## 5 Analytical Modelling
### Assessment of Analytical Modelling Capabilities  
The results obtained by applying the stochastic cracking model to the investigated sets of TRC systems are displayed in the following, with reference to M1- and M2-based composites reinforced with Fabrics F1 (Figure 16), F2 (Figure 17), F4 (Figure 18), F6 (Figure 19) and F7 (Figure 20). Average sample dimensions were considered and the analytical responses were cut at the maximum average stress value \(\sigma_{TRC,max}\) (see Table 7). The analytical curves were compared with the experimental results in terms of composite stress \(\sigma_{TRC}\) vs. strain \(\varepsilon\). Experimental strains were evaluated from the LVDT outputs, as the ratio between the average crack opening displacement COD and the measured gauge lengths GL. Please note that each curve in the figures is broken to the loss of the first transducer. Figure 16: Comparison between the predicted analytical curves and the experimental responses in terms of nominal stress vs. strain for M1-based (**a**) and M2-based (**b**) composites reinforced with Fabric F1. Figure 15: Fabric average tensile responses in terms of fabric stress on effective filaments \(\sigma_{f}^{*}\) vs. normalised displacement in the warp directions (**a**) and in the weft directions (**b**). Figure 19: Comparison between the predicted analytical curves and the experimental responses in terms of nominal stress vs. strain for M1-based (**a**) and M2-based (**b**) composites reinforced with Fabric F6. Figure 17: Comparison between the predicted analytical curves and the experimental responses in terms of nominal stress vs. strain for M1-based (**a**) and M2-based (**b**) composites reinforced with Fabric F2. Figure 18: Comparison between the predicted analytical curves and the experimental responses in terms of nominal stress vs. strain for M1-based (**a**) and M2-based (**b**) composites reinforced with Fabric F4. From the comparison, it was possible to notice that the analytical curves seem to properly fit the experimental results in almost any of the investigated TRCs. It was observed that:  
* The stochastic model should fit the sequence of cracking stress values in the stable propagation phase. In all the cases investigated, the choice of a different tensile strength (type \"B\" analysis in place of type \"A\") allowed containing estimation errors. Moreover, it could be concluded that: (i) the two-parameter Weibull distribution function worked adequately; (ii) the \(m\) parameter, despite being obtained from flexural results, satisfactorily simulated the slope of direct tension curves; (iii) as expected, the effect of dispersed microfibers was not captured; and (iv) differences between the analytical and the experimental cracking stresses may be connected to mixed tensile-bending stress fields (unbalanced shrinkage could lead to a loss of planarity of the specimen). * The transition point between the second and the third branch was satisfactorily caught by the analytical simulations. The ability of a simplified model to fit this response range was really important, bearing in mind that one of the main advantages of TRC--as an example in the case of retrofitting applications--is its energy dissipation capacity (Section 4.3). In these preliminary analyses, the use of mean saturated crack distances \(X\) obtained from experimental observation seemed an adequate compromise solution and again confirmed that the Weibull function can effectively describe discrete distributions of cracks. Future alternatives for the assessment of the \(X\) value might be based on the experimental evaluation of the frictional shear stress by means of pull-out tests and the use of refined approaches, such as the car parking problem solution. * The hypothesis of evaluating the effective amount of AR-glass reinforcement multiplying the fibres volume fraction \(V_{f}\) by the fabric efficiency factor \(EF_{f}\) appeared to properly fit the third-branch slopes in almost all cases, with the only exception being F7-based composites.