310
M. R. Karamooz-Ravari et al.
material regains its original, unreformed configuration. This behavior is referred to as
shape memory effect (SME). At temperatures above A f , austenite also transforms to
stress-induced martensite during loading; but, elastic and inelastic deformations are
altogether spontaneously recovered upon unloading with no need of further heating
so that the material reaches its initial configuration once the load is completely
removed. This property is called superelasticity (SE) or pseudoelasticity (PE).
To develop a microplane model for shape memory alloys, appropriate microplate
laws should be derived based on macroscopic 1-D constitutive equations. One of the
simplest, though very efficient, 1-D models was proposed by Brisnon (1993) and
is employed here to provide the basic concepts of a microplane model for shape
memory alloys. Needless to say, other 1-D SMA models can also be utilized in the
microplane modeling of an SMA (Mehrabi et al. 2012a). Neglecting thermal strain
compared to inelastic strains, Brinson’s 1-D constitutive model can be written as,
ε = ε
e
+ ε
t
=
σ
E
+ ε
∗
ξ s
(17.19)
where ε
t is transformation-induced strain, ε
∗ is the maximum recoverable strain,
and ξ s is the stress-induced martensite volume fraction whose summation with the
temperature-induced martensite volume fraction yields total martensite volume fraction. These fractions are usually stated as functions of instantaneous stress and
temperature as well as history of the alloy in conjunction with a so-called stress–
temperature phase diagram of an SMA. Moreover, the Young’s modulus depends on
total martensitic volume fraction, so it is a function of stress and temperature.
Microplane modeling of SMAs was first performed by Brocca et al. (2002), and
this approach was further extended by Mehrabi and Kadkhodaei (2013), Kadkhodaei et al. (2007a, b), Mehrabi et al. (2012a, b, 2014a, b) from various aspects. As
martensitic transformations are driven by shear (displacive) deformations, the total
inelastic strain is only due to shear strains on microplanes. A linear elastic stress–
strain relation is thus considered for the normal direction, and a 1-D SMA constitutive
law is used for the shear direction on each microplane. Accordingly, with the use of
Eq. (17.8), the strain at each point is decomposed as:
ε i j = ε
e
i j + ε
t
i j
(17.20)
where
ε
e
i j = −
ν
E
σ ss δ i j +
1 + ν
E
σ rt ·
3
2π
Ω
N rs N i j + T rs T i j
dΩ
(17.21)
ε
t
i j =
3
2π
ε
∗
ξ s
Ω
T i j dΩ
(17.22)
M. R. Karamooz-Ravari et al.
material regains its original, unreformed configuration. This behavior is referred to as
shape memory effect (SME). At temperatures above A f , austenite also transforms to
stress-induced martensite during loading; but, elastic and inelastic deformations are
altogether spontaneously recovered upon unloading with no need of further heating
so that the material reaches its initial configuration once the load is completely
removed. This property is called superelasticity (SE) or pseudoelasticity (PE).
To develop a microplane model for shape memory alloys, appropriate microplate
laws should be derived based on macroscopic 1-D constitutive equations. One of the
simplest, though very efficient, 1-D models was proposed by Brisnon (1993) and
is employed here to provide the basic concepts of a microplane model for shape
memory alloys. Needless to say, other 1-D SMA models can also be utilized in the
microplane modeling of an SMA (Mehrabi et al. 2012a). Neglecting thermal strain
compared to inelastic strains, Brinson’s 1-D constitutive model can be written as,
ε = ε
e
+ ε
t
=
σ
E
+ ε
∗
ξ s
(17.19)
where ε
t is transformation-induced strain, ε
∗ is the maximum recoverable strain,
and ξ s is the stress-induced martensite volume fraction whose summation with the
temperature-induced martensite volume fraction yields total martensite volume fraction. These fractions are usually stated as functions of instantaneous stress and
temperature as well as history of the alloy in conjunction with a so-called stress–
temperature phase diagram of an SMA. Moreover, the Young’s modulus depends on
total martensitic volume fraction, so it is a function of stress and temperature.
Microplane modeling of SMAs was first performed by Brocca et al. (2002), and
this approach was further extended by Mehrabi and Kadkhodaei (2013), Kadkhodaei et al. (2007a, b), Mehrabi et al. (2012a, b, 2014a, b) from various aspects. As
martensitic transformations are driven by shear (displacive) deformations, the total
inelastic strain is only due to shear strains on microplanes. A linear elastic stress–
strain relation is thus considered for the normal direction, and a 1-D SMA constitutive
law is used for the shear direction on each microplane. Accordingly, with the use of
Eq. (17.8), the strain at each point is decomposed as:
ε i j = ε
e
i j + ε
t
i j
(17.20)
where
ε
e
i j = −
ν
E
σ ss δ i j +
1 + ν
E
σ rt ·
3
2π
Ω
N rs N i j + T rs T i j
dΩ
(17.21)
ε
t
i j =
3
2π
ε
∗
ξ s
Ω
T i j dΩ
(17.22)