484
14 Constitutive Modelling of UHPCC Material Under Impact …
It can be observed that all of the five parameters almost increase linearly with the
increase of fiber volume fraction V f . Therefore, the following equations are proposed
to determine these five parameters.
T = 80 × V f + 6.5 (unit in MPa)
(14.14)
T 1 = 5.45 + 72 × V f (unit in MPa)
(14.15)
ε 1 = 9e
−4
(14.16)
ε 2 = 0.002 −
6e
−2
× V f
(14.17)
ε f rac = 0.0125 +
V f − 0.01
× 0.5
(14.18)
14.2.3 Parameter Calibration
Three set of parameters should be defined in the Kong-Fang model, i.e., parameters
related to the failure surface, the damage and EoS. Based on existing and our test
data, these parameters are carefully calibrated in this section.
14.2.3.1 Parameters Related to Failure Surface
As discussed in Sect. 14.2.1, parameters related to the failure surface include the
elastic modulus E, the Poisson’s ratio ν, unconfined uniaxial compressive strength f c
and tensile strength T, material constants a 1 and a 2 , tensile-to-compressive meridian
ratio ψ and dynamic increase factors for tension and compression (DIFt and DIFc).
It should be pointed out that only UHPCC with steel fibers is considered.
Usually, the f c , T and E are available in the impact and blast experiments. If
E is absent, it can be approximately determined by cwith E m and E s being the
Young’s modules of matrix and steel fibers, respectively (Wille et al. 2014). E m can
be estimated as E m = k × f
1/3
c
(both E m and f c are in MPa units) with the value of k
ranged from 8800~11,000 and E s as 210GPa. Experimental data (Ren et al. 2018b)
shows that the Poisson’s ratio of UHPCC material is around 0.23, and it is almost
unchanged with the fiber volume content, thus a fix value of 0.23 is suggested.
Then the bulk modulus K and shear modulus G can be determined according to
K = E/3(1 − 2ν) and G = E/2(1 + ν), respectively.
Material constants a 1 and a 2 should be determined by a suitable set of TXC data.
In the original Kong-Fang model, a 1 = 0.5876 and a 2 = 0.025/ f c are determined
by a large amount of TXC data of traditional concrete. Figure 14.3 presents our
14 Constitutive Modelling of UHPCC Material Under Impact …
It can be observed that all of the five parameters almost increase linearly with the
increase of fiber volume fraction V f . Therefore, the following equations are proposed
to determine these five parameters.
T = 80 × V f + 6.5 (unit in MPa)
(14.14)
T 1 = 5.45 + 72 × V f (unit in MPa)
(14.15)
ε 1 = 9e
−4
(14.16)
ε 2 = 0.002 −
6e
−2
× V f
(14.17)
ε f rac = 0.0125 +
V f − 0.01
× 0.5
(14.18)
14.2.3 Parameter Calibration
Three set of parameters should be defined in the Kong-Fang model, i.e., parameters
related to the failure surface, the damage and EoS. Based on existing and our test
data, these parameters are carefully calibrated in this section.
14.2.3.1 Parameters Related to Failure Surface
As discussed in Sect. 14.2.1, parameters related to the failure surface include the
elastic modulus E, the Poisson’s ratio ν, unconfined uniaxial compressive strength f c
and tensile strength T, material constants a 1 and a 2 , tensile-to-compressive meridian
ratio ψ and dynamic increase factors for tension and compression (DIFt and DIFc).
It should be pointed out that only UHPCC with steel fibers is considered.
Usually, the f c , T and E are available in the impact and blast experiments. If
E is absent, it can be approximately determined by cwith E m and E s being the
Young’s modules of matrix and steel fibers, respectively (Wille et al. 2014). E m can
be estimated as E m = k × f
1/3
c
(both E m and f c are in MPa units) with the value of k
ranged from 8800~11,000 and E s as 210GPa. Experimental data (Ren et al. 2018b)
shows that the Poisson’s ratio of UHPCC material is around 0.23, and it is almost
unchanged with the fiber volume content, thus a fix value of 0.23 is suggested.
Then the bulk modulus K and shear modulus G can be determined according to
K = E/3(1 − 2ν) and G = E/2(1 + ν), respectively.
Material constants a 1 and a 2 should be determined by a suitable set of TXC data.
In the original Kong-Fang model, a 1 = 0.5876 and a 2 = 0.025/ f c are determined
by a large amount of TXC data of traditional concrete. Figure 14.3 presents our