PSI - Issue 39

Rosa De Finis et al. / Procedia Structural Integrity 39 (2022) 528–545 Author name / Structural Integrity Procedia 00 (2019) 000–000

531

4

invariant and principal stresses:

̇ = − � + � 2 − 1 � � ̇ + � ( 1+ ) 2 − 1 � ̇

(1)

Where ρ is the material density , C ε is the specific heat under constant strain, T is the temperature , σ i are principal stresses, v is the Poisson modulus, s is the first stress invariant, E is the Young modulus and α is the linear thermal expansion coefficient. The material constants, a and b, are defined in (Patterson et al. (2007) , Palumbo et al. (2015) and Di Carolo et al. (2) where is the angular frequency and are the amplitude of uniaxial stress along the generic i direction. Under a generic sinusoidal loading in which the load changes between its maximum and minimum values, P min and P max , the loading ratio R can be defined as: = = , , , = 1 1 + − = with i=1,2 (3) By deriving the Eq. (2) and substituting them in Eq. (1), it leads: ̇ = �− ( 1 + 2 ) + ( − 2 1 2 + 1 2 + 2 2 ) � + [ (1 + 2 )( 1 2 + 2 2 ) + +2 1 2 ] (4) By integrating Eq. (4) , considering ΔT<