6.3. Cleavage fracture
Cleavage fracture originates from microcracks that nucleate from defects (carbides, cracks arrested at grain boundaries, etc.). The location of these defects is statistical by nature and hence modeling efforts rely on probabilistic analysis. Since the seminal work by Beremin (1983), cleavage fracture toughness estimations are based on Weibull statistics and the weakest link model, where the probability of failure equals the probability of sampling (at least) one critical fracture-triggering particle. Grounded on such approach, a novel probabilistic framework is here proposed that takes advantage of the advanced statistical tools of MATLAB to estimate all Weibull-related parameters without any a priori assumptions.
For a given Weibull stress and a threshold stress for crack growth , the cumulative probability of failure , in terms of the modulus and scaling parameter is given by:
where the Weibull stress can be defined as:
Here is a reference volume, is the volume of the material unit (finite element) in the fracture process zone experiencing a maximum principal stress and is the number of finite elements/material units in the fracture process zone. The parameter is needed due to the fact that cracks do not propagate below a certain threshold energy value. However, the concurrent estimation of the threshold, modulus and shape parameters remains a complicated task; a common approach lies in assuming a value for and estimating and from a set of experiments. Here, all three parameters (, and ) will be obtained by means of a novel iterative procedure involving least squares estimates of the cumulative distribution functions.
The capabilities of Abaqus2Matlab to model cleavage fracture will be benchmarked with an extensive experimental data set developed within the Euro toughness project (Heerens, J. & Hellmann, D., 2002). As in the experiments, a quenched and tempered pressure vessel steel DIN 22NiMoCr37 steel will be investigated; only tests where significant ductile crack growth is not observed will be considered and the reference experimental data will be that obtained at -40oC with a compact tension specimen of size 1T. Comprehensive details of the material tensile properties, specimen size and failure loads are provided in the original experimental article (Heerens, J. & Hellmann, D., 2002) and will not be reproduced here for the sake of brevity.
The algorithm methodology is described in Figure 1. First, the finite element results are computed by running the corresponding Abaqus job inside the proposed toolbox. A finite element mesh of approximately 2000 quadratic plane strain quadrilateral elements with reduced integration is employed, with the elements being progressively smaller as the crack tip is approached. The values of the volume element and the maximum principal stress are read and stored for each finite element and load level of interest. The latter is characterized through the J-integral and the pin displacement, that are also read in the Abaqus2Matlab environment. The statistical analysis is then conducted. The probability of failure for each failure load reported experimentally is first computed through the relation:
where denotes the number of experiments and the rank number. Afterwards, an iterative procedure is conducted to simultaneously estimate , and . In each iteration the Weibull stress is computed from the values of and from the previous iteration and subsequently inserted in Eq. (1) to compute the values of , and in the current iteration by fitting a univariate distribution through the least squares method. Convergence is achieved when the relative norm of the change in the solution is below an appropriate tolerance value. Therefore, taking advantage of Matlab capabilities, Weibull parameters are calculated by finding the distribution whose cumulative function best approximates the empirical cumulative distribution function of the experimental data.
Figure 1: Schematic overview of the use of Abaqus2Matlab to estimate the probability of cleavage failure
The results obtained for the particular case considered (Euro toughness data set, T1, -40oC) are displayed in Figure 2. The Figure shows the probability of failure versus the external load from the experimental study and the current statistical model. The calibrated Weibull stress parameters are also embedded in the Figure. As it can be observed, a good agreement is attained between the failure probability estimated from Eq. (1) and the experimental results.
Figure 2: Failure probability as a function of the external load. The Figure includes the experimental data for 22NiMoCr37 steel (EuroData project) and the predictions from the present statistical model for the values of , and displayed.
Results indicate that, for the particular case under consideration, a 50% probability of failure will be attained for an external load of approximately = 150 N/mm, while the 5% and 95% probability bonds are attained at = 60 N/mm and = 250 N/mm, respectively. Weibull-parameters' estimation reveals that stresses lower than = 983.7 MPa are innocuous and that a failure probability of 64% in a unit element is attained at a stress level of + = 2164.6 MPa.
More insight into the local failure probability can be gained by means of a hazard map. A hazard map highlights the areas being affected or vulnerable to a certain type of failure, providing visual information on the failure probability at each particular unit element (Muniz-Calvente M. et al., 2016). Thus, the local probability of failure (i.e., for a local ) is shown in Figure 3 in logarithmic scale. The mesh can be easily constructed by reading the nodal coordinates and the element connectivity through Abaqus2Matlab.
Figure 3: Hazard map. The legend shows the local probability of failure.
Statistical tools are indispensable to assess cleavage fracture as experimental data tends to be widely scattered; two identical specimens of the same material may have very different toughness values due to the random location of the fracture-triggering particle. Abaqus2Matlab enables the usage of Matlab's in-built statistical capabilities to estimate all Weibull parameters without any prior assumptions. This novel iterative framework allows for more precise estimations of failure probabilities, a crucial aspect in risk quantification and operational decision making in engineering applications.
Abaqus2Matlab - www.abaqus2matlab.com
Copyright (c) 2017 by George Papazafeiropoulos
If using this application for research or industrial purposes, please cite:
G. Papazafeiropoulos, M. Muniz-Calvente, E. Martinez-Paneda.
Abaqus2Matlab: a suitable tool for finite element post-processing.
Advances in Engineering Software. Vol 105. March 2017. Pages 9-16. (2017)
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