8.4 Calibration of K&C Model Parameters for UHPCC
257
d ε
p
ij =
∂g
∂σ ij
d ˜
μ =
3σ
ij
2
√
3J 2
+
ωδ ij
3
d γ
dp
d ˜
μ
(8.17)
where d ˜
μ is a proportionality constant and the effective plastic strain increment can
be derived from Eq. (8.17) as follows
d ε p =
2
3
d ε
p
ij d ε
p
ij =
1 +
2
27
ω
d γ
dp
2
d ˜
μ
(8.18)
It could also be obtained from Eq. (8.17) in the uniaxial stress state that
d ε
p
11 =
1 +
1
3
ω
d γ
dp
d ˜
μ
(8.19)
where d ε
p
11 denotes the axial plastic strain increment. Substituting Eq. (8.19) into
Eq. (8.18), the effective plastic strain increment could be expressed by the axial
plastic strain increment as
d ε p =
1 +
2
27
ω
d γ
dp
2
1 +
1
3
ω
d γ
dp
d ε
p
11
(8.20)
Substituting Eq. (8.20) into Eq. (8.9), the increment of damage function could be
obtained as
d λ =
1 +
2
27
ω
d γ
dp
2
1 +
1
3
ω
d γ
dp
d ε
p
11
r f (1 + p/f t )
b1
(8.21)
In K&C model (Malvar et al. 1997), the value of b 1 is 1.6, ω = 0.5 represents the
partially associative flow rule, and d γ /dp could be solved from the yield surface as the
function of pressure shown in Eqs. (8.4 ~ 8.6). Without considering the influences
of strain rate effect, r f is equal to 1.0. d ε
p
11 denotes the axial plastic strain increment which could be obtained from the uniaxial compressive stress–strain curve.
Therefore, the dependence of η ~λ could be determined through a given uniaxial
compressive stress–strain curve.
Attard and Setunge (1996) proposed an empirical formula to predict the complete
compressive stress–strain curve of concrete. Based on the test data of UHPCC (Ren
et al. 2018a), the specific form of formula is given in Eq. (8.22) and plotted in
Fig. 8.17. Therefore, the relationship between η and λ for UHPCC could be obtained,
as shown in Fig. 8.18. Then the 13 pairs of (λ, η) input in K&C model for UHPCC
are listed in Table 8.9.
257
d ε
p
ij =
∂g
∂σ ij
d ˜
μ =
3σ
ij
2
√
3J 2
+
ωδ ij
3
d γ
dp
d ˜
μ
(8.17)
where d ˜
μ is a proportionality constant and the effective plastic strain increment can
be derived from Eq. (8.17) as follows
d ε p =
2
3
d ε
p
ij d ε
p
ij =
1 +
2
27
ω
d γ
dp
2
d ˜
μ
(8.18)
It could also be obtained from Eq. (8.17) in the uniaxial stress state that
d ε
p
11 =
1 +
1
3
ω
d γ
dp
d ˜
μ
(8.19)
where d ε
p
11 denotes the axial plastic strain increment. Substituting Eq. (8.19) into
Eq. (8.18), the effective plastic strain increment could be expressed by the axial
plastic strain increment as
d ε p =
1 +
2
27
ω
d γ
dp
2
1 +
1
3
ω
d γ
dp
d ε
p
11
(8.20)
Substituting Eq. (8.20) into Eq. (8.9), the increment of damage function could be
obtained as
d λ =
1 +
2
27
ω
d γ
dp
2
1 +
1
3
ω
d γ
dp
d ε
p
11
r f (1 + p/f t )
b1
(8.21)
In K&C model (Malvar et al. 1997), the value of b 1 is 1.6, ω = 0.5 represents the
partially associative flow rule, and d γ /dp could be solved from the yield surface as the
function of pressure shown in Eqs. (8.4 ~ 8.6). Without considering the influences
of strain rate effect, r f is equal to 1.0. d ε
p
11 denotes the axial plastic strain increment which could be obtained from the uniaxial compressive stress–strain curve.
Therefore, the dependence of η ~λ could be determined through a given uniaxial
compressive stress–strain curve.
Attard and Setunge (1996) proposed an empirical formula to predict the complete
compressive stress–strain curve of concrete. Based on the test data of UHPCC (Ren
et al. 2018a), the specific form of formula is given in Eq. (8.22) and plotted in
Fig. 8.17. Therefore, the relationship between η and λ for UHPCC could be obtained,
as shown in Fig. 8.18. Then the 13 pairs of (λ, η) input in K&C model for UHPCC
are listed in Table 8.9.