Thermodynamically consistent non-isothermal phase-field modelling of elastocaloric effect: indirect vs direct method

Modelling elastocaloric effect (eCE) is crucial for the design of environmentally friendly and energy-efficient eCE based solid-state cooling devices. Here, a thermodynamically consistent non-isothermal phase-field model (PFM) coupling martensitic transformation with mechanics and heat transfer is developed and applied for simulating eCE. The model is derived from a thermodynamic framework which invokes the microforce theory and Coleman--Noll procedure. To avoid the numerical issue related to the non-differentiable energy barrier function across the transition point, the austenite-martensite transition energy barrier in PFM is constructed as a smooth function of temperature. Both the indirect method using isothermal PFM with Maxwell relations and the direct method using non-isothermal PFM are applied to calculate the elastocaloric properties. The former is capable of calculating both isothermal entropy change and adiabatic temperature change ($\Delta T_{\text{ad}}$), but induces high computation cost. The latter is computationally efficient, but only yields $\Delta T_{\text{ad}}$. In a model Mn-22Cu alloy, the maximum $\Delta T_{\text{ad}}$ ($\Delta T_{\text{ad}}^{\text{max}}$) under a compressive stress of 100 MPa is calculated as 9.5 and 8.5 K in single crystal (3.5 and 3.8 K in polycrystal) from the indirect and direct method, respectively. It is found that the discrepancy of $\Delta T_{\text{ad}}^{\text{max}}$ by indirect and direct method is within 10% at stress less than 150 MPa, confirming the feasibility of both methods in evaluating eCE at low stress. The results demonstrate the developed PFM herein, combined with both indirect and direct method for eCE calculations, as a practicable toolkit for the computational design of elastocaloric devices.