# An Investigation of Softening Laws and Fracture Toughness of Slag-Based Geopolymer Concrete and Mortar

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## Abstract

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## 1. Introduction

_{2}emissions produced during the production of PC, making it a greener solution [4,5]. The main hydration product of SG is C-S-H gel with a lower Ca/Si ratio than traditional PC and no zeolite or mica hydrations are found [1,2,3]. Extensive studies have demonstrated that SG can exhibit similar mechanical strength with or even perform better than PC does in many aspects, including low hydration heat, high early strength, good durability, and resistance to chemical attack [2,6]. However, several disadvantages such as quick setting, efflorescence, possibility of alkali–aggregate reaction, obvious shrinkage, and micro-cracks have also been stated [2]. In addition, SG exhibits a brittle behavior similar to that of PC.

_{F}) and the characteristic length of SG concrete and mortar and PC concrete and mortar with comparable compressive strength, and found that the G

_{F}of SG concrete was always higher than that of PC concrete given similar compressive strength, whereas the G

_{F}of the mortar system exhibited an opposite tendency. However, obviously, these fracture properties are very essential for predicting the mechanical performance of SG structure elements subjected to static and dynamic load, and for achieving safe applications of SG materials [12,13,14,15,16,17,18,19,20]. Therefore, for this paper, the authors conducted a systematic experimental study to fill in the above gap. Three compressive strength levels varying from normal strength to high strength of PC and SG concrete and mortar were tested for comparison purposes. TPB tests were conducted according to the RILEM TC50-FMC [21] recommendation. The tension softening laws of PC and SG concrete and mortar were then determined using inverse analysis based on experimental results. Furthermore, the cohesive toughness of PC and SG concrete and mortar was predicted from the tension softening curves using analytical method and compared with the experimental values. The consistence of the cohesive toughness of PC and SG concrete and mortar between the experimental results and analytical ones validated the obtained softening laws and the double-K model. In addition, comparisons of tension softening curves and fracture toughness between PC concrete and mortar and their SG counterparts were discussed.

## 2. Materials and Methods

#### 2.1. Raw Materials

_{2}to Na

_{2}O) of sodium silicate solution were 59% (by mass) and 3.7, respectively; and the sodium hydroxide (NaOH) flakes had a purity of 99%. A Grade 42.5 PC whose chemical composition is listed in Table 1 was adopted. The fine aggregate used was medium river sand with a fineness modulus of 2.47. The specific density and the water absorption of the fine aggregate were 2340 kg/m

^{3}and 2.75%, respectively. Furthermore, gravel particles from a local river with a maximum size of 10 mm were selected as coarse aggregate. The bulk specific density and the water absorption of the coarse aggregate were 2530 kg/m

^{3}and 1.83%, respectively.

#### 2.2. Mix Proportions

_{s}). All the concrete specimens had a constant sand ratio (SR) of 0.4, which was equal to the amount of fine aggregate per unit volume to that of the sum of fine aggregate and coarse aggregate.

#### 2.3. Specimen Preparation

#### 2.4. Testing Procedure

#### 2.4.1. Compressive and Splitting Tensile Strengths

_{t}was calculated using the following equation:

^{2}).

#### 2.4.2. Three-Point Bending (TPB) Tests

## 3. Experimental Results

_{u}, the initial cracking load P

_{ini}, the $CMO{D}_{c}$ and $CTO{D}_{c}$ at ultimate load P

_{u}, the modulus of elasticity E calculated from the P-CMOD curves [11], and the fracture energy G

_{F}calculated from the P-d curves [21] are summarized in Table 4, in which the average values of four identical specimens are provided. These parameters are also essential for calculating the fracture toughness in the following sessions. The initial cracking load P

_{ini}was determined using a graphical method in this study, referring to the load value where non-linearity started on the P-d curves [11]. It can be concluded from Table 4 that the initial cracking loads of concrete were around 50–65% of their ultimate loads, whereas the initial cracking loads of mortar were approximately 80–95% of their ultimate loads. Concrete usually has a higher load resistance than mortar does after the crack is formed.

_{u}of concrete and mortar beams increased with increasing compressive strength for both the SG and PC series as expected. The improvement of the average ultimate load P

_{u}with the compressive strength increasing from C30 to C70 was more significant of the PCC beams than that of the SGC specimens. In the former case, P

_{u}increased from 2.39 kN to 3.56 kN with a 49.6% increase, whereas only a 14.7% increase was observed in the latter case. Comparing the ultimate loads P

_{u}between the PCC beams and the SGC beams, it is seen that, in the case of the C30 strength grade, the average ultimate load P

_{u}of the PCC beams was 20.4% lower than its SGC counterpart. Nevertheless, with increasing compressive strength, the ultimate load P

_{u}of the SGC beams became close to that of the PCC beams. It is known that the interfacial transition zones (ITZs) between the aggregates and the matrix are generally the weakest parts in low strength concrete. As a result, cracks prefer to occur in the ITZs and thus a stronger ITZ could lead to a higher ultimate load. The microscopic observations conducted by previous researchers [24,25] revealed that the ITZs between the SG paste and aggregates are denser and more homogenous than those between PC and aggregates. This explains the higher ultimate load of SG at C30. However, with a further increase of compressive strength, cracks may pass through the aggregates directly, so that the ITZ may exhibit comparable strength with the aggregate and trans-granular fracture happens. Moreover, the ultimate loads P

_{u}of the PCM beams were always slightly higher than those of the SGM beams given the similar compressive strength grade. This could be explained by the high shrinkage of SGM [26,27,28], which would lead to more intrinsic micro-cracks and reduce the load-bearing capacity.

## 4. Determination of Softening Laws

_{t}, the kink points (${\sigma}_{s}$, ${w}_{s}$), and the crack width ${w}_{0}$ that corresponds to zero cohesive stress. Roelfstra and Wittmann [35] emphasized that for a simulation of the whole load–displacement curves of TPB tests, the most critical step is to determine the kink point of the bilinear softening law.

## 5. Determination of Fracture Toughness

#### 5.1. Experimental Approach

#### 5.2. Analytical Approach

_{c}can be measured directly by using the clip gauge holder. The detailed values are listed in Table 4.

#### 5.3. Results and Discussion

#### 5.3.1. Initial and Unstable Fracture Toughness Values (${K}_{Ic}^{ini}$ and ${K}_{Ic}^{un}$)

^{1/2}to 0.538 MPa·m

^{1/2}(i.e., a 56.8% increase) and 1.238 MPa·m

^{1/2}to 1.960 MPa·m

^{1/2}(i.e., a 58.3% increase), respectively, with the increase of compressive strength from 30 MPa to 70 MPa. In the case of SGC, the increases of ${K}_{Ic}^{ini}$ and ${K}_{Ic}^{un}$ were not as significant as those of their PC counterparts; these values increased from 0.403 MPa·m

^{1/2}to 0.432 MPa·m

^{1/2}(i.e., a 7.2% increase) and 1.586 MPa·m

^{1/2}to 1.793 MPa·m

^{1/2}(i.e., a 12.6% increase), respectively (see Figure 5a). In addition, the increases of both ${K}_{Ic}^{ini}$ and ${K}_{Ic}^{un}$ of PCM and SGM with compressive strength were less significant compared with those of concrete beams (Figure 5b).

^{1/2}, which was 21.9% lower than that of SGC at 1.586 MPa·m

^{1/2}. However, with increasing compressive strength, the difference of average ${K}_{Ic}^{un}$ between PCC and SGC was reduced. According to Equation (4), ${K}_{Ic}^{un}$ is proportional to ${P}_{\mathrm{u}}$, $\sqrt{{a}_{\mathrm{c}}}$, and ${F}_{1}\left(\raisebox{1ex}{${a}_{\mathrm{c}}$}\!\left/ \!\raisebox{-1ex}{$H$}\right.\right)$. The relationship of ${P}_{\mathrm{u}}$ between PCC and SGC with the increase of compressive strength discussed above is consistent with the trend of ${K}_{Ic}^{un}$. In addition, it can be found from Table 4 that the average critical crack length ${a}_{\mathrm{c}}$ of SGC at C30 was 63.65 mm, which is larger than that of PCC at 60.82 mm, and such values became closer with the increase of the compressive strength for both cases. Furthermore, ${F}_{1}\left(\raisebox{1ex}{${a}_{\mathrm{c}}$}\!\left/ \!\raisebox{-1ex}{$H$}\right.\right)$ is also a monotonic increasing function of the variable $\raisebox{1ex}{${a}_{\mathrm{c}}$}\!\left/ \!\raisebox{-1ex}{$H$}\right.$. Hence, higher ${P}_{\mathrm{u}}$ and ${a}_{\mathrm{c}}$ must lead to higher ${K}_{Ic}^{un}$. Regarding ${K}_{Ic}^{ini}$, Figure 5a shows that ${K}_{Ic}^{ini}$ of PCC was generally slightly higher than that of SGC. Based on Equation (3), the initial cracking load ${P}_{\mathrm{ini}}$ is the only variable that is proportional to the initial fracture toughness ${K}_{Ic}^{ini}$. Referring to Table 4, the relationship of the initial cracking load ${P}_{\mathrm{ini}}$ between SGC and PCC is consistent with the trend of initiation fracture toughness.

^{1/2}and 1.174 MPa·m

^{1/2}, which were 18.3% and 11.5%, respectively, higher than those of SGM. In contrast, given the compressive strength of M70, the average initial and unstable toughness values of PCM were significantly higher than those of SGM (i.e., 20.2% and 37.3%, respectively). The aggravation of the discrepancy of the initial and unstable fracture toughness values between SGM and PCM with the increase of strength is attributed to the utilization of a higher alkali concentration activator of SGM that generated more serious shrinkage cracks [26,28], leading to a more brittle matrix and a relatively low load resistance in the case of high strength grade.

#### 5.3.2. Cohesive Fracture Toughness

## 6. Conclusions

- For both the PC and SG series, the values of ${w}_{s}$ at kink point and ${w}_{0}$ at the stress-free point of the bilinear softening law decrease, whereas the values of ${\sigma}_{s}$ at the kink point generally increase with the compressive strength.
- The first descending slopes of the normalized bilinear softening curves of PCC and SGC are generally the same, whereas PCM has a gentler first descending branch than its SGM counterpart.
- The ${K}_{Ic}^{ini}$ and ${K}_{Ic}^{un}$ of the PC and SG concrete and mortar all increase with compressive strength increase. Moreover, both ${K}_{Ic}^{ini}$ and ${K}_{Ic}^{un}$ of SGM are lower than those of PCM given the same compressive strength.
- The ${K}_{Ic}^{ini}$ of SGC is generally lower than that of PCC except for C30. Moreover, the ${K}_{Ic}^{un}$ of SGC at C30 is significantly higher than that of PCC and then becomes similar with increasing compressive strength.
- The variation of ${K}_{Ic}^{c}$ of the PC and SG series with increasing compressive strength is similar to that of unstable fracture toughness. The ${K}_{Ic}^{c}$ calculated by analytical approach and experimental approach is similar, which also proves the correctness of the bilinear softening laws obtained by inverse analysis and the applicability of the double-K fracture model to SG concrete and mortar.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

a_{0} | initial crack length |

a_{c} | critical crack length |

B | specimen thickness |

C_{i} | initial compliance of load–CMOD curve |

$CMOD$ | crack mouth opening displacement |

$CMO{D}_{c}$ | critical crack mouth opening displacement |

$CTOD$ | crack tip opening displacement |

$CTO{D}_{c}$ | critical crack tip opening displacement |

d | mid-span deflection |

$E$ | modulus of elasticity |

f_{t} | splitting tensile strength |

H | depth of the specimen |

${H}_{0}$ | thickness of clip gauge holder |

${K}_{Ic}^{ini}$ | initial fracture toughness |

${K}_{Ic}^{un}$ | unstable fracture toughness |

${K}_{Ic}^{c}$ | cohesive fracture toughness |

${K}_{Ic}^{c,E}$ | cohesive fracture toughness by experiment |

${K}_{Ic}^{c,A}$ | cohesive fracture toughness by analytical method |

P_{u} | maximum load |

P_{ini} | initial cracking load |

S | span of the specimen |

${\sigma}_{s}$ | cohesive stress corresponding to the kink point of bilinear softening law |

$\sigma \left(x\right)$ | cohesive stress corresponding to crack length x |

$\sigma \left(CTO{D}_{c}\right)$ | critical value of cohesive force at notch tip |

${w}_{s}$ | crack width corresponding to the kink point of bilinear softening law |

${w}_{0}$ | crack width corresponding to the stress-free point |

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**Figure 1.**Configuration of the three-point bending (TPB) test beams [20].

**Figure 2.**Typical (

**a**) load vs. mid-span displacement (P-d) and (

**b**) load vs. crack mouth opening displacement (P-CMOD) curves of Portland cement (PC) and slag-based geopolymer (SG) concrete beams.

**Figure 3.**Bilinear softening laws of the PC and SG concrete and mortar. (

**a**) PCC and SGC and (

**b**) PCM and SGM.

**Figure 4.**P-d curves obtained from experiment and prediction (

**a**) PC mortar and concrete and (

**b**) SG mortar and concrete.

**Figure 5.**The ${K}_{Ic}^{ini}$, ${K}_{Ic}^{c}$, and ${K}_{Ic}^{un}$ of (

**a**) PC and SG concrete and (

**b**) PC and SG mortar.

**Table 1.**Chemical composition of the ground granulated blast furnace slag (GGBFS) and the Portland cement (PC) (wt %) [12].

CaO | Al_{2}O_{3} | SiO_{2} | SO_{3} | P_{2}O_{5} | MgO | Na_{2}O | K_{2}O | TiO_{2} | |
---|---|---|---|---|---|---|---|---|---|

GGBFS | 33.3 | 16.9 | 33.4 | 2.35 | 3.77 | 7.0 | 2.0 | 0.16 | 0.61 |

PC | 64.5 | 5.30 | 21.9 | 2.03 | – | 1.51 | 0.19 | 0.62 | – |

**Table 2.**Mix proportions of the Portland cement mortar (PCM) and the Portland cement concrete (PCC) [12].

Cement (kg/m ^{3}) | Fine Aggregate (kg/m ^{3}) | Coarse Aggregate (kg/m ^{3}) | Water (kg/m ^{3}) | w/c | SP | SR | |
---|---|---|---|---|---|---|---|

PCM30 | 600 | 1200 | – | 300 | 0.5 | – | – |

PCM50 | 700 | 1155 | – | 245 | 0.35 | 0.09% | – |

PCM70 | 850 | 1010 | – | 240 | 0.3 | 0.16% | – |

PCC30 | 350 | 776 | 1164 | 210 | 0.6 | – | 0.4 |

PCC50 | 380 | 795 | 1192 | 133 | 0.35 | 0.42% | 0.4 |

PCC70 | 420 | 782 | 1172 | 126 | 0.3 | 0.50% | 0.4 |

**Table 3.**Mix proportions of the slag-based geopolymer mortar (SGM) and the slag-based geopolymer cement (SGC) [12].

n (%) | M_{s} | Slag kg/m ^{3} | Fine Aggregate kg/m ^{3} | Coarse Aggregate kg/m ^{3} | Water kg/m ^{3} | Alkali Activator | w/b | SR | ||
---|---|---|---|---|---|---|---|---|---|---|

Sodium Silicate Solution (kg/m ^{3}) | Sodium Hydroxide (kg/m ^{3}) | |||||||||

SGM30 | 3 | 1.5 | 783 | 1174 | – | 276 | 109 | 18 | 0.44 | – |

SGM50 | 4 | 1.5 | 783 | 1174 | – | 253 | 145 | 24 | 0.44 | – |

SGM70 | 5 | 1.5 | 783 | 1174 | – | 254 | 182 | 30 | 0.44 | – |

SGC30 | 3 | 1.5 | 350 | 746 | 1120 | 127 | 49 | 8 | 0.45 | 0.4 |

SGC50 | 4 | 1.5 | 380 | 724 | 1087 | 127 | 71 | 11 | 0.45 | 0.4 |

SGC70 | 4.5 | 2 | 420 | 694 | 1041 | 117 | 117 | 11 | 0.45 | 0.4 |

_{2}O by mass of GGBFS, M

_{s}is modulus of the alkali activator referring to the mole ratio of SiO

_{2}to Na

_{2}O, and w/b is water/slag ratio, here the total water included the water added and the water in sodium silicate solution.

P_{u}(kN) | P_{ini}(kN) | P_{ini}/P_{u} | CMOD_{c}(μm) | CTOD_{c}(μm) | G_{F}(N/m) | E (GPa) | a_{c}(mm) | |
---|---|---|---|---|---|---|---|---|

PCC30 | 2.39 | 1.37 | 57.1% | 48.07 | 20.52 | 127.1 | 23.4 | 60.82 |

PCC50 | 3.24 | 1.99 | 61.2% | 57.25 | 28.41 | 173.8 | 26.6 | 62.28 |

PCC70 | 3.56 | 2.96 | 83.0% | 62.01 | 30.78 | 177.2 | 29.2 | 63.52 |

SGC30 | 2.99 | 1.54 | 51.6% | 68.49 | 32.50 | 177.3 | 22.1 | 63.65 |

SGC50 | 3.40 | 1.70 | 49.8% | 72.85 | 32.95 | 183.4 | 24.2 | 63.20 |

SGC70 | 3.43 | 1.73 | 50.3% | 67.13 | 29.39 | 207.9 | 24.9 | 62.31 |

PCM30 | 1.94 | 1.79 | 92.0% | 78.43 | 28.61 | 101.0 | 14.4 | 65.91 |

PCM50 | 2.21 | 2.02 | 91.5% | 82.76 | 35.24 | 120.8 | 16.1 | 65.88 |

PCM70 | 2.41 | 2.25 | 93.4% | 72.02 | 29.88 | 119.1 | 18.5 | 64.71 |

SGM30 | 1.79 | 1.55 | 86.4% | 83.05 | 37.75 | 125.9 | 11.3 | 64.16 |

SGM50 | 2.13 | 1.82 | 85.4% | 75.15 | 34.35 | 99.5 | 12.8 | 62.55 |

SGM70 | 2.14 | 1.88 | 87.7% | 67.81 | 30.38 | 91.9 | 13.7 | 61.09 |

f_{t} | ω_{0} | σ_{s} | ω_{s} | f_{t} | ω_{0} | σ_{s} | ω_{s} | ||
---|---|---|---|---|---|---|---|---|---|

PCC30 | 3.20 | 0.269 | 0.269 | 0.0468 | SGC30 | 3.56 | 0.284 | 0.245 | 0.0665 |

PCC50 | 4.58 | 0.245 | 0.656 | 0.0378 | SGC50 | 4.39 | 0.278 | 0.555 | 0.0416 |

PCC70 | 5.29 | 0.227 | 0.762 | 0.0290 | SGC70 | 5.15 | 0.276 | 0.709 | 0.0346 |

PCM30 | 2.16 | 0.0754 | 0.513 | 0.0679 | SGM30 | 2.06 | 0.163 | 0.564 | 0.0624 |

PCM50 | 3.40 | 0.0737 | 0.954 | 0.0429 | SGM50 | 3.45 | 0.123 | 0.866 | 0.0213 |

PCM70 | 3.83 | 0.0654 | 0.978 | 0.0404 | SGM70 | 4.27 | 0.111 | 0.960 | 0.0130 |

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**MDPI and ACS Style**

Ding, Y.; Bai, Y.-L.; Dai, J.-G.; Shi, C.-J.
An Investigation of Softening Laws and Fracture Toughness of Slag-Based Geopolymer Concrete and Mortar. *Materials* **2020**, *13*, 5200.
https://doi.org/10.3390/ma13225200

**AMA Style**

Ding Y, Bai Y-L, Dai J-G, Shi C-J.
An Investigation of Softening Laws and Fracture Toughness of Slag-Based Geopolymer Concrete and Mortar. *Materials*. 2020; 13(22):5200.
https://doi.org/10.3390/ma13225200

**Chicago/Turabian Style**

Ding, Yao, Yu-Lei Bai, Jian-Guo Dai, and Cai-Jun Shi.
2020. "An Investigation of Softening Laws and Fracture Toughness of Slag-Based Geopolymer Concrete and Mortar" *Materials* 13, no. 22: 5200.
https://doi.org/10.3390/ma13225200