A Finite Strain Constitutive Model for Martensitic Transformation in Shape Memory Alloys Based on Logarithmic Strain

摘要

Shape Memory Alloys (SMAs) are materials with the ability to recover apparently permanent deformation under specific thermomechanical loading and are thus desirable in various fields such as aerospace and automobile, among others. Most constitutive models for SMAs in the literature are developed based on the assumption of infinitesimal strain. However, such an assumption may not be proper in the presence of geometric discontinuities, such as cracks, and repeated cycling loading that has been reported to induce irrecoverable strains up to 20% due to transformation induced plasticity. In addition to finite strains, SMA-based devices may also undergo finite rotations. Thus, it is indispensable to develop a constitutive model based on finite strain to provide for accurate predictions of these actuators response. A three-dimensional phenomenological constitutive model for SMAs considering finite strains and finite rotations is proposed in this work. This model utilizes the logarithmic strain, or called natural strain, as a finite strain measure which is the only strain measure whose logarithmic rate in a corotating material frame is equal to the rate of deformation tensor. The proposed constitutive model is motivated by the earlier work of Lagoudas and coworkers based on the infinitesimal strain assumption. In the proposed model, the martensitic volume fraction and the second-order logarithmic transformation strain tensor are chosen as the internal state variables associated with the inelastic transformation process. Numerical simulations considering basic SMAs component geometries such as a bar, a beam and a torque tube are performed to test the capabilities of proposed model under both mechanically and thermally induced phase transformation. For numerical examples in which the SMA components exhibits finite strains along with finite rotations, discrepancies are observed between the responses predicted by the present model and its infinitesimal counterpart. Also, the spurious accumulated residual stress observed in infinitesimal strain model is eliminated by proposed model. This shows that the infinitesimal strain assumption is not applicable in such cases and the proposed model considering large strains and rotations is needed to provide accurate predictions. The presented model formulation will be extended in future work for the incorporation of transformation-induced plasticity.