Crystal Plasticity Finite Element Methods

Mechanics of Molecular Crystal

Molecular crystalline materials have molecules, instead of atoms, at their lattice points. Unlike ionic or covalent crystals, molecular crystals are held together by weak van der Waals forces, dipole-dipole interactions or hydrogen bounding. A variety of materials with a wide range of physical, chemical, and mechanical properties are molecular crystalline in nature. For a long time, molecular crystals had been ignored as engineering or structural materials, therefore, our understanding of the physical and chemical properties of such materials is not well developed. Recent interest in these materials has been spurred by their use in energetic, electronic, photonic, and bio-nanoporous applications. We have developed CPFE models to study the thermomechanical response as well as mechanical characteristics of these crystals.

Bionanoporous Molecular Crystals

Bionanoporous materials such as proteins crystals are highly ordered three-dimensional structures, in which the protein molecules bind to each other with specific intermolecular interactions. Recently, protein crystals have emerged as promising bionanoporous materials for different applications including highly selective biocatalysis, biosensing, bioseparation, vaccine formulation, and drug delivery. Over the past decade, significant research and development efforts have been focused on engineering protein crystals, efficacy testing, model development, and production and characterization. Despite these successes, many challenges associated with the characterization of protein crystals including stability remain. The environmental working conditions require the protein crystals to be both chemically and mechanically stable. The chemical stability of protein crystals has been the subject of intense research but our understanding of mechanical stability of protein crystals under different conditions still remains largely obscure. We have developed a CPFE model to explore the anisotropic mechanical response of protein crystals with tetragonal lysozyme as a model. Our investigation reveals that the mechanical strength of the lysozyme crystal is significantly sensitive to temperature and the amount of intracrystalline water molecules (Figure 1).

Figure 1. Left subplot shows the effect of the amount of intracrystalline water molecule on critical resolved shear stresses (CRSS) of slip systems in Lysozyme crystal. The effect of temperature on critical resolved shear stresses is presented in the right panel.

Plasticity of Hydroxyapatite Single Crystals

Bone is a composite material with an organic matrix and inorganic minerals arranged in a hierarchy of structures spanning several length scales. At the nanometer length scale, for example, the structure consists of self-assembled collagen brils and inorganic hydroxyapatite nano-crystals. The mechanical properties of the organic and inorganic phases together with their hierarchical arrangement impart bone its characteristic strength and toughness. There is therefore much interest in characterizing the mechanical properties of hydroxyapatite, not only for its importance in the overall mechanical behavior of bone in disease and in health but also for its widescale application in biomaterials, regenerated hard tissue and in medicine.

We have developed a CPFE based computational model to identify the mechanical properties of hexagonal hydroxyapatite single crystal. The hardness and Young’s modulus obtained from the nanoindentation experiments are used to compute yield stress and flow parameters. It is observed that hydroxyapatite single crystals exhibit anisotropic mechanical response (Figure 2). The stress–strain curves extracted here could be used for developing constitutive models for hydroxyapatite single crystals.

Figure 2. Stress–strain response of the hydroxyapatite single crystal extracted from nanoindentation data.

Relevant Publications

1. Zamiri, A., and De, S. (2009). Modeling the mechanical response of tetragonal lysozyme crystals. Langmuir, 26(6), 4251-4257.
2. Zamiri, A., and De, S. (2011). Mechanical properties of hydroxyapatite single crystals from nanoindentation data. Journal of the Mechanical Behavior of Biomedical Materials, 4(2), 146-152.

Energetic Molecular Crystals

Molecular crystalline materials such as HMX, RDX, PETN, and Fox7 are widely used as energetic materials in solid propellants and explosives. For these high explosives, the shock sensitivity of initiation and sustained reaction is known to be controlled by processes occurring at the crystal level. For example, the threshold to reaction is greatly influenced by changes in crystal morphology, defect content and size distribution.

We have developed a continuum slip-theory based thermomechanically consistent model that is capable of predicting the highly anisotropic, orientation and size dependent response of the energetic molecular crystals at high strain rate. Using this model, we elucidate the role of anisotropic deformation in sensitivity of energetic materials such as β-HMX (Figure 3) and α-RDX (Figure 4). We further propose to engineer polycrystal texture to control the sensitivity of the energetic materials.
We further extended the single crystal model to include phase-transformation within a thermodynamically consistent continuum scale sharp- and diffuse-interface models that account for the microstructural effects and physics of high loading rate. The objective was to investigate the role of shock-induced solid-solid phase transformation on the deformation mechanisms and associated sensitivity of crystalline energetic materials. Using this model, we studied the role of shock-induced α-γ phase transformation in the sensitivity of RDX. An interesting observation was that the α-γ phase transformation requires finite time to occur and it is associated with increased temperature rise and hence the shock-sensitivity, when compared to the α-polymorph of RDX (Figure 5). In addition he showed that the shock response of α-γ phase transition of RDX shows orientation dependence.

Figure 3. Simulation results for the 100 randomly oriented grain texture of β-HMX at each integration point showing temperature (K), pressure (MPa), and average cumulative plastic slip.

Figure 4. Particle velocity profile of the α-RDX single crystal showing anisotropic and pressure dependent response. Solid lines correspond to experimental results (Hooks et al., 2006; Cawkwell et al., 2010) while dashed lines correspond to model predictions.

Figure 5. Shock induced α-γ phase transformation of RDX for impact loading on the (100) plane. Phase evolution, uniaxial compressive stress (GPa), and temperature (K) are shown at 0.6μs. For comparison the stress and temperature contours of α-RDX are also shown.

Relevant Publications

1. Josyula, K., Rahul, and De, S. (2017). A level sets approach for shock-induced α-γ phase transition of RDX. Computational Mechanics, DOI: 10.1007/s00466-017-1493-1.
2. Rahul, and De, S. (2017). A phase-field model for shock-induced α-γ phase transition of RDX. International Journal of Plasticity, 88, 140-158.
3. Josyula, K., Rahul, and De, S. (2016). Quasi-static Response and Texture Evolution of α- and γ-polymorphs of Cyclotrimethylene Trinitramine: A Comparative Study. Philosophical Magazine. 96, 1790-1808.
4. De, S., Zamiri, A. R., and Rahul (2014). A fully anisotropic single crystal model for high strain rate loading conditions with an application to α-RDX. Journal of the Mechanics and Physics of Solids, 64, 287-301.
5. Zamiri, A. R., and De, S. (2011). Multiscale modeling of the anisotropic shock response of β-HMX molecular polycrystals. Interaction and Multiscale Mechanics, 4(2), 139-153.
6. Zamiri, A. R., and De, S. (2011). Modeling the Anisotropic Deformation Response of β-HMX Molecular Crystals. Propellants, Explosives, Pyrotechnics, 36(3), 247-251.
7. Zamiri, A. R., and De, S. (2010). Deformation distribution maps of β-HMX molecular crystals. Journal of Physics D: Applied Physics, 43(3), 035404.

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Radiation Material Science

One of the most difficult challenges in computational materials science is to simulate, accurately and efficiently, the interaction of radiation with matter (particle transport or radiation transport processes). However, such understanding is essential to diffuse threats of nuclear terrorism (WMD), develop next generation (Gen IV) nuclear reactors and nuclear submaries. In this project, we adopt a multiscale computational strategy that links twenty orders of magnitude time scale and ten orders of magnitude spatial scale employing ab intio, molecular dynamic, Monte Carlo and multilevel finite element methods in conjunction as well as in isolation to address this problem.

Multiscale Modeling of the Effects of Neutron Irradiation Metals

The goal of this research is to predict the mechanical response of face-centered cubic (FCC) and body-centered cubic (BCC) metals under neutron irradiation at the continuum level. While it is known that material properties degrade as a result of neutron irradiation, recent experimental evidence indicates that FCC and BCC metals behave in different ways which are yet to be explained. The drawback of existing methods is that they assume some form of constitutive equation at the continuum level, which is not straightforward for complex nonlinear processes such as material yield and is limited to the regime of their calibration. The objective of our approach is to eliminate this major deficiency and allow seamless coupling between disparate time and length scales without assuming any explicit constitutive format at the continuum scale. To this end, we have developed a Jacobian-free Multiscale Method (JFMM) that is implemented within the framework of large deformation crystal plasticity. Using this multiscale modeling approach, we studied mechanical response of neutron-irradiated metals using a set of constitutive models that we have proposed to capture the post-irradiation deformation response of OFHC copper and molybdenum, see Figure 6.

Figure 6. Left panel compares the nominal tensile stress vs strain plots for irradiated OFHC copper with varying radiation dose. The solid line represents computational results, whereas, the markers represent experimental data from Singh et al. (2001). Right panel presents the true stress-strain curves for Molybdenum neutron-irradiated at 80 degC and tested at 100 degC at a strain rate 1x10-3s-1. Experimental data obtained from Li et al. (2008).

Relevant Publications

1. Rahul, and De, S. (2014). Multiscale modeling of irradiated polycrystalline FCC metals. International Journal of Solids and Structures, 51(23), 3919-3930.
2. Krishna, S., and De, S. (2011). A temperature and rate-dependent micromechanical model of molybdenum under neutron irradiation. Mechanics of Materials, 43(2), 99-110.
3. Krishna, S., Zamiri, A., and De, S. (2010). Dislocation and defect density-based micromechanical modeling of the mechanical behavior of fcc metals under neutron irradiation. Philosophical Magazine, 90(30), 4013-4025.