# Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced Concrete

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}footprint [12]. It is well-known how HPFRC is a composite material in which the advantages of fibre-reinforced concrete (FRC) are combined with those of a high-performance concrete (HPC), reducing the weaknesses of conventional concrete and improving its durability and mechanical performance. The addition of discontinuous fibres (e.g., steel fibres [13,14,15,16], synthetic fibres [17,18,19,20,21], natural fibres [22,23,24], basalt [25,26]), carbon and glass fibres [27]) in the HPC as well as in the concrete in general is able to significantly reduce its brittle behaviour, thus improving cracking, postcracking strength, and toughness [28,29,30] as well as its durability such as freeze–thaw resistance. Thanks to its dense microstructure, high-performance concrete also has a low permeability, resulting in a good resistance to various external agents such as chloride attacks [31] and carbonation [32] as well as freezing and thawing cycles [33,34,35,36]. Its good antifreezing property makes it suitable for application at both high altitudes and in northern areas where cyclic freeze–thaw conditions are one of the main causes of two types of concrete degradation: surface scaling, which is the loss of cement paste from the exposed surface, and internal crack growth, which makes the concrete crumble and deteriorate. Both phenomena can reduce the quality of concrete throughout its lifetime. Over the last few years, the research on evaluating the freezing resistance of HPFRC has significantly increased, with several relevant achievements having been obtained: in Feo et al. [37], the effects of 75 freeze and thaw cycles on both the dynamic moduli of elasticity, cracking and postcracking strength, as well as the toughness of HPFRC beam specimens reinforced with steel fibres were evaluated; in [38], it was reported how the incorporation of basalt fibres can reduce the influence of freeze–thaw on the damage and failure process of the beam specimen under a bending test; in [39], it was studied how mineral admixtures (e.g., blast furnace slag, fly ash, silica fume, and metakaolin) contained in the HPC matrix possess an excellent frost resistance; in [40], an experimental investigation on the freeze–thaw resistance of HPC containing air-cooled slag (AS) and water-cooled slag (WS) was conducted; in [41], it was explained how adding nanosilica to the concrete makes it durable by enhancing its properties such as impermeability, porosity, and acid resistance. However, based on our knowledge, such studies are experimental and not many predictive models have been proposed that capture the mechanical response of HPFRC, especially under freeze and thaw cycles.

## 2. Outline of the Experimental Results

_{f}/d

_{f}, equal to 65 were chosen as the reinforcement of the HPC matrix whose mix design was provided by the manufacturer [50]. For each type of HPFRC mixture, eight standard 150 × 150 × 600 mm prismatic specimens (PS) and five standard 150 × 150 × 150 mm cubic samples (CS) were cast. At the end of the curing period, the prismatic specimens for each HPFRC mixture were subjected to 75 freeze–thaw cycles according to UNI 7087-2017 [51].

_{avg}) were monitored during each test. All the cube samples were only tested in compression to evaluate, according to EN 12390-4 [53], the compressive load, F

_{u}, as well as the compressive strength, f

_{c}, for any changes of the fibres’ volume fraction.

_{lf}, of the first crack strength, f

_{(lf,avg)}, and the equivalent postcracking strengths, f

_{(eq(0–0.6),avg)}and f

_{(eq(0.6–3),avg)}, before and after the freeze–thaw cycles for each type of HPFRC mixture.

_{(1,avg)}and U

_{(2,avg)}, and the ductility indices, D

_{(0,avg)}and D

_{(1,avg)}, before and after the freeze–thaw cycles for the two types of HPFRC mixture (CM1 and CM2).

## 3. Theoretical Model

#### 3.1. Assumptions and Formulation

_{0}(Figure 1).

_{(f,k)}, y

_{(f,k)}, and z

_{(f,k)}(with 𝑘 between 1 and n

_{f}), and α

_{(y,k)}and α

_{(z,k)}are the three coordinates of the fibre centroid (G

_{(f,k)}) and the two relevant angles, respectively. In order to consider the bridging effect offered by the fibres, the total number of fibres in the midspan section, n

_{f}, was determined as a ratio between the fibres’ volume fraction, V

_{f}, and the cement matrix volume, V

_{c}.

- (i)
- The flexibility was distributed in the central part of the specimen for a length equal to “$s$” while a rigid body behaviour was exhibited by the remaining end parts (Figure 2).
- (ii)
- The midspan cross-section was discretized in ${n}_{c}$ layers as shown in Figure 3. The average axial strain of the k-th layer, ${\epsilon}_{k}$, before crack formation, and the crack-opening displacement, ${w}_{k}$, after the crack formation, can be easily expressed for the k-th layer (k = 1, …, ${n}_{c}$) as in Equations (1) and (2).

- (iii)
- Consequently, the average value of the axial stress, ${\sigma}_{c,k}$, at k-th strip can be determined as a function of the axial deformation, ${\epsilon}_{k}$, before cracking, or a function of the crack-opening displacements, ${w}_{k}$, after cracking. The stress–strain and stress–displacement relationships assumed in this paper are reported in Section 3.2.2.
- (iv)
- A transition length, ${l}_{t}$, was introduced in the notched cross-section (Figure 4), which starts from the top of the notch to the top of the integral part of the section, in order to consider the possible microdamage phenomena produced by the notching process. The mechanical meaning of this quantity is discussed in details in a previous paper [42] and omitted herein for the sake of brevity. Therefore, a reduced value of the width, ${b}_{k}$, inside the transition zone was considered which can be evaluated with an exponential law as in Equation (3) where ${l}_{k}$ and $\alpha $ are the distance of the k-th strip from the top of the notch and the coefficient of the exponential law, respectively:

- (v)
- The bridging effect offered by the fibres was taken in to account by introducing the action, ${F}_{k,j}$, mobilised at the j-th step of the incremental analysis as in Equation (4):

_{c,j}of the neutral axis for the j-th imposed rotation φ

_{j}can be determined by solving the following equilibrium equation along the longitudinal axis which can be written as follows:

_{c}is the number of layers into which the midspan section is discretized (Figure 3) and n

_{t}is the number of layers of reduced width [42].

_{0}(Figure 3).

_{j}can be obtained as follows:

_{j}value of can be obtained from Equation (2) by just replacing the generic value of z

_{k}with the position of the crack tip (z

_{k}= −h/2 + h

_{0}from Figue 2).

_{j}, a couple (CTOD

_{j}, P

_{j}) can be determined, and then, the Force-CTOD graph can be incrementally determined up to failure.

#### 3.2. Constitutive Laws Assumed in the Present Study

#### 3.2.1. Stress–Strain Relationships for Concrete in Compression and in Tension

#### 3.2.2. Modified Bond-Slip Model for Short Steel Fibres

- a linear-elastic behaviour up to the stress level corresponding to matrix tensile strength, identified by the two parameters ${s}_{el}$, ${\tau}_{el}$;
- a hardening behaviour, characterized by the formation of many microcracks in the HPFRC mix, identified by the two parameters ${s}_{R}$ and ${\tau}_{R}$;
- a constant behaviour defined by the two parameters ${s}_{u}$ and ${\tau}_{u}$.

## 4. Inverse Identification of the Relevant Material Laws

_{el}< s

_{R}< s

_{u}, τ

_{el}< τ

_{R}and τ

_{R}= τ

_{u}) intended at respecting the mechanical consistency of the model. In order to evaluate how the values of these parameters depend on the effect of freeze–thaw cycles, several numerical simulations were carried out. In particular, three groups of 100 simulations each, assuming ${n}_{c}=50$ and $s$ = 300 mm, were run as described below.

- In the first one, the cylindrical compression strength, ${f}_{cm}$, the transition length ${l}_{t}$, and the exponential parameter, $\alpha $, were calibrated on the flexural response of the conditioned CM0 specimens (labelled CM0-FT). Experimentally, a 21% reduction in the cylindrical compression strength, ${f}_{cm}$, was observed on the conditioned specimens compared to not conditioned ones. This reduction was taken in account to calibrate the value of the transition length, ${l}_{t}$, whose value, in the present model, was assumed equal to 85 mm (with an increase of 21% compared to that used in the previous model [42] in which the flexural behaviour of unconditioned CM0 specimens (labeled CM0-NFT) was predicted with a transition length, ${l}_{t}$, equal to 70 mm). In both models, the coefficient of the exponential law, $\alpha $, was considered constant and equal to 0.40 (Table 3). Figure 7 shows both the average experimental $P-CTO{D}_{,avg}$ curve (light-blue line) and the average numerical $P-CTO{D}_{,avg}$ curve (pink line) obtained with the present model employed for the CM0-FT specimens.
- In the second one, the six parameters of the bond-slip law (i.e., ${s}_{el},{s}_{R},{s}_{u},{\tau}_{el},{\tau}_{R},{\tau}_{u}$) were calibrated on the flexural response of conditioned CM1 specimens (labelled CM1-FT). A 13% reduction in the parameter ${\tau}_{el}$ was adopted in the calibration of the conditioned specimens compared to the unconditioned ones.
- In the last one, only the parameter, ${\tau}_{el}$, was calibrated again on the flexural response of conditioned CM2 specimens (labelled CM2-FT) while all the other parameters were considered constant. A 19% reduction in the parameter ${\tau}_{el}$ was adopted for the conditioned specimens compared to the unconditioned ones. Moreover, as in [42], a 20% reduction in the fibres’ volume fraction, ${V}_{f}$, was considered in order to take into account the nonuniform fibre distribution.

## 5. Results

## 6. Conclusions

- The freeze–thaw cycles effect the cylindrical compression strength, ${f}_{cm}$, the transition length ${l}_{t}$, and the bond-slip law of fibres, which confirms their significance as relevant parameters controlling the resulting response of HPFRC specimens;
- Table 3 shows that the compressive strength f
_{cm}undergoes a substantial reduction (in the order of 20%) as a result of the degradation processes induced by the FT cycles; - as for the transition zone, which is a peculiar aspect of the considered model, a moderate increase in its the depth (from 70 mm to 85 mm) can be identified after the FT cycles, whereas its shape (controlled by the exponent α) does not change;
- Table 4 points out that the bond-slip law of fibres is also affected by the FT cycles, as, specifically, the elastic limit stress, τ
_{el}, (and, consequently, the initial elastic stiffness of the same law) reduces by about 15%, with no changes in the other parameters; - however, under the designers’ standpoint (and besides the specific values obtained in the present study), it should be noted that this change affects both serviceability and ultimate limit states in the structural response.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic representation of the 3D HPFRC beam: adapted from [43].

**Figure 2.**Kinematics of the cracked-hinge model: adapted from [43].

**Figure 4.**Notched cross-section with exponential law of the reduced width, ${b}_{k}$, in the transition zone, ${l}_{t}$: (

**a**) linear expression ($\alpha $ = 1) of ${b}_{k}$, (

**b**) exponential expression with $\alpha >1$, (

**c**) exponential expression with $\alpha <1$.

**Figure 6.**Schematic graph of the stress–strain relation for uniaxial tension: (

**a**) for ${\epsilon}_{k}\le 0.00015$, the stress–strain behaviour is described by a bilinear relation; (

**b**) for ${\epsilon}_{k}>0.00015$, the stress–strain behaviour is described by a softening constitutive stress–crack opening law.

**Figure 7.**The average experimental $P-CTO{D}_{,avg}$ curve (light-blue line) versus the average numerical $P-CTO{D}_{,avg}$ curve (pink line) for conditioned CM0 specimens (labelled CM0-FT) obtained with the present model.

**Figure 9.**(

**a**). The average experimental $P-CTO{D}_{,avg}$ curve (violet line) versus the average numerical $P-CTO{D}_{,avg}$ curve (green line) for unconditioned CM1 specimens (labelled CM1-NFT) obtained with the previous model of Ref. [42]. (

**b**). The average experimental $P-CTO{D}_{,avg}$ curve (violet line) versus the average numerical $P-CTO{D}_{,avg}$ curve (green line) for unconditioned CM2 specimens (labelled CM2-NFT) obtained with the previous model of Ref. [42].

**Figure 10.**(

**a**). The average experimental $P-CTO{D}_{,avg}$ curve (light-blue line) versus the average numerical $P-CTO{D}_{,avg}$ curve (pink line) for conditioned CM1 specimens (labelled CM1-FT) obtained with the present model. (

**b**). The average experimental $P-CTO{D}_{,avg}$ curve (light-blue line) versus the average numerical $P-CTO{D}_{,avg}$ curve (pink line) for conditioned CM2 specimens (labelled CM2-FT) obtained with the present model.

**Figure 11.**Correlation between the experimental and theoretical average values of the equivalent postcracking strengths after the freeze–thaw cycles, ${f}_{eq\left(0\u20130.6\right),avg}^{FT}$ and before the freeze–thaw cycles of Ref. [42], ${f}_{eq\left(0\u20130.6\right),avg}^{NFT}$ for the CM1 and CM2 mixtures.

**Figure 12.**Correlation between the experimental and theoretical average values of the equivalent postcracking strengths after the freeze–thaw cycles, ${f}_{eq\left(0.6\u20133\right),avg}^{FT}$, and before the freeze–thaw cycles of Ref. [42], ${f}_{eq\left(0.6\u20133\right),avg}^{NFT}$, for the CM1 and CM2 mixtures.

**Figure 13.**Correlation between the experimental and theoretical average values of the working capacity indices after the freeze–thaw cycles, ${U}_{1,avg}^{FT}$, and before the freeze–thaw cycles of Ref. [42], ${U}_{1,avg}^{NFT}$, for the CM1 and CM2 mixtures.

**Figure 14.**Correlation between the experimental and theoretical average values of the working capacity indices after the freeze–thaw cycles, ${U}_{2,avg}^{FT}$, and those before the freeze–thaw cycles of Ref. [42], ${U}_{2,avg}^{NFT}$, for the CM1 and CM2 mixtures.

**Table 1.**The average values of the first crack load, P

_{lf}, of the first crack strengths, f

_{(If,avg)}, and of the equivalent postcracking strengths, f

_{(eq(0–0.6),avg)}and f

_{(eq(0.6–3),avg)}, before (NFT) and after the freeze–thaw cycles (FT) for each type of HPFRC mixture (CM0, CM1, and CM2).

Mix. | ${P}_{If,avg}^{NFT}$ | ${P}_{If,avg}^{FT}$ | ${f}_{If,avg}^{NFT}$ | ${f}_{If,avg}^{FT}$ | ${f}_{eq\left(0\u20130.6\right),avg}^{NFT}$ | ${f}_{eq\left(0-0.6\right),avg}^{FT}$ | ${f}_{eq\left(0.6\u20133\right),avg}^{NFT}$ | ${f}_{eq\left(0.6\u20133\right),avg}^{FT}$ |
---|---|---|---|---|---|---|---|---|

[kN] | [kN] | [MPa] | [MPa] | [MPa] | [MPa] | [MPa] | [MPa] | |

CM0 | 11.213 | 9.105 | 3.05 | 2.477 | - | - | - | - |

CM1 | 14.489 | 12.538 | 4.013 | 3.475 | 6.617 | 5.435 | 7.99 | 6.845 |

CM2 | 18.595 | 16.175 | 5.06 | 4.327 | 9.15 | 7.537 | 11.473 | 9.255 |

**Table 2.**The average values of the work capacity indices, U

_{(1,avg)}and U

_{(2,avg)}, and ductility indices, D

_{(0,avg)}and D

_{(1,avg)}, before (NFT) and after the freeze–thaw cycles (FT) for the two types of HPFRC mixture (CM1 and CM2).

Mix. | ${U}_{1,avg}^{NFT}$ | ${U}_{1,avg}^{FT}$ | ${U}_{2,avg}^{NFT}$ | ${U}_{2,avg}^{FT}$ | ${D}_{0,avg}^{NFT}$ | ${D}_{0,avg}^{FT}$ | ${D}_{1,avg}^{NFT}$ | ${D}_{1,avg}^{FT}$ |
---|---|---|---|---|---|---|---|---|

[kNmm] | [kNmm] | [kNmm] | [kNmm] | [-] | [-] | [-] | [-] | |

CM1 | 14,283.43 | 11,742.40 | 69,226.73 | 59,175.15 | 1.647 | 1.565 | 1.265 | 1.265 |

CM2 | 20,508.47 | 16,895.83 | 102,877.93 | 82,992.00 | 1.837 | 1.747 | 1.257 | 1.250 |

**Table 3.**Calibration of the input data for unconditioned CM0 specimens (labelled CM0-NFT) and for conditioned CM0 specimens (labelled CM0-FT) used in the previous model of Ref. [42] and in the last one, respectively.

Specimen Designation | ${f}_{cm}$ | ${l}_{t}$ | $\alpha $ | Model |
---|---|---|---|---|

[MPa] | [mm] | [-] | ||

CM0-NFT | 53.0 | 70.0 | 0.4 | Ref. [42] |

CM0-FT | 42.0 | 85.0 | 0.4 | Present paper |

**Table 4.**Calibration of the six parameters in the local bond-slip law for unconditioned CM1 and CM2 specimens (labelled, respectively, CM1-NFT and CM2-NFT) as well as for conditioned CM1 and CM2 specimens (labelled, respectively, CM1-FT and CM2-FT) used in the previous model of Ref. [42] and in the last one, respectively.

Series | ${s}_{el}$ | ${s}_{R}$ | ${s}_{u}$ | ${\tau}_{el}$ | ${\tau}_{R}$ | ${\tau}_{u}$ | Model |
---|---|---|---|---|---|---|---|

[mm] | [mm] | [mm] | [MPa] | [MPa] | [MPa] | ||

CM1-NFT | 0.10 | 8.00 | 10.00 | 8.00 | 21.50 | 21.50 | Ref. [42] |

CM1-FT | 0.10 | 8.00 | 10.00 | 7.00 | 21.50 | 21.50 | Present paper |

CM2-NFT | 0.10 | 8.00 | 10.00 | 8.00 | 21.50 | 21.50 | Ref. [42] |

CM2-FT | 0.10 | 8.00 | 10.00 | 6.50 | 21.50 | 21.50 | Present paper |

**Table 5.**Comparison between the experimental and theoretical average values of the two equivalent postcracking strengths after the freeze–thaw cycles, ${f}_{eq\left(0\u20130.6\right),avg}^{FT}$ and ${f}_{eq\left(0.6\u20133\right),avg}^{FT}$, and those before the freeze–thaw cycles of Ref. [42], ${f}_{eq\left(0\u20130.6\right),avg}^{NFT}$ and ${f}_{eq\left(0.6\u20133\right),avg}^{NFT}$, for the CM1 and CM2 mixtures.

Results | CM1 | CM2 | ||||||
---|---|---|---|---|---|---|---|---|

${f}_{eq\left(0\u20130.6\right),avg}^{NFT}$ | ${f}_{eq\left(0\u20130.6\right),avg}^{FT}$ | ${f}_{eq\left(0.6\u20133\right),avg}^{NFT}$ | ${f}_{eq\left(0.6\u20133\right),avg}^{FT}$ | ${f}_{eq\left(0\u20130.6\right),avg}^{NFT}$ | ${f}_{eq\left(0\u20130.6\right),avg}^{FT}$ | ${f}_{eq\left(0.6\u20133\right),avg}^{NFT}$ | ${f}_{eq\left(0.6\u20133\right),avg}^{FT}$ | |

[MPa] | [MPa] | [MPa] | [MPa] | [MPa] | [MPa] | [MPa] | [MPa] | |

Experimental | 6.617 | 5.435 | 7.990 | 6.845 | 9.150 | 7.537 | 11.473 | 9.255 |

Theoretical | 6.179 | 5.411 | 7.672 | 6.871 | 8.486 | 7.125 | 11.532 | 9.871 |

Percentage difference (%) | 6.61 | 0.45 | 3.98 | 0.39 | 7.26 | 5.47 | 0.51 | 6.66 |

**Table 6.**Comparison between the experimental and theoretical average values of the two working capacity indices after the freeze–thaw cycles, ${U}_{1,avg}^{FT}$ and ${U}_{2,avg}^{FT}$, and those before the freeze–thaw cycles of Ref. [42], ${U}_{1,avg}^{NFT}$ and ${U}_{2,avg}^{NFT}$, for the CM1 and CM2 mixtures.

Results | CM1 | CM2 | ||||||
---|---|---|---|---|---|---|---|---|

${U}_{1,avg}^{NFT}$ | ${U}_{1,avg}^{FT}$ | ${U}_{2,avg}^{NFT}$ | ${U}_{2,avg}^{FT}$ | ${U}_{1,avg}^{NFT}$ | ${U}_{1,avg}^{FT}$ | ${U}_{2,avg}^{NFT}$ | ${U}_{2,avg}^{FT}$ | |

[kNmm] | [kNmm] | [kNmm] | [kNmm] | [kNmm] | [kNmm] | [kNmm] | [kNmm] | |

Experimental | 14,283.43 | 11,742.40 | 69,226.73 | 59,175.15 | 20,508.47 | 16,895.83 | 102,877.93 | 82,992.00 |

Theoretical | 13,625.51 | 11,930.62 | 67,667.93 | 60,606.61 | 18,711.24 | 15,709.63 | 101,710.39 | 87,061.56 |

Percentage difference (%) | 4.61 | 1.60 | 2.25 | 2.42 | 8.76 | 7.02 | 1.13 | 4.90 |

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**MDPI and ACS Style**

Penna, R.; Feo, L.; Martinelli, E.; Pepe, M.
Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced Concrete. *Materials* **2022**, *15*, 6122.
https://doi.org/10.3390/ma15176122

**AMA Style**

Penna R, Feo L, Martinelli E, Pepe M.
Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced Concrete. *Materials*. 2022; 15(17):6122.
https://doi.org/10.3390/ma15176122

**Chicago/Turabian Style**

Penna, Rosa, Luciano Feo, Enzo Martinelli, and Marco Pepe.
2022. "Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced Concrete" *Materials* 15, no. 17: 6122.
https://doi.org/10.3390/ma15176122