**Continuum Models of Dislocations and Plasticity**

The network of dislocations (together with all microstructural defects) essentially determines the irreversible deformation of crystalline solids. Dislocations are nucleated on the crystal's microscale upon plastic deformation, they slide and climb through the crystal lattice to accommodate plastic flow, they attract, repel or annihilate each other, they interact with other lattice defects such as vacancies or interstitials, and they form complex networks and pile up at grain boundaries and obstacles. Models to describe those myriad interactions must find a compromise between low computational expenses on the one hand and a high degree of accuracy (as well as transferability and predictiveness) on the other hand. Among the available methods to describe and simulate the formation and subsequent evolution of the complex dislocation network upon plastic deformation, we employ - among others - continuum theories which employ methods of continuum mechanics and are based on thermodynamic energy principles to model the microstructural network.

One approach involves the use of continuum dislocation theory. We describe the plastic deformation in terms of the plastic slip on specific, active slip systems of the crystal, from which one can obtain Nye's dislocation density tensor that contains information about the dislocation density at a material point. The energetic model is based on the principles of minimum potential energy and of minimum dissipation potential (which, for our purposes, is equivalent to Biot's law). To this end, we introduce the stored energy density as the sum of elastic energy and energy of the dislocation network, and we seek novel representations of the dislocation network energy. In the past, we have made use of the formulation proposed by Berdychevksi (2003) for the energy of the dislocation network, which contains a saturation behavior: the dislocation density cannot grow arbitrarily but is bounded by a saturation dislocation density. Our physics-based energetic model falls within the class of strain-gradient plasticity approaches. The model has been applied successfully to various interesting problems of crystal plasticity.

Analytical models for the pile-up of dislocations at the boundaries of single-crystal thin strips and bicrystals have been derived, which are capable of reproducing a clear size effect (typical to problems of crystal plasticity), Bauschinger translational work hardening, and a characteristic hysteresis behavior upon cyclic loading. Numerical simulations have investigated the pile-up of dislocations at the grain boundaries in polycrystals and confirmed a size effect of Hall-Petch type for various grain geometries and arrangements. The continuum dislocation approach has also been applied to model the dynamic continuous recrystallization e.g. during the process of equal channel angular extrusion. Finally, the model has been applied to deformation twinning, where the onset of plastic flow and of twinning have been determined analytically for simple model cases, and the complete stress-strain behavior has been simulated and studied. For hcp materials such as magnesium, we have coupled the continuum dislocation approach to a phase field description for deformation twinning in order to study the complex deformation mechanisms in Mg and Mg alloys.

Current research investigates the complex interaction mechanisms of dislocations in crystals from several viewpoints (both on the atomistic scale and on the continuum level), where the exact representation of the energy stored in the dislocation network is of essential interest. Furthermore, the interaction of dislocations with grain boundaries, twin boundaries and other defects is the subject of ongoing research with a particular focus on magnesium.

#### Schematic dislocation animations:

A few schematic animated illustrations of edge dislocation motion through a crystal lattice can be found
**here**.

#### References:

- Y. Chang, D. M. Kochmann.
A variational constitutive model for slip-twinning interactions in single- and
polycrystalline magnesium,
*Int. J. Plasticity***73**(2015), 39-61. - B. Klusemann, D. M. Kochmann.
Microstructural pattern formation in crystal plasticity during cyclic loading: relaxation vs. gradient plasticity,
*Comput. Methods Appl. Mech. Engrg.***278**(2014), 765-793. - D. M. Kochmann, K. C. Le,
**An energetic approach to deformation twinning**, in: K. Hackl (Ed.),*Variational Concepts with Applications to the Mechanics of Materials*, IUTAM Bookseries, Springer (2010), pp. 141-155. - D. M. Kochmann, K. C. Le,
**Continuum dislocation theory and related size effects in crystal plasticity**, in: H. Geelvinck, S. Reynst (Eds.),*Advances in Condensed Matter and Materials Research*, Vol. 7, Nova Science Publishing (2010), pp. 537-591. - D. M. Kochmann, K. C. Le,
**Plastic deformation of bicrystals within continuum dislocation theory**,*J. Math. Mech. Solids***14**(2009), 540-563. - D. M. Kochmann, K. C. Le,
**A simple model for dynamic recrystallization during severe plastic deformation**,*Arch. Appl. Mech.***79**(2009), 579-586. - D. M. Kochmann, K. C. Le,
**A continuum model for initiation and evolution of deformation twinning**,*J. Mech. Phys. Solids***57**(2009), 982-1002. - D. M. Kochmann, K. C. Le,
**Dislocation pile-ups in bicrystals within continuum dislocation theory**,*Int. J. Plasticity***24**(2008), 2125-2147.