Statistics of Fracture David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA 02139 March 30 2001 Introduction One particularly troublesome aspect of fracture especially in high strength and brittle materials is its variability The designer must be able to cope with this and limit stresses to those which reduce the probability of failure to an acceptably low level Selection of an acceptable level of risk is a difficult design decision itself obviously being as close to zero as possible in cases where human safety is involved but higher in doorknobs and other inexpensive items where failure is not too much more than a nuisance The following sections will not replace a thorough study of statistics but will introduce at least some of the basic aspects of statistical theory needed in design against fracture The text by Collins1 includes an extended treatment of statistical analysis of fracture and fatigue data and is recommended for further reading Basic statistical measures The value of tensile strength f cited in materials property handbooks is usually the arithmetic mean simply the sum of a number of individual strength measurements divided by the number of specimens tested f N 1 X f i N i 1 1 where the overline denotes the mean and f i is the measured strength of the ith out of N individual specimen Of course not all specimens have strengths exactly equal to the mean some are weaker some are stronger There are several measures of how widely scattered is the distribution of strengths one important one being the sample standard deviation a sort of root mean square average of the individual deviations from the mean v u u s t N 1 X f x i 2 N 1 i 1 2 The significance of s to the designer is usually in relation to how large it is compared to the mean so the coefficient of variation or C V is commonly used 1 Collins J A Failure of Materials in Mechanical Design Wiley 1993 1 C V s f This is often expressed as a percentage Coefficients of variation for tensile strength are commonly in the range of 1 10 with values much over that indicating substantial inconsistency in the specimen preparation or experimental error Example 1 In order to illustrate the statistical methods to be outlined in this Module we will use a sequence of thirty measurements of the room temperature tensile strength of a graphite epoxy composite2 These data in kpsi are 72 5 73 8 68 1 77 9 65 5 73 23 71 17 79 92 65 67 74 28 67 95 82 84 79 83 80 52 70 65 72 85 77 81 72 29 75 78 67 03 72 85 77 81 75 33 71 75 72 28 79 08 71 04 67 84 69 2 71 53 Another thirty measurements from the same source but taken at 93 C and 59 C are given in Probs 2 and 3 and can be subjected to the same treatments as homework There are several computer packages available for doing statistical calculations and most of the procedures to be outlined here can be done with spreadsheets The Microsoft Excel functions for mean and standard deviation are average and stdev where the arguments are the range of cells containing the data These give for the above data f 73 28 s 4 63 kpsi The coefficient of variation is C V 4 63 73 28 100 6 32 The normal distribution A more complete picture of strength variability is obtained if the number of individual specimen strengths falling in a discrete strength interval f is plotted versus f in a histogram as shown in Fig 1 the maximum in the histogram will be near the mean strength and its width will be related to the standard deviation Figure 1 Histogram and normal distribution function for the strength data of Example 1 2 P Shyprykevich ASTM STP 1003 pp 111 135 1989 2 As the number of specimens increases the histogram can be drawn with increasingly finer f increments eventually forming a smooth probability distribution function or pdf The mathematical form of this function is up to the material and also the test method in some cases to decide but many phenomena in nature can be described satisfactorily by the normal or Gaussian function 1 X 2 f X exp 2 2 X f f s 3 Here X is the standard normal variable and is simply how many standard deviations an individual specimen strength is away from the mean The factor 1 2 normalizes the function so that its integral is unity which is necessary if the specimen is to have a 100 chance of failing at some stress In this expression we have assumed that the measure of standard deviation determined from Eqn 2 based on a discrete number of specimens is acceptably close to the true value that would be obtained if every piece of material in the universe could somehow be tested The normal distribution function f X plots as the bell curve familiar to all gradeconscious students Its integral known as the cumulative distribution function or Pf X is also used commonly its ordinate is the probability of fracture also the fraction of specimens having a strength lower than the associated abscissal value Since the normal pdf has been normalized the cumulative function rises with an S shaped or sigmoidal shape to approach unity at large values of X The two functions f X and F X are plotted in Fig 2 and tabulated in Tables 1 and 2 of the Appendix attached to this module Often the probability of survival Ps 1 Pf is used as well this curve begins at near unity and falls in a sigmoidal shape toward zero as the applied stress increases Figure 2 Differential f X and cumulative Pf X normal probability functions One convenient means of determining whether or not a particular set of measurements is normally distributed involves using special graph paper a copy is included in the Appendix whose ordinate has been distorted to make the sigmoidal cumulative distribution Pf plot as a straight line Sometimes it is easier to work with straight lines on curvy paper than curvy lines on straight paper Experimental data are ranked from lowest to highest and each assigned a rank based on the fraction of strengths having higher values If the ranks are assigned as i N 1 where i is the position of a datum in the ordered list and N is the number of specimens the ranks are always greater than zero and less than one this facilitates plotting The degree to which these rank strength data plot as straight lines on normal probability paper is then a visual measure of how well the data are described by a normal distribution The 3 best fit straight line through the points passes the 50 cumulative fraction line at the sample mean and its slope gives the standard