mit.edustat.dviStatistics of FractureDavid RoylanceDepartment of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridge, MA 02139March 30, 2001IntroductionOne 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 whichreduce the probability of failure to an acceptably low level. Selection of an acceptable level ofrisk is a difficult design decision itself, obviously being as close to zero as possible in cases wherehuman safety is involved but higher in doorknobs and other inexpensive items where failure isnot too much more than a nuisance. The following sections will not replace a thorough studyof statistics, but will introduce at least some of the basic aspects of statistical theory neededin design against fracture. The text by Collins1includes an extended treatment of statisticalanalysis of fracture and fatigue data, and is recommended for further reading.Basic statistical measuresThe value of tensile strength σfcited in materials property handbooks is usually the arithmeticmean, simply the sum of a number of individual strength measurements divided by the numberof specimens tested:σf=1NNXi=1σf,i(1)where the overline denotes the mean and σf,iis 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 thedistribution of strengths, one important one being the sample standard deviation, asortofrootmean square average of the individual deviations from the mean:s =vuut1N − 1NXi=1(σf− σx,i)2(2)The significance of s to the designer is usually in relation to how large it is compared to themean, so the coefficient of variation, or C.V., is commonly used:1Collins, J.A., Failure of Materials in Mechanical Design, Wiley, 1993.1C.V. =sσfThis is often expressed as a percentage. Coefficients of variation for tensile strength are com-monly in the range of 1–10%, with values much over that indicating substantial inconsistencyin the specimen preparation or experimental error.Example 1In order to illustrate the statistical methods to be outlined in this Module, we will use a sequence ofthirty measurements of the room-temperature tensile strength of a graphite/epoxy composite2.Thesedata (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◦Cand-59◦C, are given in Probs. 2and 3, and can be subjected to the same treatments as homework.There are several computer packages available for doing statistical calculations, and most of theprocedures to be outlined here can be done with spreadsheets. The Microsoft Excel functions for meanand standard deviation are average() and stdev(), where the arguments are the range of cells containingthe data. These give for the above dataσf=73.28,s=4.63 (kpsi)The coefficient of variation is C.V.= (4.63/73.28) × 100% = 6.32%.The normal distributionA more complete picture of strength variability is obtained if the number of individual specimenstrengths falling in a discrete strength interval ∆σfis plotted versus σfin a histogram as shownin Fig. 1; the maximum in the histogram will be near the mean strength and its width will berelated to the standard deviation.Figure 1: Histogram and normal distribution function for the strength data of Example 1.2P. Shyprykevich, ASTM STP 1003, pp. 111–135, 1989.2As the number of specimens increases, the histogram can be drawn with increasingly finer∆σfincrements, eventually forming a smooth probability distribution function,or“pdf”. Themathematical form of this function is up to the material (and also the test method in somecases) to decide, but many phenomena in nature can be described satisfactorily by the normal,or Gaussian, function:f(X)=1√2πexp−X22,X=σf−σfs(3)Here X is the standard normal variable, and is simply how many standard deviations an indi-vidual specimen strength is away from the mean. The factor 1/√2π normalizes the function sothat its integral is unity, which is necessary if the specimen is to have a 100% chance of failingat some stress. In this expression we have assumed that the measure of standard deviation de-termined 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 grade-conscious students. Its integral, known as the cumulative distribution function or Pf(X), isalso used commonly; its ordinate is the probability of fracture, also the fraction of specimenshaving a strength lower than the associated abscissal value. Since the normal pdf has been nor-malized, the cumulative function rises with an S-shaped or sigmoidal shape to approach unityat large values of X. The two functions f (X)andF(X) are plotted in Fig. 2, and tabulatedin Tables 1 and 2 of the Appendix attached to this module. (Often the probability of survivalPs=1−Pfis used as well; this curve begins at near unity and falls in a sigmoidal shape towardzero 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 isnormally distributed involves using special graph paper (a copy is included in the Appendix)whose ordinate has been distorted to make the sigmoidal cumulative distribution Pfplot as astraight line. (Sometimes it is easier to work with straight lines on curvy paper than curvy lineson straight paper.) Experimental data are ranked from lowest to highest, and each assigneda rank based on the fraction of strengths having higher values. If the ranks are assigned asi/(N + 1), where i is the position of a datum in the ordered list and N is the number ofspecimens, 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 probabilitypaper is then a visual measure of how well the data are described by a normal