Tensile softening (σ-w) curve is a critical parameter necessary to model the non-linear fracture behavior of concrete structures by the finite element method. The uniaxial tensile test is the most direct approach to determine the softening curve, but it is a challenge due to its insufficient stiffness of the loading machine, eccentricity, asymmetric fracture modes, and multiple cracking. Alternatively, a softening curve can be derived via minimizing the deviation between the predicted and experimental results based on the load-crack mouth open displacement (F-l_CMOD) curve measured by the bending or wedge-splitting tests. Inverse analysis methods are commonly used to derive σ-w curve include the J-integral, poly-linear, and global optimization. Although the J-integral method is computationally least demanding, it requires conducting the tests on two specimens with different notch sizes for deriving each σ-w curve, yielding a lower precision and a higher dispersion. For the given test data, the poly-linear method can produce a unique solution without any pre-assumptions regarding the softening curve's shape. However, it is sensitive to arbitrarily small measurement errors, yielding a softening curve with severe oscillation, and accumulating all previous errors in the current analysis step. The global optimization technique is not easy to converge when there are many fitting variables, and it needs to pre-assume the shape of the softening curve. There is no guarantee that the outputs of this technique will be globally optimal. Therefore, this paper proposed a self-modifying inverse analysis method to determine the softening curve of concrete after combining global best-fitting with an automatic correction technique.

The self-modifying inverse analysis method belongs to the global optimization technique with more additional constraints. This method consists of two modules, i.e., global best fitting and parameter self-modifying. The former mainly focuses on the optimum fitting of the F-l_CMOD curve to minimize the cumulative deviation between the simulated and measured values. For this purpose, a computational program was developed a software named MATLAB. In this program, the ability to define search boundaries for each control variable increases the robustness of the fitting algorithm and lessens the likelihood of finding a pseudo-optimal solution. Despite, the optimal solution may be still the local minimum in some situations, so it is also necessary to check whether the local error requirement is satisfied at each discrete point and to find out the maximum load deviation as the second criterion for evaluating the prediction accuracy. Thus, the role of the second module is to automatically modify the optimal value of the fitting parameter when the second criterion is not met. After providing the corrected parameter values (including the boundaries) to the first module, the global optimization procedure is executed again until the load tolerance error is satisfied. The current model cannot accurately simulate the crack propagation process if the load condition cannot be satisfied after the maximum number of iterations. It is required to add a new line segment to the current softening model and conduct the global fitting procedure again with the model containing additional variables until the load criterion is satisfied or the maximum number of line segments is reached. In this algorithm, it is possible to output a precise global optimal solution even when the initial estimate of fitting parameters differs from the ideal solution or the initial softening model is relatively simple.

For the sample, in all regions of the F-l_CMOD curve, the simulated responses match well with the experimental results, and the maximum relative error between the calculated and measured loads is less than 5%. This indicates that the developed method can accurately simulate the crack propagation process of concrete, and this method is highly versatile, applying to different specimen forms and various concrete materials. The fracture energy determined by the F-l_CMOD curve is in a reasonable agreement with that by the σ-w curve, and their deviation does not exceed 2%, further confirming the rationality of the recommended technique. Concrete exposed at different humidities exhibits a similar shape to the softening curve, consisting of a fast descending, a slow descending, and a tail. The tensile strength (f_t) increases with decreasing age while critical crack width (w_c), finally tending to their stabilizing values. For a given age, f_t and w_c decrease with reducing ambient humidity, and this decrease is more significant in the later stages, which is due to the lack of sufficient water in the pores for hydration reaction, the non-uniform shrinkage deformation between aggregate and mortar, and the disturbance of local structure of C-S-H.

Although there are multiple "optimal" solutions at different initial fitting parameters, the ideal σ-w curve is rather narrow, implying that the algorithm converges to the similar solutions. The determined softening curve is independent of the initial estimate, and the algorithm is robust. In the case of different initial models, the optimal σ-w curve is similar with only some minor differences in the tails. This demonstrates that the softening model's initial form has little effect on the optimization outputs, and trustworthy optimal solutions can be obtained even by a simple beginning model.

The self-modifying inverse analysis method could consider the effect of local response on the softening curve, allow the definition of searching boundaries of the fitting parameters, and eliminate the need to predefine the final shape of the softening curve, reducing the probability of obtaining a "pseudo-optimal solution" and weakening the dependence of the optimization output on the initial guesses. In all the region of the F-l_CMOD curve, the calculated load was correlated well with the experimental result, and the fracture energy determined by the F-l_CMOD curve differed by no more than 2% one by the σ-w curve. The proposed method was highly robust and versatile. It could accurately determine the σ-w curve of various concrete materials with different specimen configurations, and the resultant optimal solution was independent of the initial estimate of fitting parameters and the initial shape of the softening model. Concrete exposed at different humidities exhibits a similar shape to the softening curve. The f_t increased with decreasing age while critical crack width w_c, finally tending to their stabilizing values. f_t and w_c decreased as the ambient humidity decreased, and this reduction was more pronounced in the later age.