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I.

INTRODUCTION

MANY fatigue

crack growth theories postulate that the material at the tip of a crack accumulates damage up to a critical amount after which the crack grows into the fatally damaged material. Those theories which use plastic displacement or plastic work as the critical damage parameter have been shown to yield crack growth laws which have a fourth power stress intensity dependence. ~ Crack growth laws developed by Weertman 2 and Mura and Lin 3 are typical examples of theories which accumulate plastic displacement and plastic work, respectively. Their crack growth laws are given by: da

l~g 4

-A-dN /ztr0U

[11

where/x is the shear modulus, or0 is the yield stress, AK is the stress intensity factor, da/dN is the crack growth rate, and U is the energy required to produce a unit area of crack surface. The major contribution to U is from the cyclic plastic work of deformation in the plastic zone of the crack. Accumulated damage models which use plastic strain or plastic work density as the critical damage parameter give a crack growth law of the form: ~ da

dN

AK 2 -

B -

[21

p,oroec

where ec is a critical strain. This crack growth law is typical of second power laws in that there is no U term. However, Weertman 4 has shown by dimensional analysis that U is proportional to AK 2 for materials which obey a second power growth law. U =

constant

U = C(AK) 2

n =

4

n = 2

[3]

TODD S. GROSS, formerly with the Materials Science and Engineering Department and Materials Research Center, Northwestern University, Evanston, IL, is now Assistant Professor, Department of Metallurgical Engineering and Materials Science, University of Kentucky, Lexington, KY 40506. JOHANNES WEERTMAN is Walter P. Murphy Professor of Materials Science and Engineering and Materials Research Center, and Professor of Geophysics, Department of Geological Science, Northwestern University, Evanston, IL 60201. Manuscript submitted November 19, 1981.

METALLURGICAL TRANSACTIONS A

Knowledge of the dependence of U on M~ in different environments and under different loading histories is necessary to determine the nature of different crack growth mechanisms. Unfortunately, U is a difficult parameter to measure due to the small size of the plastic zone. Fine and his co-workers measured U by measuring the local work of deformation in the plastic zone with a micro strain gage) They found that for materials where U is a constant (n = 4) that U is inversely proportional to the cyclic yield stress. 6 They also reported that U is proportional to AK 2 for a powder metallurgical aluminum alloy with a crack growth exponent n = 2.3. 7 A drawback of that particular study was that only two points were used to imply a parabolic relationship. No error bars were reported. The strain gage technique for measuring U has several serious shortcomings. The strain gage measures only the surface work. It has been shown for several different metals that the surface strain is substantially different from the interior strain in both magnitude and character, s'9 S