# Coherent Multiscale Hierarchical Modeling Scheme (CMHM) for Hierarchically Structured Titanium-Boron Based Armor System

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## Coherent Multiscale Hierarchical Modeling Scheme (CMHM) for Hierarchically Structured Titanium-Boron Based Armor System

### Problem Statement

A critical factor in the design of protective armor systems is the ratio of the weight of the armor to its impact absorbing capacity. For some armor applications, maximizing this design variable is the most important consideration in design optimization. While a variety of steels have been used in armor systems successfully over the years, the weight of Iron as the primary material has limited absorption to weight ratio. Changing the base material to titanium can improves the energy absorption to weight ratio. Further improvements to the energy absorption ration can be made by implementing a hierarchical structural design.

### LENS Fabrication Process

The LENS fabrication process can be applied to achieve a hierarchical structure. Laser Engineered Net Shaping (LENS) is a fabrication process whereby material is added to a part by melting a pool (called a weld pool) on the surface with a focused laser beam and then injecting a stream powdered filler material into the molten pool at the focal point of the beam. The laser continues to heat the weld pool enough to mix with and melt the injected powder. As the laser moves to heat another location, the weld pool cools and solidifies along with the added material. A computer numerical controlled (CNC) table moves the part beneath the focal point of the laser beam and powder stream allowing precise placement the new material. In addition to controlling the placement of new material on the part, the LENS system can also control the type of material added by selecting which powder to inject into the weld pool. This capability of selectively controlling filler material can be used to vary the composition of a part from location to location by changing the powder.

### Titanium Boron System

The addition of boron to titanium results in the production of the chemical phase Titanium Boride (TiB) which possesses an orthorhombic crystal structure. TiB has a higher elastic modulus than the Titanium matrix in which it forms while maintaining similar density. Because the acoustic speed of a material is proportional to the radical of the ratio of the Elastic modulus to the density, an increase in the volume fraction of TiB leads to an increase in the acoustic speed of the material. For armor applications, variations in the acoustic speed within a material can control the flow of energy during a dynamic impact event, thus affecting energy dissipation. Some patterns of spatial variation in acoustic speed will be deleterious to impact energy absorption while others may be beneficial. By combining the composition control capabilities of the LENS process with the Titanium and Titanium Boride material system, a material can be formed that exhibits a controlled spatial variation of TiB volume fraction. The resulting material will consequently have spatial variations in acoustic speeds.

### Multiscale Analysis Proposal

To predictively analyze performance of a structural scale armor assembly composed of a controlled hierarchical distribution of TiB volume fraction, a multiscale analysis is proposed to obtain the necessary modeling parameters to fully describe a system and predict its behavior under impact loading conditions.

#### Electron Scale

DFT calculations will be performed on the electronic scale to obtain elastic moduli for each constituent phase: α-Titanium (HCP), β-Titanium (BCC), Titanium Boride ^{[1]} (orthorhombic), and Titanium Diboride ^{[2]} (HCP). The interatomic potentials will be handed up to the atomic scale simulation. The resulting moduli will be available to higher length scales for subsequent calculations of bulk properties applying the rule of mixtures. The generalized stacking fault energy will be used to determine the crystallographic anisotropy of the yield surface for each phase. The surface formation energy will be used for determining the evolution of damage at the micromechanical scale.

#### Nanoscale

For the third and seventh length scale bridge, the nanoscale simulation will use the code microMegas and will model the dislocation interaction within the constituent phases. The simulation needs stacking fault energies, moduli, and surface formation energies from the atomistic scale simulation. The result will be the hardening parameters for the plastic deformation for each phase. This information will be handed up to the next scale to model polycrystalline hardening and damage evolution.

#### Mesoscale

For the fourth and eighth length scale bridge, the mesoscale simulation will use the visco-plastic self-consistent (VPSC) model to evaluate the polycrystalline plasticity behavior of variable combinations of volume fractions of all phases.

#### Macroscale

Bridge ten represents the continuum level Internal State Variable (ISV) model comprised of model constants obtained from Bridges five through nine. The ISV model can then be implemented in a macroscale ABAQUS FEA calculation to model the behavior of layers containing weld passes of varying TiB volume fraction.

#### Structural Scale

Finally, bridge eleven provides ISV model constants for structural scale FEA calculations using the ABAQUS code.

#### References

- ↑ Panda, K. B. and K. S. R. Chandran (2006). "First principles determination of elastic constants and chemical bonding of titanium boride (TiB) on the basis of density functional theory." Acta Materialia 54(6): 1641-1657.
- ↑ Panda, K. B. and K. S. Ravi Chandran (2006). "Determination of elastic constants of titanium diboride (TiB2) from first principles using FLAPW implementation of the density functional theory." Computational Materials Science 35(2): 134-150.