Alternative plasticity theory I know that there exist several continuum-mechanical models of plasticity. But maybe there exist some plasticity and fracture models based on atomistic equations. One can assume a model where an atom of mass $m$ lies in a potential $V(x)$ that has the shape of a wave (one can Fourier transform it). The equation of motion is
$m \ddot{x} = - \nabla V(x)$
and it is nonlinear in general. It can be shown that for small displacements $x$ harmonic oscillator equations can be obtained; thus, elasticity would be included in this model. Is such a model plausible to describe plasticity and even fracture (this would be the case, if the atom leaves the region where $V \neq 0$) of metallic materials?
I think it is also straightforward to quantize this equation. Effects such as tunneling will play a role (cannot be treated with any continuum-mechanical model). Can on this way quantum mechanics and material mechanics be unified?
 A: There are indeed many theories of plasticity, but they all have three common ingredients: (1) elastic behavior below a certain strain level, (2) yield surface and (3) plastic flow law. For example, linear elasto-plastic solid without hardening has equations:
$$\sigma_{ij} = \sum_{k,l} C_{ijkl}(\varepsilon_{kl}-\varepsilon_{kl}^{(p)})$$
where $\sigma_{ij}$ are stress components, $\varepsilon_{ij}$ strain components, $\varepsilon_{kl}^{(p)}$ plastic strain components, and $C_{ijkl}$ elastic constants (depending on the type of anisotropy of the solid). The previous equation describes the elastic/reversible regime. For the irreversible or plastic regime you need to specify a yielding surface:
$$\phi(\sigma_{ij},\varepsilon_{ij}^{(p)},\dots) \le 0$$
For $\phi < 0$ you are in the elastic region, for $\phi = 0$ you need to specify a plastic flow rule $\dot{\varepsilon}_{ij}^{(p)} = f_{ij}(\sigma_{ij},\varepsilon_{ij}^{(p)},\dots)\dot{\phi}$
At some point of strain you reach the yield surface, then thermodinamically irreversible transformations occur. This yielding is thermodinamically irreversible, so entropy increases, this can not be achieved by means of a mechanical potential, because mechanical energy is NOT conserved (general energy > mechanical energy), only in the elastic regime you have conversation of mechanical energy. Because of this, for sure you can not obtain a plastic theory in the way you suggests, much less to construct a quantum theory of plasticity.
