Differential and integrated form consistency in 1-D phenomenological models for shape memory alloy constitutive behavior

Shape memory alloys are being explored increasingly for developing smart structures and devices in aerospace, automotive and other application areas. The material behavior is highly nonlinear with coupled thermomechanical response involving temperature and/or stress induced phase transformations. Modeling the constitutive behavior of these materials poses several challenges and a few phenomenological models exist that provide a quick and reasonable approach to assess their behavior. Due to phenomenological approach, several assumptions are made in order to simplify the model and some of them introduce inconsistencies or anomalies into the model. In this paper, a frequently used approach, namely, Brinson [Brinson, L.C., 1993. One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4(2), 229–242.] model, is investigated. The constitutive equation is usually expressed at the outset in the differential form and the integrated form of the same is obtained. It is shown that the two forms of equations are not consistent in the Brinson [Brinson, L.C., 1993. One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4(2), 229–242.] model, given the assumed form of material functions. In the present work, the nature and implications of the inconsistency are highlighted. The cause of incompatibility is the inconsistent material definitions. A modified consistent constitutive model is proposed by redefining the material function which satisfies the compatibility condition. The advantages in using the proposed modified model are highlighted with numerical case studies involving hysteretic stress–strain behavior.