# What is the difference betwee a failure criterion and a yield condition?

You may meet natural and engineering scientists who blame their colleagues from social sciences or humanities for working unscholarly, not adhering to an explicit and unique terminology but substituting scientific cognition by adopting novel terms. Those sitting in a glasshouse should not throw stones, however. Imprecise terminology and hazy definitions are not at all a “privilege” of social scientists. When I started learning fracture mechanics, I discovered that nearly every anomaly in the real failure behaviour of components which did not fit into the common concept was attributed to “constraint” - but few people had a precise idea what constraint actually is and how to quantify it. The multifarious usage of “damage” in the current literature is an actual example, and “plasticity” is another.

Though von Mises, Drucker, Hill and many others established a precise foundation of phenomenological plasticity, it has become a bad habit to call any inelastic, nonlinear mechanical behaviour “plastic”. One will find applications of the Mises-Prandtl-Reuss equations to polymers, and the authors do not even query, much less justify this approach. In my previous blog, #4, I criticised Mäkelä and Östlund (*Engineering Fracture Mechanics*, Vol. 79, 2012) for modelling the deformation of paper by means of plasticity. One year later I find an “application” to wood.

Henrik Danielsson and Per Johan Gustafsson: A three dimensional plasticity model for perpendicular to grain cohesive fracture in wood, *Engineering Fracture Mechanics* Vol. 98 2013, pp.137–152.

The authors’ misconception is a different one. The deformation behaviour of wood is considered as linear elastic and, of course, orthotropic. But they add a new facet to the term “plasticity”, namely the irreversible and unstable material softening in some process zone: “*Initiation of softening, i.e. the formation of a fracture process zone, is determined by an initial yield function F according to the Tsai–Wu failure criterion*”. This is a failure criterion, correct, and the respective limit surface in the stress space may be assumed as convex as the yield surface in the theory of plasticity. For the sake of a thermodynamically consistent theory, one may also define a corresponding damage potential, but this is not a plastic potential! Once again: The theory of plasticity deals with the stress-strain relationship of ductile materials, having metals in mind, where plastic flow occurs by sliding along crystallographic planes or by twinning. “A physical theory of plasticity starts with these microscopic details and attempts to explain why and how plastic flow occurs” (Khan & Huang: Continuum Theory of Plasticity, Wiley, 1995, p. 310). Following Drucker, classical phenomenological plasticity describes stable, i.e. strain-hardening, material behaviour.

The authors continue “*The change in size of the yield surface f is described by the softening parameter K which is a function of an internal variable that memorizes the plastic loading and determines the softening behavior*”, and they introduce a “*dimensionless deformation δeff*“, as internal variable, which is „*related to the plastic straining of the material*” (wood?), whatever this is supposed to mean. It is not just the “size” of the failure surface that changes, by the way, as Fig 2 shows. In the context of cohesive models, *δ* is commonly called “separation”, i.e. a jump in the discontinuous displacement field, and Fig 3 is a typical traction-separation law. So why introduce a terminology divergent from the established one?

Roberto Balarini stated in a blog http://www.imechanica.org/node/7622 : "Cohesive models are linear elasticity”. In contrast, the present authors apparently assert that cohesive models “are” plasticity. What is so difficult in understanding the model of a cohesive zone? Cohesive models “are” neither elasticity nor plasticity. They describe the nonlinear decohesion process in a continuum that obeys any kind of constitutive equations, for instance plasticity, visco-plasticity or, as in the present case, orthotropic elasticity.

More generally: What is so complicated in applying a unique terminology which is established in the scientific community, and how about the reviewers of manuscripts like this: Are they not aware of the correct terminology themselves or do they just don’t care about it? Remember: The corruption of reasoning starts with a false handling of language!

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