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Table 3 Equivalent displacements and stresses corresponding to different failure modes

From: Modelling of Damage Evolution in Braided Composites: Recent Developments

Failure mode I

Equivalent displacement

Equivalent stress

L t, σ 1  ≥ 0

\( {X}_{eq}^{Lt}=l\sqrt{<{\varepsilon}_{11}{>}^2+{\varepsilon}_{12}^2+{\alpha \varepsilon}_{31}^2} \)

\( l\left(<{\sigma}_{11}><{\varepsilon}_{11}>+{\sigma}_{12}{\varepsilon}_{12}+{\alpha \sigma}_{13}{\varepsilon}_{13}\right)/{X}_{eq}^{Lt} \)

L c, σ 1  < 0

\( {X}_{eq}^{Lc}=l<-{\varepsilon}_{11}> \)

\( l\left(<-{\sigma}_{11}><-{\varepsilon}_{11}>\right)/{X}_{eq}^{Lc} \)

Y t, σ 2  ≥ 0

\( {X}_{eq}^{Yt}=l\sqrt{<{\varepsilon}_{22}{>}^2+{\varepsilon}_{12}^2+{\alpha \varepsilon}_{23}^2} \)

\( l\left(<{\sigma}_{22}><{\varepsilon}_{22}>+{\sigma}_{12}{\varepsilon}_{12}+{\alpha \sigma}_{23}{\varepsilon}_{23}\right)/{X}_{eq}^{Yt} \)

Y c, σ 2  < 0

\( {X}_{eq}^{Yc}=l<-{\varepsilon}_{22}> \)

\( l\left(<-{\sigma}_{22}><-{\varepsilon}_{22}>\right)/{X}_{eq}^{Yc} \)

Z t, σ 3  ≥ 0

\( {X}_{eq}^{Zt}=l\sqrt{<{\varepsilon}_{33}{>}^2+{\varepsilon}_{23}^2+{\varepsilon}_{31}^2} \)

\( l\left(<{\sigma}_{33}><{\varepsilon}_{33}>+{\sigma}_{23}{\varepsilon}_{23}+{\sigma}_{13}{\varepsilon}_{13}\right)/{X}_{eq}^{Zt} \)

Z c, σ 3  < 0

\( {X}_{eq}^{Zc}=l<-{\varepsilon}_{33}> \)

\( l\left(<-{\sigma}_{33}><-{\varepsilon}_{33}>\right)/{X}_{eq}^{Zc} \)

<x > = (x + |x|)/2