In-plane and flexural thermo-elastic behavior

II.1.7.1 In-plane and flexural thermo-elastic behavior

The stresses induced in plies for a thermo-elastic loading are calculated assuming that the material strain components are all constrained to zero. Equation (II.1.21) becomes:

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

In laminate axes, the equation is rewritten:

{σ}_{lam} = - Δ T {[C]}_{lam} \cdot {α}_{lam} .

Substituting the assumed temperature profile in the equation gives:

{σ}_{lam} = - [(T_{0} - T_{ref}) + z T_{, z}] {[C]}_{lam} \cdot {α}_{lam} .

The corresponding laminate in-plane force tensor is obtained by integrating the Cauchy stress tensor along the thickness:

\begin{align} {N}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} {σ}_{lam} d z, \\ = - \int_{- h ∕ 2}^{h ∕ 2} [(T_{0} - T_{ref}) + z T_{, z}] {[C]}_{lam} \cdot {α}_{lam} d z, \\ = - (\int_{- h ∕ 2}^{h ∕ 2} {[C]}_{lam} \cdot {α}_{lam} d z) (T_{0} - T_{ref}) - (\int_{- h ∕ 2}^{h ∕ 2} z {[C]}_{lam} \cdot {α}_{lam} d z) T_{, z}, \\ = - {α E h}_{lam} (T_{0} - T_{ref}) - {α E h^{2}}_{lam} T_{, z} . \end{align}

In the previous expression, two new symbols have been introduced that are calculated as follows:

\begin{align} {α E h}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} {[C]}_{lam} \cdot {α}_{lam} d z, \\ = \sum_{k = 1}^{N} {[C]}_{lam}^{k} \cdot {α}_{lam}^{k} (z_{k} - z_{k - 1}) . & (II.1.56) \end{align}

\begin{align} {α E h^{2}}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} z {[C]}_{lam} \cdot {α}_{lam} d z, \\ = \sum_{k = 1}^{N} {[C]}_{lam}^{k} \cdot {α}_{lam}^{k} (\frac{z_{k}^{2} - z_{k - 1}^{2}}{2}) . & (II.1.57) \end{align}

Similarly the bending moment tensor is obtained by integrating the Cauchy stress tensor multiplied by

z

along the thickness:

\begin{align} {M}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} z {σ}_{lam} d z, \\ = - \int_{- h ∕ 2}^{h ∕ 2} z [(T_{0} - T_{ref}) + z T_{, z}] {[C]}_{lam} \cdot {α}_{lam} d z, \\ = - (\int_{- h ∕ 2}^{h ∕ 2} z {[C]}_{lam} \cdot {α}_{lam} d z) (T_{0} - T_{ref}) - (\int_{- h ∕ 2}^{h ∕ 2} z^{2} {[C]}_{lam} \cdot {α}_{lam} d z) T_{, z}, \\ = - {α E h^{2}}_{lam} (T_{0} - T_{ref}) - {α E h^{3}}_{lam} T_{, z} . \end{align}

In the previous expression, one new symbol has been introduced:

\begin{align} {α E h^{3}}_{lam} & = \int_{- h ∕ 2}^{h ∕ 2} z^{2} {[C]}_{lam} \cdot {α}_{lam} d z, \\ = \sum_{k = 1}^{N} {[C]}_{lam}^{k} \cdot {α}_{lam}^{k} (\frac{z_{k}^{3} - z_{k - 1}^{3}}{3}) . & (II.1.58) \end{align}

Because of the linearity of all the equations, the thermo-elastic loading may be considered as an additional loading applied to the laminate, and if an additional imposition of average in-plane strain and curvature is considered, the laminate in-plane forces and bending moments are given by:

\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} {α E h}_{lam} \\ {α E h^{2}}_{lam} \end{matrix}} (T_{0} - T_{ref}) - {\begin{matrix} {α E h^{2}}_{lam} \\ {α E h^{3}}_{lam} \end{matrix}} T_{, z} . & (II.1.59) \end{matrix}

Using relation (II.1.33), the previous expression is reversed as follows:

\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} {[a]}_{lam} & {[b]}_{lam} \\ [b] T_{lam} & {[d]}_{lam} \end{matrix}] \cdot {\begin{matrix} {α E h}_{lam} \\ {α E h^{2}}_{lam} \end{matrix}} (T_{0} - T_{ref}) \\ + [\begin{matrix} {[a]}_{lam} & {[b]}_{lam} \\ [b] T_{lam} & {[d]}_{lam} \end{matrix}] \cdot {\begin{matrix} {α E h^{2}}_{lam} \\ {α E h^{3}}_{lam} \end{matrix}} T_{, z} . & (II.1.60) \end{matrix}

In the last expression, four new quantities can be identified:

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

(II.1.61)

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

(II.1.62)

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

(II.1.63)

{α_{1}^{κ}}_{lam} = [b] T_{lam} \cdot {α E h^{2}}_{lam} + {[d]}_{lam} {α E h^{3}}_{lam} .

(II.1.64)

So that finally, the “compliance” equation is:

\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} {α_{0}^{ϵ}}_{lam} \\ {α_{0}^{κ}}_{lam} \end{matrix}} (T_{0} - T_{ref}) + {\begin{matrix} {α_{1}^{ϵ}}_{lam} \\ {α_{1}^{κ}}_{lam} \end{matrix}} T_{, z} . & (II.1.65) \end{matrix}

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