# Size Effect in Plasticity and in LEFM

Scaling laws are the most fundamental aspect of every physical theory. As discussed in Chapter 1, the problem of scaling law and size effect in the theories of structural failure has received considerable attenlion, particularly with regard to distributed damage and nonlinear fracture behavior. The necessity of using theories that correctly predict the size effect was emphasized in Section 1.2.5 and the basis for the general analysis together with some simple size effect derivations and examples was presented in Section 1.4. It was shown that for plasticity theory (and for allowable-stress analysis, too) no size effect was predicted, but that for LEFM the nominal structural strength decreased with increasing size as D-1/2. The main objective of the present section is to derive these properties from the basic theories.

To do so, we first study the implications of limit analysis for size effect. Then we set up the general forms that the expressions for К/ and Q must take, and derive the size effect for LEFM. We end the section
with a brief discussion of the effect of structure size on the strength of structures containing relatively very small flaws whose size is independent of the size of the structure.

We remember from Section 1.4 that, unless otherwise stated, the size effect is defined by comparing geometrically similar structures of different sizes (in the case of notched or fractured structures, the geometric similarity, of course, means that the notches or initial traction-free cracks are also geometrically similar). In this section, our interest is in analyzing the effect of the size on the nominal strength a^u which was defined in Section 1.4.1. Wc recall that its general form for plane problems is

Pu

а.’и—-ск— (2.3.1)

bu

where Cjv depends only on geometrical ratios and thus is a constant for geometrically similar structures.

We also recall from Section 1.4 that, flncc any two consistently defined nominal stresses are related by a constant factor, the general trend of the size effect is independent of the exact choices for and D. However, for quantitative analysis and, specially, for comparison of results from various sources, it is useful to make a consistent choice throughout.