PSI - Issue 61

Necdet Ali Özdür et al. / Procedia Structural Integrity 61 (2024) 277–284 N.A. O¨ zdu¨r et al. / Structural Integrity Procedia 00 (2024) 000–000

280

4

after). Simultaneously, instantaneous force values are recorded using the analog output of the load cell (label 6 ). At the end of the experiment, the light is turned back on and a final macro-DIC image is recorded to depict the final deformation. Sensor noise in the form of spurious jumps appearing in raw temperature readings are eliminated by subtracting the average temperature change values of the dummy sample. A Savitzky-Golay filter is then applied to smooth the temperature signal. The total strain rate along the gage section does not stay constant throughout the experiment, even though the loading is applied in a displacement-controlled manner. Therefore, the displacement rate could not be assumed constant and inferred directly from the macro-DIC measurement at the end of the loading. Thus, the strain history of the specimen is measured by DIC analysis on IR images themselves, by treating the entire gage section as a single deformation subset. Since the IR images are low resolution, and the surface pattern is sub-optimal due to the matte black paint, IR-DIC measurements are validated by comparing the final strain value to the macro-DIC strain average inside the gage section (Fig. 2). (1) Here, the left-hand side of the equation represents change in thermal energy, div( q ) represents heat losses due to conduction, r is the external heat supply, and D int is the internal (or intrinsic) dissipation. Under the postulate of existence of a Helmholtz free energy function, W ( F , T , Z ), where Z denote a suitable set of internal variables (e.g. plastic deformation, F p , plastic slip on the slip system α , γ α ), the above equation for energy balance can be reformulated into the following local form given in Stainier (2013): − T ∂ 2 W ∂ T 2 ˙ T = D int − div( q ) + r + T ∂ 2 W ∂ F ∂ T : ˙ F + T ∂ 2 W ∂ Z ∂ T · ˙ Z (2) The two newly occurring terms in the right-hand side in Eq. 2 correspond to the thermo-elastic part and thermal couplings for the internal variables, respectively. Following the variational framework of Ortiz and Stainier (1999), the constitutive equations can be written as 2.4. Thermal Modeling The energy conservation in terms of entropy rate, ˙ η , is given by ρ 0 T ˙ η = D int − div( q ) + r where Y k are the thermodynamic driving forces for the internal variables Z k . Here, ∆ ∗ ( ˙ Z ; Z , T ) denotes the dual form of the dissipation pseudo-potential, which incorporates rate dependency relations. Together with rate of Helmholtz free energy, they form the stress power, ˙ W +∆ ∗ . In the variational framework, the minimum (or stationary point) of stress power with respect to internal variables gives the evolution law of internal variables. Furthermore, note that following the work of Taylor and Quinney (1934), the internal dissipation is traditionally related to the total plastic work through a coe ffi cient, β , also referred to as the Taylor-Quinney coe ffi cient. D int = Y · ˙ Z = β S : D p (4) Here, S , and D p denote Mandel stress and plastic flow rate tensors, respectively. Combining Eqs. 3, 4 with Eq. 2, and defining the heat capacity as ρ 0 C = − T ∂ 2 W /∂ T 2 , the local thermal equilibrium equation can be written as ρ 0 C ˙ T = β S : D p − div( q ) + r + T ∂ P ∂ T : ˙ F − T ∂ Y ∂ T · ˙ Z . (5) For the experimental measurements in this work, further simplifications are made to Eq. 5 using the following assumptions: P = ∂ W ∂ F ∂ W ∂ Z k ∂ W ∂ T Y k = − ρ 0 η = − = ∂ ∆ ∗ ∂ ˙ Z k (3)