# Where is quantum plasticity theory?

Plasticity theory (the theory of irreversible deformations) is in the world of engineering, but not in the world of Theoretical Physics is my opinion. It is therefore an entirely classical theory that is formulated in terms of deformation gradients. But are there alternative theories of plasticity theory?

I would say yes. The Hamilton operator for a crystal that consists on $N$ elements (atoms) can be viewed as

$H = \sum_{i=1}^N \frac{p_i^2}{2m_i}+ \sum_{i,j=1, i<j}^N V(x_i-x_j) + \sum_{i \in I}J_ix_i$.

The first term is the total kinetic energy operator, the second term the interatomic potential that vanishes for large differences $x_i-x_j$ (such that fracture can also be described by this theory) and the third term is the external load potential energy; $J_i$ is the external force and $I$ the set of all atoms where the external force acts on. Commutator relations:

$[p_i,p_j] = [x_i,x_j] = 0, [x_i,p_j] = i\hbar \delta_{ij}$.

The Hamiltonian is quite general, so I can use a Taylor expansion of the interatomic potential up to a certain order. If there is no fracture (atoms are only slightly displaced from ich rest position), we can expand e.g. to the 4th order; we can e.g. have terms like $kx^2+ \lambda x^4$ with a sufficiently small anharmonic constant $\lambda$. Is such an approximation enough to describe plastic deformations or do these occur on really high orders of displacements? There results a system of harmonic oscillators with additional terms (often related to scattering of phonons if some performs a perturbation theory).

For a nonperturbative treatment of the potential we can derive a theory of fracture mechanics. Nonperturbative methods are more general; there exists:

• Monte-Carlo method
• Dyson-Schwinger equations

I am interested in the Dyson-Schwinger equations. Can I use that to describe a theory of quantum plastodynamics/ Quantum fracture mechanics?

The interesting point is whether quantum effects like tunneling effect material plasticity. Engineering theories respect tunneling absolutely nothing. Are there scientific results that examined the effect of tunneling in mechanical material behavior?