9.4 Numerical Simulation
297
σ
vp
ij = (1 − γ ) × σ
T
ij + σ
P
ij , γ = ((t/η)/(1 + t/η)
(9.21)
This interpolation γ depends on the effective fluidity coefficient η and the time
step t. In addition, the coefficient η can be estimated by four user input parameters
given in Eq. (9.22), such as the fluidity parameters (η t , η s , η c ) in uniaxial tensile stress,
shear stress, and uniaxial compressive stress. The dynamic tensile and compressive
strength f
t,d and f
c,d can be obtained according to Eq. (9.23).
η t = η 0t /˙ ε
N t , η c = η 0c /˙ ε
N c , η s = Srate × η t
(9.22)
f
c,d = f
c + E ˙
εη D , f
t,d = f
t + E ˙
εη B
(9.23a)
η D = η s + trans D (η c − η s ), η B = η s + trans B (η t − η s )
(9.23b)
where the parameter ˙
ε is the strain rate, η 0t and η 0c are the strain rate effect parameters
for uniaxial tension and compression, respectively. N t and N c are the strain rate
effects power for uniaxial tension and compression, respectively. Srate is the ratio
of effective shear stress to tensile stress fluidity parameters. Therefore, the dynamic
increasing factors (DIF t and DIF c ) for direct tension and unconfined compression
can be expressed as
DIF t = f
t,d /f
t = 1 + E ˙
εη 0t /(f
t ˙
ε
N t )
(9.24a)
DIF c = f
c,d /f
c = 1 + E ˙
εη 0c /(f
c ˙
ε
N c )
(9.24b)
Previously, we have examined the strain rate effect of UHPCC under uniaxial
tension and compression based on the Hopkinson bars impact test (Ren et al. 2018b;
Wu et al. 2018), and presented the following Eq. (9.25) to estimate the DIF t and
DIF c of UHPCC.
DIF t =
(˙ ε/˙ ε 0t )
0.018
˙
ε ≤ ˙
ε TR
11.561 × log ˙
ε − 10.446 ˙
ε > ˙
ε TR
(9.25a)
DIF c =
(˙ ε/˙ ε 0c )
0.014
˙
ε ≤ ˙
ε TR
0.5103 × (log ˙
ε)
2
− 1.2301 × log ˙
ε + 1.6804 ˙
ε > ˙
ε TR
(9.25b)
where ˙
ε 0t is the strain rate for static tensile loading (1.0 × 10
–6 s
−1 ) and ˙
ε 0c is the
strain rate for static compression loading (3.0 × 10
–5 s
−1 ); and the inflection points
˙
ε TR in Eq. (9.25a) and (9.25b) are 10.5 s
−1 and 90 s
−1 , respectively. Therefore, the
strain rate formula Eq. (9.24) of CSC model can be determined based on the data
from Eq. (9.25), as shown in Fig. 9.22. The corresponding strain rate parameters can
be obtained as
297
σ
vp
ij = (1 − γ ) × σ
T
ij + σ
P
ij , γ = ((t/η)/(1 + t/η)
(9.21)
This interpolation γ depends on the effective fluidity coefficient η and the time
step t. In addition, the coefficient η can be estimated by four user input parameters
given in Eq. (9.22), such as the fluidity parameters (η t , η s , η c ) in uniaxial tensile stress,
shear stress, and uniaxial compressive stress. The dynamic tensile and compressive
strength f
t,d and f
c,d can be obtained according to Eq. (9.23).
η t = η 0t /˙ ε
N t , η c = η 0c /˙ ε
N c , η s = Srate × η t
(9.22)
f
c,d = f
c + E ˙
εη D , f
t,d = f
t + E ˙
εη B
(9.23a)
η D = η s + trans D (η c − η s ), η B = η s + trans B (η t − η s )
(9.23b)
where the parameter ˙
ε is the strain rate, η 0t and η 0c are the strain rate effect parameters
for uniaxial tension and compression, respectively. N t and N c are the strain rate
effects power for uniaxial tension and compression, respectively. Srate is the ratio
of effective shear stress to tensile stress fluidity parameters. Therefore, the dynamic
increasing factors (DIF t and DIF c ) for direct tension and unconfined compression
can be expressed as
DIF t = f
t,d /f
t = 1 + E ˙
εη 0t /(f
t ˙
ε
N t )
(9.24a)
DIF c = f
c,d /f
c = 1 + E ˙
εη 0c /(f
c ˙
ε
N c )
(9.24b)
Previously, we have examined the strain rate effect of UHPCC under uniaxial
tension and compression based on the Hopkinson bars impact test (Ren et al. 2018b;
Wu et al. 2018), and presented the following Eq. (9.25) to estimate the DIF t and
DIF c of UHPCC.
DIF t =
(˙ ε/˙ ε 0t )
0.018
˙
ε ≤ ˙
ε TR
11.561 × log ˙
ε − 10.446 ˙
ε > ˙
ε TR
(9.25a)
DIF c =
(˙ ε/˙ ε 0c )
0.014
˙
ε ≤ ˙
ε TR
0.5103 × (log ˙
ε)
2
− 1.2301 × log ˙
ε + 1.6804 ˙
ε > ˙
ε TR
(9.25b)
where ˙
ε 0t is the strain rate for static tensile loading (1.0 × 10
–6 s
−1 ) and ˙
ε 0c is the
strain rate for static compression loading (3.0 × 10
–5 s
−1 ); and the inflection points
˙
ε TR in Eq. (9.25a) and (9.25b) are 10.5 s
−1 and 90 s
−1 , respectively. Therefore, the
strain rate formula Eq. (9.24) of CSC model can be determined based on the data
from Eq. (9.25), as shown in Fig. 9.22. The corresponding strain rate parameters can
be obtained as