A Unified Thermodynamic Constitutive Model for SMA and Finite Element Analysis of
Active Metal Matrix Composites
Dimitris C. Lagoudas, Zhonghe Bo and Muhammad A. Qidwai
Center for Mechanics of Composites
Aerospace Engineering Department
Texas A&M University
College Station, Texas 77843-3141
ABSTRACT
A unified thermodynamic constitutive model for Shape Memory Alloy (SMA) materials,
derived based on the thermodynamic frame proposed by Boyd and Lagoudas (1995a), is presented
in this paper. The specific selections for the form of Gibbs free energy associated with the
martensitic volume fraction are identified for several earlier models. The thermal energy released
or absorbed during the forward or reverse transformation predicted by the different models is
compared with the experimental data from calorimetric measurements. The constitutive model is
implemented in a finite element analysis scheme using a return mapping integration technique for
the incremental formulation of the model. Finally, the constitutive model is utilized to analyze the
thermomechanical response of an active metal matrix composite with embedded SMA fibers. Both
tetragonal and hexagonal periodic arrangements of SMA fibers are considered in the calculation and
the results are compared.
1. INTRODUCTION
Shape memory alloy (SMA) materials (Buehler and Wiley, 1965, Wayman, 1983) are being
used as active phases in active or "smart" composite structures ( Taya et al., 1993, Boyd and
Lagoudas, 1994, Lagoudas et al., 1994a, Lagoudas et. al., 1994b). Due to the interactions of active
1
SMA and non-active phases and the interactions among SMA particles themselves for high volume
fractions of SMA, complex 3-dimensional stress states result even when uniaxial loading is applied
to the composite. Therefore, accurate modeling of the thermomechanical response of SMA under
complex loading paths is necessary.
Extensive work has been done on the thermomechanical constitutive modeling of SMA
materials. In general, there have been two approaches: (1) the direct approach: the evolution laws
are obtained by either considering transformation micro-mechanisms (Tanaka, 1986, Sato and
Tanaka, 1988), or by directly matching experimental results (Liang and Rogers, 1990, 1991,
Graesser and Cozzarelli, 1991, Barrett, 1994); and (2) the thermodynamic approach: it starts with
constructing a free energy and then, by utilizing a dissipation potential in conjunction with the
second law of thermodynamics, the evolution laws for the internal state variables, i.e., the volume
fraction of the various forms of martensite, are derived (Berveiller et al., 1991, Ortin and Planes,
1989, 1991, Raniecki et al., 1994, Sun and Hwang, 1993a, 1993b, and Boyd and Lagoudas,
1995a,1995b). In the thermodynamic approach, the second law of thermodynamics applies
constraints to material constitution, but these constraints are usually weak. Hence, if a constitutive
model does not violate the thermodynamic constraints, usually it can also be derived by the
thermodynamic approach. It will be shown in this paper that several earlier models which were
derived based on different approaches are all related to each other under the thermodynamic
formulation.
Metal Matrix Composites (MMC) with SMA fibers are referred to as Active MMC
(AMMC) due to the incorporation of the active SMA phase. When SMA fibers are embedded in
elastoplastic metal matrix, they enhance the overall yield and hardening characteristics of the
composite at elevated temperatures, if they are properly prestrained at low temperature (Furuya et.
al., 1993, Taya et. al., 1993, Lagoudas et. al., 1994b). The thermomechanical response of AMMC
is influenced by the martensitic phase transformation in the SMA, which can be activated by applied
stresses and it occurs within a small temperature range. In this paper the effective thermomechanical
response of AMMC will be studied using the finite element method on a proper unit cell for a
periodic active composite. The purpose of this work is to investigate the interaction effects between
transforming SMA fibers and the elastoplastic matrix and study their impact on the effective
thermomechanical properties of the composite.
2
In Section 2, a general form of a thermodynamic model, which is derived based on the
formalism of Boyd and Lagoudas (1995a) is given. The unification of two other models (Tanaka,
1986, Liang and Rogers, 1990) under the framework of the present thermodynamic model and
comparisons of these models with the experimental data is briefly discussed. In Section 3, a
numerical procedure for implementing the thermodynamic constitutive model is described in detail.
Finally, in Section 4 the development of a unit cell for periodic fibrous composites with tetragonal
and hexagonal periodic arrangements and the appropriate boundary conditions for the unit cell are
described, together with some selected results for the overall thermomechanical response and shape
memory characteristics of AMMC.
2. A UNIFIED THERMODYNAMIC CONSTITUTIVE MODEL FOR SMA
2.1 Derivation of the thermodynamic constitutive model
Based on the formulation of Boyd and Lagoudas (1995a), the total specific Gibbs free
energy, G , of a polycrystalline SMA is assumed to be equal to the mass weighted sum of the free
energy G ,
M or A , (superscript " M " refers to martensitic phase and superscript " A " refers to
the austenitic phase, respectively) of each phase plus the free energy, G mix , of mixing:
t
G( ij,T, , ij)
G A( ij,T)
G M( ij,T)
G A( ij,T)
t
G mix( ij,T, , ij)
(1)
The free energy of each species is written as
G ( ij,T)
1 1
Sijkl
2
1
ij kl
ij ij(T
T0) c (T T0) Tln(
T
)
To
so T uo
(2)
and the free energy of mixing is assumed to be given in this work by
t
G mix( ij,T, , ij)
3
t
f ( , ij)
(3)
where , T,
ij,
t
ij
are martensitic volume fraction, temperature, stress tensor and transformation
, Sijkl,
strain tensor, respectively, and
ij ,
c , so , uo are mass density, elastic compliance tensor,
thermal expansion coefficient tensor, specific heat, specific entropy at a reference state, and specific
internal energy at the reference state of the " " phase, respectively. The generic function f( ,
t
ij )
physically represents the elastic strain energy due to the interaction between martensitic variants and
the surrounding parent phase, and among the martensitic variants themselves. Reorientation
(detwinning) effects have been omitted for simplicity (Boyd and Lagoudas, 1995a).
Note that the present formulation does not account for permanent changes in microstructure
of SMA materials, i.e., after a complete thermomechanical loading cycle the material returns to its
original state without increase of any local entropy (Ortin and Planes, 1989). For modeling of cyclic
responses of SMA with changing of internal microstructures, the work by Bo and Lagoudas (1995)
can be referenced.
The Gibbs free energy defined above is related to the internal energy u by the following
equation
te
u( ij , s, ,
where
te
ij
t
ij
ij
per unit mass, and
t
ij )
G
1
Ts
is the thermoelastic strain tensor,
ij
te
ij ij
(4)
is the total strain tensor, s is the entropy
is the mass density of the SMA, assumed to be independent of the martensitic
volume fraction. Infinitesimal strains will be assumed throughout the paper, which is a reasonable
assumption for polycrystalline SMA materials undergoing constraint deformations as part of the
microstructure of the AMMC. By performing standard calculations (Malvern, 1969), the strong
form of the second law of thermodynamics (Truesdell and Noll, 1965) can be written in the
following form for the local internal dissipation rate
T
(
te
ij
G
ij
)
ij
(s
G
)T
T
4
(
G
ij
t
ij
)
t
ij
G
0
(5)
where
is the local dissipation rate (it does not include the entropy production rate due to heat
conduction) and the “dot” above is a symbol that indicates the increment of the corresponding
quantity. Since
ij
and T are independent state variables, and G is independent of
ij
and T , the
following set of constitutive relations can be obtained by identically satisfying inequality (5)
G
te
ij
Sijkl
(T
kl
T0)
ij
ij
G
T
s
1
cln(
ij ij
(6)
T
To
)
so
where
Sijkl
A
M
Sijkl
ij
( Sijkl
A
ij
(
M
ij
A
Sijkl ) ,
c
cA
(c M c A)
A
ij )
so
so
A
(so so )
,
M
A
(7)
The local internal dissipation rate is then given by
T
where the effective stress
eff
ij
f
ij
G
eff t
ij ij
t
ij
0
acts as a thermodynamic force conjugate to
(8)
t
ij .
To simplify the formulation given above for the current case in which only transformation,
without reorientation of martensitic variants, is considered, the following assumption can be
introduced to relate the evolution of the transformation strain to the evolution of the martensitic
volume fraction
(Bondaryev and Wayman, 1988, Boyd and Lagoudas, 1995a):
t
ij
The transformation tensor
ij
ij
is assumed to have the following form
5
(9)
3
H ¯eff
2
1 eff
ij
,
> 0
(10)
ij
t
H ¯
t max
where H
eff
ij
eff
ij
1
3
eff
kk ij ,
1 t
ij
,
< 0
is the maximum uniaxial transformation strain, ¯
t
and ¯
2 t t
3 ij ij
1
2
eff
3
2
eff
ij
eff
ij
1
2
,
.
Using equation (9), the local dissipation rate given by equation (8) can be rewritten as
T
eff
ij
where
G
ij
eff
ij
G
0
ij
(11)
is the thermodynamic force conjugate to . Inequality (11) is the
thermodynamic constraint to the SMA material system. For the forward phase transformation (when
> 0 ) the driving force
(when
< 0)
must be greater than zero, and for the reverse phase transformation
must be less than zero.
To obtain the evolution equation for the internal state variable , an additional assumption
must be introduced, either utilizing Edelen's formalism (Edelen, 1974) of dissipation potential theory
or directly assuming that
satisfies a certain criterion during the phase transformation (Fischer et
al., 1994). The latter approach is a special case of Edelen's formulation. Following Edelen's
formalism, a dissipation potential,
( ; ij, T, ,
t
ij ) ,
can be introduced such that the evolution of the
internal state variable, , for the rate independent case, is given by (Boyd and Lagoudas, 1995a)
t
( ; ij, T, , ij)
6
(12)
where
satisfies the Kuhn-Tucker conditions
0 ,
Y ,
(
Y) 0
(13)
and Y is a material parameter. During phase transformation ( > 0 ),
the consistency condition, i.e.,
Y
0 . If the potential
can be evaluated by using
is assumed to be the convex quadratic
function
1
2
2
(14)
then by Kuhn-Tucker condition given in equation (13) for the case of
eff
ij
1
ij
2
Sijkl
ij kl
ij ij
T
c T Tln
T
T0
s oT
> 0 , we get
f( )
uo
±Y
(15)
where
Sijkl
so
Y
M
Sijkl
M
A
Sijkl ,
A
so
so ,
ij
uo
M
ij
M
uo
A
ij ,
A
uo ,
c
cM
T
T
cA
T0
(16)
2Y
To satisfy the thermodynamic constraint given in equation (11), the plus sign in equation (15) should
be used for the forward phase transformation, while the minus sign should be used for the reverse
phase transformation. Equation (15) can be alternatively used to obtain the evolution equation for
the internal state variable , without utilizing evolution equation (12), since in the present case the
internal state variable
is only a scalar.
Note that the material constant
thermodynamic force
Y
can be interpreted as the threshold value of the
for the onset of the phase transformation. It is also a measure of the
internal dissipation due to microstructural changes during the phase transformation. Using equations
(11) and (15), the total internal dissipation during a complete forward and reverse isothermal phase
transformation cycle is given by
7
1
T
Td
d
0
Y d
0
( Y )d
2Y
(17)
1
2.2. Unification of several SMA constitutive models
Different constitutive models are distinguished intrinsically by different selections of internal
state variables and their evolution equations. Since the interaction elastic strain energy, represented
by f( ,
t
ij ) ,
describes transformation induced strain hardening in the SMA material, if the internal
state variables are selected to be the same for different SMA constitutive models, then they can
possibly be unified under the current thermodynamic framework. This can be done by selecting
different forms of the function f to represent the transformation induced strain hardening as
predicted by these models.
In determination of the hardening function, f , it is assumed that f is independent of
t
ij ,
which physically can be interpreted as the absence of kinematic transformation hardening. Then for
f( ) the following properties are assumed to be preserved: (1) parent austenitic phase is stress free
if no external mechanical loading is applied. This condition is satisfied by selecting f( ) to be zero
at the fully austenitic state, i.e., f(0)
0 ; (2) the function f( ) must be non-negative since it
represents part of the elastic strain energy stored in the material. This requirement can be satisfied
by appropriately choosing material constants; and (3) the function f( ) must be continuous during
the phase transformation, including return points, for all possible loading paths (a return point
during the phase transformation is characterized by a change in the sign of ). Due to the formation
of different microstructures during the forward and reverse phase transformations, f( ) may have
different functional forms during the forward and reverse phase transformations, but these functions
must be continuously connected at return points.
Applying the above constraints, a form for the function f( ) can be selected as follows:
8
f( )
f M( ) ,
>0
f A( ) ,
<0
(18)
where
1
f M0( )
f M( )
f A( R)
R
1
f M0( R)
(19)
A
A0
f ( )
and
R
M
f ( )
R
R
f ( )
A0
R
f ( )
is the martensitic volume fraction at the return point. The range of
R
transformation is given by
given by 0
R
1 , while the range of
for the forward phase
for the reverse phase transformation is
. This selection of f( ) satisfies all of the properties from (1) to (3) given
above, provided f M0(0)
f A0(0)
0 , f M0( )
0 , f A0( )
1 , and f M0(1)
0 for 0
f A0(1) .
The functions f M0( ) and f A0( ) are selected differently for different models. For Tanaka's
exponential model (Sato and Tanaka, 1986), they can be selected as
so
f M0( )
M
(1
)ln(1
e
)
ae
so
A0
f ( )
A
e
(µ1
µ2)
(20)
ln( )
1
ae
e
(µ1
e
µ2)
while for Liang-Rogers' cosine model (Liang and Rogers, 1990), they take the form
so
f M0( )
0
M
ac
cos 1(2
1) d
c
(µ1
c
µ2 )
(21)
so
f A0( )
0
A
ac
cos 1(2
9
1) d
c
( µ1
c
µ2 )
M
A
M
A
where ae , ae , ac , ac , and µ1 where
e, c are material constants, and the parameter µ2 can
be determined from the continuity condition f M0(1)
f A0(1) .
The equivalence between the above selection of function f and the corresponding original
models can be illustrated as follows by taking Liang-Rogers’ cosine model as an example:
Substituting equation (18) into equation (15), and using equations (19) and (21), for the forward
phase transformation the criterion given by equation (15) can be written explicitly as follows
eff
ij
1
ij
2
Sijkl
s0
M
ij kl
cos 1(2
ij ij
T
c T
c
1)
c
µ1
ac
Tln
µ2
For the case of a complete loading-unloading cycle,
f A( R)
f M0( R)
T
s oT
T0
f A( R)
1
R
(22)
f M0( R)
uo
R
Y
0
0 , then using equations (19) and (21) ,
0 . Equation (22) can then be rearranged to obtain
explicitly in terms of
temperature and applied stress as follows
1
M
cos ac (T
2
M
ac
of
M )
M
C H
ij
ij
1
S
2 ijkl
ij kl
ij ij
T
1
(23)
c
where Table 1 and the expression for µ2 from below (paragraph following equation (24)) are
utilized, and
c
0 is assumed. The constant C M
s0
H
is called the martensite stress
influence coefficient in the original models. If one neglects the effect due to the terms
and
ij ij
1
S
2 ijkl
ij kl
T , equation (23) is exactly the same as the one that has been proposed by Liang and
Rogers (1990), except for the term
1
H
ij
ij
that is used here as the effective driving stress, instead
10
of only the applied stress in the original model. The corresponding expressions for Tanaka’s
exponential model as well as for the case of the reverse phase transformations for both models can
be obtained similarly.
The unification attempted in this work captures the main hardening features predicted by the
corresponding models, and it also gives a form of generalization to these models. The effect of the
change of the elastic compliances and thermal expansion coefficients is introduced naturally in the
current thermodynamic formalism by the terms,
current model the quantity
1
H
eff
ij
ij
1
S
2 ijkl
ij kl
, and
ij ij
T . Note also that in the
(generalized effective Von Mises stress) is utilized as the
driving force to the stress-induced phase transformation, instead of ¯eff (effective Von Mises stress).
As a result, during the reverse phase transformation the stress applied to the opposite direction of
t
ij ,
the accumulated transformation strain,
will help the reverse phase transformation. This
phenomenon has been observed in experiments (Graesser and Cozzarelli, 1991).
Table 1. Evaluation of material constants for given transformation temperatures.
Data given: M os, M of, A os and A of .
Exponential
A
ae
M
ae
e
1
2
Cosine
ln(0.01)
A os A of
A
ac
ln(0.01)
M os
A of
c
M os
1
2
so A of
A os
bM
so M os
M of
M of
so M os
11
bA
A os
M
ac
M of
so M os 2A of A os
Polynomial
A of
p
1
2
so M os
A of
1
2
Ye
so A os
M os
1
2
Yc
so
2ln(0.01)
M os M of A of A os
1
4
so A of
M os
Yp
1
4
so M os M of A of A os
1
2
so A of
M os
so M os M of A of A os
For a polynomial representation of the function f( ) including up to quadratic terms, the
different hardening effects during the forward and reverse phase transformations can be accounted
for by the following selection
1
f M0( )
2
1
A0
f ( )
2
2
( µ1
p
µ2 )
A 2
p
( µ1
p
µ2 )
bM
b
p
(24)
where b M and b A are linear isotropic hardening moduli for the forward and reverse phase
transformations, respectively. If the hardening parameters b M and b A are selected to be the same,
p
then µ2
0 and the original form of f( ) reported by Boyd and Lagoudas (1995a) is recovered,
p
except for the additional linear term, µ1 , in equation (22).
The parameter µ2 is introduced to enforce the continuity condition at
of
µ2 for each model is given by
s0
c
µ2
4
1
1
M
ac
A
ac
s0
e
µ2
2
p
for the cosine model, and µ2
1
1
aA
aM
1
bA
4
1 . The evaluation
for the exponential model,
b M for the polynomial model,
respectively. The material constant µ1 is used to describe the accumulation of elastic strain energy
at the onset of the forward phase transformation. It is shown by Salzbrenner and Cohen (1979) that
12
for polycrystalline SMA the forward and reverse phase transformation zones shift to lower
temperature, compared to their single crystal counterpart. This phenomenon indicates that
f(0)
0 , and suggests that the function f( ) must contain the linear term.
Using the data given in Table 2, the uniaxial stress-strain curves predicted by the different
models are plotted in Fig. 1. The exponential law is plotted for two different ways of calibrating the
material constants (refer to Section 2.3). It is shown that the stress-strain curves of all three models
are similar, and they give the same amount of total hysteresis in a complete loading-unloading cycle,
except for the curve corresponding to the exponential model with input data given transformation
temperatures.
Table 2. Material constants for aluminum and SMA.
Material Constants for Aluminum at Room Temperature
E = 69x103 MPa
= 0.33
α = 23.6x10-6 /oC
Yield stress = 70 MPa
Material Constants for (Polynomial Hardening Law) SMA
13
EA = 70.0 x 103 MPa
EM = 30.0 x 103 MPa
A
= 0.33
M
= 0.33
Α
α = 10.0x10-6 /oC
αΜ = 10.0x10-6 /oC
H = 0.05
ρ∆c = 0.0
ρ∆so = -0.35 MPa/°C
Mos = 3°C
Mof = 18°C
Aos = 22°C
Aof = 42°C
ρbA = 7.0 MPa
ρbM = 5.25 MPa
γp = -10.5 Mpa
Yp* = 3.7625 MPa
To compare further the strain hardening behavior predicted by different models, the
experimental data from calorimetric measurements can be utilized. In order to perform the
comparison, the excess specific heat due to the phase transformation predicted by the models is
investigated first. The first law of thermodynamics reads
u
Q
ij ij
(25)
where Q is the heat input rate. By substituting equation (4) into (25) and using the constitutive
relations given by equation (6) the following equation can be derived
Q
cT
T
s
ij
( T
s
)
ij
Table 3. Material constants for all three models.
14
g(T, ij)T
hij(T, ij)
ij
(26)
Material constants in common for all three models
E A, E M,
A
M
,
A
,
M
,
, H,
c,
so, Y
Material constants different
Exponential
A
M
a e , ae ,
Cosine
A
e
M
Polynomial
b A,
c
a c , ac ,
b M,
p
Table 4. Evaluation of material constants for given difference of transformation
temperatures, their average and the value of hysteresis.
M sf, A sf, T̄
Data given:
AM
and Y .
Exponential
ln(0.01)
A
ae
ln(0.01)
M
M os
1
2
1 AM
T̄
2
1 1
2 aM
A of
1 AM
T̄
2
1 1
2 aM
AM
M
A sf
Y
so
c
M os
1 ln(0.01)
aA
Y
so
1 ln(0.01)
aA
A of
bA
a 4 A fs
bM
a 4 M sf
A sf
ac
M sf
so T̄
Polynomial
A
ac
A sf
ae
e
Cosine
M sf
1
2
so T̄
1 AM
T̄
2
1
M sf
4
AM
Y
so
p
M os
A sf
1 AM
Y
T̄
2
so
1
M sf
A sf
4
A of
For the stress free case, equation (26) can be written in the following form
15
1
2
so T̄
1 AM
T̄
2
1
M sf
4
AM
Y
so
A sf
1 AM
Y
T̄
2
so
1
M sf
A sf
4
Q
cT
(27)
g(T)T
where, by utilizing equation (6b) and (15), the function g(T) predicted by the current model can be
written as
g(T)
c ln(
T
)
T0
s oT
16
Y
T
(28)
The
g(T)
function
17
can
be
called the excess specific heat per unit mass due to the phase transformation.
The term on the left hand side of equation (28) can be measured by a calorimetric
measurement at constant temperature rate. Since T is constant, the function g(T) can be obtained
using (27). Thus theoretical predictions can be compared directly with experimental results. The
curves of the function g(T) computed by using equation (28) for different models are plotted in Fig.
3, and an experimental curve from a calorimetric measurement is also plotted in the same figure.
It is shown that these different models give different predictions for the excess specific heat during
forward phase transformation (similar results are valid for the reverse phase transformation).
One observation from Figs. 2 and 3 is that, even though the curves of the excess specific heat
due to phase transformation are predicted to be remarkably different for the three models, the stressstrain curves are similar, because the former curves are approximately proportional to the derivatives
of the latter ones.
2.3 Determination of material constants used in the SMA constitutive equations
For the polycrystalline SMA it can be reasonably assumed that the material is isotropic.
Then, the elastic compliance tensor for both phases can be represented by the Young’s modulus and
poission ratio E A,
A
and E M,
M
, respectively, and the thermal expansion coefficients tensor for
both phases can be reduced to two scalar constants
E A, E M , Poisson ratios
A
,
M
A
and
M
, respectively. Young’s moduli
, and thermal expansion coefficients
A
and
M
can be obtained by
performing standard tests at low temperature for martensitic constants and at high temperature for
austenitic constants, respectively.
The maximum uniaxial phase transformation strain H can be obtained from a uniaxial test
at a temperature below austenitic start temperature A os by measuring the residual strain retained
after the specimen has been fully loaded to martensitic phase followed by complete unloading. The
specific heat of the two phases c A and c M can be determined from a calorimetric measurement test.
In the current model only the difference of the specific heat
18
c is used. Many calorimetry
experiments show that the specific heat of both phases is almost equal, henceforth it is assumed that
c
0 . The selection of the remaining six material parameters is discussed next.
For the case of uniaxial tension, the stress-temperature transformation curves represented
by equation (15) for fixed martensitic volume fraction are plotted in Fig. 3. These curves can be
obtained experimentally (Miyazaki et al., 1981, Shaw and Kyriakides, 1995). The entropy difference
so per unit volume between the phases can then be determined phenomenologically by the slope
of these curves at zero stress (Fig. 3). To analytically obtain the slopes, we first take the differential
of equation (15) for constant martensitic volume fraction , and after rearrangement of terms, the
slope of the curves can be obtained by the following equation:
d
dT
so
H
S
T
(29)
Equation (29) is usually called the Clausius-Clapeyron equation (Ortin and Planes, 1989).
Substituting zero stress and neglecting the thermal term in the denominator in the above equation,
the slope
so
d
of these curves is found out to be
dT
H
. This slope is predicted to be the same for
both forward and reverse phase transformations. This is consistent with experimental observations
(McCormick and Liu, 1994, Stalman et. al., 1992, Miyazaki, et. al., 1981).
The constant Y can be obtained from an isothermal uniaxial pseudo-elasticity test (stress
induced martensitic phase transformation fully reversible to the parent austenitic phase upon
unloading (Wayman, 1983)). The area enclosed by the stress-strain curve of a complete isothermal
pseudo-elasticity loading cycle can be calculated by integrating the stress over strain for a complete
cycle, i.e.,
d
ij
ij
ijd ij .
Utilizing equations (6) and (15) the integral can be explicitly integrated as
2Y . Therefore, Y is directly correlated to the total hysteresis area enclosed by a stress-
strain curve which can be measured from an isothermal pseudo-elasticity test.
Following the arguments given by Salzbrenner and Cohen (1979), the phase equilibrium
temperature T eq of single crystal SMA materials can be obtained by measuring the martensitic start
19
and austenitic finish temperatures, M̃
constant
uo can be obtained as
polycrystalline SMA materials
os
of
and à , and assuming T eq
1
2
( M̃
os
of
à ) . Then the
s0 T eq (Raniecki and Lexcellent, 1994). For
uo
uo can be obtained by performing tests on its single crystal
counterpart as described above. However, to evaluate the stress increment for given strain and
temperature increments, it is not necessary to determine
individually, but rather
µ1
uo and
µ1 where
e, c, p
uo , since they both contribute to the linear part of the free
energy with respect to . Thus, there are three remaining material constants to be determined for
the three models, as shown on the bottom of Table 3.
After the above ten material constants are evaluated, the remaining three are determined
indirectly by specifying the total amount of hardening for forward and reverse phase
transformations, which is represented by
M sf
M os
M of , A sf
A of
A os , and by
prescribing the average phase transformation temperature for the polycrystalline SMA,
T̄
AM
1
2
(M os
A of ) . The pertinent evaluations of the remaining constants for the three models
are shown in Table 2. Note that since Y
is utilized as input, the resulting transformation
temperatures for the three models are not identical, as it can be seen from Table 4.
An alternative way of calibrating these three remaining constants in the case that all four
transformation temperatures, i.e., M os, M of, A os and A of , are utilized as input is shown in Table
1. These temperatures are obtained from the stress-temperature transformation lines, shown in Fig.
3, by calculating the intersection points of these lines with the temperature axis. However, as Table
1 shows, the evaluation of Y for the exponential model is different from its evaluation for other
two models. This means that the total hysteresis area predicted by the exponential model is different
from the other two models.
Using the experimental data given by Jackson et. al. (1972), Howard (1995), Stalmans et.
al. (1992) and the way of determining the material constants discussed above, the material constants
needed for implementing the current model are determined as given in Table 2. The uniaxial stress-
20
strain curves predicted by the different models are plotted in Fig. 1. The exponential law is plotted
for the two different ways of calibrating the material constants described above. It is shown that if
phase transformation start and finish stresses are specified to be the same, then the hysteresis loop
predicted by the exponential model will be different from the hysteresis loops predicted by the other
models. Conversely, if the hysteresis area is enforced to be the same for all models, then the
transformation start and finish stresses predicted by exponential model will be different from the
predictions of the other two models.
3. NUMERICAL IMPLEMENTATION OF THE CONSTITUTIVE MODEL
In typical finite element analysis, a displacement formulation is utilized, and for solving
nonlinear problems an incremental Newton-Raphson integration method is a usual choice
(Zienkiewicz, 1977). In implementing the SMA constitutive model the tangent stiffness tensor and
the stress tensor at each integration point of all elements should be updated in each iteration for given
increments of strain and temperature. In order to utilize established numerical schemes used in
plasticity theory, we define the transformation function as follows
Y ,
>0
Y ,
(30)
<0
The thermomechanical transformation zone described by Kuhn-Tucker condition, equation (13) can
equivalently be characterized by letting
> 0 and 0
1 . The transformation function
takes
the similar role as the yield function in plasticity theory, but in the current case, an additional
constraint for
must also be satisfied.
To derive the tangent stiffness tensor, using the definition of the thermal elastic strain tensor
te
ij ,
the stress-strain equations of state (6a) can be written in the following rate form
ij
Sijkl
T
kl
ij
21
Qij
(31)
where
Qij
Sijkl
ij(T
kl
T0)
(32)
ij
0 reads
The consistency condition
ij
T
ij
T
0
Equation (31) and (33) can be used to eliminate
(33)
and obtain the relationship between stress
increment and strain and temperature increments as
ij
Lijkl
kl
lijT
(34)
where the tangent stiffness tensor Lijkl and tangent thermal moduli lij are defined by
1
Qij
Lijkl
Sijkl
kl
,
lij
Lijkl Qkl
(35)
kl
±Qij , where the plus sign is used for the forward phase transformation and the minus
Since
ij
sign is for the reverse one, the tangent stiffness tensor is always symmetric, i.e., Lijkl
Lklij .
To calculate the increment of stress for given strain and temperature increments, a return
mapping integration algorithm proposed by Ortiz and Simo (1986) has been used. Because the
present formulation is time independent, equation (31) can be written in the following incremental
form
22
1
Sijkl
ij
kl
kl
T
Qkl
The elastic predictor is calculated in the first step by letting
1
ij
Sijkl
1
kl
kl
0 , i.e.
1
ij
T ,
(36)
0
ij
1
ij
(37)
An iterative scheme is then carried out to obtain the transformation corrector from equation (36) by
assuming
ij
0 , and
T
0 , i.e., during the pth iteration
p 1
ij
p
Sijkl( p) 1Qkl
p 1
(38)
The transformation function, equation (30), is expanded into a Taylor series about the current value
of state variables, denoted by superscript " p ", and is truncated at the linear part as shown below:
p
( ij , )
(
p
p p
ij , )
p
p 1
ij
p 1
(39)
ij
Tp
where temperature, T , is fixed, and thus
by equation (13c) for the case of
and
tp
ij
1
1
0 . By applying the Kuhn-Tucker condition given
0 during phase transformation, and using equation (38),
p+1
can be obtained as follows:
p
p 1
p
p
sijkl 1Qkl
p
,
ij
The state variables
p 1
and
tp
ij
1
can then be updated as follows
23
tp
ij
1
p
ij
p 1
(40)
p 1
p
p 1
tp
ij
,
1
tp
ij
tp
ij
1
(41)
and the stress can then be obtained by using the constitutive equation (6a)
p 1
ij
The iterative procedure ends if
Sijkl(
p 1
p 1
)
1
kl(
kl
p 1
) T
tp
kl
1
(42)
is less than a specified tolerance. If the convergence criterion
is not satisfied, calculations given by equations (38) through (42) are repeated until convergence is
achieved. For typical cases it takes only about three iterations to satisfy the tolerance of 10 8 .
4. FINITE ELEMENT ANALYSIS FOR AMMC COMPOSITES
Due to the nature of fabrication techniques, one expects to have a certain periodic pattern of
SMA fiber placement in AMMC composites. In the present paper the tetragonal and hexagonal
arrangements, as shown in Fig. 4a and Fig. 4b, of SMA fibers will be investigated. The basic unit cells
for these two cases are tetragonal and hexagonal respectively (shown in the figure by dashed lines).
Assuming that the applied loads do not violate the geometric symmetry of the unit cells, the smallest
representative unit cells which can be used to predict the composite overall behavior, corresponding
to the tetragonal and hexagonal cases, are given in the Fig. 4c and Fig. 4d, respectively. In the
tetragonal case, Fig. 4c, the cross section of the representative cell is a square with length a, while
in the hexagonal case, Fig. 4d, the cell is a rectangle with length a, and width b, and b=0.866a. The
stress and strain in the unit cells will completely represent the whole
24
composi
t
e
response to the external applied loading because of the periodicity and symmetry conditions of the
composite geometry, material parameters, as well as the applied load.
To reduce the cost of FEM analysis, it is assumed that the temperature field within the
AMMC specimen is uniform. The three main reasons for this assumption are as follows: (1)
Consideration of heat conduction at the unit cell level, will destroy the periodicity of the
microstructure, thus rendering the whole analysis costly; (2) The temperature gradient in the specimen
will be small, if the thermomechanical loading processes are assumed to be slow (the meaning of slow
will be defined better at the end of this paragraph); (3) The main purpose of the current analysis is
to investigate the characteristics of the effective thermomechanical response of AMMC specimens,
and not their detailed response under temperature gradients. In heat transfer analysis that we
25
performed using FEM on a cylindrical unit cell, which is close to that of the hexagonal case, with a
cylindrical SMA fiber of 2mm in diameter at the center of the unit cell and the volume fraction of the
SMA fiber of 30% , we found that a temperature differential of ten degrees between the center of
the SMA fiber and the external adiabatic boundary of the unit cell, equilibrates to less than 0.1
degrees difference in temperature in less than 0.1 second. Although the geometry of the actual unit
cells for the tetragonal and hexagonal cases are different, it is expected that the time scale will remain
in the same order of the magnitude given above.
4.1 Boundary conditions for the unit cells
To derive the boundary conditions on the boundaries of the unit cells, let us first consider the
symmetry conditions on the surfaces represented by the lines AB, BC, CD, and AD in Fig. 4. On all
of these surfaces for the tetragonal case and the surfaces AB, BC, and AD for the hexagonal case,
the reflection symmetries require that the shear stresses vanish at all points and the normal
displacements are constant. These conditions can be mathematically expressed as follows for the
tetragonal case:
u1(0,x2,x3)
u2(x1,0,x3)
12(o,x2,x3)
ū1(0) ,
ū2(0) ,
u1(a,x2,x3)
u2(x1,a,x3)
12(a,x2,x3)
12(x1,0,x3)
ū1(a) ,
ū2(a) ,
12(x1,a,x3)
(43)
0
(44)
and for the hexagonal case:
a
,x ,x )
2 2 3
u2(x1, b,x3)
u1(
a
) ,
2
ū2( b) ,
ū1(
26
a
u1( ,x2,x3)
2
a
ū1( ) ,
2
(45)
(
12
a
, x ,x )
2 2 3
a
( , x2,x3)
2
(x1 , b,x3)
12
where ūi(x̄i) is the displacement of the plane xi
0
12
x̄i , i
(46)
1, 2, 3 , in ith direction, and x̄i is a
constant. For the hexagonal case there is a 180 rotational symmetry about point O, which is at the
middle of line CD. The displacement boundary conditions on this surface are given by
u1( x1,0,x3)
u1(x1,0,x3) ,
u2( x1,0,x3)
u2(x1,0,x3)
(47)
22(x1,0,x3)
(48)
the traction boundary conditions satisfy the following equations
12(
x1,0,x3)
12(x1,0,x3)
,
22(
x1,0,x3)
In the longitudinal, x3 , direction, a generalized plane strain state is assumed, and the
dimension of the unit cell in x3 direction is taken to be unit length. The boundary condition on the
two surfaces can be expressed by the following
u3(x1,x2,0)
ū3(0) ,
u3(x1,x2,1)
ū3(1)
(49)
and
13(x1,x2,0)
23(x1,x2,0)
13(x1,x2,1)
23(x1,x2,1)
0
(50)
In implementing the above in a finite element analysis the boundary conditions stated in
equations (43) to (47) and (49) to (50) are standard. Neglecting rigid body motion, it can be set that
ū1(0)
ū2(0)
ū3(0)
0 for the tetragonal case and ū1(
a
)
2
ū2( b)
ū3(0)
0 for the
hexagonal case. The rest of the kinematic boundary conditions given in equations (43), (45), (47),
and (49) can be applied by using an equation type of constraints provided by most of commercially
available FEM software, such as ABAQUS. The boundary conditions given by equation (48),
however, are constraints for the components of the traction vector, and in general they can not be
explicitly enforced by existing FEM software. A detailed study and comparison with our in-house
27
numerical FEM code, has shown that these force constraints are implicitly enforced by applying the
kinematic boundary conditions given by equation (47), due to the way that the kinematic boundary
conditions are applied by operating on the global stiffness matrix.
4.2 Results and discussions
The results that follow the unit cell and boundary conditions discussed in Section 4.1 have
been used, while the constitutive response for the SMA fibers, presented in Section 2.2, have been
utilized, with material parameters given in Table 2. For the elastoplastic matrix a Von Mises yield
criterion with an associative flow rule and Prager-Ziegler kinematic hardening have been used with
material parameters, given in Table 2, corresponding to those of 6061 aluminum. The temperature
dependence of the material properties of aluminum has been considered for completeness (Military
Handbook, 1994). For the results presented here, the ABAQUS finite element program has been
utilized with an appropriate user supplied subroutine (UMAT) for the thermomechanical constitutive
response of the SMA fibers.
The temperature and mechanical loading history considered in the present work for finite
element analysis is schematically shown in Figs. 5 & 6, which is similar to the experimental processes
performed by Taya et al. (Taya et al. 1993, Furuya et al. 1993). Both axial and transverse loadings
are considered for tetragonal and hexagonal periodic arrangements. In the actual processing of the
composite residual stresses appear by annealing (austenite phase) at high temperature, i.e., 500 oC,
and then cooling to the point just before the forward transformation into martensite takes place, at
40 C. The effect of this process on the composite response, is also investigated in this study as can
be seen from loading step 1-2 in Fig. 6. Step 2-3 of the loading is to prestrain the composite. During
this step the SMA fibers fully transform into the martensitic phase. Residual stresses are expected
after unloading, at the end of step 3-4 due to the incompatible plastic deformation of aluminum matrix
and the transformation induced deformation (strains) of SMA fibers accumulated during loading step
2-3. Step 4-5 is introduced to simulate the elevated temperature service environment, at a
temperature above the austenitic finish temperature. Finally the composite is loaded mechanically at
elevated temperature in step 5-6.
28
In the results that follow, numbers on the graphs indicate the loading steps, while lower case
letters indicate the state of the SMA or aluminum matrix. The following letters are used: (a) for
aluminum yielding during loading, (b) for initiation of the forward stress induced transformation from
29
austenite to martensite, ( c) for the end of the forward transformation, (d) for aluminum yielding
during the composite unloading, (e) for initiation of the reverse transformation from martensite to
austenite, (f) for aluminum yielding at elevated temperature, and (g) for initiation of the forward stress
induced transformation from austenite to martensite at elevated temperatures.
The overall axial stress vs. strain curve for axial applied loading is shown in Fig. 7. Upon cooling
to 40 C from 500 C, the aluminum matrix develops residual tensile stresses whereas the SMA fibers
develop compressive stresses, due to the mismatch in the thermal expansion coefficients (
13.6x10-6 /oC), as shown in Fig. 8 (points 2), for the average
33
(
zz
-
Al
SMA
=
) stress component for both
aluminum and SMA. This state of stress in SMA fibers causes stress induced phase transformation
at much higher temperature, about 39.7 C, than the stress-free martensitic start temperature, which
is about 18 C. The tensile residual stresses in aluminum matrix cause its early yielding at zero applied
axial stress due to residual tensile stresses (point a in Fig. 7). Stress induced martensitic phase
transformation in the SMA fibers starts at about 100 MPa overall applied stress (point b). The phase
transformation ends at about 170 MPa (point c), followed by elastic loading of the SMA fibers and
linear kinematic hardening of the aluminum matrix. Unloading of the composite takes place elastically
until aluminum yielding starts at about 160 MPa (point d). The reverse transformation back to
austenite starts at about 30 MPa (point e) but due to low temperature is only partially completed. The
temperature is then increased to 85 C and the SMA fibers fully transform into austenite at about 61
o
C, instead of the stress free finish temperature of 42 o C. The recovery of the transformation strain
in the SMA fibers induces axial compressive stress in the aluminum matrix, enhancing therefore the
required overall stress for composite yielding. When the composite is axially loaded, its overall yield
stress is increased to about 120 MPa (point f), which is much higher than the initial aluminum yield
stress value of 69 MPa (Table 2).
Contour plots of Von Mises stress and martensitic volume fraction, , for the applied axial loading
case are shown in Figs. 9 and 10, during step 2-3 at a point where the overall applied stress is about
140 MPa. Both the Von Mises stress and martensitic volume fraction in the SMA fibers are almost
uniform.
30
31
Fig. 11 is similar to Fig. 7 but for overall transverse loading applied to the composite. As seen
from Fig. 8, both the yield and the phase transformation points are not sharply defined in this case due
to highly nonuniform stress and inelastic strain fields in aluminum matrix. The contour plots of Von
Mises stress and martensitic volume fraction, , for the transverse loading case are shown in Figs. 12
and 13, during step 2-3 at a point where the overall applied stress is about 140 MPa. In the transverse
loading case it is observed that the Von Mises stress in the matrix is not uniform.
The comparison of the overall axial and transverse responses for both tetragonal and hexagonal
periodic arrangements in Figs. 14 and 15 reveal similar responses, however, tetragonal response is
stiffer compared to hexagonal response, for same fiber volume fraction. This can be explained by
observing the geometry of both arrangements. It can be seen that the placement of fibers in a
tetragonal periodic composite provides lines of stiffer resistance to applied force, unlike a hexagonal
periodic composite where the arrangement of fibers allows the softer matrix to contribute more to
the effective response, hence, a more compliant composite.
32
Finally, in Fig. 16 we compare the overall axial stress vs. overall axial strain for the loading case
described in Fig. 7, when the three SMA constitutive models discussed in Section 2 and whose
uniaxial response is plotted in Fig. 1 are utilized. One observes that the differences among the three
models during the forward and reverse transformations are less pronounced, with respect to the
monolithic SMA response shown in Fig. 1. This occurs mainly due to the fact that the aluminum
matrix has the same hardening response in all three cases, thus reducing the impact of the differences
among the SMA constitutive models to the overall composite response. Since the overall axial loading
is fiber dominated, the transverse loading case, which is matrix dominated in terms of deformations,
is expected to result to even less pronounced differences in the overall composite response for the
different SMA models.
The results obtained above agree qualitatively with the experimental data given by Taya et al.
(1993). The results acquired for the fiber volume fractions of 0.04 and 0.09, show a percentage
increase of about 52 % and 112 % respectively, in the effective yield stress of the composite relative
to the initial aluminum yield stress of 33 MPa, for the case of applied axial loading. The same trend
is observed in the present study, where for 0.3 fiber volume fraction, both tetragonal and hexagonal
periodic arrangements yield an increase of about 79 %, compared to the initial aluminum yield stress
of 70 MPa, for the case of applied axial loading.
5. CONCLUSIONS
A general thermodynamic constitutive model has been proposed based on the work done by
Boyd and Lagoudas (1994). Several earlier models can be identified by selecting different forms of
the function f( ) in the mixing free energy. The thermal energy output rates during forward (from
austenite to martensite) transformation predicted by these different models are compared with the
experimental data obtained from a calorimetric measurement. The results show that the different
models give substantially different predictions for the thermal energy output, even though the stress
strain curves predicted by the different models are similar to each other.
The effective thermomechanical response of Active Metal Matrix Composites (AMMC) with
SMA fibers has been modeled in this work. A 3-D constitutive model for SMA was employed in a
33
34
unit cell FEM analysis for a periodic AMMC to obtain the effective response of the elastoplastic
matrix / active fiber composite. A detailed procedure for implementing the constitutive model in the
finite element analysis scheme was given for both tetragonal and hexagonal periodic arrangements.
In addition to the overall thermomechanical response, the overall shape memory behavior of AMMC
was analyzed, and in particular its effect on the yield strength of the composite.
For all cases studied, it was demonstrated that both the axial and transverse prestraining have the
same strengthening effect for the final response of the composite, provided that the bonding between
matrix and SMA fibers is perfect. However, this condition may not be true for real composites
undergoing transverse loading due to the stress concentrations and large phase transformation strains.
Due to mechanical interactions, the phase transformation temperature range in AMMC has, finally,
been found to be much larger than the monolithic SMA.
35
36
6. ACKNOWLEDGMENTS
37
The authors acknowledge the financial support of the Army Research Office, contract No.
DAAL03-92-G-0123, monitored by Dr. G.L. Anderson.
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List of Figures
Fig. 1 Stress-temperature diagram for transformation start and finish lines.
Fig. 2 Uniaxial stress-strain behavior for different hardening laws.
Fig. 3 Heat energy output rate during austenite to martensite phase transformation.
Fig. 4 Tetragonal and hexagonal periodic arrangement with respective unit cells.
Fig. 5 Schematic representation of the loading history used in the analysis.
Fig. 6 Applied load and temperature history used in the analysis.
Fig. 7 Overall axial stress vs. axial strain for the case of applied axial loading.
Fig. 8 SMA and matrix axial average stress vs. overall axial strain for the case of applied axial
loading.
Fig. 9 Contour plot of Von-Mises stress for the case of applied axial loading at 140 MPa and 44 oC.
Fig. 10 Contour plot of martensitic volume fraction for the case of applied axial loading at 140 MPa
and 44 oC.
Fig. 11 Overall transverse stress vs. transverse strain for the case of applied transverse loading.
Fig. 12 Contour plot of Von-Mises stress for the case of applied transverse loading at 140 MPa and
44 oC.
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Fig. 13 Contour plot of martensitic volume fraction for the case of applied transverse loading at 140
MPa and 44 oC.
Fig. 14 Comparison of overall axial stress vs. axial strain for the case of applied axial loading for
tetragonal and hexagonal pariodic arrangements.
Fig. 15 Comparison of overall transverse stress vs. transverse strain for the case of applied axial
loading for tetragonal and hexagonal pariodic arrangements.
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