where the system is subject to a uniform axial strain ε and a uniform transverse stress σT. The equilibrium equations (4.20)–(4.22) for the fibre and matrix, assuming symmetry about the fibre axis so that stress components are independent of θ and the shear stresses σrθ and σθz are zero, then reduce to the form

      StartFraction partial-differential sigma Subscript r r Baseline Over partial-differential r EndFraction plus StartFraction partial-differential sigma Subscript r z Baseline Over partial-differential z EndFraction plus StartFraction sigma Subscript r r Baseline minus sigma Subscript theta theta Baseline Over r EndFraction equals 0 comma(4.23)

      StartFraction partial-differential sigma Subscript z z Baseline Over partial-differential z EndFraction plus StartFraction partial-differential sigma Subscript r z Baseline Over partial-differential r EndFraction plus StartFraction sigma Subscript r z Baseline Over r EndFraction equals 0 period(4.24)

      In regions away from the loading mechanism it is reasonable to assume that

      where ε is the axial strain applied to the composite. A solution is now sought of the following classical Lamé form

      where Af, Am and ϕ are constants to be determined.

      4.3.2 Solution for an Isolated Fibre Perfectly Bonded to the Matrix

      The boundary and interface conditions that must be satisfied are