The term yield refers to the onset of inelastic behaviour such as described in the preceding chapter. In this chapter we will try to make a precise description of yielding. In particular, we will try to establish a set of mathematical conditions for yielding that will be referred to as the yield criterion.
There have been many different yield criteria suggested by different researchers and engineers. Coulomb set down the first useful yield criterion in 1773. It forms one of the cornerstones of our understanding of the way soils behave and it will be considered in detail later in this chapter. First, however, we will investigate some of the yield criteria suggested for ductile metals. Metals are a bit simpler than geomaterials, and many of the basic ideas can be developed in a simpler context.
A yield criterion can be visualised as a mathematical function. We will represent it by f. The arguments of f might be almost anything to do with the state of the body at the onset of plastic behaviour, but the most obvious candidates for arguments would be the components of stress or strain or both. Modern developments in plasticity accept that the most appropriate arguments are the individual components of the stress matrix. Realising that there are only six independent stress components, we can write our prototype yield criterion as follows:
f (oxx, °yy, Оzz, °xy, °yx, Оzx) k C3.1)
where k represents a constant. It may be zero, but in many cases it will be convenient to have a non-zero constant. As for the left-hand side of (3.1) we will think of this as simply some function of the stress components which is, as yet, undetermined. Yielding is signalled when this function becomes equal to the constant k. Also, there may be functional arguments other than stress that we might wish to add later, but stress will be the central criterion for yielding in most modern plasticity models.
Next, we are aware from our study of Mohr’s circle that it may not be necessary to know all the components of stress. We can recreate the stress components in any coordinate frame using the principal stresses, o1,o2,o3, provided we know the respective orientations of the principal directions. This would suggest that the six stress components in (3.1) could be replaced by the three principal stresses, plus some information concerning the orientation of the principal directions. At this point, the developments that follow will benefit from the introduction of an important material characteristic that removes all dependence on orientation of the principal directions. The material characteristic is isotropy. For an isotropic material, there can be no dependence of material response on a specified direction. Thus, for isotropic bodies, (3.1) can be rewritten, without any loss of generality, in terms of just principal stresses:
f (01,02,03) = k (3.2)
Here k is still a constant, although its form may differ from that in (3.1).
Alternatives to the principal stresses are the principal invariants I1,12,13. Equation (1.19) gave the relations between the invariants and the principal stresses. Obviously we could recast (3.2) as
f (I1, I2, I3) = k (3.3)
In some circumstances this form for the yield criterion may be more convenient than (3.2), but in most cases we will find (3.2) to be the more useful description.
Much of what follows in this chapter will be directed towards visualising various yield criteria. In this effort, the form (3.2) will be most useful. Because of this it will be convenient to digress for a moment and introduce a new threedimensional space with coordinates proportional to the values of the principal stresses themselves.