Hygrometric behavior of laminates

The results for thermo-elastic behavior are transposed below, 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} .