Out-of-plane shear thermo-elastic behavior

II.1.7.2 Out-of-plane shear thermo-elastic behavior

Starting with the out-of-plane shear constitutive equation (II.1.28) and the expression defining the out-of-plane shear force vectors ${Q_{s}}_{lam}$ , developments similar to those of section II.1.7.1 are carried out, defining the following quantities:

\begin{align} {α_{s} G h}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} {[g]}_{lam} \cdot {α_{s}}_{lam} d z, \\ = \sum_{k = 1}^{N} {[g]}_{lam}^{k} \cdot {α_{s}}_{lam}^{k} (z_{k} - z_{k - 1}) . & (II.1.66) \end{align}

\begin{align} {α_{s} G h^{2}}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} z {[g]}_{lam} \cdot {α_{s}}_{lam} d z, \\ = \sum_{k = 1}^{N} {[g]}_{lam}^{k} \cdot {α_{s}}_{lam}^{k} (\frac{z_{k}^{2} - z_{k - 1}^{2}}{2}) . & (II.1.67) \end{align}

They are used in the expression:

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

(II.1.68)

Correspondingly, the laminate out-of-plane shear CTE vectors are defined:

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

(II.1.69)

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

(II.1.70)

These two expressions allow one to write the inverse of (II.1.68):

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

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