458
13 Experimental and Numerical Studies on Dynamic Behavior …
where K is the elastic bulk modulus, μ crush is the ultimate volumetric strain, P crush
is the hydrostatic pressure corresponding to μ crush , E is the elastic modulus and v is
the Poisson’s ratio.
The second is the transition stage (μ crush ≤ μ ≤ μ plock ), in which the air voids are
gradually compressed out of the concrete, and the plastic volumetric strain produces
the compaction damage until reaching the point (μ plock , P lock ). The loading path in
this phase is expressed as:
P = P crush + K lock (μ − μ crush ), P crush ≤ P ≤ P lock
(13.15)
where K lock = (P lock − P crush )/(μ plock − μ crush ) is the slope, and μ plock is the volumetric strain at the compacting pressure P lock . The unloading path is expressed
as:
P = P 0 − [(1 − F)K + F K 1 ](μ 0 − μ)
(13.16)
where F = (μ 0 − μ crush )/(μ plock − μ crush ) is the interpolation factor, and (1 −
F)K + F K 1 represents the slope of the unloading path. P 0 is the pressure whesn the
volumetric strain reaches μ 0 before unloading.
The third is the non-porous compact stage, in which all air voids are removed
from concrete. During the loading state, the non-linear relationship is as follow:
P = K 1 ¯
μ + K 2 ¯
μ
2
+ K 3 ¯
μ
3
, P ≥ P lock
(13.17)
where ¯
μ = (μ − μ lock )/(1 + μ lock ) is the modified volumetric strain, and μ lock =
ρ grain /ρ 0 − 1 is the locking volumetric strain. ρ grain is the grain density of the material
without air voids. K 1 , K 2 , and K 3 are the material constants. During the unloading
state, the relationship is as follow:
P = K 1 ¯
μ
(13.18)
By fitting the hydrostatic pressure test data of high strength concrete, the relationship between the uniaxial compressive strength f c and P crush can be obtained. The test
data of P crush are the elastic limit pressure from the hydrostatic pressure-volumetric
strain curves. In this study, the P crush of UHPCC with uniaxial compressive strength
of 125 MPa is 159 MPa, as shown in Fig. 13.12a. According to Eqs. (13.13) and
(13.14), the first stage of EOS is determined. Besides, the second and third stages
of EOS are determined by curve fitting of the existing flyer-plate-impact test data
according to Eqs. (13.15) and (13.17), as shown in Fig. 13.12b. The related EOS
parameters are listed in Table 13.6. Based on Table 13.6 and Eqs. (13.13) ~ (13.18),
the tabulated EOS input data of UHPCC are obtained and given in Table 13.7.
13 Experimental and Numerical Studies on Dynamic Behavior …
where K is the elastic bulk modulus, μ crush is the ultimate volumetric strain, P crush
is the hydrostatic pressure corresponding to μ crush , E is the elastic modulus and v is
the Poisson’s ratio.
The second is the transition stage (μ crush ≤ μ ≤ μ plock ), in which the air voids are
gradually compressed out of the concrete, and the plastic volumetric strain produces
the compaction damage until reaching the point (μ plock , P lock ). The loading path in
this phase is expressed as:
P = P crush + K lock (μ − μ crush ), P crush ≤ P ≤ P lock
(13.15)
where K lock = (P lock − P crush )/(μ plock − μ crush ) is the slope, and μ plock is the volumetric strain at the compacting pressure P lock . The unloading path is expressed
as:
P = P 0 − [(1 − F)K + F K 1 ](μ 0 − μ)
(13.16)
where F = (μ 0 − μ crush )/(μ plock − μ crush ) is the interpolation factor, and (1 −
F)K + F K 1 represents the slope of the unloading path. P 0 is the pressure whesn the
volumetric strain reaches μ 0 before unloading.
The third is the non-porous compact stage, in which all air voids are removed
from concrete. During the loading state, the non-linear relationship is as follow:
P = K 1 ¯
μ + K 2 ¯
μ
2
+ K 3 ¯
μ
3
, P ≥ P lock
(13.17)
where ¯
μ = (μ − μ lock )/(1 + μ lock ) is the modified volumetric strain, and μ lock =
ρ grain /ρ 0 − 1 is the locking volumetric strain. ρ grain is the grain density of the material
without air voids. K 1 , K 2 , and K 3 are the material constants. During the unloading
state, the relationship is as follow:
P = K 1 ¯
μ
(13.18)
By fitting the hydrostatic pressure test data of high strength concrete, the relationship between the uniaxial compressive strength f c and P crush can be obtained. The test
data of P crush are the elastic limit pressure from the hydrostatic pressure-volumetric
strain curves. In this study, the P crush of UHPCC with uniaxial compressive strength
of 125 MPa is 159 MPa, as shown in Fig. 13.12a. According to Eqs. (13.13) and
(13.14), the first stage of EOS is determined. Besides, the second and third stages
of EOS are determined by curve fitting of the existing flyer-plate-impact test data
according to Eqs. (13.15) and (13.17), as shown in Fig. 13.12b. The related EOS
parameters are listed in Table 13.6. Based on Table 13.6 and Eqs. (13.13) ~ (13.18),
the tabulated EOS input data of UHPCC are obtained and given in Table 13.7.