modelling shape memory alloy

1. Most of these models describe quasi-static (thermodynamic equilibrium), one-way shape memory
behavior under uniaxial loading, and these are broadly classified into three categories:
1. phenomenology based macro-mechanics models,
2. thermodynamic-based micro-mechanics models,
3. micromechanics-based macroscopic models.
• For structural analysis, the model has to be simple and applicable in standard stress-strain mechanics
analyses.
• It should also incorporate realistic physics and be applicable over a wide range of temperatures and
stresses to capture both shape memory effect and pseudoelasticity.
• It should be adaptable to a wide range of materials and textures in both single crystals and
polycrystals.
• The crystallographic symmetry of the austenite phase is higher than that of the martensite phase, and
as a result, one can get a number of symmetry-related variants of martensite evolved around the load
and temperature history.

2. 1. Macroscopic Phenomenological Models
These models are based on phenomenological thermodynamics and are mostly defined using
experimental data (curve-fitting). These are simple and capture adequate physics. They are quite
amenable to inclusion in engineering analyses. Inmost of these models, the strain, temperature and
martensite volume fraction are the only state variables.
2. Microscopic Thermodynamics Models
These models depend on micro-scale thermodynamics to describe phenomena and as a result are
quite involved. These are less amenable to inclusion in engineering analyses. They are beneficial for
explaining phenomena at the micro-scale, such as nucleation, interface motion, and growth of a
martensite phase.
3. Micromechanics-Based Hybrid MacroscopicModels
These are hybrid between the first two categories. They capture key details from micro-scale
thermodynamics and incorporate several simplifying assumptions to describe phenomena at the
macroscopic level. They estimate the interaction energy due to phase transformation of the material at
themicrostructure level using a group of important variants. They may be amenable for inclusion in
engineering analyses.

3. • Typically, the volume fraction of the martensite phase is used as the internal variable, and most of
these models are perfected for uniaxial loading.
• Under the first category, some of the models are due to Tanaka [19], Liang and Rogers [20],
Brinson [21], Boyd and Lagoudas [22], and Ivshin and Pence [23].
• In these models, it is assumed that strain, temperature, and the martensite volume fraction are the
only state variables.
• One of the pioneering models is due to Tanaka, which was derived from second law of
thermodynamics expressed in Helmholtz free energy format, and in which the variation of
martensite volume fraction with stress and temperature is expressed in exponential form.
• It is based on the Clausius-Duhem inequality. Liang and Rogers made a change to the
development of martensite volume fraction from exponential form (Tanaka) to cosine format.

4. Quasi-Static Macroscopic Phenomenological Constitutive Models
This section describes four quasi-static constitutive models that have been proposed to describe
the material behavior in the one-way shape memory effect.
These models are chosen due to their common approach and wide applicability to a range of
operating conditions.
To accommodate the large variations in SMA material properties due to manufacturing,
composition, training, heat treatment, and other factors, these models use material parameters
that are determined experimentally.
For simplicity, only uniaxial loading behavior is considered, and quasi-static deformation is
assumed, resulting in an isothermal condition.

5. In these models, it is assumed that strain, temperature, and the martensite volume
fraction are the only state variables.
Most of the constitutive models are developed for quasi-static loading, and as such,
it is assumed that the material at each instant is in thermodynamic equilibrium
Tanaka [35] developed an exponential expression to describe the stress and
temperature, rather than determining the free energy expression.
Liang and Rogers [36] presented a model which is based on the rate form of the
constitutive equation developed by Tanaka. In theirmodel,
Tanaka’s equation is integrated with respect to time and it is assumed that the
coefficients in the equation are constant.

6. Figure 4: Hysteresis curve for a thermoelastic
martensitic transformation. denotes the fraction of
martensite in the material. 0 and T0 are prescribed
initial conditions. They represent a condition
that the material contains some martensite (0) and
some austenite (1 − 0) at a temperature T0.

9. Generally, it is assumed that = and Mf , Ms, Af and As
are critical temperaturesat which phase changes occur
without stress ( = 0).
With applied external stress ( = 0), higher
temperatureswill be needed for a phase change.
The increase is linear, with critical temperatures
increasing with applied stress.
For M A transformation with ( = 0) we use

19. The designer of shape memory actuators must balance several interdependent variables. For instance, a
change in mean coil diameter will effect the free length, sensitivity, and hysteresis loop, initial winding
pitch, maximum load output, and "start to move" temperature.
There is no substitute for experience when trying to resolve these interrelated aspects of shape memory
design. The final step, of course, to prove any shape memory actuator design is construction of a
prototype.
Electrically excited shape-memory actuators can offer unique advantages, but the design scenario
requires detailed understanding and often complicated trade-offs. A good approach for collecting
data includes:
• Measuring mechanical data (stroke vs. load) for different material processes.
• Measuring hysteresis data (stroke vs. current) at various ambient temperatures.
• Measuring temporal data (stroke vs. time) for different ambient temperatures.
• Repeating the above as required for different materials and dimensions.
Design Principles For Ni-Ti SMA Actuators

20. The aim of conventional helical spring design is to produce a mechanical element that will store
energy by generating the desired forces at given deflections. The designer uses standard formulae
based on linear elastic theory to determine wire diameter, spring diameter and number of turns.
From a design standpoint, the behavior of a shape memory spring is best described in terms of
material rigidity. Ni-Ti exhibits a large change in shear modulus over relatively narrow temperature
range, increasing from low to high temperature.
This produces a concomitant increase in spring rate, since spring rate is directly proportional to
shear modulus . The change in modulus with temperature is the result of a reversible martensite to
austenite solid state phase transformation and related pretransformational phenomena

21. Hysteresis Behavior
An important characteristic of the heating and cooling behavior of shape memory springs is the
hysteresis that occurs. As shown in Figure 3 at a temperature below Mf the spring is 100 % martensite
and fully extended. During heating the the spring begins contracting at As and completes its motion at Af
when the spring is 100 %
austenite. During cooling the spring does not begin to extend (reset) until Ms is reached. Extension is
complete at Mf.
Transformation temperatures are alloy dependent and can be altered by changing alloy composition.
Hysteresis width is also alloy dependent. Transformation temperatures also depend on stress level,
increasing with shear stress (Figure 4), according to the Clausius-Clapeyron relation
Effect of stress on transformation temperatures