Out-of-plane shear properties

II.1.3.4 Out-of-plane shear properties

One makes developments similar to those in the previous section. The out-of-plane shear constitutive equations are written as follows:

{\{τ_{s}\}}_{ply} = {[g]}_{ply} \cdot {\{γ_{s}\}}_{ply} - Δ T {[g]}_{ply} \cdot {\{α_{s}\}}_{ply} - Δ H {[g]}_{ply} \cdot {\{β_{s}\}}_{ply},

(II.1.27)

{\{γ_{s}\}}_{ply} = {[g^{- 1}]}_{ply} \cdot {\{τ_{s}\}}_{ply} + Δ T {\{α_{s}\}}_{ply} + Δ H {\{β_{s}\}}_{ply} .

(II.1.28)

If $ξ$ is the angle of the ply in the laminate, the previous relations can be written in laminate axes by rotating them by an angle $- ξ$ . For example:

{\{γ_{s}\}}_{lam} = {[S_{+}]}_{(- ξ)} \cdot {\{γ_{s}\}}_{ply},

{\{τ_{s}\}}_{lam} = {[S_{+}]}_{(- ξ)} \cdot {\{τ_{s}\}}_{ply},

{\{α_{s}\}}_{lam} = {[S_{+}]}_{(- ξ)} \cdot {\{α_{s}\}}_{ply},

{\{β_{s}\}}_{lam} = {[S_{+}]}_{(- ξ)} \cdot {\{β_{s}\}}_{ply},

{\{γ_{s}\}}_{ply} = {[S_{-}]}_{(- ξ)} \cdot {\{γ_{s}\}}_{lam} = {[S_{+}]}_{(ξ)} \cdot {\{γ_{s}\}}_{lam},

{\{τ_{s}\}}_{ply} = {[S_{-}]}_{(- ξ)} \cdot {\{τ_{s}\}}_{lam} = {[S_{+}]}_{(ξ)} \cdot {\{τ_{s}\}}_{lam},

{\{α_{s}\}}_{ply} = {[S_{-}]}_{(- ξ)} \cdot {\{α_{s}\}}_{lam} = {[S_{+}]}_{(ξ)} \cdot {\{α_{s}\}}_{lam},

{\{β_{s}\}}_{ply} = {[S_{-}]}_{(- ξ)} \cdot {\{β_{s}\}}_{lam} = {[S_{+}]}_{(ξ)} \cdot {\{β_{s}\}}_{lam} .

Then, one makes consecutive transformations of relations (II.1.27) as follows:

\begin{matrix} {[S_{+}]}_{(ξ)} \cdot {\{τ_{s}\}}_{lam} = {[g]}_{ply} \cdot {[S_{+}]}_{(ξ)} \cdot {\{γ_{s}\}}_{lam} \\ - Δ T {[g]}_{ply} \cdot {[S_{+}]}_{(ξ)} \cdot {\{α_{s}\}}_{lam} - Δ H {[g]}_{ply} \cdot {[S_{+}]}_{(ξ)} \cdot {\{β_{s}\}}_{lam}, & (II.1.29) \end{matrix}

\begin{matrix} {\{τ_{s}\}}_{lam} = {[S_{+}]}_{(- ξ)} \cdot {[g]}_{ply} \cdot {[S_{+}]}_{(ξ)} \cdot {\{γ_{s}\}}_{lam} \\ - Δ T {[S_{+}]}_{(- ξ)} \cdot {[g]}_{ply} \cdot {[S_{+}]}_{(ξ)} \cdot {\{α_{s}\}}_{lam} \\ - Δ H {[S_{+}]}_{(- ξ)} \cdot {[g]}_{ply} \cdot {[S_{+}]}_{(ξ)} \cdot {\{β_{s}\}}_{lam}, & (II.1.30) \end{matrix}

{\{τ_{s}\}}_{lam} = {[g]}_{lam} \cdot {\{γ_{s}\}}_{lam} - Δ T {[g]}_{lam} \cdot {\{α_{s}\}}_{lam} - Δ H {[g]}_{lam} \cdot {\{β_{s}\}}_{lam},

where one introduced:

{[g]}_{lam} = {[S_{+}]}_{(- ξ)} \cdot {[g]}_{ply} \cdot {[S_{+}]}_{(ξ)} .

One says that tensor $[g]$ is rotated by matrix ${[S_{+}]}_{(- ξ)}$ which corresponds to the expression of the shear stiffness tensor in a new coordinate system obtained by rotating the previous one by an angle $- ξ$ .

The transformation of the out-of-plane shear compliance tensor by the same angle $- ξ$ is made with the same expression as for the stiffness tensor:

{[g^{- 1}]}_{lam} = {[S_{+}]}_{(- ξ)} \cdot {[g^{- 1}]}_{ply} \cdot {[S_{+}]}_{(ξ)} .

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