308
M. R. Karamooz-Ravari et al.
constraint is derived since both static and kinematic constraints yield the same constitutive equations. As a result, for microplane modeling of any responses, the volumetric–deviatoric split is recommended to be applied (Kadkhodaei et al. 2007a) so
that macroscopic elastic moduli are applicable for microplane laws and that double
constraint formulation is obtained for elastic responses.
By developing proper microplane laws based on 1-D macroscopic constitutive
equations, microplane models for several kinds of materials and responses (Badnava
et al. 2016; Brocca and Bažant 2000; Kadkhodaei et al. 2007a, b; Ožbolt et al. 2001;
Bažant and Zi 2003; Bažant et al. 2000; Zreid and Kaliske 2016; Prat and Bažant
1991; Salviato et al. 2016; Carol et al. 1991; Bažant and Prat 1987; Kuhl 2001; Bažant
and Di Luzio 2004; Caner et al. 2007; Chang and Sture 2006; Caner and Carol 2006;
Li et al. 2017; Kirane et al. 2015; Chatti et al. 2019; Steinke et al. 2019; Etse et al.
2003; Jin 2016) have been so far presented. This theory has been extended for large
deformations as well (Bažant et al. 2000; Carol et al. 2004; Indriyantho et al. 2019).
The main advantage of microplane theory over conventional modeling approaches is
that only 1-D constitutive equations are required to derive a 3-D constitutive model;
however, appropriate adjustments of 1-D macroscopic models to obtain microplane
laws are challenging aspects of this technique (Kadkhodaei et al. 2007a).
Prior to proposition of a microplane model for shape memory alloys (SMAs),
one more improvement to conventional microplane formulations is required to be
developed. It can be shown (Kadkhodaei et al. 2007a, b) that expressing shear stress
within each microplane by two resolved components on the plane causes inaccurate
results and even may lead to prediction of shear strain during pure axial loading under
certain conditions or axial strain during pure shear loading of isotropic materials. The
main reason for this discrepancy is believed to be the so-called directional bias due
to the same nonlinear stress–strain laws for the shear directions m and l. To avoid
this, shear stress on each microplane can be described by using the resultant shear
stress vector on that plane, instead of resolving into two components on arbitrary
directions m and l perpendicular to each other. For instance, when static constraint
is used, one shear stress on each microplane is considered as is shown in Fig. 17.2.
Shear stress within each microplane is characterized by its resultant value on the
plane, and it can be expressed in the form of:
σ T = T i j σ i j
(17.16)
where T i j =
n i t j + n j t i
/2 in which,
t i =
σ ik n k − σ N n i
σ jr σ js n r n s − σ
2
N
(17.17)
represents Cartesian components of the unit vector t along the direction of resultant
shear stress on the plane. Accordingly, the principle of complementary virtual work
yields the following relation for strain tensor:
M. R. Karamooz-Ravari et al.
constraint is derived since both static and kinematic constraints yield the same constitutive equations. As a result, for microplane modeling of any responses, the volumetric–deviatoric split is recommended to be applied (Kadkhodaei et al. 2007a) so
that macroscopic elastic moduli are applicable for microplane laws and that double
constraint formulation is obtained for elastic responses.
By developing proper microplane laws based on 1-D macroscopic constitutive
equations, microplane models for several kinds of materials and responses (Badnava
et al. 2016; Brocca and Bažant 2000; Kadkhodaei et al. 2007a, b; Ožbolt et al. 2001;
Bažant and Zi 2003; Bažant et al. 2000; Zreid and Kaliske 2016; Prat and Bažant
1991; Salviato et al. 2016; Carol et al. 1991; Bažant and Prat 1987; Kuhl 2001; Bažant
and Di Luzio 2004; Caner et al. 2007; Chang and Sture 2006; Caner and Carol 2006;
Li et al. 2017; Kirane et al. 2015; Chatti et al. 2019; Steinke et al. 2019; Etse et al.
2003; Jin 2016) have been so far presented. This theory has been extended for large
deformations as well (Bažant et al. 2000; Carol et al. 2004; Indriyantho et al. 2019).
The main advantage of microplane theory over conventional modeling approaches is
that only 1-D constitutive equations are required to derive a 3-D constitutive model;
however, appropriate adjustments of 1-D macroscopic models to obtain microplane
laws are challenging aspects of this technique (Kadkhodaei et al. 2007a).
Prior to proposition of a microplane model for shape memory alloys (SMAs),
one more improvement to conventional microplane formulations is required to be
developed. It can be shown (Kadkhodaei et al. 2007a, b) that expressing shear stress
within each microplane by two resolved components on the plane causes inaccurate
results and even may lead to prediction of shear strain during pure axial loading under
certain conditions or axial strain during pure shear loading of isotropic materials. The
main reason for this discrepancy is believed to be the so-called directional bias due
to the same nonlinear stress–strain laws for the shear directions m and l. To avoid
this, shear stress on each microplane can be described by using the resultant shear
stress vector on that plane, instead of resolving into two components on arbitrary
directions m and l perpendicular to each other. For instance, when static constraint
is used, one shear stress on each microplane is considered as is shown in Fig. 17.2.
Shear stress within each microplane is characterized by its resultant value on the
plane, and it can be expressed in the form of:
σ T = T i j σ i j
(17.16)
where T i j =
n i t j + n j t i
/2 in which,
t i =
σ ik n k − σ N n i
σ jr σ js n r n s − σ
2
N
(17.17)
represents Cartesian components of the unit vector t along the direction of resultant
shear stress on the plane. Accordingly, the principle of complementary virtual work
yields the following relation for strain tensor: