Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced Concrete
Abstract
:1. Introduction
2. Outline of the Experimental Results
3. Theoretical Model
3.1. Assumptions and Formulation
- (i)
- The flexibility was distributed in the central part of the specimen for a length equal to “” while a rigid body behaviour was exhibited by the remaining end parts (Figure 2).
- (ii)
- The midspan cross-section was discretized in layers as shown in Figure 3. The average axial strain of the k-th layer, , before crack formation, and the crack-opening displacement, , after the crack formation, can be easily expressed for the k-th layer (k = 1, …, ) as in Equations (1) and (2).
- (iii)
- Consequently, the average value of the axial stress, , at k-th strip can be determined as a function of the axial deformation, , before cracking, or a function of the crack-opening displacements, , after cracking. The stress–strain and stress–displacement relationships assumed in this paper are reported in Section 3.2.2.
- (iv)
- A transition length, , 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, , inside the transition zone was considered which can be evaluated with an exponential law as in Equation (3) where and 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, , mobilised at the j-th step of the incremental analysis as in Equation (4):
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 , ;
- a hardening behaviour, characterized by the formation of many microcracks in the HPFRC mix, identified by the two parameters and ;
- a constant behaviour defined by the two parameters and .
4. Inverse Identification of the Relevant Material Laws
- In the first one, the cylindrical compression strength, , the transition length , and the exponential parameter, , were calibrated on the flexural response of the conditioned CM0 specimens (labelled CM0-FT). Experimentally, a 21% reduction in the cylindrical compression strength, , 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, , 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, , equal to 70 mm). In both models, the coefficient of the exponential law, , was considered constant and equal to 0.40 (Table 3). Figure 7 shows both the average experimental curve (light-blue line) and the average numerical 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., ) were calibrated on the flexural response of conditioned CM1 specimens (labelled CM1-FT). A 13% reduction in the parameter was adopted in the calibration of the conditioned specimens compared to the unconditioned ones.
- In the last one, only the parameter, , 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 was adopted for the conditioned specimens compared to the unconditioned ones. Moreover, as in [42], a 20% reduction in the fibres’ volume fraction, , 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, , the transition length , 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 fcm 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, τel, (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
References
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Mix. | ||||||||
---|---|---|---|---|---|---|---|---|
[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 |
Mix. | ||||||||
---|---|---|---|---|---|---|---|---|
[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 |
Specimen Designation | Model | |||
---|---|---|---|---|
[MPa] | [mm] | [-] | ||
CM0-NFT | 53.0 | 70.0 | 0.4 | Ref. [42] |
CM0-FT | 42.0 | 85.0 | 0.4 | Present paper |
Series | 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 |
Results | CM1 | CM2 | ||||||
---|---|---|---|---|---|---|---|---|
[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 |
Results | CM1 | CM2 | ||||||
---|---|---|---|---|---|---|---|---|
[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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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
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 StylePenna, 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