A Multi-Phase Modeling Framework Suitable for Dynamic Applications

Abstract

Under dynamic loading conditions and the associated extreme conditions many metals will undergo phase transformations. The change in crystal structure associated with solid–solid phase transformations can significantly alter the subsequent mechanical response of the material. For the interpretation of experiments involving dynamic loading it is beneficial to have a modeling framework that captures key features of the material response while remaining relatively simple. We introduce a candidate framework and apply it to the metal tin to highlight a range of behaviors that are captured by the model. We also discuss potential extensions to capture additional behaviors that could be important for certain materials and loading scenarios. The model is useful for analysis of results from dynamic experiments and offers a point of departure for more complex model formulations.

1. Introduction

With a variety of experimental platforms now able to probe dynamic phase transformations there is significant interest in modeling frameworks that can facilitate interpretation of experiments. Widely varying dynamic loading rates and ranges of conditions can be accessed in experiments driven by gas-guns, lasers, pulsed-power platforms, dynamic diamond anvil cells, and other drivers—for example [1,2,3,4,5,6,7,8,9]. Here we outline a modeling framework that can be used to analyze dynamic experiments with phase transformations, including an approach for tracking the state of the material through transformation. Experimental observations are influenced by material response characteristics including the thermo-elastic response of the material in each phase, which can be modeled in terms of an equation of state (EOS) and a shear modulus, the strength in each phase, and the kinetics for transforming among the various phases. Strength of non-ambient phases can be particularly challenging to infer from dynamic experiments, and additional emphasis is placed here on the strength-related aspects of the modeling framework.

A subset of the modeling framework is described in an earlier report [10], with the report facilitating collaboration among national laboratories working in this area. Given the purpose of supporting collaborative work, the modeling framework was described in [10] as a “Common Model of Multi-phase Strength and Equation of State” or CMMP, and we retain that designation here. That same multi-institution team previously focused on an experimental study of tantalum, which remains in the body-centered cubic phase across a wide range of conditions. Thus, a diverse set of experimental platforms could be brought to bear on a given phase of the material, providing insights into the strength of tantalum over a wide range of conditions [11]. The framework presented here is meant to help support extension of the multi-experimental-platform approach to examination of materials that undergo phase transformations. Even with a multi-platform approach data informing non-ambient phase behavior may remain sparse, and we do not expect that the data will be sufficiently detailed to suggest model forms. The plan is thus to produce a simple framework with few parameters and then to add complexity as needed in the future to capture experimental observations. More elaborate models are certainly possible, particularly if insights into atomic-scale mechanism of transformation are available, as for example in [12,13,14]. Under quasi-static loading conditions, detailed diffraction data can provide relevant information [15], but under dynamic loading experimental diagnostic options are more limited.

While some experiments may be sensitive principally to non-ambient phase strength [16], we are not always able to clearly separate aspects of the material response. As shown in [7], Bayesian model calibration provides one means of evaluating correlations in the inferences about the various portions of the physical model. Once sufficient data are collected for a given material, it may be possible to use approaches such as Bayesian cross-validation to quantitatively compare various functional forms, as has been done for strength models evaluated against quasi-static data [17]. Furthermore, the framework outlined here could provide a starting point for such studies.

The framework is based on pressure and temperature equilibrium of all co-existing phases, combined with deviatoric stress averaging through a volume fraction weighted flow stress and a volume fraction weighted shear modulus. Either phase transformation kinetics or equilibrium phase fractions can be used, with equilibrium phase fractions being appropriate if the transformation kinetics are fast compared to the loading condition. We describe a method for transferring state among the phases. While a similar approach has been used in previous work [18], details of the approach have not previously been provided. After showing example results for application to tin, we discuss potential directions for refinement of the modeling framework.

2. Methods

The CMMP modeling framework is meant to be suited to the simulation of dynamic loading conditions, and it is formulated for use in a “hydrocode” setting [19]. That is, in explicit codes meant to solve transient problems involving large deformations. What makes CMMP simple is that (among other approximations below) it is couched in a traditional framework in which the deviatoric deformation, associated with material strength, and the volumetric deformation, associated with the equation of state, are handled in a weakly coupled manner. As described below, within this weakly coupled setting we employ a multi-phase shear modulus, G, and a multi-phase flow stress Y. The multi-phase representation of the material is centered around the calculation of these variables in terms of evolving phase fractions, assumptions about decomposition of the plastic deformation rate, and methods for evolving phase-specific hardening variables. In Section 4, we will discuss potential extensions to the modeling framework that would produce stronger coupling between the deviatoric and volumetric aspects of the material response.

For each time step and material location, the role of the material model is to update the Cauchy stress, $\sigma$, and any requisite material state variables, $\mathcal{S}$, such as phase fractions and phase-specific effective plastic strain. The host code provides to the material model the velocity gradient, $\mathit{L}=\partial \mathit{v}/\partial \mathit{x}$, zone-level homogenized density, $\rho$, and zone-level specific internal energy per unit mass, e.

2.1. Traditional Model Decomposition

CMMP is based on a typical isototropic, hypoelastic, $J_2$ plastic material modeling approach in conjunction with an EOS for the pressure and temperature response given material density and specific internal energy. This is entirely consistent with conventional strength model implementations into hydrocodes. Aspects of the associated theory are presented briefly for completeness.

Consider the velocity gradient $\mathit{L}$, and its additive decomposition into symmetric $\mathit{D}$, and skew $\mathit{W}$, components

$$\mathit{L}=\frac{\partial \mathit{v}}{\partial \mathit{x}}=\mathit{D}+\mathit{W}$$

(1)

where $\mathit{D}=\frac{1}{2}\left(\mathit{L}+{\mathit{L}}^{\mathrm{T}}\right)$ and $\mathit{W}=\frac{1}{2}\left(\mathit{L}-{\mathit{L}}^{\mathrm{T}}\right)$. The deformation rate $\mathit{D}$ can be further decomposed into deviatoric and volumetric parts

$$\mathit{D}={\mathit{D}}^{\prime }+\frac{1}{3}\frac{\dot{V}}{V}\mathit{I}$$

(2)

where $V=1/\rho$ is the specific volume per unit mass and $\frac{\dot{V}}{V}=\nabla ·\mathit{v}=\mathrm{Trace}\left(\mathit{D}\right)$. Similarly, the total stress tensor is decomposed into

$$\sigma ={\sigma }^{\prime }-P\mathit{I}$$

(3)

where ${\sigma }^{\prime }$ is deviatoric stress and where $P=-\frac{1}{3}\mathrm{Trace}\left(\sigma \right)$ is the pressure related to specific volume and temperature (or internal energy) through an equation of state. The constitutive equations that define the deviatoric stress evolution are [20]

$${\stackrel{\odot }{\sigma }}^{\prime }=2G\left({\mathit{D}}^{\prime }-{\mathit{D}}_{\mathrm{p}}^{\prime }\right)+\frac{\dot{G}}{G}{\sigma }^{\prime }$$

(4)

with ${\stackrel{\odot }{\sigma }}^{\prime }$ being a suitable objective stress rate such as the Jaumann rate [19] commonly used in hydrocode settings. Here G is the effective multi-phase shear modulus and ${\mathit{D}}_{\mathrm{p}}^{\prime }$ is the plastic rate of deformation tensor. Note that the last term in the above expression may be important if there is a change in stiffness from phase transformation with very little deviatoric deformation.

The plastic rate of deformation tensor is isochoric by construction and defined by the flow rule

$${\mathit{D}}_{\mathrm{p}}^{\prime }=\dot{\lambda }\frac{\partial \phi }{\partial {\sigma }^{\prime }}$$

(5)

which represents associated flow that is normal to a surface, $\phi \left(\sigma \right)$, in stress space. We assume a yield surface or flow potential of the form

$$\phi \left({\sigma }^{\prime }\right)=\tau \left({\sigma }^{\prime }\right)-Y$$

(6)

where Y is the effective multi-phase flow stress and the deviatoric effective stress is defined as $\tau ={\left(\frac{3}{2}{\sigma }^{\prime }:{\sigma }^{\prime }\right)}^{1/2}$. The constitutive equations are closed by imposing the Kuhn-Tucker consistency equations that $\dot{\lambda }=0$ for $\phi <0$ (elastic deformation) and $\dot{\lambda }>0$ for $\phi =0$ indicative of plastic loading. While there is some additional complexity from the computation of G and Y in a multi-phase setting, the above equations are commonly employed in hydrocodes.

2.2. Multi-Phase Description of Shear Modulus and Flow Stress

We denote the total volume of a portion of the multi-phase material as $\mathsf{\Omega }={\sum }_k{\mathsf{\Omega }}_k$, where ${\mathsf{\Omega }}_k$ is the volume occupied by the kth phase. Likewise, the total mass is $M={\sum }_k{M}_k$, where $M_k$ is analogous to ${\mathsf{\Omega }}_k$, but for phase mass. We denote the volume fraction of the kth phase as ${\xi }_k={\mathsf{\Omega }}_k/\mathsf{\Omega }$ and the mass fraction ${\chi }_k=M_k/M$. From these definitions we have that ${\sum }_k{\chi }_k=1$ and $0\le {\chi }_k\le 1$, and analogously for ${\xi }_k$. The relationship between volume and mass fraction of the kth phase is

$${\chi }^k=\frac{{\rho }_k{\xi }_k}{\rho }.$$

(7)

The effective shear modulus, G, and flow stress, Y, are then defined as volume fraction weighted averages of the corresponding phase-specific values according to

$$G=\sum _k{\xi }_k{G}_k\left(T,{\rho }_k\right)$$

(8)

and

$$Y=\sum _k{\xi }_k{Y}_k\left({ϵ}_{\mathrm{p},k},{\dot{ϵ}}_{\mathrm{p}},T,{\rho }_k\right)$$

(9)

where we have introduced phase-specific values of the shear moduli, $G_k$, flow stress, $Y_k$, and effective plastic strain ${ϵ}_{\mathrm{p},k}$. Averaging G and Y in terms of volume fractions is consistent with $G_k$ and $Y_k$ being defined in the current configuration of the material and with the upper-bound-like assumption for the plastic strain rate partitioning among the phases described below in Section 2.4. Values of the phase fractions, phase-specific density, temperature, and the pressure are obtained from the phase-specific equations of state in combination with expressions for the transition kinetics, and those aspects of the model framework is described below in Section 2.5.

2.3. Phase-Specific Shear Modulus and Strength Models

The framework could be used with a variety of shear modulus and strength model forms for the specific phases. For simplicity we describe only the Burakovsky-Greeff-Preston (BGP) form for the phase-specific reference cold shear modulus and phase-specific reference melt temperature, and the Preston-Tonks-Wallace (PTW) forms for the phase-specific strength models. The PTW model is described elsewhere [21] and details are not reproduced here. To obtain the shear modulus at finite temperature, the cold shear modulus $G_0\left(\rho \right)$ is assumed to be linear in the homologous temperature, as in [22],

$$G_k\left({\rho }_k,T\right)=G_{0,k}\left({\rho }_k\right)\left[1-{\alpha }_k\frac{T}{T_{\mathrm{m},k}\left(\rho \right)}\right]$$

(10)

where $T_{\mathrm{m},k}$ is the melt temperature of the kth phase at a particular density, and ${\alpha }_k$ is a phase-specific material parameter that captures the variation of shear modulus with respect to temperature (at fixed density). For most metals Equation (10) agrees well with experimental observations away from 0 K. We note that “cold shear modulus” is somewhat of a misnomer, because Equation (10) does not hold for temperatures approaching 0 K. Furthermore, we also note that the melt temperature $T_{\mathrm{m},k}$ may in some sense be a virtual or reference melt temperature, because at the given density the material may not in fact melt from the phase in question.

The expression for the variation of melt temperature with density is based on Burakovsky-Greeff-Preston model [23]:

$$T_{\mathrm{m},k}\left({\rho }_k\right)=T_{\mathrm{ref},k}{\left(\frac{{\rho }_k}{{\rho }_{\mathrm{m},k}}\right)}^{1/3}exp\left[f_T\left(\rho \right)\right]\mathrm{where}{f}_T\left(\rho \right)=\sum _{j=1,3}\frac{2{\gamma }_{k,j}}{q_{k,j}}\left(\frac{1}{{\rho }_{\mathrm{m}}^{q_{k,j}}}-\frac{1}{{\rho }^{q_{k,j}}}\right).$$

(11)

While the original model developed in [23] results in a similar expression for the cold shear modulus using the same expression for Grüneisen parameter as in the melt temperature, in practice it has been effective to employ a separate parameterization pertaining to the cold shear modulus, such that

$$G_{0,k}\left({\rho }_k\right)=G_{\mathrm{ref},k}{\left(\frac{{\rho }_k}{{\rho }_{0,k}}\right)}^{4/3}exp\left[f_G\left(\rho \right)\right]\mathrm{where}{f}_G\left(\rho \right)=\sum _{j=1,2}\frac{2{\gamma }_{k,j}}{q_{k,j}}\left(\frac{1}{{\rho }_0^{q_{k,j}}}-\frac{1}{{\rho }^{q_{k,j}}}\right).$$

(12)

That is, the series expression for the Grüneisen parameter is assumed to have a different truncation term applicable to the shear modulus than for melt temperature. Together, Equations (10)–(12) constitute a complete analytical model for the shear modulus and melt temperature for the kth phase across a range of density and temperature. We note that the PTW strength model makes direct use of the melt temperature, and that it is thus desirable to have a phase-specific melt temperature as in Equation (11) rather than a single global melt temperature.

For single-phase materials, the flow stress model is typically parameterized by adjusting model parameters to best match stress versus strain behavior from quasi-static and split Hopkinson pressure bar experiments, combined with adjustments based on observations of response from other experimental platforms. For multi-phase materials, it is not typically possible to obtain direct information about stress versus strain in high-pressure phases. The paucity of data limits the number of parameters that can be calibrated in a model for the flow stress of the high-pressure phase. Thus, we propose a hybrid approach to flow stress model calibration, in which the phase-specific flow stress for phases that are difficult to measure is defined by a scaling from another phase that is experimentally accessible. The single scale factor can then be used in calibration to limited data. As demonstrated in [24], the scaling can utilize the ambient-pressure phase of the same material. Or, as in [25], the strength of a high-pressure phase can be based on other materials having similar crystal structure. Multi-scale modeling can also provide a basis for modeling a high pressure phase, as in [18]. As we develop a more complete understanding of the uncertainties associated with multi-scale modeling, we may develop robust approaches to the combined calibration of multi-scale-informed models to both experimental data and sub-scale computational results. In all of these approaches, the strength model of a given phase employs a phase-specific shear modulus model informed by ab initio calculations. Section 3 below provides a specific example in which we utilize the PTW strength model.

2.4. Phase-Specific Accumulated Effective Plastic Strain

Macroscopic flow stress models are often in the form $Y=\widehat{Y}\left({ϵ}_{\mathrm{p}},{\dot{ϵ}}_{\mathrm{p}},G,T,P,\mathcal{S}\right)$ where $\mathcal{S}$ are additional state variables. For example, the PTW [21] and Steinberg-Guinan [26] strength models that are commonly used in simulations of high-pressure loading conditions, use this form and do not include any additional state variables $\mathcal{S}$. In such models, ${ϵ}_{\mathrm{p}}$ serves as a convenient proxy for tracking evolving state, even though ${ϵ}_{\mathrm{p}}$ is not directly associated with a measurable feature of the material state. Other models have state variables such as the mechanism-specific threshold stresses [27] or dislocation density [28,29] for tracking evolving material response. An important aspect of a multi-phase mixture is how plastic strain rate is partitioned among the phases and how the variables, ${ϵ}_{\mathrm{p}}$ and $\mathcal{S}$, evolve within each of the phases. In the simple CMMP framework outlined here, the flow stress does not include any additional state variables $\mathcal{S}$, thus we are focused on describing the plastic strain rate and accumulated plastic strain in each of the phases. Additional scalar state variables can be handled in a manner similar to that described here for the plastic strain. Although, as in [18], physical considerations related to the specific nature of the variables may factor into how they are evolved.

There are a variety of assumptions that could be made regarding the relationship among the plastic strain rates in the various phases. Here, as in [18,24], it is assumed that at a given macroscopic location all of the phases present at the sub-scale are undergoing the same plastic strain rate. This assumption is not strictly consistent with the approximation made in the development of equation of state and transition kinetics where it is instead assumed that each phase is at the same temperature and pressure. A generalization of this latter assumption would be that of uniform stress, including both pressure and deviatoric portions of the stress, in which case the plastic strain rates would not be identical. The uniform constituent stress approach has been previously used, for example in [12,30,31]. However, the uniform plastic strain rate assumption is simple and computationally efficient. Whether a more elaborate approach is warranted may depend on the specific application.

The relationship among values of phase-specific accumulated plastic strain requires a further and separate assumption. The effective plastic strain is a path-dependent variable which characterizes some aspect of the deformation history of the material. One could assume that upon transformation to a new phase, there is no influence of the deformation history from the parent phase on the new phase. A physical motivation of this assumption could be based on the elimination of the dislocation substructure in the parent phase as the material transforms to the product phase. While this scenario may be unlikely, it is can be useful to explore such extreme assumptions as a vehicle for testing model sensitivities.. Under this assumption, the plastic strain in a newly transformed region should be zero to represent a virgin state of that phase. On the other hand, one could assume that the dislocation substructure persists through the transformation [18]. Such an assumption would be more natural to represent within a physically-based flow stress model that includes, for example, dislocation density within $\mathcal{S}$. As a starting point for exploring sensitivity to such considerations, we consider two cases: plastic strain goes to zero in newly transformed material, or plastic strain is carried through the phase transformation.

2.4.1. Evolution Method

In the case of plastic strain going to zero upon transformation we cannot track a single overall plastic strain variable for the collection of phases. Because the entire volume fraction of material does not transform from one phase to another simultaneously (that is, the volume fractions evolve continuously over the interval $\xi \in \left[0,1\right]$), the entire volume fraction of a phase will not have accumulated the same amount of effective plastic strain. A newly transformed portion of material should have zero plastic strain, while the volume fraction that had already transformed may have been subjected to deformation such that it has a non-zero plastic strain. A detailed approach would be to evaluate the flow stress over separate sub-volume fractions of the transformed phase and then volume average the heterogeneous values of Y. However, such an approach would be complex in terms of representing sub-volumes with a distribution of plastic strain levels and correspondingly of flow stresses.

A simpler approach is to effectively average the accumulation of plastic strain in the transformed phase. The approach described here was used in [18], but that publication only describes a subset of details that were specific to a particular use-case. Some notes pertinent to the following description of the algorithm include:

• The approach here is described in terms of volume fractions ${\xi }_{\mathit{k}}$ with k being the phase index, but the same approach holds for quantities that should be weighted by mass. One just uses the mass fractions instead of the volume fractions.
• We assume that volume fractions are available at the beginning and end of the time step, ${\xi }_{\mathit{k}}^{\ominus }$ and ${\xi }_{\mathit{k}}^{\oplus }$. That does not amount to utilizing all of the transformation rates among the various phases. In many cases for polycrystalline metals, only two phases are exchanging mass at a given spatial location such that having the volume fractions provides complete information. The algorithm based on volume (or mass) fractions rather than rates can be simpler to implement.
• Cutoffs and thresholds can be important to numerically robust behavior of the implementation. These are particularly important in an Eulerian or Arbitrary Lagrangian–Eulerian (ALE) hydrocodes [19] given the action of the “advection” (or remap) step on the history variables. The monotonicity enforcement mentioned in the following section is similar to the monotonicity enforcement in the fixed-time mesh remap step in a hydrocode, but here the monotonicity is enforced during the time-evolution step. It can be useful to skip the contribution of phases with a tiny volume fraction, less than say ${10}^{-10}$.

2.4.2. Evolution Algorithm

Under the simplifying assumptions here, if the volume fraction of a phase is going down then we do not need to adjust the state in that phase beyond the evolution that would happen as a result of the deformation condition experienced by that phase. That is, the transfer of volume fraction influences the state reached in the product phase but not that reached in the parent phase. This is revisited briefly in the following section.

For phase k, let ${\dot{h}}_{\mathit{k}}$ be the “local” state evolution rate in the absence of phase transformation for a generic state or history variable h. Furthermore, let $h_{\mathit{k}}^{\mathrm{f}}$ be the end-of-time-step state that is induced by the transfer of volume fractions among the phases. We then assume that the overall end-of-time-step state value can be computed as

$$h_{\mathit{k}}^{\oplus }=h_{\mathit{k}}^{\mathrm{f}}+{\dot{h}}_{\mathit{k}}\mathsf{\Delta }t=h_{\mathit{k}}^{\mathrm{f}}+h_{\mathit{k}}^{\mathrm{e}}-h_{\mathit{k}}^{\ominus }$$

(13)

with $h_{\mathit{k}}^{\mathrm{e}}$ and $h_{\mathit{k}}^{\ominus }$ being, respectively, the end-of-step and beginning-of-step values that associated with time evolution in the absence of phase transformation. As mentioned above, considerably more complex choices could be made. However, Equation (13) has the advantage of being simple, and, as described in the next section, capturing desirable limiting behaviors.

Now define a “bath” value $h^{\mathrm{b}}$

$$h^{\mathrm{b}}=\frac{{{\displaystyle \sum _{\left\{k|{\xi }_{\mathit{k}}^{\oplus }<{\xi }_{\mathit{k}}^{\ominus }\right\}}}}^{({\xi }_{\mathit{k}}^{\ominus }-{\xi }_{\mathit{k}}^{\oplus })\left(h_{\mathit{k}}^{\ominus }\right)}}{{{\displaystyle \sum _{\left\{k|{\xi }_{\mathit{k}}^{\oplus }<{\xi }_{\mathit{k}}^{\ominus }\right\}}}}^{({\xi }_{\mathit{k}}^{\ominus }-{\xi }_{\mathit{k}}^{\oplus })}}.$$

(14)

That is, the weighted summation over phases for which the volume fraction is going down. This bath value is used for the influx to phases whose volume fractions are increasing.

For newly formed phases (${\xi }_{\mathit{k}}^{\oplus }>0$, ${\xi }_{\mathit{k}}^{\ominus }=0$), $h_{\mathit{k}}^{\ominus }$ has not been established and we use $h_{\mathit{k}}^{\ominus }=h^{\mathrm{b}}$ as the starting point for computation of $h_{\mathit{k}}^{\mathrm{e}}$.

For a phase k in which the volume fraction is increasing, one has

$$h_{\mathit{k}}^{\mathrm{f}}=\frac{h_{\mathit{k}}^{\ominus }{\xi }^{\ominus }+\left(h^{\mathrm{b}}\right)({\xi }^{\oplus }-{\xi }^{\ominus })}{{\xi }^{\oplus }}.$$

(15)

To meet “monotonicity” requirements (that the output should fall within the range of the inputs) we subsequently restrict $h_{\mathit{k}}^{\mathrm{f}}$ to fall within the interval between $h_{\mathit{k}}^{\ominus }$ and $h^{\mathrm{b}}$. Finally, if Equation (eqn:histAll) gives $h_{\mathit{k}}^{\oplus }\le 0$ for a state variable that should be positive, then $h_{\mathit{k}}^{\mathrm{f}}$ is used instead. This can happen in rare cases if the local evolution of h has extreme behavior.

2.4.3. Specific Examples

Instead of Equation (eqn:DCBath), one can use a more detailed equation that includes specific fluxes of mass among phases. Suppressing the per-phase index for convenience, we can write:

$$h^{\mathrm{f}}{\xi }^{\oplus }=h^{\ominus }{\xi }^{\ominus }+\left(\sum _{\mathrm{in}}{\dot{\xi }}^{\mathrm{in}}{h}^{\mathrm{in}}-\sum _{\mathrm{out}}{\dot{\xi }}^{\mathrm{out}}{h}^{\ominus }\right)\mathsf{\Delta }t.$$

(16)

If there is no influx for the phase in question then $-\mathsf{\Delta }t{\sum }_{\mathrm{out}}{\dot{\xi }}^{\mathrm{out}}={\xi }^{\oplus }-{\xi }^{\ominus }$ and Equation (eqn:fluxes) produces $h^{\mathrm{f}}=h^{\ominus }$. This is in line with the comment in the preceding section that Equation (eqn:DCBath) is applied when the volume fraction of a given phases is increasing, and that the local evolution is not modified when the volume fraction is decreasing.

For either Equation (15) or (16) if there is no mass leaving the phase in question and the mass coming in has zero state variable value ($h^{\mathrm{in}}=0$ in Equation (eqn:fluxes) or $h^{\mathrm{b}}=0$ in Equation (eqn:DCBath)), then we get $h^{\mathrm{f}}=h^{\ominus }\frac{{\xi }^{\ominus }}{{\xi }^{\oplus }}$. Thus, the simpler model in Equation (15) captures the behavior that the state variable value can decrease in a given phase due to the influx of newly transformed material.

2.5. Multi-Phase Equation of State (EOS) and Kinetics of Transformation

In traditional single-phase material models, the equation of state is a potential function that defines the relationship between the thermodynamic state variables $\rho$, T, P, s where s is the specific entropy (per unit mass). For example, a typical representation of an EOS is defined by the specification of a state function for the specific Helmholtz free energy F of the specific volume (per unit mass) $V=1/\rho$ and temperature T, $F=\widehat{F}\left(V,T\right)$ which completely defines the pressure and entropy through the thermodynamic state relations ${P=-\partial F/\partial V|}_T$ and ${s=-\partial F/\partial T|}_V$, where ${|}_x$ indicates that the derivative is to be taken along constant x. Given a description of $F=\widehat{F}\left(V,T\right)$, any other thermodynamic potential (such as specific internal energy e or Gibbs free energy g) can be obtained through partial Legendre-Fenchel transforms, so that the thermodynamic state is entirely defined via the state variables V and T. For our purposes (in the context of an implementation that executes mass and energy conservation) it is convenient to represent this as a functional relationship in density and specific internal energy that can be queried $P=\widehat{P}\left(\rho ,e\right)$ and $T=\widehat{T}\left(\rho ,e\right)$. Note that these two functional relations need to be combined with a thermodynamic potential, that is, we need a complete EOS in order to determine the specific entropy needed to fully describe the state.

For a multi-phase material model, the equation of state defines these thermodynamic relationships for each phase, $F_k={\widehat{F}}_k\left(V_k,T_k\right)$, and via a closure assumption for the mixture (such as $P_k=P$ and $T_k=T$) also defines the equilibrium volume fractions, ${\xi }_k$, of the individual constituents. In this setting “kinetics” refers to the relationship between the thermodynamic state and the actual rate at which the transformation from one phase to another can occur. If this rate is sufficiently fast in comparison with the deformation rate, then the volume fractions of each phase will be near their equilibrium values. Under such cases and with other restrictions to be noted below, the equation of state alone can describe the process. However, when the loading rate is fast relative to the time required for transformation (that is, for slow kinetics), then the equation of state must be combined with a kinetic model for the transformation rate, ${\dot{\xi }}_k=\mathcal{K}\left(P,T,{\xi }_m\right)$, which captures the irreversible, path-dependent history of the trajectory through state space. This implies that there are two different modeling approaches that should be considered as part of the CMMP, namely (1) using equilibrium phase fractions and (2) using finite transformation rate kinetics. These are described briefly in the following subsections. In both cases we assume local pressure and temperature equilibrium among the phases ($P_k=P$ and $T_k=T$), with the corresponding assumptions that local heat transport among the phases is rapid and that surface energy contributions are negligible.

Finally, it is worth noting that in general the function providing ${\dot{\xi }}_k$ could depend on other quantities such as state variables associated with the number of nucleation sites that have been activated or driving force contributions associated with the deviatoric part of the stress. We will return to this topic below in Section 4.

2.5.1. Multi-Phase EOS with Equilibrium Phase Fractions

Multi-Phase equation of state tables such as SESAME 2161 and 2162 for tin [32,33], include a “flattened,” or homogenized, table that represents the effective EOS of the equilibrium mixture, that is of $F=\widehat{F}\left(V,T\right)$. In addition, separate tables for each individual phase EOS are available, based on $F_k={\widehat{F}}_k\left(V_k,T_k\right)$. Using the infrastructure in the hydrocodes, it is relatively straightforward to evaluate the “flattened” table to compute $P=\widehat{P}\left(\rho ,e\right)$ and $T=\widehat{T}\left(\rho ,e\right)$. The mass fractions and volume fractions can also be obtained from the flattened table, with relationships as in Equation (7) providing the relationship between mass and volume fractions. The mass and volume fractions are fully determined by the complete EOS and can thus be tabulated once and for all. Employing the assumption of uniform pressure and temperature within each of the constituent phases, the individual phase densities and specific internal energies can be obtained as ${\rho }_k={\widehat{\rho }}_k\left(P,T\right)$ and $e_k={\widehat{e}}_k\left(P,T\right)$. While this approach is relative straightforward in some regards and provides all of the information needed by the multi-phase strength description above, it can require iterative solutions to compute, for example, ${\widehat{\rho }}_k\left(P,T\right)$ from the phase-specific EOS tables.

2.5.2. Multi-Phase EOS with Finite Rate Kinetics

Rather than assuming an equilibrium mixture of phases, there is a class of multi-phase EOS models that define a transformation rate as a function of the difference in Gibbs energies in each of the phases. For example, Ref. [34] has developed a model and implementation algorithm for the rate of transformation between constituent phases which can be used to define the irreversible evolution of phase fractions. It is also possible to include physical effects such as nucleation and growth in the kinetics model, and models with such effects have been used to examine pressure-driven solidification [35]. Here we use the master equations

$${\dot{\chi }}_k=\sum _{j\ne k}\left({\chi }_j{R}_{jk}-{\chi }_k{R}_{kj}\right)$$

(17)

with a simple phenomenological transformation rate matrix, $R_{kj}$, defined as

$$R_{kj}=\left\{\begin{array}{ccc}\frac{{\nu }_{kj}\mathsf{\Delta }{g}_{kj}}{B_{kj}}exp\left({\left(\frac{\mathsf{\Delta }{g}_{kj}}{B_{kj}}\right)}^2\right)\hfill & \mathrm{when}& \mathsf{\Delta }{g}_{kj}>0\hfill \\ 0\hfill & \mathrm{when}& \mathsf{\Delta }{g}_{kj}\le 0\hfill \end{array}\right.$$

(18)

where $\mathsf{\Delta }{g}_{kj}=g_k-g_j$, and ${\nu }_{kj}$ and $B_{kj}$ are material parameters [34]. The rate equation is dependent on the present state of the phase mixture via the specific Gibbs free energy differences, $\mathsf{\Delta }{g}_{kj}$, between phases k and j, whether or not these phases are currently present in the mixture. The requirement that we also consider the specific Gibbs free energy differences including an absent phase is, of course, due to the possibility of populating the absent phase as the mixture evolves through time. The Gibbs free energy of an absent phase is most likely in the metastable or unstable regions of the phase EOS, and this requires some extra considerations. Specifically, valid regions of a phase-specific EOS can be specified using $\rho$ and e windows, in a manner analogous to the practice commonly employed in construction of an equilibrium multi-phase EOS. As before, closure relations are required to define a system of equations to fully compute the thermodynamic state, and subsequently update the history dependent variables, ${\chi }_k$. Again, we use the uniform pressure and temperature over phases conditions in addition to the constraints on the phase-specific internal energies and densities, derived from the fact that the total mass and internal energy are the sums of the individual phase quantities:

$$\begin{array}{cc}\hfill {\widehat{P}}_k\left({\rho }_k,e_k\right)& =P\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\widehat{T}}_k\left({\rho }_k,e_k\right)& =T\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \sum {\chi }_k{e}_k& =e\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \sum {\rho }_k{\xi }_k& =\rho .\hfill \end{array}$$

(22)

These equations are valid for any phase mixture where all phases are assumed to have the same pressure and temperature, that is, also for the equilibrium phase fractions that we discussed in the previous section. The master equation in Equation (17) describes the change of mass of a phase k as the difference of mass that is transformed from all other phases to this phase k and the mass that is transformed from this phase k to all other phases. The full set of master equations thus describe mass conservation. While in principle Equation (17) enforces $\sum {\chi }_k\le 1$ and $0<{\chi }_k\le 1$ by construction [34], care is required in numerical implementations due to effects from finite time step size and numerical stability considerations. One approach is to evolve the mass fractions using a smaller time step size than that used in the time advance of the host hydrocode. Another approach is to use a more implicit (backward-Euler) time integration scheme. In the solution of an implicit set of equations, one can condense out a degree of freedom so that the solution of the time-discretized equations once again satisfies mass conservation by construction (see Section 3 in [12]).

With e, $\rho$, and $\mathsf{\Delta }t$ known inputs from the hydrocode, Equations (17) and (19)–() define a system of $3N+2$ equations in the $3N+2$ unknowns—${\rho }_k$, $e_k$, ${\chi }_k$, P, and T. These equations can be solved using a general implicit solution scheme or using the semi-implicit scheme of [34] which uses a forward-Euler update of the phase fractions and an implicit solution for the specific volume and internal energy of each phase.

3. Results

To highlight specific behaviors of the model, we examine the metal tin, which undergoes phase transformations at relatively accessible pressures and is of ongoing interest [7,24,33,36,37,38,39,40,41,42]. In this section, we describe simulations of a pulsed-power ramp-release experiment that accesses the $\beta$ and $\gamma$ phases of tin, with the material beginning in the $\beta$ phase, transforming to the $\gamma$ as the pressure ramps up, and then reverting to the $\beta$ phase as the pressure falls off. Simulations use the SESAME 2162 multi-phase EOS for tin [33]. Details of recent experiments focused on the $\beta$ phase and an associated PTW strength model calibration are detailed in [42], where quasi-static, split Hopkinson pressure bar (SHPB), and Taylor cylinder experimental results are examined. For completeness,

Table A1. PTW calibration for $\beta$-tin as in [42].

Parameter	Value	Parameter	Value
$\theta$	0.0597	p	2.82
$s_0$	0.0256	$y_0$	0.00239
$s_{\infty }$	0.000675	$y_{\infty }$	0.000313
$\kappa$	0.186	$y_1$	0.0355
$\gamma$	0.00000136	$y_2$	0.45
$\beta$	0.45

provides the $\beta$-phase PTW parameters as obtained in [42]. Here the principal focus is on the model formulation rather than on specifics of the calibration.

That said, the phase-specific shear modulus model based on ab initio calculations is important in the scaling approach described in Section 2, and the phase-specific shear modulus calibration has not been published elsewhere.

Table 1. Parameters of the BGP model for phase-specific shear modulus models. For fits with non-zero ${\gamma }_1$, the exponent $q_1$ equals $1/3$. Units of the $\gamma$ values are consistent with density units (g/cm${}^3$) and the corresponding q. See Equations (10)–(12) for the relevant functional forms.

Phase	${{\rho }}_0$	${{G}}_{\mathit{ref}}$	${{\rho }}_{\mathit{m}}$	${{T}}_{\mathit{ref}}$	${{\gamma }}_1$	${{\gamma }}_2$	${{q}}_2$	${{\gamma }}_3$	${{q}}_3$	${\alpha }$
	g/cm${}^3$	GPa	g/cm${}^3$	K	*	*	-	*	-	-
$\beta$	7.40	25.4	7.18	505.1	0	600	3.2	800	3.2	0.53
$\gamma$	7.82	26.8	7.585	595.0	0	820	3.2	2300	3.2	0.7
$\delta$	10.50	23.0	9.425	1900.0	2.125	$5.5\times {10}^8$	8.0	$1.1\times {10}^8$	8.0	0.2

includes the parameters for the BGP shear modulus model employed here. Additional information about the $\beta$ phase shear modulus is provided in [42].

Following the scale-factor-based approach, the strength in the non-ambient $\gamma$ phase is based on a scale factor $X_{\gamma }$ and the strength of form of the $\beta$ phase using the shear modulus for the $\gamma$ phase:

$$Y_{\gamma }=X_{\gamma }{\widehat{Y}}_{\beta }\left({ϵ}_{\mathrm{p},\gamma },{\dot{ϵ}}_{\mathrm{p}},{\widehat{G}}_{\gamma }\left({\rho }_{\gamma },T\right),T,P\right).$$

(23)

For models such as Steinberg-Guinan [26] in which the shear modulus appears only as a prefactor to the overall flow strength, the above equation could be rewritten

$$Y_{\gamma }=X_{\gamma }{\widehat{G}}_{\gamma }\left({\rho }_{\gamma },T\right)\frac{{\widehat{Y}}_{\beta }\left({ϵ}_{\mathrm{p},\gamma },{\dot{ϵ}}_{\mathrm{p}},{\widehat{G}}_{\beta }\left({\rho }_{\beta },T\right),T,P\right)}{{\widehat{G}}_{\beta }\left({\rho }_{\beta },T\right)}$$

(24)

however, the PTW model does not fall into this category given the use of the shear modulus in the evaluation of the strain rate normalization factor (see Equation (2) in [21]). With the $\gamma$ phase strength described by Equation (23) we are left with a single calibration parameter $X_{\gamma }$ for the strength in that phase.

To highlight various features of the modeling framework as applied to tin, Figure 1 shows results of a sensitivity study. As noted above, these results are for a ramp-release experiment. Specifically, we examine the experiment shown in Figure 7 in [7]. In the experiment, velocimetry is used to record the motion of the back face of the target, and pulse-shaping capabilities at the experimental facility allow for control of the history of the axial stress applied on to the tin target. The experiment was conducted at the Thor pulsed power facility at Sandia National Laboratories [43]. Flow of current through an electrode produces a magnetic field and associated Lorentz forces, with the pressure in the electrode then driving the sample of interest. Drive measurements are collected to facilitate experimental interpretation. Given the stringent requirements for drive conditions probing materials that exhibit complex response including phase transformation, the Thor facility was developed with flexible pulse shaping capabilities. The experiment involves one-dimensional wave propagation and the target is not in a uniform state, but various physical mechanisms have identifiable influences on features in the velocity trace.

Specifically in Figure 1 we show the effects of: forward transformation rate, reverse transformation rate, scale-factor $X_{\gamma }$ for the $\gamma$ phase strength, and whether the plastic strain is carried through the transformation or set to zero for newly transformed material. The parameter values are selected to illustrate a range of physically plausible transformation behaviors and variations in the non-ambient strength. We set the parameters $B_{\beta \gamma }=B_{\gamma \beta }={10}^4$ J/kg. As described in [44], the ratios ${\nu }_{\beta \gamma }/B_{\beta \gamma }$ and ${\nu }_{\gamma \beta }/B_{\gamma \beta }$ have the physical interpretation of the rate of the transformation per difference in the per-phase Gibbs energies. Given an experimental configuration, one can usually estimate this quantity to within a few orders or magnitude; based on the Thor pulsed power experiments, ${\nu }_{\beta \gamma }/B_{\beta \gamma }$ and ${\nu }_{\gamma \beta }/B_{\gamma \beta }$ are set to the values ${10}^2$, ${10}^3$, and ${10}^4$ 1/(J/kg s). As a simple example of independently adjusting per-phase strength parameters, the non-ambient flow strength is scaled by a factor $X_{\gamma }$. A value of $X_{\gamma }=1$ means that the strength is the same for both phases. The values $0.1$, $0.5$, and 1 were selected to illustrate the effects of the non-ambient strength on the velocity trace of the pulsed power experiment. We emphasize that although only a single strength parameter is varied in this example, the proposed model and implementation strategy is sufficiently general that all the parameters of each per-phase model could be independently varied. Or a wholly different functional form could be used for the strength of the non-ambient phase. For comparison purposes, Figure 1 also contains a reference case (dashed black curve) using equilibrium phase fractions and the same strength model for both the ambient and non-ambient phases.

While the plots in Figure 1 show examination of specific effects one at a time, the physical effects in the modeling framework have overlapping influences on the observed velocity trace. These combined effects will be explored further in future work. A feature associated with the $\beta \to \gamma$ transformation is clearly visible in the rise of the velocity, as is a feature associated with the $\gamma \to \beta$ transformation during the release. Phase kinetics influences the shapes of these features, and the underlying EOS influences where the transitions appear. Compressibility of the material in the phases influences the rise and fall of the velocity, and strength and anelastic effects are relatively prominent just after the peak in the velocity trace at the start of the reversal. Anelastic effects on the reversal are potentially related to the Bauschinger effect under quasi-static loading [45], and anelasticity is not discussed further here given that it was a focus in [7]. Setting the plastic strain to zero as material transforms influences the subsequent strength response and changes the shape of the simulated velocity trace, particularly near the reversal where the velocity has large sensitivity to strength. Experiments involving large amounts of plastic strain after the transformation, such as Rayleigh-Taylor instability growth experiments with instability growth predominantly in the product phase [16], may be less sensitive to this detail of the material response. Figure 1 highlights a significant subset of the experimentally observable variations in behavior captured by the CMMP framework. Despite a number of simplifying assumptions, the framework is a promising point of departure for ongoing studies of multi-phase material response.

4. Discussion

Moving forward, there are interesting questions related to the degree to which we can uniquely calibrate a model given a collection of experimental data. A subset of these questions was explored in [7] in the context of Bayesian model calibration, although that study utilized equilibrium phase fractions and thus did not examine the important effects of phase kinetics. As noted in the introductory section, studies involving multiple experimental platforms such as [11] can help in distinguishing functional dependencies by probing a range of conditions and loading paths. As more data are collected, one could consider adding features to the model to capture additional effects.

One potential effect to be considered is the refinement of the grain size during transformation, due to the formation of multiple product-phase variants inside of each grain in the parent phase. It is possible that dynamically driven phase transformations produce a more dramatic reduction in effective grain size than quasi-static phase transformations. If has for example been speculated that refinement of the effective grain size is responsible for the high apparent strength when iron is dynamically driven into the $ϵ$ phase [46]. Indeed, in situ diffraction observations suggest that under laser-driven conditions the $ϵ$ phase of iron may become nanocrystalline [47]. Models that include a nucleation and growth character to their kinetics might be able to inform a grain size in the product phase.

Another set of model features to be considered in the future involves the coupling between the phase transformation and the deviatoric aspects of the material response. Such coupling has been observed in quasi-static diamond anvil cell experiments [15], with deviatoric stresses influencing the selection of specific product-phase variants. At an atomic scale, phase transformations tend to involve both volume change and shear deformation [33], and the variant selection is believed to be associated with the action of the deviatoric stress on the shear portion of the deformation during transformation. Molecular dynamics simulations have provided additional evidence of interactions between plasticity and phase transformation [48]. The crystal-mechanics-based model described in [12] has been used to investigate these coupled effects, with the model tying variant selection at a given spatial location to the local stress state and with corresponding relaxation of the local stress driven by the deformation associated with the specific variant selection. Simulation results for the $\alpha \to ϵ$ transformation in iron are shown in Figure 2.

These simulations use the model as described in [12], but with elastic moduli adapted from [49] and with variations in the dislocation glide kinetics to explore the interaction between deviatoric stress and the transformation. In the “slow glide” case the dislocation-mediated plasticity is made to be sluggish, so that plasticity does not relax the deviatoric stresses developing in the polycrystal, and thus the deviatoric stresses accumulate until they begin to relax at the onset of the $\alpha \to ϵ$ transformation at an axial of about 3%. In the “fast glide” case, the dislocation-mediated plasticity in the $\alpha$ phase develops early in the loading, resulting in yielding at an axial strain of less than 1%, as shown in Figure 2. In the simulations the polycrystal is compressed in the vertical direction at a strain rate of roughly ${10}^6$/s and the polycrystals are confined laterally so that the pressure increases. The plots in the figure show the deviatoric stress falling off during the transformation. This is due to the amount of shear associated with the transformation [50,51,52] and the corresponding effect of the transformation relaxing the deviatoric stress. The deviatoric stress begins to climb again once most of the material has transformed from $\alpha$ to $ϵ$. We also note that the transformation begins sooner in the slow glide case, due to the higher deviatoric stresses helping to drive onset of transformation. The larger deviatoric stresses in the slow glide case more strongly drive variant selection and as seen in Figure 2 this influences the $ϵ$ phase morphology, producing a more refined transformed domain structure and more planar phase interfaces. As noted in [53], the crystal-mechanics-based formulation does not require planar interfaces but they tend to emerge as an outcome of the long-range field interactions.

Crystal-mechanics-based formulations are attractive in that this rich set of behaviors emerges naturally. However, the models can be complicated to calibrate and are computationally expensive. Simulations with crystal-mechanics-based modeling indicate that the crystal-scale coupling between deviatoric and volumetric aspects of the material response has manifestations at the macroscale, and in the future it may be desirable to capture these macroscale manifestations in modeling frameworks such as the one described here. The influence of shear on the phase transformation may be approximated as providing a driving force contribution of ${\gamma }_{kj}\sqrt{{\sigma }^{\prime }:{\sigma }^{\prime }}$ where ${\gamma }_{kj}$ captures the amount of shear strain associated with the transformation between phases k and j. Correspondingly, it may be desirable to have a rate of distortional deformation associated with transformation in the presence of deviatoric stress that would driver preferred variant selection and thus net shear deformation. In general the deviatoric portion of the strain energy also factors into the driving force for transformation, and this can be a dominant effect in special cases [54]. To capture this effect, our description of the thermodynamic potential needs to be expanded to include deviatoric energy. The contribution to Gibbs energy from deviatoric strain of the lattice is

$$g_{\mathrm{dev},k}\left({\sigma }^{\prime }\right)=-V_k\frac{{\sigma }_k^{\prime }:{\sigma }_k^{\prime }}{4G_k}.$$

(25)

When the material is plastically flowing in all phases, the expression

$$g_{\mathrm{dev},k}\left({\sigma }^{\prime }\right)=-V_k\frac{Y_k^2}{6G_k}$$

(26)

can be used to estimate the contribution to Gibbs energy, and the difference in energy between two phases would then enter into the driving force for transformation. It is possible that the relatively coarse assumptions made in the deviatoric kinematic relationships among phases will not be sufficient to describe all salient of the crystal-scale behaviors; however, simple relationships may provide a useful starting point for further evaluation.

A principal goal of this work has been to develop a relatively simple framework that can be used at the macroscopic scale to simulate dynamically driven experiments involving phase transformations. As shown in Section 3, The framework is able to capture a variety of experimentally observed behaviors. Furthermore, as describe above, it may be possible to expand the framework to include additional multi-phase effects as they become of interest for interpretation of experiments. In that wave propagation and dynamic material response are significantly influenced by phase transformations, it is expected other aspects of material response, such as damage and failure, would be influenced by the multi-phase behavior of materials that are prone to phase transformation.