Pure rate effect on compressive strength of concrete

Dynamic Increase Factor (DIF) has been used to consider the compressive strength enhancement of concrete at the high and intermediate strain rates. However, DIF formulae suggested until now include the inertia effects as well as the rate effect because the DIF formulae has been assumed as a function of only the strain rate and the inertia effects cannot be avoided in tests at the high and intermediate strain rate region. Therefore, applying the DIF to design or analysis of social infrastructures may be dangerous because the resistance by the inertia effects are considered repetitively. In this study, an apparent DIF formula, which includes the inertia effects, was proposed by introducing terms related with the strain acceleration, which represent the axial and radial inertia effects. Then, a nonlinear regression analysis was conducted to determine the coefficients in the apparent DIF formula with results of Split Hopkinson Pressure Bar (SHPB) tests for concrete. Finally, the DIF formula excluding the axial and radial inertia effects was proposed for compressive strength of concrete at the high and intermediate strain rates.


Introduction
When a concrete structure under extreme events like explosion, car, ship, and aircraft crush is analysed or designed, Dynamic Increase Factor (DIF) has been used to consider the compressive strength enhancement of concrete in a constitutive equation. In general, DIF is obtained by conducting Split Hopkinson Pressure Bar (SHPB) test, but the inertia effects cannot be avoided in SHPB test for two reasons. First of all, the material properties of specimen tested are unknown, so a desirable incident wave minimizing the inertia effects cannot be determined. In addition, although the desirable incident wave is known, it is too difficult to obtain the incident wave intended precisely. As a result, test data of SHPB includes the inertia effects as well as the rate effect, so present DIF formulae of concrete compressive strength also include both effects. However, because the inertia effects are already considered in the equation of motion in Finite Element Analysis (FEA) for a structures under extreme loads, the analysis with DIF including the inertia effects, as called as apparent DIF, may give unconservative results by overestimating the dynamic strength of concrete. In this study, an apparent DIF formula, which includes the inertia effects, was proposed by introducing terms related with the strain acceleration, which represents the axial and radial inertia effects, and then, DIF by pure rate effect was proposed.

Inertia effects in apparent DIF
The inertia effects in SHPB test were investigated analytically and experimentally, and it is found out that the axial strain acceleration, dimension, and density of a specimen are the key factors influencing the inertia effects [1][2][3][4]. Therefore, the inertia effects in SHPB test for concrete can be assumed by Eq. 2 k , and 3 k denote the axial and radial inertia effects in SHPB, axial strain acceleration, density, diameter, length of a specimen, and coefficients for each inertia effect respectively.
And then, apparent DIF can be written such as Eq. (2) Meanwhile, the rate effect can be assumed by a power function of loading rate like Eq.
Therefore, apparent DIF can be expressed such as Eq. (6)

Specimen preparation
In order to obtain apparent DIF data for regression of Eq. (6), concrete SHPB tests were conducted with 76.2 mm SHPB of Extreme Performance Testing Center (EPTC) in Seoul National University. Total 72 data for concrete SHPB test was obtained, and the variables of tests are listed on Table 1. I.D. of specimens is defined as Sxx-Dyy-Lzz, and xx, yy, and zz denote static strength from standard cylinder compressive test, diameter, and length of a specimen respectively. Specimens of SHPB test were placed in D50×100 and D75×150 moulds, and were cut and ground on age of 28 days. Table 2 shows concrete mix proportion. Maximum aggregate size is 13 mm for representability of small specimens.

Test procedure
Annular pulse shapers were used to reduce high frequency wave components due to the radial inertia effect of pulse shaper [9] as shown as Fig. 1. (b), and for each test, each annular pulse shaper was designed by pre-analyses using LS-DYNA. In order to minimize friction effect, the impact faces and interfaces of a pulse shaper and a specimen were lubricated by petroleum jelly as shown as Fig. 1. (b) and (c). Diameter of each bar component is 76.2 mm, and the length of incident and transmitted bars is 5.5 m. The striker length is ranged from 200 mm to 600 mm, and the impact velocity was ranged from 12 m/s to 20 m/s.

Test results
Dynamic equilibrium was checked, and the raw data was transformed to stress and strain by 3 waves post-process. DIF was determined from the static strength of D150×300 standard cylinder specimen. Test results are shown in Fig. 2 and 3.

Regression analysis
Nonlinear regression analysis was conducted for Eq. (6) to concrete SHPB test data. The objective function was the least square error such as Eq. (7). It was confirmed that there is the unique minimum point. Apparent DIF and DIF by pure rate effect were obtained such as Eq. (8) and (9) respectively. For dimensionless k 1 , k 2 , and k 3 , units of variables were considered as Table 3.  Fig. 4 shows the proposed DIF by pure rate effect. The inertia effects become dominant as strain rate becomes high. First of all, DIF in fib MC2010 below transition strain rate is similar to proposed DIF, and it means that the steep increase of DIF is due to the inertia effects. In the case of DIF in ACI 349-13, the upper limit is just equal to 1.25, so it is similar to proposed DIF at the 10 -1,000 s -1 of strain rate region. For DIF of ACI 370R-14 also, because the transition strain rate is higher than those of the other DIF and the rate effect is underestimated, overestimation of the dynamic strength is started beyond about 100 s -1 . Therefore, both DIFs of ACI are conservative relatively. Finally, DIF of UFC 3-340-02 always overestimates the dynamic strength.

Conclusion
The inertia effects in apparent DIF by SPHB test were investigated, and DIF by pure rate effect was proposed. And also, following conclusions were derived.
1. DIF by pure rate effect was proposed for application in constitutive equation. 2. DIF of fib MC2010 is unconservative beyond transition strain rate, and this steep increase of DIF in fib MC2010 is due to the inertia effects.