# Potential surface and yield surface

Hello,

Would you please tell me the basic difference between potential plastic surface and yield surface with a physical meaning, if possible. I know that if both the surfaces match, then it is associated flow-rule, otherwise it is a non-associative flow rule.

I know that this is a very basic question, but I am facing difficulty in completely understanding it.

-Shivani

### Associate/Non-associate plasticity

Dear Shivanirani,

The question is basic but not elementary.

In the phenomenological theory of plasticity, the yield surface represents the boundary of the elastic (or, in case elasticity is neglected, rigidity) domain. On the other hand, the plastic potential is employed to represent the direction of plastic flow (the rate of plastic deformation). While the yield surface has a direct root in observation (at least for metals, hence being a true phenomenological concept), the plastic potential may be regarded as a mathematical convenience (it simplifies the set of constitutive elements required for the writing of the stress-strain relationship in rate form). One calls the flow rule associate (to the yield surface) when a further simplification is adopted: the plastic potential is taken to be identical to the yield surface (and to be more precise, this is called the normality rule, because the rate of plastic deformation is collinear with the exterior normal to the yield surface). Within a purely phenomenological theory of plasticity there is no decisive argument for an associate flow rule (one could mention Druker's stability postulate; opinions vary; to me, it looks like an interpretation of the normality rule, rather than being a motivation).

If we restrict the discussion to metals, and consider their physical structure (as aggregates of single crystals), we can obtain further, decisive insight into the structure of the constitutive system (and also find an an answer to your question). The first hint in support of the associate flow rule may be found in Hill's book (Mathematical theory of plasticity), Ch. 3.1, "The plastic potential". The first rigorous argumentation, although restricted to rigid-plastic constituents, can be found in papers Hill coauthored with Bishop (for example: 1951. A theory of the plastic distortion of a polycrystalline aggregate under combined stresses. Phil. Mag. 42, 414-427), where it was shown that at the origin of the normality rule is the Schmid activation criterion of the slip systems of a single crystal. Later, in the 1960's and the 1970's, the discussion was extended to elastic-plastic behavior of single crystals, but the conclusion for the overall behavior of an aggregate did not seem decisive. Today, it can be shown, under quite general conditions, that a slightly generalized form of an associated flow rule always characterizes the overall behavior of any metal polycrystal (provided Schmid rule is valid, within certain limits defined by non-Schmid effects).

Thus, getting back to your question: from a physical standpoint, the plastic potential always coincides with the yield surface, or, more generally, the direction of the rate of plastic deformation is always related to the yield surface.

Best

### Hi Shivani,

Hi Shivani,

The yield surface defines the onset of plasticity, but the plastic potential determines your flow rule.

If your plastic potential and yield function are the same you have associated plasticiy and if they are not you have non associated plasticity.

The associated flow rule is usually used for metals and alloys, while the non associated is used for some ceramics such as concrete and clay.

Best