In-plane properties

II.1.3.3 In-plane properties

One considers the properties of the ply in a plane parallel to the laminate. Then the constitutive equation (16) reduces to:

(\begin{matrix} ϵ_{11} \\ ϵ_{22} \\ γ_{12} \end{matrix}) = (\begin{matrix} c_{1111} & c_{1122} & c_{1112} \\ c_{2211} & c_{2222} & c_{2212} \\ c_{1211} & c_{1222} & c_{1212} \end{matrix}) \cdot (\begin{matrix} σ_{11} \\ σ_{22} \\ τ_{12} \end{matrix}) + Δ T (\begin{matrix} α_{11} \\ α_{22} \\ α_{12} \end{matrix}) + Δ H (\begin{matrix} β_{11} \\ β_{22} \\ β_{12} \end{matrix}) .

(II.1.19)

The indices in this notation are integers and indicate that the corresponding properties are given in ply coordinate system. The equation (II.1.19) is written more shortly as follows:

{\{ϵ\}}_{ply} = {[c]}_{ply} \cdot {\{σ\}}_{ply} + Δ T {\{α\}}_{ply} + Δ H {\{β\}}_{ply} .

(II.1.20)

One introduces in (II.1.20) the material in-plane compliance matrix ${[c]}_{ply}$ . In order to avoid too complicated notations, one uses the same notations as for the $6 \times 6$ full material compliance matrix introduced in (II.1.17). This will be done systematically for the in-plane matricial and vectorial quantities in the rest of the document ( $[c]$ , $[C]$ , $\{α\}$ , $\{β\}$ , $\{ϵ\}$ ,...

The inverse of expression (II.1.20) is noted:

{\{σ\}}_{ply} = {[C]}_{ply} \cdot {\{ϵ\}}_{ply} - Δ T {[C]}_{ply} \cdot {\{α\}}_{ply} - Δ H {[C]}_{ply} \cdot {\{β\}}_{ply} .

(II.1.21)

In (II.1.21) one introduces the in-plane stiffness matrix ${[C]}_{ply} = {[c]}_{ply}^{- 1}$ .

Plies are characterized by their orientation in the laminate. Let $ξ$ be the angle of the ply in the laminate axes. Then, the laminate axes are obtained by rotating the ply axes by an angle $- ξ$ . Equations (II.1.20) and (II.1.21) are expressed in the laminate coordinate system as follows:

{\{ϵ\}}_{lam} = {[T_{+}^{'}]}_{(- ξ)} \cdot {[c]}_{ply} \cdot {[T_{+}]}_{(- ξ)} \cdot {\{σ\}}_{lam} + Δ T {[T_{+}^{'}]}_{(- ξ)} \cdot {\{α\}}_{ply} + Δ H {[T_{+}^{'}]}_{(- ξ)} \cdot {\{β\}}_{ply},

\begin{matrix} {\{σ\}}_{lam} = {[T_{-}]}_{(- ξ)} \cdot {[C]}_{ply} \cdot {[T_{-}^{'}]}_{(- ξ)} \cdot {\{ϵ\}}_{lam} \\ - Δ T {[T_{-}]}_{(- ξ)} \cdot {[C]}_{ply} \cdot {\{α\}}_{ply} - Δ H {[T_{-}]}_{(- ξ)} \cdot {[C]}_{ply} \cdot {\{β\}}_{ply} . & (II.1.22) \end{matrix}

This leads to the new expression in laminate axes:

{\{ϵ\}}_{lam} = {[c]}_{lam} \cdot {\{σ\}}_{lam} + Δ T {\{α\}}_{lam} + Δ H {\{β\}}_{lam},

{\{σ\}}_{lam} = {[C]}_{lam} \cdot {\{ϵ\}}_{lam} - Δ T {[C]}_{lam} \cdot {\{α\}}_{lam} - Δ H {[C]}_{lam} \cdot {\{β\}}_{lam},

where one introduces new notations for in-plane ply properties rotated by an angle $- ξ$ (in laminate axes):

{[c]}_{lam} = {[T_{+}^{'}]}_{(- ξ)} \cdot {[c]}_{ply} \cdot {[T_{+}]}_{(- ξ)},

(II.1.23)

{\{α\}}_{lam} = {[T_{+}^{'}]}_{(- ξ)} \cdot {\{α\}}_{ply},

(II.1.24)

{\{β\}}_{lam} = {[T_{+}^{'}]}_{(- ξ)} \cdot {\{β\}}_{ply},

(II.1.25)

{[C]}_{lam} = {[T_{-}]}_{(- ξ)} \cdot {[C]}_{ply} \cdot {[T_{-}^{'}]}_{(- ξ)},

(II.1.26)

{[C]}_{lam} \cdot {\{α\}}_{lam} = {[T_{-}]}_{(- ξ)} \cdot {[C]}_{ply} \cdot {\{α\}}_{ply},

{[C]}_{lam} \cdot {\{β\}}_{lam} = {[T_{-}]}_{(- ξ)} \cdot {[C]}_{ply} \cdot {\{β\}}_{ply} .

When a matrix is transformed as in (II.1.23) or a vector as in (II.1.24), one says that they are rotated with ${[T_{+}^{'}]}_{(- ξ)}$ rotation matrix.

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