476
14 Constitutive Modelling of UHPCC Material Under Impact …
Case model (K&C model) (Malvar et al. 1997) and its modified versions (Kong et al.
2017a; Hong et al. 2017). Reviews of these models mentioned above were made by
Tu and Lu (2009) and Kong et al. (2018a), in which their advantages and limitations
were clearly pointed out. However, these material models were originally developed
for traditional concrete and cannot be applied to UHPCC material directly, which
results from their different mechanical properties, especially the strain-rate effect,
EoS and dynamic tensile behavior mentioned above. Frequently, the strain-rate effect
is incorporated into the material model by defining the dynamic increase factor (DIF)
at different strain-rates. Review of dynamic increase factor for tension (DIFt) of
UHPCC material demonstrates that DIFt of UHPCC is higher than that of traditional
concrete at a high strain-rate, and effects of fiber volume content, fiber type and fiber
distribution on DIFt are negligible (Thaomas and Sorensen 2017). For the EoS that
strongly affects the compressive behavior under high pressure, the porosity in materials has an obvious effect on the so-called transition stage where the air voids are
gradually compressed out, beyond which all air voids are removed and materials are
in a densification state. In the densification stage, flyer-plate-impact experiments indicate that UHPCC and traditional concrete have a similar EoS (beyond a pressure about
3GPa) mainly because they have a similar condense density (Neel 2018). However,
since the porosity of UHPCC is always small, in the transition stage UHPCC should
have a higher volumetric stiffness when compared with the traditional concrete.
In particular, the tensile behavior especially the post-peak (softening) response of
UHPCC is far different from the traditional concrete. The tensile-softening behavior
of the traditional concrete has been extensively studied and a power function known
as the “Hordijk-Reinhard expression” is used to define the softening curve (Kong
et al. 2018a; Weerheijm and van Doormaal 2007). However, the tensile-softening of
UHPCC exhibits multiple complex stages mainly due to the fiber bridging mechanism, which involves the first softening stage due to matrix cracking, the second
platform where stress remains almost unchanged with the increase of crack width
mainly due to the fiber bridging effect between the disconnected crack surfaces and
the final failure softening stage associated with the pull-out or break of fibers (Kang
et al. 2010). Modelling the tensile-softening behavior of UHPCC in a material model
is still a challenging topic and such model has not been fully established (Othman
and Marzouk 2018).
Considering the different material properties between traditional concrete and
UHPCC discussed above, existing studies on the material model of UHPCC have
been mostly conducted by calibrating parameters of concrete material models based
on corresponding test data. Mao et al. (2014) modified the tensile-softening curve
of the K&C model by adjusting the value of b 2 , while holding other parameters
unchanged. This modification leaded to a reasonable description of tensile behavior
of UHPCC, but the other parameters, such as the strength surface parameters and
EoS parameters still used the automatic generated ones originally suggested for
the traditional concrete with an unconfined compressive strength (f c ) of 45.4 MPa.
Recently, Xu et al. (2020) presented a systematic calibration of the K&C material
model for UHPCC material, including the strength surface parameters based on
triaxial compression data, the EoS parameters based on hydrostatic compression test
14 Constitutive Modelling of UHPCC Material Under Impact …
Case model (K&C model) (Malvar et al. 1997) and its modified versions (Kong et al.
2017a; Hong et al. 2017). Reviews of these models mentioned above were made by
Tu and Lu (2009) and Kong et al. (2018a), in which their advantages and limitations
were clearly pointed out. However, these material models were originally developed
for traditional concrete and cannot be applied to UHPCC material directly, which
results from their different mechanical properties, especially the strain-rate effect,
EoS and dynamic tensile behavior mentioned above. Frequently, the strain-rate effect
is incorporated into the material model by defining the dynamic increase factor (DIF)
at different strain-rates. Review of dynamic increase factor for tension (DIFt) of
UHPCC material demonstrates that DIFt of UHPCC is higher than that of traditional
concrete at a high strain-rate, and effects of fiber volume content, fiber type and fiber
distribution on DIFt are negligible (Thaomas and Sorensen 2017). For the EoS that
strongly affects the compressive behavior under high pressure, the porosity in materials has an obvious effect on the so-called transition stage where the air voids are
gradually compressed out, beyond which all air voids are removed and materials are
in a densification state. In the densification stage, flyer-plate-impact experiments indicate that UHPCC and traditional concrete have a similar EoS (beyond a pressure about
3GPa) mainly because they have a similar condense density (Neel 2018). However,
since the porosity of UHPCC is always small, in the transition stage UHPCC should
have a higher volumetric stiffness when compared with the traditional concrete.
In particular, the tensile behavior especially the post-peak (softening) response of
UHPCC is far different from the traditional concrete. The tensile-softening behavior
of the traditional concrete has been extensively studied and a power function known
as the “Hordijk-Reinhard expression” is used to define the softening curve (Kong
et al. 2018a; Weerheijm and van Doormaal 2007). However, the tensile-softening of
UHPCC exhibits multiple complex stages mainly due to the fiber bridging mechanism, which involves the first softening stage due to matrix cracking, the second
platform where stress remains almost unchanged with the increase of crack width
mainly due to the fiber bridging effect between the disconnected crack surfaces and
the final failure softening stage associated with the pull-out or break of fibers (Kang
et al. 2010). Modelling the tensile-softening behavior of UHPCC in a material model
is still a challenging topic and such model has not been fully established (Othman
and Marzouk 2018).
Considering the different material properties between traditional concrete and
UHPCC discussed above, existing studies on the material model of UHPCC have
been mostly conducted by calibrating parameters of concrete material models based
on corresponding test data. Mao et al. (2014) modified the tensile-softening curve
of the K&C model by adjusting the value of b 2 , while holding other parameters
unchanged. This modification leaded to a reasonable description of tensile behavior
of UHPCC, but the other parameters, such as the strength surface parameters and
EoS parameters still used the automatic generated ones originally suggested for
the traditional concrete with an unconfined compressive strength (f c ) of 45.4 MPa.
Recently, Xu et al. (2020) presented a systematic calibration of the K&C material
model for UHPCC material, including the strength surface parameters based on
triaxial compression data, the EoS parameters based on hydrostatic compression test