Integration along the laminate thickness

II.1.11.4 Integration along the laminate thickness

The laminate thermal in-plane conductivity, thermal out-of-plane conductivity and thermal capacity are considered separately.

II.1.11.4.1 In-plane conductivity

To calculate the laminate in-plane thermal conductivity properties, it is assumed that temperature is constant along the laminate thickness. Consequently, the temperature gradient does not depend on $z$ . The thermal flux ${q}_{x y}$ however depends on $z$ because the thermal conductivity does:

{q}_{x y} (z) = {[λ^{T}]}_{x y} (z) \cdot {\nabla T}_{x y} .

The laminate in-plane thermal conductivity is calculated as follows:

\begin{align} {Q}_{x y} & = \int_{- h ∕ 2}^{+ h ∕ 2} {q}_{x y} (z) d z, \\ = \int_{- h ∕ 2}^{+ h ∕ 2} {[λ^{T}]}_{x y} (z) \cdot {\nabla T}_{x y} d z, \\ = (\int_{- h ∕ 2}^{+ h ∕ 2} {[λ^{T}]}_{x y} (z) d z) \cdot {\nabla T}_{x y}, \\ = (\int_{- h ∕ 2}^{+ h ∕ 2} {[S_{+}]}_{(θ)} (z) \cdot {[λ^{T}]}_{12} (z) \cdot {[S_{-}]}_{(θ)} (z) d z) \cdot {\nabla T}_{x y}, \\ = {[Λ^{T}]}_{x y} \cdot {\nabla T}_{x y} . \end{align}

In the previous equation, the laminate in-plane thermal conductivity is introduced:

\begin{align} {[Λ^{T}]}_{x y} & = \int_{- h ∕ 2}^{+ h ∕ 2} {[S_{+}]}_{(θ)} (z) \cdot {[λ^{T}]}_{12} (z) \cdot {[S_{-}]}_{(θ)} (z) d z, \\ = \sum_{k = 1}^{N} {[S_{+}]}_{(θ_{k})} \cdot {[λ_{k}^{T}]}_{12} \cdot {[S_{-}]}_{(θ_{k})} e_{k}, \end{align}

where $e_{k}$ is the thickness of ply $k$ .

II.1.11.4.2 Out-of-plane conductivity

The out-of-plane thermal flux is assumed constant across the laminate thickness. Then, as thermal conductivity depends on $z$ , so does the out-of-plane gradient of temperature:

\nabla T_{z} = \frac{q_{z}}{λ_{33}^{T}} .

The integration across the thickness gives the difference of temperature between upper and lower laminate surfaces:

\begin{align} T_{sup} - T_{inf} & = \int_{- h ∕ 2}^{+ h ∕ 2} \nabla T_{z} d z, \\ = \int_{- h ∕ 2}^{+ h ∕ 2} \frac{q_{z}}{λ_{33}^{T}} d z, \\ = (\int_{- h ∕ 2}^{+ h ∕ 2} \frac{1}{λ_{33}^{T}} d z) q_{z}, \\ = R_{33}^{T} q_{z} . \end{align}

In the previous expression, the out-of-plane thermal resistance is introduced:

\begin{align} R_{33}^{T} & = \int_{- h ∕ 2}^{+ h ∕ 2} \frac{1}{λ_{33}^{T}} d z, \\ = \sum_{k = 1}^{N} \frac{1}{{(λ_{33}^{T})}_{k}} e_{k} . \end{align}

II.1.11.4.3 Thermal capacity

To estimate the laminate thermal capacity, a temperature constant across the laminate thickness is again assumed. Then, the heat energy stored per unit of surface is:

\begin{align} U & = \int_{- h ∕ 2}^{+ h ∕ 2} ρ C_{p} T d z, \\ = (\int_{- h ∕ 2}^{+ h ∕ 2} ρ C_{p} d z) T, \\ = {(ρ C_{p} h)}_{lam} T . \end{align}

In the previous expression, the areal thermal capacity is introduced:

\begin{align} {(ρ C_{p} h)}_{lam} & = \int_{- h ∕ 2}^{+ h ∕ 2} ρ C_{p} d z, \\ = \sum_{k = 1}^{N} {(ρ C_{p})}_{k} e_{k} . \end{align}

Note that in Nastran, when a thermal material MAT4 or MAT5 is defined, the density $ρ$ and the heat capacity per unit of mass $C_{p}$ are defined separately. So it is the responsibility of the user to select appropriate values for these two quantities. Also, in Nastran, the thickness is defined separately in the PSHELL property card. (PCOMP or PCOMPG cards do not accept thermal materials.)

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