Hygrometric behavior of laminates

II.1.7.3 Hygrometric behavior of laminates

One transposes below the results for thermo-elastic behavior. It is done simply by replacing the CTE $α$ by the CME $β$ in the definitions and equations.

\begin{align} {\{β E h\}}_{lam} & = \sum_{k = 1}^{N} {[C]}_{lam}^{k} \cdot {\{β\}}_{lam}^{k} (z_{k} - z_{k - 1}), & (II.1.71) \end{align}

\begin{align} {\{β E h^{2}\}}_{lam} & = \sum_{k = 1}^{N} {[C]}_{lam}^{k} \cdot {\{β\}}_{lam}^{k} (\frac{z_{k}^{2} - z_{k - 1}^{2}}{2}), & (II.1.72) \end{align}

\begin{align} {\{β E h^{3}\}}_{lam} & = \sum_{k = 1}^{N} {[C]}_{lam}^{k} \cdot {\{β\}}_{lam}^{k} (\frac{z_{k}^{3} - z_{k - 1}^{3}}{3}), & (II.1.73) \end{align}

\begin{matrix} \{\begin{matrix} {\{N\}}_{lam} \\ {\{M\}}_{lam} \end{matrix}\} = [\begin{matrix} {[A]}_{lam} & {[B]}_{lam} \\ {[B]}_{lam} & {[D]}_{lam} \end{matrix}] \cdot \{\begin{matrix} {\{ϵ^{0}\}}_{lam} \\ {\{κ\}}_{lam} \end{matrix}\} \\ - \{\begin{matrix} {\{β E h\}}_{lam} \\ {\{β E h^{2}\}}_{lam} \end{matrix}\} (H_{0} - H_{ref}) - \{\begin{matrix} {\{β E h^{2}\}}_{lam} \\ {\{β E h^{3}\}}_{lam} \end{matrix}\} H_{, z}, & (II.1.74) \end{matrix}

{\{β_{0}^{ϵ}\}}_{lam} = {[a]}_{lam} \cdot {\{β E h\}}_{lam} + {[b]}_{lam} {\{β E h^{2}\}}_{lam},

(II.1.75)

{\{β_{1}^{ϵ}\}}_{lam} = {[a]}_{lam} \cdot {\{β E h^{2}\}}_{lam} + {[b]}_{lam} {\{β E h^{3}\}}_{lam},

(II.1.76)

{\{β_{0}^{κ}\}}_{lam} = [b] T_{lam} \cdot {\{β E h\}}_{lam} + {[d]}_{lam} {\{β E h^{2}\}}_{lam},

(II.1.77)

{\{β_{1}^{κ}\}}_{lam} = [b] T_{lam} \cdot {\{β E h^{2}\}}_{lam} + {[d]}_{lam} {\{β E h^{3}\}}_{lam},

(II.1.78)

\begin{matrix} \{\begin{matrix} {\{ϵ^{0}\}}_{lam} \\ {\{κ\}}_{lam} \end{matrix}\} = [\begin{matrix} {[a]}_{lam} & {[b]}_{lam} \\ [b] T_{lam} & {[d]}_{lam} \end{matrix}] \cdot \{\begin{matrix} {\{N\}}_{lam} \\ {\{M\}}_{lam} \end{matrix}\} \\ + \{\begin{matrix} {\{β_{0}^{ϵ}\}}_{lam} \\ {\{β_{0}^{κ}\}}_{lam} \end{matrix}\} (H_{0} - H_{ref}) + \{\begin{matrix} {\{β_{1}^{ϵ}\}}_{lam} \\ {\{β_{1}^{κ}\}}_{lam} \end{matrix}\} H_{, z}, & (II.1.79) \end{matrix}

\begin{align} {\{β_{s} G h\}}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} {[g]}_{lam} \cdot {\{β_{s}\}}_{lam} d z, \\ = \sum_{k = 1}^{N} {[g]}_{lam}^{k} \cdot {\{β_{s}\}}_{lam}^{k} (z_{k} - z_{k - 1}), & (II.1.80) \end{align}

\begin{align} {\{β_{s} G h^{2}\}}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} z {[g]}_{lam} \cdot {\{β_{s}\}}_{lam} d z, \\ = \sum_{k = 1}^{N} {[g]}_{lam}^{k} \cdot {\{β_{s}\}}_{lam}^{k} (\frac{z_{k}^{2} - z_{k - 1}^{2}}{2}), & (II.1.81) \end{align}

{\{Q\}}_{lam} = {[G]}_{lam} \cdot {\{Γ\}}_{lam} - {\{β_{s} G h\}}_{lam} (H_{0} - H_{ref}) - {\{β_{s} G h^{2}\}}_{lam} H_{, z},

{\{β_{0}^{s}\}}_{lam} = {[G]}_{lam}^{- 1} \cdot {\{β G h\}}_{lam}

(II.1.82)

{\{β_{1}^{s}\}}_{lam} = {[G]}_{lam}^{- 1} \cdot {\{β G h^{2}\}}_{lam}

(II.1.83)

{[Γ]}_{lam} = {[G]}_{lam}^{- 1} \cdot {\{Q\}}_{lam} + {\{β_{0}^{s}\}}_{lam} (H_{0} - H_{ref}) + {\{β_{1}^{s}\}}_{lam} H_{, z} .

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