PSI - Issue 57

SEVEDE Théo et al. / Procedia Structural Integrity 57 (2024) 335–342 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3. 1D analysis approach The starting point for the analysis of a self-heating results is to solve the heat equation. Using the Helmholtz free energy = − , with “ e ” the specific internal energy of the system and “ s ” its entropy, the local form of the heat equation with the assumption of small perturbations is expressed as: ̇ − ( ) = = + + 2 : ̇ + 2 : ̇ With ( ρ ) the density of the material, = − 2 2 the specific heat, (q) the heat flux and the heat source which is composed of the intrinsic dissipated energy ( ) , the thermal radiation ( ) and the thermodynamic couplings source ( 2 : ̇ + 2 : ̇) . Where ( ) is the elastic strain tensor and ( ) a set of internal variables. Theoretically, this equation can be used to reconstruct the thermal source term field from temperature field measurement. However, it is a complex equation to solve, since several problems are superimposed,such as the spatial-temporalevolution of heat exchange, and the time dependence of the mechanical and thermal fields. It is therefore necessary to make simplifying assumptions. In cases where temperature variations are very small or at least sufficiently small, it can be assumed that: - Specific heat C and density ρ are independent of temperature (over the range of temperature variation during the test); - Convection term is negligible compared to the temporal evolution of temperature, so the particle derivative of temperature is defined only by its temporal component thus ̇= / ; - The radiation source term is constant in time, and reasoning by temperature difference with respect to a reference noted T 0 (The initial temperature at the start of our test), the temperature elevation = − 0 is then independent of the thermal radiation source r ; - Only the thermoelastic coupling term ( ( 2 / ) : ̇) is considered. The heat equation in its 1D form can then be written as explained in [4]: ̇ ( , ) + ′ ( , ) − ′ 2 ( , ) 2 − ′ ( , ) ( ) 1 ( ) = ( , ) 3.1. Spectral basis First, the temperature field can be expressed in a spectral basis independently of the boundary conditions. Thus, neglecting conduction heat transfer and thermal source terms, the 1D heat equation becomes: ̇ ( , ) + ( , ) ′ − ′ 2 ( , ) 2 = 0 Therefore, the temperature field is expressed in the form of two functions: one depending only on time t and the other on position x ( , ) = ( ). ( ) . The equation can then be written: ′ ̇( ) ( ) + ′ ′ = 1 ( ) 2 ( ) 2 =− By choosing =√ = 2 the solution can be expressed as follows: ( ) = ∑( sin ( . ) + cos( . ) ) Where A i and B i are heat-source dependant parameters.