After completing the 4 weeks of sustained loading in service conditions, the beams were loaded until the specified CMOD at ultimate conditions, w_{u} = 0.5 mm, was reached. The beams were then submitted to a sustained loading corresponding to 60 % of P_{u} for 7 days, after which the sustained loading was increase to 75 % for 14 days and finally to 90 % for 7 days (Table 2).

3.2.1 Load-Deflection Responses

The load-deflection curves of the two beams analysed in this paper are presented in Fig. 5. The loads P_{u} of beams P07-D and P07-S measured at w_{u} are respectively

Fig. 5 Load-deflection response at ultimate conditions

42.5 and 47.2 kN. The average load-deflection static bending response and its corresponding 95 % confidence intervals, calculated with the Student law and applied on 6 characterization tests, are also plotted in Fig. 5. The confidence intervals represent the static behaviour envelope of the characterization beams. In Fig. 5, a cross-mark indicates the failure point obtained during the ultimate loading conditions. The results show that the failure of beams occured as their deflection reached the static behaviour envelope defined by the 95 % confidence intervals. The complete experimental program revealed that this observation is true regardless of the CMOD (w_{u}) and the loading history (% P/P_{u}) applied on beam specimens [6, 13].

3.2.2 Deflection-Time and CMOD-Time Responses

The deflection-time and the CMOD-time responses of beams for the loading at ultimate conditions are presented in Fig. 6. The sustained load levels of each beam expressed in percentages of P_{u} are presented above the curves. The results show that beam P07-D and P07-S respectively failed upon the unloading-reloading cycle performed on the 49th and 56th day to reach 90 % of P/P_{u}. Although beam P07-D failed 7 days earlier than beam P07-S, Fig. 6 shows that the evolution of the deflection and the crack opening are comparable. As SFRC is an heterogeneous material, it is reasonable to assume that failure time of various specimens should vary under identical conditions. This phenomenon was also noted in studies by [5, 14].

Figure 6a, b were used to calculate the deflection and CMOD rates in the secondary creep phase for all sustained load intervals. The secondary rates were calculated for the time interval corresponding to the linear portion of each sustained load interval. In order to do so, mathematical functions best fitting the experimental results were defined for each sustained loading 7-day interval. Using these

Fig. 6 Deformation at ultimate conditions

Fig. 7 Secondary rates per unit load versus load level curves [6]

nonlinear functions, time intervals for which the second derivative approaches zero were selected. This method allowed determining accurately the boundaries of the linear portion of each sustained loading interval used to calculate the secondary creep deflection and CMOD rates.

The secondary deflection and secondary CMOD rates per unit load are plotted in Fig. 7 as a function of the sustained load level expressed in percentages of P_{u}. This figure includes the results obtained on five of the ten beams that were tested with the aforementioned procedure (Fig. 2) with various w_{u} and P/P_{u} ratios. A complete description of the results obtained for these five flexural creep tests may be found in [6, 13]. Figure 7 shows that an exponential relation exists between both the secondary creep deflection and secondary CMOD rates and sustained load levels greater or equal to 60 %. Yet, the deflection and the CMOD represent measurements at the global and at the local scale respectively. The deflection provides

Fig. 8 Compliance rate versus secondary rates per unit load curves for high load levels [6]

global information on the beam’s damage state, whereas the CMOD provides localised information of the macrocrack. Therefore, the increase in both the secondary creep deflection and secondary CMOD rates indicates that the same instigative mechanism of creep is involved at both the global and local scales.

The elastic compliance rate represents the difference of compliance between two unloading-reloading cycles divided by the elapsed time between those two cycles. The results are presented in Fig. 8 in function of the secondary deflection and secondary CMOD rates per unit load for sustained load levels greater than 60 % of the same series of specimens. This figure shows that a proportional relationship exists between the relative evolution of the compliance per unit load and the secondary creep deflection rate per unit load. The same remark applies to the secondary CMOD rates per unit load. In fracture mechanics, the elastic compliance corresponds to the overall damage state of the specimen. Hence, this correlation between the overall damage state and the secondary rates at high sustained load levels confirms that the propagation of the macrocrack governs the evolution of the long-term deflection of the SFRC beams. This conclusion was also made by Rossi et al. [15] in a similar study on ordinary concrete. Therefore, this propagation phenomenon must be directly linked to the cementitious matrix and not the fibres absent in ordinary concrete.

3.2.3 Significance

The observation on the beams’ failure produced by high sustained load levels within the static behavior envelop goes along with the work of Walkinshaw [16] who suggested that a failure criteria based on a critical deflection could be applied for flexural creep tests performed on ordinary concrete. It means that a state of damage defined by the static behaviour envelope must be attained to cause the failure of the beams under high creep loads.

The previous correlations between deflection, crack opening and compliance under high sustained loads can be explained by the mechanism proposed by Rossi et al. [15, 17] for ordinary concrete. According to the authors, sustained loading leads to the creation and propagation of microcracks at the macrocrack tip. The creation of new microcracks generates local moisture shocks resulting in water and vapour pressure gradients. These gradients induce water movements from the capillaries surrounding the microcracks to the voids created by these microcracks, which results in additional tensile stresses due to this internal shrinkage. The additional tensile stresses lead to stress redistributions at the macrocrack tip that may lead to the creation of new microcracks or the propagation of the macrocrack, hence contributing to the widening of the macrocrack. This can be considered as the source of a significant portion of the long-term behaviour of creep.

In short, creep of concrete would be mainly due to microcracking, and the kinetics would depend on the water gradients. The physical explanation of this mechanism can be adapted to SFRC.

Test results, more precisely the proportional relation between the compliance and the secondary creep deflection and secondary CMOD rates, have shown the mechanism is directly linked to crack propagation. As fibres are present in the process zone and represent additional heterogeneity, it is reasonable to assume that their interfaces with the cement matrix are also affected by this phenomenon. New microcracks and the coalescence or propagation of microcracks in the vicinity of the fibres progressively deteriorate the fibre-matrix interface and may favor the macrocrack propagation, thus creep. However, creep of the fibre-matrix interface, acting at the local scale, does not play a significant role in the evolution of the compliance, which is mainly governed by the propagation of the macrocrack (structural scale). Therefore, the propagation of the macrocrack, favoured by the deterioration of the fibre-matrix interface, governs the secondary creep phase, ultimately leading to the beam’s failure at high sustained load levels.