Temperature diffusion in laminates

II.1.11 Temperature diffusion in laminates

The thermal conservation equation in a solid material is written as follows:

ρ C_{p} \frac{\partial T}{\partial t} = - \nabla \cdot q + r,

where $T$ is the temperature, $C_{p}$ is the heat capacity of the material and vector $q$ is the heat flux. Note that $C_{p}$ and $ρ$ are used for transient thermal calculations only. Generally, the heat flux is related to the gradient of temperature by Fourier’s law:

q = - λ^{T} \cdot \nabla T,

where $λ^{T}$ is the tensor of thermal conductivity coefficients. We consider that thermal conductivity tensor is symmetric. A justification of this symmetry can be found in [LLK $+$ 86] with part of the demonstration in [LL13].

When thermal conductivity calculations are performed with laminates, two homogenized quantities must first be calculated:

The laminate global conductivity tensor,
The laminate global thermal capacity.

These two quantities are obtained by integrating material properties along laminate thickness.

II.1.11.1 Material thermal parameters
II.1.11.2 In-plane and out-of-plane components
II.1.11.3 In-plane rotations of vectorial and tensorial properties
II.1.11.4 Integration along the laminate thickness
II.1.11.4.1 In-plane conductivity
II.1.11.4.2 Out-of-plane conductivity
II.1.11.4.3 Thermal capacity

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