One-Dimensional Thermomechanical Constitutive Relations for Shape Memory Materials

Abstract

The use of the thermoelastic martensitic transformation and its reverse transformation has recently been proposed and demonstrated for several active control applications. However. the present constitutive models have lacked several important fundamental concepts that are essential for many of the proposed intelligent material system applications such as shape memory hybrid composites. A complete, unified, one-dimensional constitutive model of shape memory materials is developed and presented in this paper. The thermomechanical model formulation herein will investigate important material characteristics involved with the internal phase transformation of shape memory alloys. These characteristics include energy dissipation in the material that governs the damping behavior, stress-strain-temperature relations for pseudoelasticity, and the shape memory effect. Some numerical examples using the model are also presented.

Modeling and characterization of shape memory alloy springs with water cooling strategy in a neurosurgical robot

“…The state variables in our model are therefore axial force ( F ), axial spring deflection ( δ ), and martensite volume fraction ( ξ ). Tanaka (Tanaka, 1986), Liang and Rogers (Liang and Rogers, 1990), Brinson (Brinson, 1993), Boyd and Lagoudas (Boyd and Lagoudas, 1996), and Frémond (Frémond, 1996) initially proposed constitutive models for SMA wire. Tobushi and Tanaka (Tobushi and Tanaka, 1991), Liang and Rogers (Liang and Rogers, 1997), Aguiar et al.…”

Development of a Meso-Scale SMA-Based Torsion Actuator for Image-Guided Procedures

“…For a torsion spring, the maximum normal stress (σ) caused by external torque (τ) is given by [22], [23]: $$\sigma =C_2{K}_c\tau$$ where $C_2=\frac{32}{\pi d^3}$ and K c is the stress-concentration factor approximately equal to one. To model the nonlinear characteristics of the SMA torsion spring, the Liang-Rogers model [24] is used, which has been proved to be capable of modeling SMA-based bending actuators in our group’s prior work [25]. By substituting (1) and (2) into the Liang-Rogers model, the constitutive model for the SMA torsion spring is given by [22], [23]: $$C_{2}false(\tau -{\tau }_{0}false)=C_{1}Efalse(\theta -{\theta }_{0}false)-C_1{\theta }_{L}Efalse(\xi -{\xi }_{0}false)$$ where E , ξ, and θ L denote the Young’s modulus, martensite volume fraction, and maximum recoverable torsion angle, respectively.…”

“…Thermal expansion is neglected, because the strain change caused by thermal expansion is much smaller than that caused by phase transformation. By substituting (1) and (2) into the expressions for ξ in the Liang-Rogers model [24], ξ is rewritten for the SMA torsion spring as [22], [23]: $${\xi }_{M\to A}=\frac{{\xi }_0}{2}false\{\text{cos}false[\frac{\pi }{A_f-A_s}false(T-A_s-\frac{C_2\tau }{C_A}false)false]+1false\}$$ and $${\xi }_{A\to M}=\frac{1-{\xi }_0}{2}\text{cos}false[\frac{\pi }{M_s-M_f}false(T-M_f-\frac{C_2\tau }{C_M}false)false]+\frac{1+{\xi }_0}{2}$$ where T is the spring temperature, A s and A f are the start and end temperatures for M→A, respectively, M s and M f are the start and end temperatures for A→M, respectively, and C A and C M are the coefficients representing the influence of stress on the transformation temperatures for these two processes, respectively. By substituting (1) and (2) into the expression of torsion spring stiffness ( K = τ/θ), K can be rewritten as: K = C 1 E/C 2 .…”

Toward a Meso-Scale SMA-Actuated MRI-Compatible Neurosurgical Robot

“…Tanaka [22] developed an exponential expression to describe the martensite volume fraction as a function of stress and temperature. Liang and Rogers [23] presented a model which is based on the rate form of the constitutive equation developed by Tanaka. The difference between these two models is in the modeling of the martensite volume fraction.…”