Analytical Study On Stress-induced Martensitic Transformation In Geometrically Graded Shape Memory Alloys

Due to their unique mechanical properties,shape memory alloys(SMA)attract the scholars' attention from different areas.The relevant research works indicate that the mechanism behind the mechanical properties is the martensitic transformation happening in the SMA specimens.Thus,it is very important to figure out the stress-induced phase transformation in order to be master of the properties of SMA.In this paper,an analytical approach is proposed to study the stress-induced phase transitions in the geometrically graded shape memory alloy(SMA)layers.First,within a twodimensional setting,the governing partial differential equation(PDE)system is formulated,which contains the mechanical equilibrium equations,stress-free boundary conditions and the phase transition criteria.By using the Coupled Series-Asymptotic Expansion(CSAE)method,the equilibrium equations and stress-free boundary conditions can be reduced into one single ordinary differential equation(ODE),which contains the leading-order term of the axial strain and the phase state functions.By further using the phase transition criteria and CSAE,the phase state functions can be asymptotically expressed in terms of the axial displacement for the pure loading and pure unloading processes.Substituting the phase state functions into the single ODE,then the asymptotic equations with variable coefficients corresponding to austenite,phase transformation and martensite are arrived.For the three kinds of SMA layers with different shapes,the asymptotic equation is simplified and both the WKB method and the method of variation of parameters are 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.In addition,the inhomogeneous deformations of the SMA layers during the phase transition process can be simulated.Besides,the effects of width ratio on the responses of the SMA layers are discussed and an explicit expression of the slops of stress-strain cures during the stage of phase transformation in terms of the width ratio is given.Except that,based on the asymptotic ODE system,the problems on loading procedures and boundary conditions are also studied in this paper.To begin with,the obtained analytical results are used to study the inner loops of the stress-strain curves.Corresponding to the inner loops of the stress-strain curves,the configurations and phase states of the SMA layers with the different shapes are simulated.Then,the effects of boundary conditions are studied,where the free end boundary conditions,the mixed boundary conditions and the clamped boundary conditions are taken into account.The corresponding analytical solutions are derived and the responses of the SMA layers are simulated,which are compared with each other to reveal the effects of boundary conditions.The results obtained from these two studies further demonstrate the validity of the analytical approach for studying the phase transformation in geometrically graded SMA specimens.