Neil McCartney

Properties for Design of Composite Structures

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that the effective bulk modulus, thermal expansion coefficient and shear modulus may be expressed as a mixtures estimate plus a correction term, so that

      mu Subscript eff Baseline equals upper V Subscript p Baseline mu Subscript p Baseline plus upper V Subscript m Baseline mu Subscript m Baseline minus StartFraction left-parenthesis mu Subscript p Baseline minus mu Subscript m Baseline right-parenthesis squared upper V Subscript p Baseline upper V Subscript m Baseline Over upper V Subscript p Baseline mu Subscript m Baseline plus upper V Subscript m Baseline mu Subscript p Baseline zero width space zero width space zero width space zero width space plus mu Subscript m Superscript asterisk Baseline EndFraction comma(3.58)

      where

      3.6 Bounds for Two-phase Isotropic Composites

      It follows from Hashin and Shtrikman [5], and the review by Hashin [1], that bounds for the effective thermal conductivity of a two-phase composite, valid for arbitrary reinforcement geometries leading to statistically isotropic effective properties, may be expressed in the form

      upper V Subscript p Baseline kappa Subscript p Baseline plus upper V Subscript m Baseline kappa Subscript m Baseline minus StartFraction left-parenthesis kappa Subscript p Baseline minus kappa Subscript m Baseline right-parenthesis squared upper V Subscript p Baseline upper V Subscript m Baseline Over upper V Subscript p Baseline kappa Subscript m Baseline plus upper V Subscript m Baseline kappa Subscript p Baseline plus 2 kappa Subscript min Baseline EndFraction less-than-or-equal-to kappa Subscript eff Baseline less-than-or-equal-to upper V Subscript p Baseline kappa Subscript p Baseline plus upper V Subscript m Baseline kappa Subscript m Baseline minus StartFraction left-parenthesis kappa Subscript p Baseline minus kappa Subscript m Baseline right-parenthesis squared upper V Subscript p Baseline upper V Subscript m Baseline Over upper V Subscript p Baseline kappa Subscript m Baseline plus upper V Subscript m Baseline kappa Subscript p Baseline plus 2 kappa Subscript max Baseline EndFraction comma(3.60)

      where now kappa Subscript max Baseline equals max left-parenthesis kappa Subscript p Baseline comma kappa Subscript m Baseline right-parenthesis comma kappa Subscript min Baseline equals min left-parenthesis kappa Subscript p Baseline comma kappa Subscript m Baseline right-parenthesis period(3.61)

      Walpole [7, Equation (26)] has derived rigorous bounds for the effective bulk modulus, which can for a two-phase composite be expressed in the following two equivalent forms

      StartFraction upper V Subscript p Baseline Over k Subscript p Baseline EndFraction plus StartFraction upper V Subscript m Baseline Over k Subscript m Baseline EndFraction minus StartFraction left-parenthesis 1 slash k Subscript p Baseline minus 1 slash k Subscript m Baseline right-parenthesis squared upper V Subscript p Baseline upper V Subscript m Baseline Over upper V Subscript p Baseline slash k Subscript m Baseline plus upper V Subscript m Baseline slash k Subscript p Baseline plus 3 slash left-parenthesis 4 mu Subscript max Baseline right-parenthesis EndFraction less-than-or-equal-to StartFraction 1 Over k Subscript eff Baseline EndFraction less-than-or-equal-to StartFraction upper V Subscript p Baseline Over k Subscript p Baseline EndFraction plus StartFraction upper V Subscript m Baseline Over k Subscript m Baseline EndFraction minus StartFraction left-parenthesis 1 slash k Subscript p Baseline minus 1 slash k Subscript m Baseline right-parenthesis squared upper V Subscript p Baseline upper V Subscript m Baseline Over upper V Subscript p Baseline slash k Subscript m Baseline plus upper V Subscript m Baseline slash k Subscript p Baseline plus 3 slash left-parenthesis 4 mu Subscript min Baseline right-parenthesis EndFraction comma(3.63)

      where mu Subscript max Baseline equals max left-parenthesis mu Subscript p Baseline comma mu Subscript m Baseline right-parenthesis comma mu Subscript min Baseline equals min left-parenthesis mu Subscript p Baseline comma mu Subscript m Baseline right-parenthesis period(3.64)

      if (kp−km)(μp−μm)(αp−αm)≥0: