# A Multi-Length Scale Approach to Capturing the Effects of Shear Deformation for a Ductile Crystalline Material.

## Contents |

### Background

The purpose of this model is to accurately capture the effects of shear deformation for a multi-layered metallic structure during a ballistic impact event. The particular deformation mechanisms of interest are void growth and fracture initiation due to shear and strain localization due to shear mechanisms. A multi-scale approach is employed to deliver pertinent information from lower length scales to the macroscale model.

### Downscaling Requirements

The goal for the model is the ability to predict macroscale observable state variable (OSV) behavior, namely the material stress – strain relationship. The Mississippi State Plasticity – Damage model ^{[1]} employs internal state variable (ISV) constitutive relations to define lower length scale effects on the macroscale stress – strain material behavior. In order to define the macroscale damage and plasticity evolution, information is required from mesoscale deformation mechanisms. The mesoscale properties of interest are strain hardening for plasticity effects and void growth for damage mechanisms. In order to accurately capture the mesoscale plasticity and damage mechanisms, the behavior of microscale dislocations must be considered. In order to model dislocation motion, dislocation mobility calculations are necessary from the atomic scale. Atomistic scale calculations and simulations require interatomic energy relationships from the electron scale.

### Upscaling Information and Methods

*Electron Scale to Atomistics*

Atomic interfacial energy information is required to accurately describe atomic motion within a material. Density Functional Theory (DFT) calculations can be used to determine atomic equilibrium energy, lattice parameter, and modulus properties ^{[2]}. The electron scale information appears at the macroscale model in the form of the elastic moduli parameters. Software, such as VASP ^{[3]}, can be used to calculate the pertinent electron scale parameters of interest for an atomic configuration. VASP simulations can also generate numeric interatomic energy and generalized stacking fault energy data for a material. The software package LAMMPS^{[4]} uses Modified Embedded Atom Method (MEAM)^{[5]} principles to calibrate atomic parameters to fit interatomic energy and generalized stacking fault data. The atomic parameters can be subsequently used in dislocation core displacement simulations.

*Atomistics to Dislocation Dynamics*

The pertinent information for material deformation at the atomic length scale is the dislocation core mobility. The LAMMPS software package can be used to generate dislocation core displacement data. The dislocation core displacement information is used to calculate dislocation drag and dislocation mobility^{[6]}. With the dislocation mobility information acquired, dislocation dynamics simulations can be run using MDDP^{[7]} to generate microscale stress–strain curves. The microscale stress–strain behavior is pertinent to material hardening.

*Dislocation Dynamics to Crystal Plasticity*

The microscale stress–strain information from MDDP simulations can be used to calibrate sets of parameters that describe material hardening^{[6]}. Other properties of interest at the microscale are strain localization, void growth, and void coalescence. Nonlocal ISV formulation in a thermodynamically coupled model has been used to accurately capture strain localization and dislocation density evolution^{[8]}^{[9]}. Crystal plasticity simulations using Abaqus software^{[10]} and experiments can be performed to produce microscale stress–strain information.

*Crystal Plasticity to Macroscale Model*

The stress – strain information generated from the crystal plasticity scale can be used to calibrate parameters in the Mississippi State Plasticity-Damage model to describe a material’s macroscale deformation behavior. The effects of strain localization and other heterogeneous deformation modes can be brought into the model through non-local microscale ISV’s and full thermodynamic coupling of stress– like and strain–like terms^{[8]}^{[9]}. The effects of void growth under shear can be captured by considering the ratio of the second and third deviatoric stress invariants in the void growth ISV formulation^{[11]}^{[12]}.

### References

- ↑ Bammann, D. J., Chiesa, M. L., Horstemeyer, M. F., and Weingarten, L. I., “Failure in Ductile Materials Using Finite Element Methods,” Elsevier, Amsterdam, 1993.
- ↑ Morse, P.M., “Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels,” Physical Review, Vol. 34, 1929.
- ↑ Vienna Ab initio Simulation Package. [Online]. Available: http://www.vasp.at
- ↑ LAMMPS Molecular Dynamics Simulator Documentation. [Online]. Available: http://lammps.sandia.gov/doc/Manual.html
- ↑ Baskes, M. I., “Application of the Embedded-Atom Method to Covalent Materials: A Semiempirical Potential for Silicon,” Phys. Rev. Let, Vol. 59, No 23, 1987, pp. 2666-2669.
- ↑
^{6.0}^{6.1}Horstemeyer, M.F., “Case Study: Conducting a Structural Scale Metal Forming Finite Element Analysis Starting From Electronics Structures Calculations Using ICME Tools,” Integrated Computational Materials Engineering (ICME) for Metals, A John Wiley & Sons, Inc, Hoboken, NJ, 2012, pp 379-409. - ↑ Zbib, H.M., Rhee, M., and Hirth, J.P., “3D Simulation of Curved Dislocations: Discretization and Long Range Interactions” Advances in Engineering Plasticity and Its Applications, eds. Abe and T, Tsuta, Pergamon, NY, 1996, pp. 15-20
- ↑
^{8.0}^{8.1}Voyiadjis, G.Z., Abu Al-Rub, R.K., Palazotto, A.N., “Non-Local Coupling of Viscoplasticity and Anisotropic Viscodamage For Impact Problems Using The Gradient Theory,” Archives of Mechanics, Vol. 55, No. 1, 2003, pp. 39-89. - ↑
^{9.0}^{9.1}Voyiadjis, G.Z., Abu Al-Rub, R.K., Palazotto A.N., “Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for localization problems using gradient theory,” International Journal of Plasticity, Vol. 20, 2004, pp. 981-1038. - ↑ Abaqus. [Online]. Available: http://academy.3ds.com/software/simulia/abaqus-student-edition/.
- ↑ Horstemeyer M.F., Gokhale, A.M., “A void-crack nucleation model for ductile materials,” International Journal of Solids and Structures, Vol 36., 1999, pp. 5029-5055.
- ↑ 12. Nashon, K., Hutchinson, J.W., “Modification of the Gurson Model for Shear Failure,” European Journal of Mechanics A/Solids, vol. 27, 2008, pp. 1-17.