Analytical study on stress-induced phase transitions in geometrically graded shape memory alloy layers. Part I: Asymptotic equation and analytical solutions

In this paper, an analytical approach is proposed to study the stress-induced phase transitions in geometrically graded shape memory alloy (SMA) layers. First, within a two-dimensional setting, the governing PDE system is formulated, which contains the mechanical equilibrium equations and the phase transition criteria. By using the coupled series-asymptotic expansion method, the equilibrium equations can be reduced into one single ODE, which contains the leading-order term of the axial strain and the phase state functions. Besides that, the function representing the varying width of the SMA layer is also involved in this ODE. By further using the phase transition criteria, the phase state functions can be expressed in terms of the axial displacement for the pure loading and pure unloading processes. Then one asymptotic equation with variable coefficients is obtained. For linearly tapered SMA layers, the asymptotic equation is simplified and the WKB method is adopted to solve this equation. The obtained analytical solutions can capture the key features of the mechanical responses of the SMA layers and reveal the underlying mechanisms. The inhomogeneous configurations of the SMA layers during the phase transition process can also be simulated.