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Without reduction, direct application of this in Mathcad gives
Princ1(a,r):=
Princ2(a,r):=
0" xx (a,r) + O"yy (a,r)
2
Now, the task is to find the contours determined in polar coordinates and then locate these same points on a Cartesian representation. We first establish an angular range for a as 0 to 180° in increments of 10° using
and
1:=1..19
7t 1O·7t theta:= 1·_·--
[ 18 18 with the origin for subscripts set to 1 under the menu choice of built-in-variabIes. If values of R are chosen in a range of 1-20 cm using the above equations
J = 1..20
and
r):= O.Ol·J
we can establish a threshold for a particular principal stress contour using the "if"function
thresholdtJ: = if [(1.1 X 106 . Pa
Princ1 (theta!'r))) < O·Pa, 1,0]
to give a stress threshold at 1.1 MPa. Then the matrix with rows in terms of
0.1.--------------,
= 0.05 11 ~ .~
~ 0
~I 8
- -D.OS
-{).1 L....._--L __ ---J.. __ ....L..._----l
o 0.02 0.04 0.06 0.08 (oonlou'I·O.Ol) .cos(thetal)
Figure 3. A plot of a stress contour for 1.1 MPa, starting at the right-hand tip of the crack shown in Figure 1.
1996 March • JOM
a and columns in terms of r appears as in Figure 2. By counting from the right in Figure 2, the radial distance in centimeters at which the stress first falls below 1.1 MPa is indicated by a zero. Now, the task is to translate this into a contour. By simply summing row-wise, the point approximating the contour position can be determined from the threshold matrix with the simple function.
contour[:= [ t threshold[,) ]
giving
7
9
10
10
10
8
7
6
5
contour = 5
5
6
7
8
10
10
10
9
7
Translating the angular values and the limits set by contour back into Cartesian positioning results in Figure 3.
The rather explicit method to show an individual contour hopefully provides a way to grasp the meaning behind the contours. Although this is not the prettiest or most elegant contour that can be constructed, the example should be simple enough to devise strategies for plotting multiple contours. A similar strategy can be employed to calculate any of the stresses around a stress raiser.
ACKNOWLEDGEMENTS Despite the use of Mathcad for these
examples, other mathematics packages can readily perform these calculations with the only differences in the details of the interface and capabilities. Purdue students and readers of prior editions of this series have provided great encouragement for the continuation of these examples.
COMING SOON The next installment of this series, Part
VII, will examine the probability of brittle failure and the Weibull modulus.
References 1. w.o. Callister, Jr., Materials Science and Engineering: An Introduction, 3rd ed. (New York: Wiley, 1994). 2. W.P. Riley and L.W. Zachary, Introduction to Mechanics of Materials (New York: Wiley, 1989).
Keith ]. Bowman is an associate professor of materials engineering at Purdue University.
For more information, contact K.J. Bowman, School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907; (317) 494-6316; fax (317) 494-1204; e-mail kbowman@ecn. purdue.edu.
Book Review Strengtll and Fracture of Engineering Solids, 2nd ed. By David K. Fe/beck and Anthony G. Atkins. ISBN ()'13-856113-3. Prentice HaII,N$wJersey. 1996.552 pages. LisI$60.75.
As indicated in the preface tt> thrs book, strength and fracture are imporlanttopics to praclicing engineers, many technraal1s,some lawyers, and stu· dents preparingrorlhese.le~.IJIJfOrluna1eIy, some of the topics included here (despite claims that the book is intended to foflm a basit rourse in materi· als science) provide a rattIer perverse view of the fundamentals of materiaisscience. As with many similar efforts, the adoQlioh ofmaterials science as faculty elCpertisewithiri mechanIcAl, indUstrial, aeronautical, and chemical engineering programs and materials chemistry provl~ asehse that, wHhout better definHion, the field of matanals science and engineering may have a limi~lite, If the materials content of this book b4oomll$~edas reason· able, there is little hope ror .materials science and engineering as a distirtct field.
The first three ~p\'OVide introductory material on stress, strain, mecllallical testing, and information common to mec:hanids. ThrOOghoutthe text, explanations rorphenomena ereeilher myste. rious because of thewritil)g {j!J$11ty or simply incorrect. This is regrettab18 siliCe the detalI the authors attempt to include is (niSsinQ rrom many texts. Ft>r example, in Rgure~'3QII, ~rOpyof rOiled 1100 series aluminum is shown will\EII!E!XpIanation [nthe caption that the half:hard CQnditlon involves a cold reductionol ~1 percen1 inllreaafler the last anneal. What might be more conhl$ing to students is that the troe. stress·true straill ~$lart al a finite stress and strain with initially linear stopes that do not pass through the origin.
Chapters 4-12 exprore.the basics of materials science with many incorrect descriptions. Begin· ning with a violation of tne f.leisenberg uncertainty principle (at absolute zero aH atoms and molecules are at rest}, chapter four conlinues as a roller coaster ride surrounding the topic of crysta"ography withoot ever meeting it. Under the topic 01 space lattices, six are defil'llld wHh rhombooedral given as a speCial case of triclinic.
Althoogh the chaplers on strength and fracture are a bit cleaner,lh&y do reterto concepts such as metallurgical error and the notion that plane stress fracture in thin ductile metals should be classified as brittle. Continued drscussion on truly brittle materials like ceramics incorporates tha cOnfusing writing style that appears ttlrooghou! the. text.
I can compliment theaulho~ on attempting to place detailed explanations in.the I~xt; Many of the correcl descriptions Ulurninata the phenomena bet· ter than tht>se in many. materials texts. I would recommend that the authors try again. but with more competentedftors and reviewers.
Keith J. Bowman Purdue University
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