The Lecture Contains: Shape Memory Alloy Constitutive Relationship Tanaka Model file:///d /chitra/vibration_upload/lecture34/34_1.htm[6/25/2012 12:42:36 PM]
Shape Memory Alloy (SMA) based Actuators Shape memory effect (SME) is the property of Shape Memory Alloys (SMA) by virtue of which these can recover apparent permanent strains to revert back to its original shape when they are heated above a certain temperature. SMA's have two stable phases: The high temperature phase called austenite (parent phase). Austenite has cubic crystal structure. The low temperature martensite phase. Martensite has monoclinic crystal structure SME occurs because of phase transformation of Martensite to Austenite beyond re-crystallization temperature. In the absence of any applied load, upon heating an SMA, the Martensite phase simply gets transformed to Austenite. During this phase transformation, though there is change in the microstructure of SMA, macroscopic shape of the SMA does not change. Again, on cooling the SMA, a reverse Martensite transformation takes place; however, the Austenite now gets transformed to twinned Martensite. This transformation is called self-accommodating transformation. Figure 34.1 shows the phase transformations in SMA. Figure 34.1: Phase tranformation of SMA Now, upon loading SMA in the Martensite phase, de-twinning of Martensite takes place resulting in large straining of SMA. The strain produced in SMA during complete transformation of twinned Martensite to de-twinned Martensite is known as transformation strain [Λ t ], and is the maximum strain file:///d /chitra/vibration_upload/lecture34/34_2.htm[6/25/2012 12:42:36 PM]
that SMA can recover upon heating. Further loading of SMA causes permanent strain which cannot be recovered. Recovered transformation strain of SMA is directly proportional to the degree of transformation of twinned Martensite to de-twinned Martensite. file:///d /chitra/vibration_upload/lecture34/34_2.htm[6/25/2012 12:42:36 PM]
There are four critical temperatures for SMA which characterize its behavior. These are A s, A f, M s and M f with A and M indicating Austenite and Martensite phases. The subscripts, s and f denote start and finish of transformation, respectively. These temperatures are stress dependent and their values change depending on the loading conditions. The Shape Memory Effect (SME) can be summarized as shown in Fig.34.2. At a temperature below M fo, SMA has twinned Martensitic microstructure (B). On loading SMA, the de-twinning of Martensite starts and it ends at some critical value of the stress (C). Further loading of SMA results in permanent plastic strain in SMA. On complete unloading, SMA still retains strain in it (D), which is recovered if it is heated beyond A fo (E). At A so, Martensite phase starts transforming to Austenite. A fo is the temperature for completing this transformation at zero stress level. Percentage of transformation strain recovered by SMA is proportional to Austenite volume fraction. On cooling SMA, Martensite transformation starts from M so and for temperatures below M fo Austenite gets completely transformed to twinned Martensite (B). C m and C A are material dependent parameters of SMA providing the relationship between temperature and stress for Martensite and Austenite phases, respectively. Their units are Pa / 0 K. C M is typically of higher magnitude than C A. Figure 34.2: Shape memory effect - Cubic crystal structure. Austenite phase - Twinned martensite crystals. Twinned martensite phase file:///d /chitra/vibration_upload/lecture34/34_3.htm[6/25/2012 12:42:36 PM]
- De-twinned martensite crystals. De-twinned martensite phase file:///d /chitra/vibration_upload/lecture34/34_3.htm[6/25/2012 12:42:36 PM]
Lecture 35: Shape Memory Alloy based Actuators SME constitutes both stress induced transformation (twinned Martensite to de-twinned Martensite) and temperature induced transformation (de-twinned Martensite to Austenite). SME is a useful property of SMA for applications in the field of linear actuators as the displacement range of an actuator can be easily controlled by controlling the temperature of the SMA element. To understand this complicated behavior and to optimally design smart actuating elements, various constitutive models have been proposed. These models may be classified into two types: The first group uses a micromechanical approach to follow closely the crystallographic phenomena of the SMAs by using the thermodynamics laws for describing the transformation. These models consider the martensitic variant as a transformation inclusion and use micromechanics to calculate the interaction energy due to the phase transformation in the material. Stresses and strains are obtained as averages over a volume in which many inclusions are considered representing possible variants. Most of the micromechanics-based constitutive models include implicit variables such as free energy and predict the material behaviors of the SMAs qualitatively. These models cannot be easily used for engineering application as variables such as free energy are not readily quantifiable. The alternate macroscopic (phenomenological) model is built on phenomenological thermodynamics and/or curve fitting of experimental data. Transition regions in the common phase diagram of SMA are experimentally determined by plotting the stress-temperature diagram. The first macroscopic constitutive model was proposed by Tanaka. This model is for one dimensional SMA and is obtained from general three dimensional theory based on the energy balance equation and the Clausius-Duhem inequality. file:///d /chitra/vibration_upload/lecture34/34_4.htm[6/25/2012 12:42:36 PM]
Tanaka's Model Tanaka's model is based on three state variables, which are strain (e), Temperature (T ) and Martensite volume fraction ( ). The relationship is given by (34.1) where is the Piola-Kirchhoff stress, is the Green strain, E is the Young's modulus, is the thermal expansion coefficient and is the transformation modulus. The dot at the top denotes differentiation with respect to time. Further, the de-twinned Martensite volume fraction ( ) could be expressed as exponential function of stress and temperature as given below. For Austenite to Martensite transformation (34.2) where, and For Martensite to Austenite transformation (34.3) where and. The slopes C M and C A are indicated in Fig. 34.2. Tanaka's model describes the behaviour of SMA qualitatively and is not suitable for engineering applications. Liang and Rogers improved this model to quantitatively describe the SMA behaviour. For avoiding the singularity appearing in the exponential form of Tanaka's Martensitic fraction model, they modified the expression by replacing the exponential with the cosine function. Brinson further improved Liang's model by introducing the fractions of stress-induced and temperature-induced Martensite as, (34.4) where ξ S is the fraction of stress-induced de-twinned Martensite with a single Martensite variant and ξ T is the fraction of temperature-induced twinned Martensite with multiple variants. The cosine model is commonly referred as Liang-Brinson model and is given in the next slide. file:///d /chitra/vibration_upload/lecture34/34_5.htm[6/25/2012 12:42:36 PM]