Dislocation Dynamics(DD) Background

The formation and interaction of dislocations in metals gives rise to plastic behavior in the crystalline structure. By simulating the behavior of dislocations metal plasticity can be modeled and the yield and hardening in plastic flow can be understood. A robust model that can capture yield and hardening of an engineering part has countless applications within industry and cannot be understated in terms of applicability to every industrial sector involved in design and manufacturing.

Traditionally direct experimental measurement of the macroscopic response of the material was considered the only way to understand crystalline plasticity. This largely comprises destructive testing or relating standard tensile specimens that would not capture geometry specific yielding. Interest in developing and modeling this data with constitutive laws and equations of state has led to the ability to simulate multiple dislocations. As the relevance of crystal plasticity has increased, so too has the level of interest in developing a modeling system. DD is still a relatively recent development in computational materials science^[1].

DD computational modeling begins with a representative volume of the material on the order of a few micrometers within a single crystal. The material considered is assumed to be elastic, homogeneous, and isotropic which should be considered in the validation of the model. In fact, these key assumptions can be logical sources of error when comparing simulation outputs to experimental data.

There are other notable limitations within the current methods of modeling DD. Dislocations are modeled as discrete 2D discontinuities in an elastic continuum and neglect the time-dependent nature of elastic materials. This leads to inadequate simulations of plastic relaxation led by the assumption that the material instantaneously acquires the shape and magnitude predicted^[2].

Inputs to DD Model

An accurate plasticity model requires the inclusion of dislocation mechanics to develop accurate hardening principles. Dislocation mechanics is also important for the determination of other mechanical properties such as ductility, fracture, fatigue, and creep ^[1]. Dislocation mobility is modeled by a phenomenological law which involves local parameters related to dislocations, such as slip system, line character, stress, temperature, and composition influencing the steady-state velocity of a dislocation ^[2]. LAMMPS provides the capability to determine dislocation mobility on the atomic level. Information from simulations from LAMMPS is bridged as in input into a model for multiple dislocations.

Driving Equations of State

Within the DD model, a collection of arbitrarily curved dislocation lines is initialized and randomly oriented. The positions of the lines and connected nodes move over time based on the Newtonian-like equation of motion given by the equation^[1].

Where:

Variable	Description
$m_s$	Effective mass of the dislocation line segment which is a function of transverse and longitudinal sound speed of the material, dislocation segment velocity, shear modulus, Berger’s vector, and internal and external cutoff radii for nonlocal continuum mechanics
${\dot{v}}_s$	Acceleration of the dislocation line segment
$\frac{1}{M_s}$	Viscous drag coefficient which is determined from MD simulations and is a result of electron and phonon drag
$v$	Dislocation line segment velocity
$F_{\text{Peierls}}$	Force opposing dislocation motion due to lattice friction
$F_{\text{dislocation}}$	Net force from all other dislocations
$F_{\text{self}}$	Force from 2 neighboring dislocations segments
$F_{\text{external}}$	Force due to externally applied load
$F_{\text{obstacle}}$	Force due to dislocation and obstacle interaction
$F_{\text{image}}$	Boundary condition or elasticity discontinuity image force

To fully understand the 'black box' approach the software is built upon, a background understanding of the model and the assumptions on which it is built is particularly useful.

DD simulations are complex and time dependent due to their motions and interactions. A robust model needs to account for slip systems involved, dislocation line senses, and approach trajectories. This information feeds into resulting interactions including annihilation, junctions, formation of jogs and dipoles, and trapping and pinning^[1]. The model is based on sets of rules derived from observed behaviors within MD simulations^[1].

Forest Hardening

Because plastic hardening is governed largely by dislocation interactions and DD models rely upon randomly distributed Frank-Read sources, a significant amount of time must first pass before the material enters into what is known as the Forest Hardening regime^[1]. Forest dislocations are important because they have the greatest effect on strength and represent the interaction of gliding dislocations and dislocations that cut through the glide plane^[3]. This becomes the dominant interaction that is responsible for yield strength and is shown to effect yield strength.

BCC Metals

When modeling BCC metals in DD it is important to understand that the properties of the metal are greatly affected by temperature and caution should be used when evaluating results. Since much of the DD model code is considered a ‘black box’ for the user, sometimes anomalous outputs can result. For BCC metals, at high temperatures properties will resemble FCC metals, however, at lower temperatures flow properties are governed by screw dislocations which have a high intrinsic Peierls barrier and low mobility^[3]. Previous length scale simulations may not account for these relationships.

The behavior of BCC metals further is complicated by modeling assumptions like isotropic properties. It is known that BCC metals have asymmetric core structures and that dislocation-dislocation interactions will vary significantly between slip systems^[4]. It is also knowns that the dislocation line tension is not constant nor isotropic and the distribution of dislocations is also not random^[3]. As such, there are a number of key assumptions that are built into the model which will require careful calibration and validation in order to have confidence in the output.

Additionally, there are distinct differences between BCC and FCC metals that should be considered:

1. Slip occurs in the closest pack <-1,1,1> direction; however, identifying slip planes in which slip occurs is not straightforward^[5]
2. No stable stacking faults are found in BCC metals and twinning and adiabatic shear banding occur on specific planes at low temperature and high strain rates^[5]
3. Yield strength and flow properties are more highly dependent upon temperature and strain rate in BCC metals when compared to FCC^[5]

Ductile Brittle Transition Temperature

Additionally, it is relevant to consider model assumptions and known properties of materials to determine how appropriate model inputs align with physical properties of the element. There are a number of noteworthy assumptions in each of the modeling code that should be compared to literature values but perhaps one of the most relevant is the ductile brittle transition temperature (DBTT) of the material^[6].

For example, the element Chromium (Cr), has a very high DBTT which is unexpected since it is a metal^[6]. This physical response is not well characterized and is heavily influenced by impurities in the material. The DBTT of a single crystal of Cr is more similar to a traditional metal (-78 to -196 °C) and would likely be appropriate, however, for unalloyed recrystallized chromium with commercial purity the DBTT is 150°C. This is known to be a result of the structure itself as well as impurities, however there is not agreement within the field as to which impurity is contributing to the increased DBTT. The high DBTT has severely limited the usability of Cr as a pure material as well as an alloy. This, however, does explain the use of Cr as an alloying element in steel as well as a plating material to increase the materials hardness.

When evaluating the models available for dislocation dynamics, one of the prerequisites must be the material is inherently ductile. Materials with a high DBTT would not be appropriate materials to study in this formalism.

Bridge to Next Length Scale

The output of this step is a direct feed of the Voce hardening parameters into finite element model for crystal plasticity. This model will allow for relationships due to temperature and particles, but this data would need to be collected in order to fit and calibrate to the DMGfit software. Due to the limitations within the MSU DMG model, there likely would be better models available that can capture the physical properties of materials with a high DBTT. Since this model is based on determining plasticity from dislocations which are properties of ductile materials, this is likely not an appropriate methodology to bridge to the continuum of Cr. Further work in this field might include a gate that is based upon the DBTT of the material in question to determine applicability with the model so that the physics of the material can be consistent with model parameters. However, in the case of Cr, there have been many theories to describe the brittle fracture of Cr at ambient temperature, but it remains an unresolved issue. As a result, it will be difficult to create a basis in a continuum model on plasticity when the mechanisms of brittle fracture are much debated and not agreed upon by investigators^[6].

References

1. ↑ M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. Wiley, 2012. [Online]. Available: http://books.google.com/books?id=KCRLnYj1YiYC
2. ↑ B. Gurrutxaga-Lerma, D. S. Balint, D. Dini, D. E. Eakins, and A. P. Sutton, “Dynamic discrete dislocation plasticity,” in Advances in Applied Mechanics, in Advances in Applied Mechanics, vol. 47. Academic Press Inc., 2014, pp. 93–224. doi: 10.1016/B978-0-12-800130-1.00002-3.
3. ↑ J. Morris, “Dislocation Plasticity : Overview,” 2008.
4. ↑ M. Tang, M. Fivel, and L. P. Kubin, “From forest hardening to strain hardening in body centered cubic single crystals: simulation and modeling,” Materials Science and Engineering: A, vol. 309–310, pp. 256–260, Jul. 2001, doi: 10.1016/S0921-5093(00)01764-0.
5. ↑ C. R. Weinberger, B. L. Boyce, and C. C. Battaile, “Slip planes in bcc transition metals,” International Materials Reviews, vol. 58, no. 5, pp. 296–314, Jun. 2013, doi: 10.1179/1743280412Y.0000000015.
6. ↑ Y. F. Gu, H. Harada, and Y. Ro, “Chromium and chromium-based alloys: Problems and possibilities for high-temperature service,” JOM, vol. 56, no. 9, pp. 28–33, Sep. 2004, doi: 10.1007/s11837-004-0197-0.