The term shape memory alloy (SMA) normally refers to alloys that can undergo large strains, effect called
superelasticity or
pseudoelasticity; and also to materials that after undergoing large deformations can remember their original shape when heated above a certain temperature.
Hot NiTi alloys are composed by a pure austenite phase. The martensite phase develops upon cooling the alloy below the martensite start temperature,
Ms. The martensite volume fraction
ξM will increase until the cooling temperature reaches the
martensite finish temperature,
Mf, below which the alloy microstructure will be pure martensite.
The reverse process has different transition temperatures: A pure martensite alloy will develop an austenite microstructure if it is heated above the austenite start temperature,
As. The austenite phase will increase upon heating the material to the
austenite finish temperature,
Af, above which the alloy microstructure becomes 100% austenite.
here, c is the heat capacity at constant pressure,
K and
G are the bulk and shear moduli,
εvol is the volumetric strain,
Hk is the hardening modulus, and
I(
εtr) is the
indicator function for the strain limit constraint. The
equivalent transformation strain εtre is used as a measure of the transformation strain tensor
The term ψch = f(T)|εtr| is denoted as the chemical energy density due to the thermally induced martensite transformation. The function
f(
T) corresponds to the temperature-dependent martensite to austenite equilibrium stress, defined from the slope of the limit curve
β and the reference temperature
T*
Here, the operator < > denotes the Macaulay brackets.
The conjugated thermodynamic stress σtr associated to the transformation strain variable is
The evolution of the transformation strain εtr is given by the so-called
limit function, which takes the same form as the yield function for metal plasticity.
where the plastic multiplier λp is solved with the Kuhn–Tucker conditions, as done for plasticity, see
Plastic Flow for Small Strains.
For Lagoudas model, Gibbs free energy density depends on two state variables: the total stress tensor σ and the temperature field
T. Additional internal variables are used to compute the transformation strain tensor
εtr and the martensite volume fraction
ξ (
Ref. 35)
Here, c is the heat capacity at constant pressure,
s0 is the specific entropy at reference state,
S is the compliance matrix,
εth is the thermal strain tensor,
εtr is the transformation strain tensor,
u0 is the specific internal energy at reference state, and
f(
ξ) is the
transformation hardening function. The compliance matrix
S, is obtained by a volume average of the elastic properties of martensite and austenite
where ΔS = SM − SA. Also, other material parameters are averaged this way.
where the normalized transformation tensor Λ changes principal directions depending on the direction of the martensitic transformation.
The maximum transformation strain in Lagoudas model can be considered constant, or stress-dependent as described in Ref. 36. A stress-dependent maximum transformation strain can be used at low stress levels, where the martensite turns into detwinned structures.
The variable for the direction of the martensitic transformation, , is calculated from the previous state to determine the expected increment or decrement of the transformation strain tensor εtr. This calculation is computational expensive and it can lead to convergence issues. Since in many applications the transformation direction
is known a priori (for instance, mechanical loading or unloading, or temperature increment/decrement) a user input enables to set the transformation direction manually to 1 or -1, thus speeding up the computational time.
Consider a NiTi alloys at constant temperature composed by 100% of austenite volume fraction. Upon loading, the slope in the stress-strain curve would be the Young’s modulus of austenite, EA. The martensite phase starts to develop when the axial stress reaches the
martensite start stress,
σMs. If the alloy is further loaded, the slope reduces, entering a region called
loading plateau. Above the
martensite finish stress,
σMf, the microstructure becomes 100% martensite. This process is commonly called the
forward transformation.
The reverse process has different transition stresses: If the 100% martensite alloy is unloaded, the slope follows the martensite Young’s modulus EM. When the axial stress reaches the
austenite start stress,
σAs, the austenite volume fraction
ξA starts to develop until the axial stress falls below the
austenite finish stress,
σAf, at which level the alloy microstructure is 100% austenite. This process is commonly called the
reverse transformation.
Figure 3-26 illustrates this process.
here, Tσ is the constant temperature at which the stress-strain curve was measured.
As plasticity is rate independent, the transformation dissipation density Wtr is obtained after integrating an extra variable in the plastic flow rule.