2.1. Experimental Characterization
Previous investigations of rubberized concrete were mostly focused on durability and mechanical properties of rubberized concrete. It was concluded that with the increase of rubber content in concrete mixes, compressive strength and elastic modulus decrease, and strain at fracture increases [
27,
28,
29]. However, there are not enough investigations of the effect of rubber particles on the uniaxial compressive stress-strain behavior of rubberized concrete.
Compressive stress-strain behavior of rubberized concrete with 12.5%, 25.0%, 37.5%, and 50.0% was investigated by Khaloo et al. [
4], to determine the influence of replacing fine (F) and coarse (C) aggregate with crumb and chipped rubber, respectively. They executed compressive displacement controlled tests of hardened concrete cylindrical specimens to obtain stress-strain curves, utilizing a speed of 0.005 mm/s. The curves indicated that the behavior of rubberized concrete was more nonlinear compared to that of plain concrete (P) [
4], which might have been accounted to lower compressive strength of these mixes. A comparison between investigated mixes revealed that the behavior was more nonlinear for mixes where coarse aggregate is substituted with chipped rubber than for mixes where fine or both fine and coarse aggregate were replaced with crumb and chipped rubber.
L. Li et al. [
18] performed an investigation on low-volume rubberized concrete with five different rubber volume content levels (2%, 4%, 6%, 8%, 10%) and five different particle sizes (0.173 mm, 0.221 mm, 0.535 mm, 2 mm, and 4 mm). The loading process was controlled by displacement with a loading speed of 0.003 mm/s. By comparing peak and ultimate strains researchers concluded that the ultimate strain of rubberized concrete was higher for larger rubber content and smaller particle size. The capability of crack prevention and plastic deformation was higher as small rubber particles were not only distributed on the interface between the aggregate and the cement matrix, but also scattered in the interface within the cement matrix. On the contrary, larger rubber particles were mainly in the interface between aggregates and cement matrix, thus having almost no influence on the deformation of the cement matrix, which led to the concentration of plastic strain in concrete.
Xie et al. [
19] substituted 4%, 8%, 12%, and 16% volume of sand with crumb rubber (CR), added 1% of steel fibers (SFs) by volume and replaced 100% of natural coarse aggregate (NCA) with recycled coarse aggregate (RCA) in concrete mixes. Thus two reference mixes were produced, one with NCA (marked with * in
Table 1) and the other with 100% replacement of NCA with RCA. Axial load was applied at a displacement rate of 0.003 mm/s. It was observed that 100% replacement of NCA by RCA resulted in larger strain at peak stress and a smoother and straighter descending branch of the stress-strain curve. It was concluded that rubberized steel fiber recycled aggregate concrete (RSFRAC) has greater ductility than NCA concrete, although RSFRAC exhibited a reduced strength and stiffness. The effect of rubber content in the stiffness of RSFRAC was not clear before the peak stress but became apparent after the peak stress.
Noaman et al. [
25] conducted a similar investigation on the compression toughness of rubberized concrete (RC) with 5%, 10%, and 15% crumb rubber, and rubberized steel fiber concrete (RSFC) with 0.5% steel fibers amounts of crumb rubber as in RC. Cylindrical specimens of both mixtures were subjected to an axial load rate of 0.3 MPa/s, after 28 days of curing. Researchers came to the conclusion that the addition of rubber leads to the increase in ductility and strain capacity as shown in
Figure 1a,b. Additionally, it can be seen that steel fibers had an impressively larger effect on stress and increase in capacity in comparison with RC.
D.V. Bompa et al. [
3] investigated the compressive stress-strain response of a traditional concrete (R00) compared to the response of concretes with 20% (R20), 40% (R40), and 60% (R60) of rubber aggregate. To determine the complete stress-strain behavior including the post peak response, cylindrical specimens were tested under uniaxial compression utilizing displacement control at a rate of 0.001 mm/s. Recorded average stress-strain curves are presented in
Figure 1c [
3]. The stress-strain curves included both pre-peak and post-peak behavior as recorded in tests. The pre-peak behavior of the concrete tests was strongly influenced by the rubber replacement of the mineral aggregates.
Aslani et al.’s [
21,
22] study investigated the effect of crumb rubber on the mechanical properties of self-compacting rubberized concrete (SCRC) with the addition of chemical admixtures (fly ash, slag, and silica fumes) and pre-treated rubber. In three different concrete mixes they replaced 10%, 20%, 30%, and 40% fine natural aggregates with crumb rubber (CR) of 2 mm and 5 mm size, and coarse natural aggregates with crumb rubber of 10 mm size. Cylindrical specimens were tested at 28 days and loaded under compression until failure to obtain the stress-strain behavior. Axial loads were applied at a displacement rate of 0.003 mm/s. As the percentage of rubber increased for each of three SCRC series, the overall peak strain decreased. Results also indicated that higher strains were generated at lower stress as the percentage of rubber replacement was increased. However, results of this investigation were not conclusive due to the missing descending portion of curves.
D. Li et al. [
23] investigated compressive stress-strain behavior of normal concrete (NC) and crumb rubber concrete (CRC) with 6%, 12%, and 18% crumb rubber (CR) aggregates. They tested three unconfined cylindrical specimens for each concrete mix under uniaxial displacement-controlled compression loading at a rate of 0.001 mm/s. Test results of the experiment are shown in
Figure 1d where it can be seen that the initial part of the curve is linear, after which stiffness reduces up to the peak stress. At higher strains, the descending portion tended to reach a constant stress level.
A summary of described previous research is presented in
Table 1 from which a conclusion can be made that replacement of natural fine and coarse aggregates with rubber allows a more uniform crack development and enables slower crack propagation. Considering the stress-strain curves, rubberized concrete specimens exhibited larger deformations compared to plain concrete specimens under same loading conditions. Quasi-plastic behavior of rubberized concrete is noticeable on the post-cracking part of the stress strain curve with small change in deformation without losing load bearing capability. Hence, obtained curves support the assertion that rubber particle usage in concrete results in concrete failure with larger deformations and higher energy dissipation.
2.2. Uniaxial Stress-Strain Constitutive Models
To mathematically simulate concrete’s behavior under uniaxial load, constitutive uniaxial models were developed. It should be noted that there are also other constitutive models which describe the behavior of concrete under various stress states, and for various purposes [
30], but due to lack of experimental data have not been developed for rubberized concrete. Experimental results show that differences in concrete’s mixture proportions, additives, etc., affect concrete behavior, namely the shape of the uniaxial curve. Thus, one constitutive model cannot fit all concretes, and for rubberized concrete only a few have been proposed. These models take into account rubber content and size of rubber particle, which can be seen in a condensed form in
Table 2. To obtain a stress-strain curve as in
Figure 2, input data in models are peak compressive stress
, peak strain
and elastic modulus
of normal concrete (NC) which are used to calculate peak compressive stress
, peak strain
and elastic modulus
of rubberized concrete (RC). However, researchers considered different parameters while developing constitutive models described with a stress factor
.
L. Li et al. [
18] presented an improved constitutive model based on the one given in the Chinese design code for concrete structures (GB50010-2002) [
31]. To improve the existing model, they considered different rubber mixture methods, absolute value of rubber content
, rubber particles size
, sand rate reduction factor
and concrete’s compressive strength. With constitutive parameters
and
, ascending and descending parts of the stress-strain curve could be obtained. Aslani [
20] developed a stress-strain relationship for rubberized concrete based on Aslani and Nejadi’s [
32] model and an experimental results database from several studies. Based on these studies he proposed different coefficients for peak compressive stress (
) and elastic modulus of elasticity (
). In his proposed model he used modified material parameters
and
for obtaining ascending and descending parts of the stress-strain curve, and two coefficients of linear equation
and
, which D. Li et al. [
23] later modified. Bompa et al. [
3] presented a constitutive model for rubberized concrete which uses equations that they proposed to estimate the elastic modulus and peak compressive stress
. The constitutive model considers the volumetric rubber ratio
up to 0.65 (65% of volumetric rubber replacement with mineral aggregate) and size of replaced mineral aggregate
with factor
. The constitutive model has three parts; first up to the proportionally limit
(1), second up to the peak strain
(ascending part of the curve) (2), and third after the peak strain (descending part of the curve) (3) which depends on the post-peak crushing energy
.