Rheological yield condition

Summary One of the main problems in Rheology is the determination of the yield condition for which no satisfactory form seems to exist. If a start is made with the stress tensor field, one soon runs into a number of difficulties. It is therefore better to deal with the geometry of the field. The change from elastic to plastic deformation can be interpreted as a mapping of one space into another, and the yield as an asymptotic sub-space. If the elastic strain field ise _ij , whose invariants areJ ₁,J ₂,J ₃, then an asymptotic behaviour may be represented by the existence of a functional relationf(J ₁, J₂,J ₃) = 0, between theJ's, which are independent of one another in the normal part of the field.This does not fix the nature of the functionf, for which we can invoke the additional geometric condition that yielding can result from infinite contraction or expansion of a macro-element. Thus theJacobian of the mapping must take on the singular values zero or infinite. These concepts give rise to the yield condition in the strain tensor field in the form 8J ₃ – 4J ₂ + 2J ₁ 1.If generalized measure of strain is used, which is necessary for creep problems, this takes the formn ³ J ₃ –n ² J ₂ +nJ ₁ 1.n being a real constant, which is equal to 2 for theAlmansi measure. These conditions do not depend on either the isotropicity or homogeneity of the field, and hence should be used for all types of yield conditions.