Transformation and regression statistics of the size-effect method for determining fracture energy and process zone size in quasi-brittle materials

This paper investigates the regression statistics of the size-effect method to obtain fracture parameters of quasi-brittle materials. The correct nonlinear regression model and assumptions are established and verified using a large dataset of size-effect tests extracted from the literature. The effect of model transformation on the change in error structure is then investigated. Three different transformations are considered, including the one leading to the linear regression recommended by RILEM (Mater Struct 23:461-465, 1990). The behavior of the nonlinear least squares estimators of the fracture parameters corresponding to the untransformed space, i.e., peak load P versus specimen size D, and to each of the three transformations are discussed. Monte Carlo simulations on generated data show that the transformations lead to the violation of model assumptions and to highly skewed error distributions prone to artificial outliers. The paper also shows that the estimator corresponding to the RILEM recommendation is asymptotically biased. The estimators corresponding to the other transformations are found either asymptotically biased or do not possess the minimum variance property. Finally, simulations show that the least squares point estimates of the unknown fracture parameters differ when a model transformation is used, and that the difference is statically significant. The fitting of the fracture parameters through the size-effect method should only be obtained in the space (P vs. D) for which the nonlinear least squares estimator is asymptotically unbiased, mean square consistent, and has minimum variance. The linear regression plot suggested by RILEM should be avoided for the statistical inverse problem of the size-effect method.