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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      where the strain parameters ε¯A, ε¯T, ε^A and ε^T, the through-thickness stress σt and the temperature difference ΔT are assumed for the moment to be known. From (2.143), (2.209) and (2.211), the in-plane strains and shear stress are given by

      The expression for ε22 in (2.209) is first solved for the stress component σ22 so that

      sigma 22 equals nu Subscript normal upper A Baseline StartFraction upper E Subscript normal upper T Baseline Over upper E Subscript normal upper A Baseline EndFraction sigma 11 plus nu Subscript normal t Baseline sigma Subscript normal t Baseline plus upper E Subscript normal upper T Baseline left-parenthesis epsilon overbar Subscript normal upper T Baseline plus ModifyingAbove epsilon With caret Subscript normal upper T Baseline x 3 right-parenthesis minus upper E Subscript normal upper T Baseline alpha Subscript normal upper T Baseline upper Delta upper T period(2.213)

      It then follows from (2.209) that

      epsilon 11 equals StartFraction sigma 11 Over upper E overTilde Subscript normal upper A Baseline EndFraction minus StartFraction nu overTilde Subscript normal a Baseline Over upper E overTilde Subscript normal upper A Baseline EndFraction sigma Subscript normal t Baseline plus alpha overTilde Subscript normal upper A Baseline upper Delta upper T minus nu Subscript normal upper A Baseline StartFraction upper E Subscript normal upper T Baseline Over upper E Subscript normal upper A Baseline EndFraction left-parenthesis epsilon overbar Subscript normal upper T Baseline plus ModifyingAbove epsilon With caret Subscript normal upper T Baseline x 3 right-parenthesis comma(2.214)

      where

      In addition, from (2.209)

      where

      StartFraction 1 Over upper E overTilde Subscript t Baseline EndFraction equals StartFraction 1 Over upper E Subscript t Baseline EndFraction en-dash StartFraction left-parenthesis nu Subscript t Baseline right-parenthesis squared Over upper E Subscript upper T Baseline EndFraction comma alpha overTilde Subscript t Baseline equals alpha Subscript t Baseline plus nu Subscript t Baseline alpha Subscript upper T Baseline period(2.217)

      2.18.2 Stress and Displacement Fields

      The immediate objective is to determine the displacement component u3 and corresponding stresses σ11, σ12 and σ12 in terms of the mechanical loading parameters ε¯A,ε^A,ε¯T,ε^T,σt and the temperature difference ΔT. From (2.209), (2.210) and (2.212), the in-plane stresses must satisfy the relations

      On solving (2.218) and (2.219) for the stresses, it follows that