8.4 Calibration of K&C Model Parameters for UHPCC
249
unconfined compression strength of the concrete. Guo et al. (2018) derived that,
based on the automated generated parameters, K&C model has low accuracy in
predicting the behavior of UHPCC under low-velocity impact. In this section, after
brief introduction of K&C model, the constitutive model parameters of which for
UHPCC are further calibrated by a series of static and dynamic tests of UHPCC
mainly conducted by authors (Ren et al. 2016, 2018a, b; Wu et al. 2018).
8.4.1 Brief Introduction of K&C Model
Three independent strength surfaces of compressive meridians, namely, the initial
yield strength surface σ y , the maximum strength surface σ m and the residual
strength surface σ r are applied in K&C model. The formations of above three
surfaces can be expressed as (Malvar et al. 1997):
σ y =
⎧
⎨
⎩
a 0y + p/(a 1y + a 2y p)
p ≥ 0.15f c
1.35f t + 3p/(1 − 3f t /f c ) 0 ≤ p ≤ 0.15f c
1.35(p + f t )
p ≤ 0
(8.3)
σ m =
⎧
⎨
⎩
a 0 + p/(a 1 + a 2 p) p ≥ f c /3
1.35/ψ(p + f t )
0 ≤ p ≤ f c /3
3(p/η + f t )
p ≤ 0
(8.4)
σ r = a 0f + p/(a 1f + a 2f p)
(8.5)
where a i , a iy and a if (i = 0, 1, 2) are strength constants, which can be determined by
suitable triaxial compression data. f c and f t are the unconfined uniaxial compressive
and tensile strengths, respectively. p is the pressure and ψ denotes the tensile-tocompressive meridian ratio. Tu and Lu (2009) introduced the advantages of above
strength surfaces represented by a segmented form in detail, especially at p ≤ f c /3.
Figure 8.11 shows the typical normalized strength surfaces.
Current failure surface γ is determined as follows:
γ =
3J 2 =
r
η
σ m − σ y
+ σ y
Strain hardening
r
[η(σ m − σ r ) + σ r ] Strain softening
(8.6)
where J 2 is the second deviatoric stress invariant. r
is the ratio of the current meridian
to the compressive meridian. η is the yield scale factor referring to the damage
function λ, which is defined as
249
unconfined compression strength of the concrete. Guo et al. (2018) derived that,
based on the automated generated parameters, K&C model has low accuracy in
predicting the behavior of UHPCC under low-velocity impact. In this section, after
brief introduction of K&C model, the constitutive model parameters of which for
UHPCC are further calibrated by a series of static and dynamic tests of UHPCC
mainly conducted by authors (Ren et al. 2016, 2018a, b; Wu et al. 2018).
8.4.1 Brief Introduction of K&C Model
Three independent strength surfaces of compressive meridians, namely, the initial
yield strength surface σ y , the maximum strength surface σ m and the residual
strength surface σ r are applied in K&C model. The formations of above three
surfaces can be expressed as (Malvar et al. 1997):
σ y =
⎧
⎨
⎩
a 0y + p/(a 1y + a 2y p)
p ≥ 0.15f c
1.35f t + 3p/(1 − 3f t /f c ) 0 ≤ p ≤ 0.15f c
1.35(p + f t )
p ≤ 0
(8.3)
σ m =
⎧
⎨
⎩
a 0 + p/(a 1 + a 2 p) p ≥ f c /3
1.35/ψ(p + f t )
0 ≤ p ≤ f c /3
3(p/η + f t )
p ≤ 0
(8.4)
σ r = a 0f + p/(a 1f + a 2f p)
(8.5)
where a i , a iy and a if (i = 0, 1, 2) are strength constants, which can be determined by
suitable triaxial compression data. f c and f t are the unconfined uniaxial compressive
and tensile strengths, respectively. p is the pressure and ψ denotes the tensile-tocompressive meridian ratio. Tu and Lu (2009) introduced the advantages of above
strength surfaces represented by a segmented form in detail, especially at p ≤ f c /3.
Figure 8.11 shows the typical normalized strength surfaces.
Current failure surface γ is determined as follows:
γ =
3J 2 =
r
η
σ m − σ y
+ σ y
Strain hardening
r
[η(σ m − σ r ) + σ r ] Strain softening
(8.6)
where J 2 is the second deviatoric stress invariant. r
is the ratio of the current meridian
to the compressive meridian. η is the yield scale factor referring to the damage
function λ, which is defined as