17 Microplane Modeling for Inelastic Responses …
305
where N i j = n i n j and n i represent the Cartesian components of the unit normal
vector n to a microplane.
Shear stress on each microplane is characterized by its components along two
perpendicular directions M and L on the plane. The corresponding unit vectors are
denoted by m and l with components m i and l i . These shear stresses can be expressed
as,
σ M = M i j σ i j , σ L = L i j σ i j
(17.2)
in which M i j =
n i m j + n j m i
/2 and L i j =
n i l j + n j l i
/2 (Bažant and Prat
1988a, b). The principle of complementary virtual work yields:
4π
3
ε : δσ = 2
Ω
(ε N δσ N + ε M δσ M + ε L δσ L )dΩ
(17.3)
where Ω is the surface of a unit hemisphere representing all possible orientations at
a point. This equation states that the virtual work done by macroscopic stress and
strain tensors over the volume of a unit hemisphere is equal to the virtual work done
by the microplane stresses and strains over its surface (Caner et al. 2019). More
details about the basics of Eq. (17.3) is provided in Bažant et al. (1996).
Substituting Eqs. (17.1) and (17.2) into (17.3), by taking into account the
independence of individual components of virtual stress tensor, yields:
ε i j =
3
2π
Ω
ε N N i j + ε M M i j + ε L L i j
dΩ
(17.4)
To use microplane formulation with static constraint, stress components on each
plane passing through a material point are first calculated by the projection rule
stated in Eqs. (17.1) ad (17.1). Then, 1-D microplane laws are required to determine
normal as well as shear strains. Equation (17.4), which is in fact homogenization of
all microplane strains, is finally utilized to obtain macroscopic strains. Compared to
conventional modeling approaches, where strain tensor is directly either an explicit
or an implicit function of stress tensor, macroscopic strains in microplane modeling
with static constraint are indirectly obtained from the macroscopic stress components.
Dual formulation of static constraint is kinematic constraint, in which projection rule
is applied for strain as:
ε N = N i j ε i j , ε M = M i j ε i j , ε L = L i j ε i j
(17.5)
Similar to Eqs. (17.3) and (17.4), the principle of virtual work is applied to obtain
the following homogenization for determination of macroscopic stress tensor:
305
where N i j = n i n j and n i represent the Cartesian components of the unit normal
vector n to a microplane.
Shear stress on each microplane is characterized by its components along two
perpendicular directions M and L on the plane. The corresponding unit vectors are
denoted by m and l with components m i and l i . These shear stresses can be expressed
as,
σ M = M i j σ i j , σ L = L i j σ i j
(17.2)
in which M i j =
n i m j + n j m i
/2 and L i j =
n i l j + n j l i
/2 (Bažant and Prat
1988a, b). The principle of complementary virtual work yields:
4π
3
ε : δσ = 2
Ω
(ε N δσ N + ε M δσ M + ε L δσ L )dΩ
(17.3)
where Ω is the surface of a unit hemisphere representing all possible orientations at
a point. This equation states that the virtual work done by macroscopic stress and
strain tensors over the volume of a unit hemisphere is equal to the virtual work done
by the microplane stresses and strains over its surface (Caner et al. 2019). More
details about the basics of Eq. (17.3) is provided in Bažant et al. (1996).
Substituting Eqs. (17.1) and (17.2) into (17.3), by taking into account the
independence of individual components of virtual stress tensor, yields:
ε i j =
3
2π
Ω
ε N N i j + ε M M i j + ε L L i j
dΩ
(17.4)
To use microplane formulation with static constraint, stress components on each
plane passing through a material point are first calculated by the projection rule
stated in Eqs. (17.1) ad (17.1). Then, 1-D microplane laws are required to determine
normal as well as shear strains. Equation (17.4), which is in fact homogenization of
all microplane strains, is finally utilized to obtain macroscopic strains. Compared to
conventional modeling approaches, where strain tensor is directly either an explicit
or an implicit function of stress tensor, macroscopic strains in microplane modeling
with static constraint are indirectly obtained from the macroscopic stress components.
Dual formulation of static constraint is kinematic constraint, in which projection rule
is applied for strain as:
ε N = N i j ε i j , ε M = M i j ε i j , ε L = L i j ε i j
(17.5)
Similar to Eqs. (17.3) and (17.4), the principle of virtual work is applied to obtain
the following homogenization for determination of macroscopic stress tensor: