In-plane and flexural thermo-elastic behavior

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

One calculates the stresses induced in plies for a thermo-elastic loading 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} .

One substitutes in the equation the assumed temperature profile:

{\{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 one considers an additional imposition of average in-plane strain and of a curvature, 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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