Rigid-plastic flow

The attached notes are written for a course on plasticity.  When I update the notes, I will post a link on my twitter account:  https://twitter.com/zhigangsuo.

Toothpaste retains its shape when stress is small, but flows when stress is sufficiently large. Thus, the toothpaste does not drip or spread under its own weight, but extrudes from a tube or brushes onto the teeth under a gentle force. The stress above which the toothpaste flows is called the yield stress.  Such a fluid is known as a yield-stress fluid, a plastic fluid, or a viscoplastic fluid (Balmforth, Frigaard, Ovarlex 2014; Coussot 2014).

Here we study plasticity in its purest form, using the rigid-plastic model.  The model is a limit of the viscoplastic model, and is also a limit of the elastic-plastic, strain-hardening model.  The rigid-plastic model characterizes a material using a single material constant, the yield stress.  The model avoids complications associated with the flow, such as viscosity, elasticity, strain hardening, and thixotropy.

Like linear viscosity and linear elasticity, the rigid-plastic model is among the most useful rheological models. The model describes toothpastes, greases, thermoplastic polymers, and metals (Hill 1950; Tanner 2000).  In particular, the rigid-plastic model describes industrial processes of metals, such as extrusion, embossing and stamping. During these processes, plastic strain is much larger than elastic strain, so that elasticity is often negligible.  For pre-strained metals, hardening is also often negligible.  Rigid-plastic flows have been analyzed for many geometries and loads (Druyanov and Nepershin 1994; Chakrabarty 2006).