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Fracture Mechanisms in Nonmetals 293

(a)

(b)

FIGURE 6.41 The “fictitious crack” model for concrete: (a) schematic stress-displacement response and (b)
damage zone ahead of a growing crack. Taken from Hillerborg, A., Modeer, M., and Petersson, P.E., “Analysis
of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements.”
Cement and Concrete Research, Vol. 6, 1976, pp. 773–782. Hillerborg, A., “Application of the Fictitious Crack
Model to Different Materials.” International Journal of Fracture, Vol. 51, 1991, pp. 95–102.

of concrete are often quite large, and the interaction between the process zone and free boundaries
can influence the behavior. Consequently, G is not a material property in general, but can depend
c
on specimen size. Fracture toughness results from small-scale tests tend to be lower than values
obtained from larger samples.
The phenomena of size effects in concrete fracture has been the subject of considerable study
[50, 53–56]. Some of the apparent size effects can be attributed to inappropriate data reduction
methods. For example, if fracture toughness is computed by substituting the load at fracture into
a linear elastic K or G relationship, the resulting value will be size dependent, because the nonlin-
earity due to the process zone has been neglected. Such an approach is analogous to applying linear
elastic K equations to metal specimens that exhibit significant plasticity prior to failure. Even
approaches that account for the process zone exhibit size dependence, however [54]. A fracture
parameter that uniquely characterizes these materials would be of great benefit.

REFERENCES

1. Johnson, W.S. (ed.), Metal Matrix Composites: Testing, Analysis, and Failure Modes. ASTM STP
1032, American Society for Testing and Materials, Philadelphia, PA, 1989.
2. Williams, J.G., Fracture Mechanics of Polymers. Ellis Horwood Ltd., Chichester, UK., 1984.