Properties for Design of Composite Structures. Neil McCartney

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Properties for Design of Composite Structures - Neil McCartney

Properties for Design of Composite Structures - Neil McCartney

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minus StartFraction 1 Over k Subscript upper T Superscript m Baseline EndFraction right-parenthesis squared upper V Subscript f Baseline upper V Subscript m Baseline comma"/>(4.184)

      nu Subscript upper A Superscript eff Baseline equals upper V Subscript f Baseline nu Subscript upper A Superscript f Baseline plus upper V Subscript m Baseline nu Subscript m Baseline minus StartFraction lamda Over 2 EndFraction left-parenthesis StartFraction 1 Over k Subscript upper T Superscript m Baseline EndFraction minus StartFraction 1 Over k Subscript upper T Superscript f Baseline EndFraction right-parenthesis left-parenthesis nu Subscript m Baseline minus nu Subscript upper A Superscript f Baseline right-parenthesis upper V Subscript f Baseline upper V Subscript m Baseline comma(4.185)

      mu Subscript upper A Superscript eff Baseline equals upper V Subscript f Baseline mu Subscript upper A Superscript f Baseline plus upper V Subscript m Baseline mu Subscript m Baseline minus StartFraction left-parenthesis mu Subscript upper A Superscript f Baseline minus mu Subscript m Baseline right-parenthesis squared upper V Subscript f Baseline upper V Subscript m Baseline Over left-parenthesis 1 plus upper V Subscript f Baseline right-parenthesis mu Subscript m Baseline plus upper V Subscript m Baseline mu Subscript upper A Superscript f Baseline EndFraction comma(4.186)

      mu Subscript t Superscript eff Baseline equals upper V Subscript f Baseline mu Subscript t Superscript f Baseline plus upper V Subscript m Baseline mu Subscript m Baseline minus StartStartFraction left-parenthesis mu Subscript t Superscript f Baseline minus mu Subscript m Baseline right-parenthesis squared upper V Subscript f Baseline upper V Subscript m Baseline OverOver upper V Subscript m Baseline mu Subscript t Superscript f Baseline plus upper V Subscript f Baseline mu Subscript m Baseline plus StartFraction k Subscript upper T Superscript m Baseline mu Subscript m Baseline Over k Subscript upper T Superscript m Baseline plus 2 mu Subscript m Baseline EndFraction EndEndFraction comma(4.187)

      upper E Subscript upper A Superscript eff Baseline equals upper V Subscript f Baseline upper E Subscript upper A Superscript f Baseline plus upper V Subscript m Baseline upper E Subscript m Baseline plus 2 lamda left-parenthesis nu Subscript m Baseline minus nu Subscript upper A Superscript f Baseline right-parenthesis squared zero width space zero width space upper V Subscript f Baseline upper V Subscript m Baseline comma(4.188)

      StartFraction 1 Over upper E Subscript upper T Superscript eff Baseline EndFraction equals StartFraction 1 Over 4 mu Subscript t Superscript eff Baseline EndFraction plus StartFraction 1 Over 4 k Subscript upper T Superscript eff Baseline EndFraction plus StartFraction left-parenthesis nu Subscript upper A Superscript eff Baseline right-parenthesis squared Over upper E Subscript upper A Superscript eff Baseline EndFraction comma StartFraction nu Subscript t Superscript eff Baseline Over upper E Subscript upper T Superscript eff Baseline EndFraction equals StartFraction 1 Over 4 mu Subscript t Superscript eff Baseline EndFraction minus StartFraction 1 Over 4 k Subscript upper T Superscript eff Baseline EndFraction minus StartFraction 4 left-parenthesis nu Subscript upper A Superscript eff Baseline right-parenthesis squared Over upper E Subscript upper A Superscript eff Baseline EndFraction comma(4.189)

      upper E Subscript upper A Superscript eff Baseline alpha Subscript upper A Superscript eff Baseline equals upper V Subscript f Baseline upper E Subscript upper A Superscript f Baseline alpha Subscript upper A Superscript f Baseline plus upper V Subscript m Baseline upper E Subscript m Baseline alpha Subscript m Baseline plus 2 lamda left-parenthesis nu Subscript m Baseline minus nu Subscript upper A Superscript f Baseline right-parenthesis left-parenthesis alpha Subscript m Baseline plus nu Subscript m Baseline alpha Subscript m Baseline minus alpha Subscript upper T Superscript f Baseline minus nu Subscript upper A Superscript f Baseline alpha Subscript upper A Superscript f Baseline right-parenthesis upper V Subscript f Baseline upper V Subscript m Baseline comma(4.190)

      table attributes columnalign left end attributes row cell alpha subscript text T end text end subscript superscript text eff end text end superscript plus nu subscript text A end text end subscript superscript text eff end text end superscript alpha subscript text A end text end subscript superscript text eff end text end superscript equals V subscript text f end text end subscript left parenthesis alpha subscript text T end text end subscript superscript text f end text end superscript plus nu subscript text A end text end subscript superscript text f end text end superscript alpha subscript text A end text end subscript superscript text f end text end superscript right parenthesis plus V subscript text m end text end subscript left parenthesis alpha subscript text m end text end subscript plus nu subscript text m end text end subscript alpha subscript text m end text end subscript right parenthesis end cell row cell space space space space space space space space space space space space space space space space space space space plus lambda over 2 left parenthesis fraction numerator 1 over denominator k subscript text T end text end subscript superscript text f end text end superscript end fraction minus fraction numerator 1 over denominator k subscript text T end text end subscript superscript text m end text end superscript end fraction right parenthesis left parenthesis alpha subscript text m end text end subscript plus nu subscript text m end text end subscript alpha subscript text m end text end subscript minus alpha subscript text T end text end subscript superscript text f end text end superscript minus nu subscript text A end text end subscript superscript text f end text end superscript alpha subscript text A end text end subscript superscript text f end text end superscript right parenthesis V subscript text f end text end subscript V subscript text m end text end subscript comma end cell end table(4.191)

      where the parameter λ is defined by

      StartFraction 2 Over lamda EndFraction equals StartFraction 1 Over mu Subscript m Baseline EndFraction plus StartFraction upper V Subscript f Baseline Over k Subscript upper T Superscript m Baseline EndFraction plus StartFraction upper V Subscript m Baseline Over k Subscript upper T Superscript f Baseline EndFraction period(4.192)

      4.10 Bounds for Two-phase Fibre-reinforced Composites

      4.10.1 Thermal Conductivity

      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

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