Out-of-plane shear properties

II.1.3.4 Out-of-plane shear properties

Developments similar to those in the previous section are carried out here. 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} .

Consecutive transformations of relations (II.1.27) are then applied 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 the following notation is introduced:

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

Tensor $[g]$ is said to be 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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