1. Shape Memory Alloys
Dr. Mohammad Tawfik

2. SMA Constitutive Relations

3. Martensite/Austenite
Transformation
• In the absence of stresses, the martensite
fraction is a function of the temperature only
• In the following we will present approximate
expressions for the fraction in terms of
temperature

4. Martensite/Austenite
Transformation
• Let’s define the fraction
of the martensite phase
in the SMA as follows:
• For the range of
 



















 15.0
fs
f
MM
MT
Cos


sf MTM 

5. Martensite/Austenite
Transformation
• Similarly, we may write an expression for the
transformation
 



















 15.0
sf
s
AA
AT
Cos


fs ATA 

6. Martensite/Austenite
Transformation
• For an arbitrary starting
point, we may write the
expression:
 



















 15.0
of
o
o
TA
TT
Cos


ff ATM 

7. Effect of Stresses
• Experiments have shown that the
characteristic temperatures of the SMA are
affected by the stresses.
• The effects are assumed linear as shown in the
figure.

8. Effect of Stresses (AM)
• Thus, the expression for the martensite
fraction may be written as:
• Where, CM is the slope of the linear relation
for the Martenite temperature
 
   


















 o
fs
Mf
o
MM
CMT
Cos 

 1
/
15.0
   fMSM MTCMTC  

9. Effect of Stresses (MA)
• And, the expression for the martensite
fraction may be written as:
• Where, CA is the slope of the linear relation
for the Austenite temperature
 



















 1
/
5.0
sf
As
o
AA
CAT
Cos


   sAfA ATCATC  

10. Linear Region
 SMlin MTC 
 
A
SM
A
lin
lin
E
MTC
E



