PSI - Issue 75

KADIRI Mounir et al. / Procedia Structural Integrity 75 (2025) 633–641 KADIRI Mounir/ Structural Integrity Procedia (2025)

636

4

measurable (the temperature) to a source term that is an intrinsic property of the material. Here, the source term is the energy dissipated during cyclic loading. The local form of the heat equation, derived from the principle of energy conservation, can be written as: ̇+ ( ⃗)= ∆+ + 2 : ̇ + 2 : ̇ = (1) The left-hand side of equation (1) defines heat absorption ( ̇) as well as thermal losses by conduction ( ( ⃗ )) where , , and ⃗ denote temperature, heat capacity, density and the heat-flux vector. The right-hand side of the equation describes the source term which we seek to identify locally. This source term is the sum of three contributions: • Intrinsic dissipation energy ∆=Σ∶ ̇ − : ̇ • External heat supply r • Coupling terms 2 : ̇ and 2 : ̇ Where Σ is the stress tensor, is the elastic strain tensor, is the plastic strain tensor, denotes a set of thermodynamic forces associated to a set of the internal variables and is the Helmholtz free energy. In order to determine the source term during self-heating tests, the heat equation must be solved. To this end, the following simplifying assumptions are made (Munier 2012): • The specific heat c is independent of temperature • Internal convection is negligible, implying • The radiation source term is constant in time • The only negligible coupling term is the thermo-elastic coupling 1 = ² : ̇ Solving the heat equation also requires knowledge of the temperature distribution over the entire volume. A “0D” approach is adopted, in which only the volume averaged temperature is considered, hence the source term is assumed spatially homogeneous. Heat losses occur by exchange with the exterior and are of two types: losses through the jaws of the testing machine and losses through the lateral faces by exchange with the ambient environment. Integrating the heat equation over the volume leads to the following expression (2): ̇ 0 ( ) + 0 ( ) 0 = 0 ( ) (2) With: • 0 the volume averaged temperature rise • 0 the characteristic time of thermal exchange • 0 the mean value of the source term over the material volume split into intrinsic dissipation and the thermoelastic coupling term 1 By integrating the source term over one cycle, the contributions of the thermoelastic coupling sum to zero and the cycle averaged source term ̅ 0 is given by: ̇ 0 ( ) + 0 ( ) 0 = ̅ 0 = ∫ 0 = ∫ ∆ = (3) With the solicitation frequency and 0 the mean value of the temperature rises per cycle Self-heating consists of a series of cyclic loading with increasing stress amplitude on the same specimen. At each stress amplitude, the cyclic loading is carried out at the same frequency and for the same number of cycles.