PSI - Issue 57

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

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In the same way, by choosing ′ = 2 ′ + 1

′ , o ne can get the expression: ( ) = (− ′ ( − 0 ))

By multiplying the two previous equations, one can obtain:

( , ) = ∑ [( ( . ) + ( . ))× (− ′ ( − 0 ))] This equation enables to estimate the spacial and time evolution of the temperature at any material point through the sample length. 3.2. Boundary conditions In steady state case, ( ) = ∑ [( ( . ) + ( . ))] . Thus, at the extremities of the specimen, i.e. at =± , = ℎ . With “ h ” the heat exchange coefficient between the sample and the amplifier, so: − =ℎ ; − ( ) = ℎ ( ) ; ( ) =− ; 2 = 0 ; ⇒ ⟶∞ However, the length of the thermal system cannot be considered infinite since there are exchanges and a rise of temperature at its extremities. As the system does not respect Fourier boundary conditions, the spectral basis needs to be completed to take these boundary conditions into account. The first method is to define eigenfunctions that respects the boundary conditions, but this requires precise knowledge of the exchange coefficients. The second method is to assume that the boundary heat exchanges have a strong influence on the mean response of the system but a negligible influence on the local variations between volume elements along the sample, as proposed by [4]. This second method is selected to complete the basis with a second-degree polynomial to be able to consider arbitrary flows at the specimen boundary. Then, the previous equation can be written as: ( ) = ( . 2 + . + ) + ∑ [( ( . ) + ( . ))× (− ′ ( − 0 ))] 3.3. Orthogonalization of the spectral basis The previously proposed spectral basis for solving the 1D problem is not orthogonal, due to the linear and quadratic terms of the second-degree polynomial. Therefore, the projection of these terms in the Fourier series is subtracted from the basis. After simplification, the final form of the temperature field can be expressed by: ( ) = ( . 2 + . + ) + ∑ [( ′ (2 . ) + ′ (2 . ))] =1 With, ′ = − . 2 . (−1) 2 . 2 et ′ = + . .(−1) . This analytical form of the temperature field, will be adopted to solve the 1D heat equation. Two methods can be used, the transient regime and the steady state regime (Figure 2). During the early stages of the test, the effects of convection can be neglected which allows to simplify the heat equation for the transient regime to: