## Simulation of complex elasto-plastic materials

For many materials of scientific and technological importance, accurately modeling them as they deform and move can pose fundamental challenges. Oftentimes, the physical basis of the material may be poorly understood, and there may be significant technical hurdles, such as dealing with multiple mechanical responses at vastly different time and length scales, or with the coupling of different phases, each with their own behavior. In my research, I have been very interested in developing new mathematical and computational techniques to address these challenges.

## Modeling of bulk metallic glasses

Snapshots of (a) effective temperature, (b) pressure, and (c) deviatoric stress in an STZ simulation of a bar being stretched. Click on each image to see a corresponding movie.

In collaboration with James Langer and Frédéric Gibou at UC Santa Barbara, I have worked on understanding the mechanical properties of bulk metallic glasses (BMGs), a certain class of metallic alloy where the atoms are arranged in an amorphous structure, as opposed to a regular crystalline arrangement that is typical of most metals. BMGs exhibit many superior properties, such as high tensile strength and excellent corrosion resistance, and are under consideration for a variety of advanced technological applications. Over the last fifteen years, Langer and co-workers have developed the shear transformation zone theory, which provides a promising framework for modeling BMGs, based on a first-principles approach utilizing techniques from statistical mechanics; one the key principles behind the model is the use of an effective temperature, which quantifies the amount of configurational disorder in the atoms that make up the BMG. The theory has been compared extensively to experimental data, but there is an important need to be able to test the theory on problems that are too complex to be analyzed theoretically. I developed an Eulerian finite-difference simulation framework to be able to test the theory in complex situations, and in an initial study, I was able to examine the underlying physical principles of the model by looking the necking instability in a bar being stretched.

Following this, I collaborated with Eran Bouchbinder to address an unsolved problem in BMG mechanics. Experiments show that BMGs can undergo a dramatic reduction in toughness due to compositional variations and heat treatments, leading to abrupt failure, severely limiting their use in structural applications. To study this, we considered crack propagation, a major vehicle for material failure. Simulating this process is challenging and cannot be carried out with conventional methods, due to the interaction of multiple disparate time scales. To address this, I devised a new numerical scheme for quasi-static elasticity, coupled to adaptive time-stepping for plasticity.

Using this method, we were able to explain the experimentally observed reduction in toughness in terms of a new instability that differentiates between two different regimes, which arises from an interplay between the mechanical response of the BMG and the geometry of the crack tip. The two images shown are snapshots of the effective temperature field in a simulation of crack initiation in a metallic glass. The metallic glass on the left is initially more relaxed, due to a longer heat treatment, than the metallic glass on the right. The very different crack tip shapes and deformation patterns under the same external conditions result in a significantly reduced breaking resistance for the more relaxed glass.

#### References

- C. H. Rycroft and F. Gibou,
*Simulations of a stretching bar using a plasticity model from the shear transformation zone theory*, J. Comput. Phys.**231**, 2155–2179 (2012). [Link] [Preprint] - C. H. Rycroft and E. Bouchbinder,
*Fracture toughness of metallic glasses: Annealing-induced embrittlement*, Phys. Rev. Lett.**109**, 194301 (2012). [Link] [arXiv preprint] *Modeling the Breaking Points of Metallic Glasses*, Berkeley Lab News Center, November 26, 2012. (Also on CRD News and EurekAlert.)- C. H. Rycroft, Y. Sui, and E. Bouchbinder,
*An Eulerian projection method for quasi-static elastoplasticity*, submitted to J. Comput. Phys. [Preprint]

## Reference map technique for finite strain elasticity and fluid–structure interaction

I worked with Ken Kamrin to develop the same Eulerian finite-difference and level set framework to carry out simulations of large-strain elasticity laws. Typically, problems of this type would be carried out using a Lagrangian, deforming mesh, but in certain cases, such as when coupling to fluid mechanics or imposing an incompressibility constraint this has limitations. To circumvent this, an additional “reference map” field can be introduced into an Eulerian simulation. The reference map is simple to implement, and it allows for the material deformation to be calculated, which can then be used to evaluate a given constitutive law. The method provides a flexible framework for carrying arbitrary non-linear elasticity, and can be adapted to study fluid–structure interaction problems.

The three figures above show example computations using the method, of deforming neo-Hookean elastic bodies immersed within a fluid flow. The colors correspond to pressure, with yellow being a positive pressure, and black being negative pressure. Click on each image to see a movie of the simulation.

#### References

- K. Kamrin, C. H. Rycroft, and J.-C. Nave,
*Reference map technique for finite-strain elasticity and fluid–solid interaction*, J. Mech. Phys. Solids**60**, 1952–1969 (2012). [Link]