Weibull statistical fracture theory for the fracture of ceramics
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PETROVIC
The Weibull statistical fracture theory is widely applied to the fracture of ceramic materials. The foundations of the Weibull theory for brittle fracture are reviewed. This theory predicts that brittle fracture strength is a function of size, stress distribution, and stress state. Experimental multiaxial loading results for A1203 tubes are compared to the stress state predictions of the Weibull theory. For the most part, the Weibull theory yields reasonable predictions, although there may be some difficulties in dealing with shear stress effects on fracture.
I.
INTRODUCTION
THEWeibull statistical fracture theory was first introduced in 1939.1.2 Since that time, this theory has become the most widely employed one for application to the fracture of ceramic materials, which typically exhibit scatter in brittle fracture strength. One major reason this is so is that the theory predicts the experimental observation that the size. The Weibull modulus, m, is widely employed in the ceramics community as an index of ceramic material strength reproducibility. For many of the current structural ceramics, the value of the Weibull modulus is approximately in the range of 10. However, for many structural ceramic applications, the desire is for the development of ceramic materials with m > 20, or equivalently, significantly lower scatter in fracture strength. The Weibull theory is most often applied to the uniaxial loading of ceramics. However, for actual structural ceramic components, multiaxial loading is the condition most often encountered. In this regard, the Weibull theory also provides predictions of multiaxial loading fracture strengths for tension-tension and tensioncompression stress states. The intent of the present discourse is to outline the Weibull theory for brittle fracture and to show how its predictions compare to experimental multiaxial loading fracture results for A1203 tubes. 3-6 The fundamental probabilistic basis of the theory will be described, as well as the basic assumptions associated with its application to uniaxial and multiaxial stress state conditions.
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Fig. 1 - - Rod of length L composed of unit rod lengths, subjected to a stretr along its length
lationship to the applied stress, and one would expect P0 t increase as o- increased. The probability of nonfracture of the rod length L is then from basic probability theory: (1 - P )
-= (1 - P0) L
[1
The length L may then be replaced by the volume V due t, their proportionality, leading to the following expression ln(l - e) = V ln(1 - Pov)
[2
where P = probability of fracture of a rod of volume V and P,,v = probability of fracture for a length of rod corre sponding to unit volume. Again, P,,,. will be some functio of stress. At this point, Weibull introduces a parameter termed th "risk of rupture" B. The risk of rupture is defined as: B = - I n ( 1 - P)
[3
For a small volume element dV, the risk of rupture is: dB = - l n ( 1 - Pov) d V
[-4
dB = N(o') d V
[5
II. F U N D A M E N T A L BASIS FOR
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