Charpy Data Fitting
Impact tests are performed to measure the response of a material to dynamic loading. Many materials exhibit a transition in fracture mode which is a strong function of test temperature. An example of such a material is ferritic steel with a body centered cubic (bcc) lattice structure. All bcc steels undergo fracture mode transition from brittle, to mixed mode, to fully ductile on the upper shelf as a result of the limited number of dislocation slip systems in these materials. As a result of this behavior, it is highly desirable to have a nonlinear data fitting algorithm so that the impact data can be fit and key indices can be quantified. A related problem is that many data sets are limited in size to save testing costs. Therefore, the nonlinear problem is complicated by the fact that most applications will involve sparse data sets. These objectives for data analysis are met by the in CharpyFitTM v2.1 software package.
The curve fitting results are given in terms of plots of Charpy energy, lateral expansion, and fracture appearance (percent shear) as functions of temperature. These plots show the data points as well as the best fit trends. Four definitions of transition temperature are typically applied to the fitted data. The four typical transition temperature definitions (which are controlled by the user), referred to as the Charpy indices, are:
30 ft-lb Charpy energy
50 ft-lb Charpy energy
35 mil lateral expansion
Fracture appearance (50% shear)
In addition, upper shelf Charpy energy and upper shelf lateral expansion can be determined.
Fitting Method
There are three models available in CharpyFit™ v2.1, and all three have the same hyperbolic tangent functional form as given by:
where y is Charpy energy (or lateral expansion or percent shear), T is temperature, and a1 are the model parameters to be determined by fitting. The parameters can be interpreted as follows: a1 is the lower shelf value of y; a2 is the upper shelf value of y; a3 is the temperature at which the hyperbolic tangent function has its inflection point; and a4 is a measure of the temperature range over which the transitional behavior occurs. The important difference between the models is the method used to weight the importance of the data points during the regression analysis. This weighting is done using the variance to weight the importance of each of the data points on the fit parameters. The most general model is the Weibull. The remaining two models are subsets of the general Weibull model, the simplest of which uses no weighting at all and fits the data to the hyperbolic tangent equation directly. The other model allows weighting which is proportional to the median fit value.
In order to improve the fitting results for sparse data sets, several of the model parameters can be specified prior to fitting based on physics and the MPM fitting data base. For example, the lower shelf can often be specified for an entire class of materials. The fitting data base is included with the package and is applicable to many materials. The data base was developed by fitting large populations and doing sensitivity studies.
Example Fits
The figure shown below is an example fit of a highly populated data set. The data set is representative of reactor pressure vessel steels. The Figure shows the data and the fit mean trend (50%) as well as 5% and 95% probability trends for the case in which the 97 point low alloy steel data set were fit using the hyperbolic tangent function.
For this fit, all a1 and b1 were obtained from the least squares algorithm.
When fitting typical data sets of 15 to 25 points, it is not reasonable to expect a reliable characterization of the statistical variation (confidence bands) to result from application of the Weibull fitting model. In these situations, it is possible to determine the median fit using constant variance weighting or variance proportional to the mean weighting. A comparison of the Weibull model results with the variance proportional to the median model fit is shown in the Figure below. As shown, the median fit for the two models are in close agreement.