**UCRL-PROP-220082**

**Equation of State and Constitutive Properties**

**Christian Mailhiot, <mailhiot1@llnl.gov>**

**Tommy Sewell, <sewell@lanl.gov>**

**John Aidun, <jbaidun@sandia.gov>**

**1. Introduction:**

The scientific underpinning of our confidence in ensuring the
performance, safety and reliability of the nuclear weapons stockpile
lies in our ability to predict  on the basis of experimentally
validated physics-based models  the dynamic response of materials under
a wide range of stress, temperature, strain, and strain rates. Under the
current moratorium on nuclear testing, there is a premium value on the
development of a fundamental understanding of the dynamic response of,
and interactions among, materials  particularly as they change over
time due to aging and/or re-manufacturing. The development of a
comprehensive set of experimentally validated predictive capabilities to
assess the effects of materials properties on stockpile margins and
uncertainties is at the heart of the stockpile stewardship program and
forms the basis of a scientific strategy to mitigate these
uncertainties.

The dynamic response of materials can be approximately categorized into
two classes of properties: *thermodynamic properties* and *engineering
constitutive properties or models*. Generally, *thermodynamic
properties* are fundamentally well defined at the quantum and atomic
scales, and include equation of state (EOS), re-solidification
(meta-stable trapping or frustration), phase boundaries and transitions
across them such as melting and solid-solid polymorphism,
thermodynamically derived transport and mechanical properties, and so
on. On the other hand, *engineering scale constitutive response*
typically is governed by phenomena occurring across length scales  from
the quantum-level scale to the continuum  and is dominated by the
collective behavior of defects and the evolution of material
microstructure. As such, practical constitutive models are usually
phenomenological due to unsolved technical challenges in incorporating
all relevant sub-grid-scale physics. (Fracture, failure, and damage more
generally, which are typically distinct from continuum-level
\[phenomenological\] constitutive models, are examples of this.) Notable
exceptions, in which constitutive response can sometimes be connected
directly to fundamental material properties, include plastic deformation
and yield strength. *Accurate physics-based constitutive models and
fundamental properties are absolutely essential to development of a
truly predictive scientific simulation capability*.

In its first decade, the ASC Academic Alliances program succeeded in
demonstrating that very large-scale science and engineering simulations
are feasible on scaleable high-performance computing platforms, and it
demonstrated several significant applications of end-to-end simulation
of systems performance. These achievements position the computational
science and engineering community to now focus intently on making
progress on the perennial issues that limit the ultimate accuracy and
predictive power of these simulations: predictive, robust, and efficient
materials response properties and process models. The emerging challenge
is to go beyond treatments of single-physics response of homogeneous
regions to describing, with quantified accuracy, the coupled,
multi-physics behavior of strongly heterogeneous media. The central goal
of enabling *predictive* simulations creates the need for quantifiable
accuracy and precision, which, in turn, demands performing a thorough
verification and validation process, and, very likely, developing
physically well founded and numerically effective multiple length- and
time-scale ("multi-scale") simulation capabilities.

**2. Thermodynamic properties of materials:**

-   ***Phase transformations, melt and solidification:***

Phase transformations have profound effects on materials response. They
occur in a wide variety and under a broad range of conditions. These two
aspects of phase transformations make developing accurate, robust, and
efficient treatments of them a prerequisite to achieving predictive
simulation capabilities. Following are selected illustrations where
improved descriptions of phase transformations are needed.

The long-standing problem in this area is simply stated: greater
knowledge is needed of real phase diagrams for pure elements, alloys,
and compounds; and likewise for the mechanisms and kinetics of
transformations across a phase boundary. Due to the influence of the
phase transformations on microstructure development and evolution,
understanding them can contribute greatly to efforts in predicting
material properties. In addition to processes for single-component
materials, the physics of phase transitions at interfaces between
un-like materials is an area of significant interest. Both atomistic-
and continuum-scale descriptions are of interest, for example, free
energy-based phase field. Significant challenges exist in the latter
case for phase transitions involving polyatomic molecular materials, due
in part to the large number of order parameters that would be required
in order to capture lattice symmetry, site symmetry, and molecular point
group in these structurally and chemically complex materials.

High energy density physics (HEDP) experiments demonstrate that the
appearance of the vapor phase can dramatically change the transport
properties of warm, dense matter. Hence, accurately simulating the low
temperature range of warm, dense matter depends on the fidelity of
modeling liquid-vapor phase transitions, including treatment of the
phase transformation kinetics, in hydrodynamic codes.

-   **Relevant research needs:** A method is needed for characterizing
    the appearance of two different phases of material, having
    potentially very large differences in specific volumes, as a fluid
    element enters the vapor dome. Currently codes treat only the mean
    properties within the vapor dome.

Phase transformations can significantly perturb material strength
behavior. Conversely, deviatoric stress can alter the stress and
temperature at which a phase transformation occurs. In addition, strain
rate and plastic deformation effects can result in persistence of
metastable phases that also change strength behavior.

-   **Relevant research needs:** Breakthroughs in understanding and
    modeling all of these aspects of the interplay between phase
    transformations and strength are needed for achieving predictive
    simulations.

Solid-liquid transformations, melting, and solidification pose many
problems in material modeling that require improved understanding and
better numerical treatments. A topic of central importance is the
prediction of the kinetic processes and ensuing microstructure evolution
when a material undergoes stress and/or temperature-driven melt or
solidification. That is, under a prescribed temperature/stress history,
competition between kinetics and thermodynamics often leads to
frustrated, or metastable, structures. This applies equally to metals
and organics. Understanding this area is relevant to process modeling
and manufacturing. Solidification determines microstructure, which, in
turn, strongly influences material properties. A central aspect of
solidification phenomena that is of interest is how it progresses under
dynamic application of stress.

-   **Relevant research needs**: Understanding and modeling efforts are
    needed to characterize the kinetics and metastability behaviors
    occurring under rapid, stress-induced solidification.

Very large-scale classical molecular dynamics (MD) simulations are now
being used routinely within the scientific community. Electronic
interactions are built into the classical interatomic potentials.
However, when the electronic contribution to the entropy difference
between two phases is large, for example solid and liquid, these methods
are generally not expected to give accurate characterizations of
melting. In contrast, quantum MD (QMD) simulations --- usually
implemented within the local density approximation (LDA) to density
functional theory (DFT) --- provide much more accurate treatments of
electronic interactions, but are severely limited in the number of atoms
that can be simulated.

-   **Relevant research needs:** Development of a method to bridge
    between the desirable aspects of classical and quantum MD would be a
    valuable breakthrough of great significance to the field of
    materials simulations. The bridge could be an efficient coupling of
    the two methods or a well-founded approach for adding the effect of
    changes in electronic character to classical potentials, guided by
    the results of QMD calculations.

In the case of non-covalent organic solids, there is a need for improved
electronic structure treatments that more accurately account for weak
intermolecular (dispersion) forces that typically dominate molecular
crystal structures and properties under mild conditions of stress and
temperature. Also, regardless of whether empirical or "quantum-based"
forces are used, non-equilibrium MD simulations of organic materials
suffer from deficiencies associated with classical (versus quantum)
statistical mechanical treatments of the vibrations. Even given a
"perfect" ground electronic state force field, this can lead to large
errors, for example, in predicted temperatures/stresses behind a
shockwave, with accompanying errors in chemical reaction rates and
mechanisms.

-   ***Equation of state (EOS):***

The physically-derived determination, prediction and
thermodynamically-consistent tabulation of the thermal properties of
materials are critical in the simulation of almost any complex system.
Of particular importance, the EOS establishes fundamental relationships
among thermodynamic variables (stress, density, temperature, etc.) and
provides closure of the hydrodynamic equations that explicitly ensure
conservation of mass, energy, and momentum. Hydrodynamic simulations
require thermodynamic input data in every (numerical) zone and for every
time step. Consequently, comprehensive and in-depth knowledge of
materials EOS is absolutely essential to all simulations of this class.
Other examples of critical thermodynamic properties of materials include
melt temperatures required to establish the domain of materials
strength. The absence of reliable EOS information over all relevant
regions of thermodynamic state space renders impossible the development
of a truly predictive capability for complex physical systems.

Of particular interest to the NNSA scientific programs, are materials
EOS that cover a wide range of conditions of stress, temperature, and
density. Such a wide range of EOS regimes typically requires multiple
models, each covering a specific phenomenon that dominates the response
over an interval of conditions. While this is often an effective
modeling strategy, it requires intricate "phase matching" and
interpolation between regions of the EOS that are represented with
different models.

-   **Relevant research needs:** The development of physically-based
    models that naturally make the needed phase matching and ensure
    thermodynamic consistency across the phases.

Though properties at standard temperature and stress (STP) conditions
are usually quite accessible to measurement, it is important for EOS to
accurately predict properties at STP for several reasons. The zero
stress crystal volume is found to strongly influence EOS accuracy at
high stress. Conversely, though an EOS may be accurate at high stress
and temperature, unless the mathematical form used is physically correct
*and* sufficient information is available to constrain its behavior
under ambient conditions, it may not provide an acceptable description
close to STP --- but this, too, can be important for the overall
accuracy of a simulation. An example where such details are important is
prediction of the temperature behind a shock wave propagating through a
high explosive. Here uncertainties in "simple" (temperature- and
stress-dependent) properties such as bulk modulus, specific heat, and
thermal expansion coefficients can lead to large uncertainties in
predicted temperatures; which, given nominally exponential dependencies
of chemical reaction rate upon temperature, can lead to large errors in
predictions of important processes such as detonation sensitivity and
initiation behavior. In addition to being accurate over wide intervals
of temperature and stress, EOS models that extend beyond simple
elasticity need to be accurate over several decades of strain rate;
viscoelastic response of polymers is one among several good examples
where this is true.

Existing EOS theories often break down in certain structural phases
exhibited by *f*-electron metals where it is known that strong
electron-electron correlation is inadequately treated by standard
mean-field electronic structure techniques --- such as density
functional theory (DFT). Of central importance to the ASC program is the
development of rigorous theoretical and computational techniques that
afford a predictive treatment of EOS in the regime where electron
correlation dominates. These goals might be achieved by using a
combination of effective Hamiltonian and quantum Monte Carlo (QMC)
techniques developed specifically for strongly-correlated electron
systems. In parallel with QMC treatment of effective Hamiltonians with
reduced numbers of orbitals per site, more approximate solutions are
being carried out both for these effective Hamiltonians as well as for
the full all-orbitals-per-site case. This enables immediate calibration
of QMC computational benchmarks to the physically relevant regime.
Recently, advanced simulations methods such as dynamical mean field
theory (DMFT) have been used to describe the properties of strongly
correlated materials. In any case, significant questions still remain
regarding the optimal path for upscaling EOS information for perfect
materials at atomistic scales into descriptions that incorporate
mesoscopic and larger scale defect structures.

-   **Relevant research needs:** The development of advanced electronic
    structure methods that afford a predictive treatment of the EOS of
    materials in the regime where electron correlation dominates.

Modern theoretical frameworks have been extremely successful in
predicting the properties of condensed matter systems at low temperature
(typically below the melt temperature, *T~m~*) and plasmas at high
temperature (typically above the Fermi temperature, *T~F~*). However,
there exists an important regime within the density-temperature phase
space where "warm" states of matter are inadequately described by either
conventional condensed matter physics-based or plasma physics-based
theories, and where the greatest uncertainties exist in our ability to
predict the thermodynamic properties of matter. From a condensed matter
physics perspective, warm dense matter refers to states of matter with
solid-like densities and temperatures comparable to *T~F~*. From a
plasma physics perspective, warm dense matter refers to states of matter
that are plasma-like, but that are too dense and/or too cold to be
adequately treated by standard plasma physics approaches. Consequently,
this warm dense matter regime --- roughly spanning the energy range
between *T~m~* and *T~F~* --- defines states of matter between solids
and plasmas. The simulation of warm dense states of matter is
particularly challenging because many competing processes are active
while none dominate.

Scientifically, the warm dense matter region of density-temperature
phase space is attracting considerable interest, both theoretically and
experimentally. Within the experimental condensed matter physics
community, interest has been stimulated by the emergence of new
techniques for generating strong shock waves in materials, and new
methods for confining and interrogating "warm" samples at high stress.
Within the plasma physics community, the development of novel sources
enabling experimental access to plasma-like states of matter at low
temperature and high density has led to emerging research opportunities
at the forefront of the field. Theoretically and computationally, recent
advances in finite-temperature *ab initio* electronic structure methods
beyond mean-field theories, coupled with the unprecedented surge in
compute power afforded by ultra-scale computing platforms, have led to
important new developments in our ability to predict the thermodynamic
properties of warm states of matter.

-   **Relevant research needs:** The development of simulation tools
    that provide a predictive treatment of the EOS of materials in the
    warm dense matter regime.

Because different EOS models are used in different physical regimes of
stress, temperature, and density, technical improvements in
interpolation schemes and numerical issues to ensure thermodynamic
consistency across models and regimes would be of significant interest.
Accurate information spanning all relevant states of stress,
temperature, and strain rate, whether obtained from theory/simulation or
experiment, is also required in order to improve the equation of state
for a given material. As typical numerical simulations of interest
involve multiple materials, as important as attaining predictive
accuracy for single material's EOS is developing reliable methods for
representing the (effective) EOS of a mixture of materials, or
equivalently, mixing EOSs in given simulation. Ultimately, the goal is
the formulation of a unified EOS model that captures all of the
important physics in all relevant regimes of thermodynamic state space.

-   ***Optical, electrical, and transport properties:***

Other materials properties of interest include optical, electrical, and
transport properties of materials under extreme conditions. Since
experimental probes often involve the measurements of the optical,
electrical, and/or transport properties of materials, the predictions of
these properties as a function of stress and/or temperature is an area
of great importance. For example, temperature in shock physics
experiments is usually extracted from the measurement of emissivity of
the materials under dynamic compression. The first-principles
development of predictive simulations of the optical properties of
materials under high-stress/high-temperature conditions is of great
interest. Similarly the design and analysis of dynamic high-stress
experiments using pulsed power technology requires physics-based models
of the materials electrical conductivity at high temperature.

-   **Relevant research needs:** The development of quantum-based
    simulation tools that provide a predictive treatment of the optical,
    electrical, and/or transport properties of materials under extreme
    conditions.

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-   ***Kinetics and metastability:***

As has already been noted, there is often a competition between
thermodynamics and kinetics that leads to geometric frustration, or
trapping of materials in non-equilibrium metastable states (geometric,
magnetic, etc.). This is true whether one considers "simple" metals
(grain size, texture) or complex fluids such as microphase separating
multi-component polymeric systems. In such cases, straightforward
atomistic simulation methods will usually fail to capture even the most
dominant morphological motifs for systems with realistic geometric
dimensions, due to limitations of accessible simulation times.
Nevertheless, it is imperative for a genuine predictive capability on
the macroscale that physics-based mesoscale models capture the essence
of the detailed chemical interactions (bonding in metals; chain
topology, chemical composition) at microscopic scales. Field theoretic
models are often useful in such situations, as they can, in principle,
provide a connection between microscopic physics and continuum
behaviors; and the "dynamics" of phase evolution can be followed for
much longer times for field models than with a particle-based model.
However, there are few examples where realistic field descriptions have
been formulated and parameterized for specific materials. Another key
ingredient is the ability to incorporate the effects of a given
materials processing history on the final morphology/properties that
result.

-   **Relevant research needs:** Development of improved models for
    composites, which may range from "standard" materials such as
    filler-reinforced polymer foams or high-explosive formulations to
    highly novel, functional nanomaterials (see white paper on *Novel
    Materials*).

**3. Constitutive properties of materials:**

Of interest to the NNSA programs are applications that anticipate and
accommodate material or structural failure occurring in the course of
normal operation because material response in a 'failed state' can
control how the entire body continues to respond in an on-going process.
Accordingly, material properties well beyond the linear response regime
need to be accurately described. This gives rise to strong interest in
plasticity and failure, the full range of phase transformations,
chemical transformations, and coupled physics behaviors, which can
become pronounced for strongly-driven conditions.

As important as the science needed to create accurate material models,
the ultimate goal of achieving predictive *simulations* cannot be
reached without algorithmic advances that enable transforming successful
materials research into robust, efficient, and codable models.
Codability would be helped by good interfaces between the host code and
material models that facilitate implementation of material models,
particularly when a host code is incompatible with the underlying
equations of the model. \[For an example, see: *MIG version 0.0: Model
interface guidelines: rules to accelerate installation of numerical
models into any compliant parent code*, R.M. Brannon, M.K. Wong.
SAND96-2000, Sandia National Laboratories, Albuquerque, NM (1996).\]

-   ***Elasticity:***

The elastic response of a pure material is a fundamental property that
can be obtained directly from appropriate derivatives of the free energy
with respect to change in volume (or applied strain). Higher derivatives
correspond to nonlinearities in the stress-strain curve, up to the
elastic limit (beyond which permanent inelastic deformation occurs). The
forms of the various terms in the elastic free energy --- second- and
third-order elastic constants, etc. --- reflect the inherent symmetry of
the material, for example, cubic, monoclinic, or triclinic. Elasticity
is generally a sensitive function of thermodynamic state (temperature,
stress, material phase), and in the case of certain materials such as
polymers strain rate effects can be large. In the case of
polycrystalline or composite materials the elastic response must be
calculated as an appropriate average over a suitably large
representative volume element (RVE) in the material that captures any
possible anisotropies due for example to texture. This generally
involves either a formal or an empirical homogenization procedure.
Knowledge of the tensoral and isotropic elastic response is a key
ingredient in specification of an EOS; which, as noted earlier, is
fundamental to the reliable description of material response. The
ability to predict and utilize the elastic behaviors of metals,
organics, piezo- and ferroelectrics, and both random and ordered
composite materials is of significant interest.

-   **Relevant research needs:** Development of simulation tools to
    compute the aforementionedproperties reliably at elevated
    temperatures and pressures, for systems large enough that
    finite-size effects do not dominate; to formulate approaches that
    account for defects and damage that serve to modify the elastic
    response from that corresponding to perfect material; and to develop
    reliable methods for satisfying the preceding needs within the
    context of structurally complex, highly anisotropic functional
    materials.

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-   ***Plasticity:***

Among the outstanding challenges in plasticity and inelasticity theory
that are of interest to NNSA include plastic behavior of metals
subjected to complex load paths and/or high strain rates; the mechanical
response of polymers used as encapsulants; the vibration and impact
response of foams -- in bulk and as encapsulants; and creep of solders
and brazes subjected to cyclic thermal loading.

The observed ability for a solid to support deviatoric stresses under
compression to high stress is of significant scientific and practical
interest. It is important in several applications, including shock,
quasi-isentropic, and isothermal compression for equation-of-state
measurements.

-   **Relevant research needs:** Theoretical developments that build on
    and unify the variety of relevant experimental observations to
    develop a basic understanding of this aspect of material response.

Such efforts would be aided by computational methods for estimating
high-stress strength properties, including molecular dynamics and
dislocation dynamics simulations, developed in tandem with mesoscopic
models that capture the essence of the microscopic physics in ways that
can be expressed in engineering-scale simulation codes (for example by
formulation of sub-grid models for inhomogeneous inelastic deformation).
Then the acquired understanding will need to be represented in
physics-based models of material strength at high stress and high strain
rates and for complicated, non-monotonic load histories.

Under dynamic loading conditions, a brittle material transitions,
suddenly, from small strains to large deformations. However, existing
methods for treating large deformations corrupt the representation of
the constitutive response. Hence, further development is needed of
numerical methods that can treat large deformations while tracking
Lagrangian state variables.

Verification remains a large need in material modeling. Renewed effort
is needed in defining "patch tests" for constitutive models. These
should include tests to match the expected single-element responses for
idealized plasticity; and, taking a rather broad view of what is
included in "constitutive behavior, failure models such as von Mises and
Mohr-Coulomb under complicated stress histories.

-   **Relevant research needs:** Established and implementation of
    procedures into frameworks for automated testing of frame- and
    basis-invariance, thermodynamic consistency, and material stability.
    Such procedures would be exercised by host codes to ensure that all
    constitutive models meet these verification tests.

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-   ***Interfacial phenomena:***

Material interfaces often dominate constitutive behavior, whether they
be grain boundaries in metals or a liquid-solid interface as for polymer
wetting. In some cases, such as metallic or intermetallic alloys,
interfaces are "sharp" -- only several atomic diameters in extent;
whereas in others, such as polymers on a surface, the range of the
"interphase" between substrate material, "surface" polymer, and "bulk"
polymer can be quite large. In addition, significant stress localization
can occur at incommensurate interfaces, and have large effects on
mechanical properties and material stability/compatibility. Thus,
understanding physical and chemical processes peculiar to interfacial
regions in materials is a key component to developing quantifiable
predictive capability. Given the importance of interface phenomena and
the current limits of predictive modeling of them, interests extends to
understanding the behavior nearly any material interface. Example
systems of interest include adhesive bonds formed by epoxies or
elastomers; adhesion of encapsulating foams to metals or integrated
circuit boards, with prediction of initiation and nature of bond
failure; adhesion and slipping between concrete and reinforcing bar.

-   **Relevant research needs:** Development of
    theoretical/computational approaches to capture the underlying
    phenomenology for the range of materials of interest (including
    interfaces among metals, insulators, organics/polymers, and liquids,
    in all combinations), and development of multi-scale approaches to
    describe the disparity between atomic dimensions and interfacial
    layer thicknesses.

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-   ***Homogenized treatment of damage, cracking, and failure:***

While localization and failure behavior entails far more than just
constitutive properties (see white papers on *Material Stability* and
*Material Failure*), a homogenized treatment of damage can provide a
useful local description of the material state approaching or in failure
on a spatial scale that cannot be resolved in practical simulations, and
is therefore often an important "sub model" that appears explicitly or
implicitly within a constitutive formulation. Making damage models
predictive requires establishing direct connection of their
phenomenology to underlying degradation mechanisms and ensuring the
models' thermodynamic consistency. This is inherently a complex,
multiscale problem.

-   **Relevant research needs:**Establishment of physically based,
    quantifiable and measurable connections between atomistic
    simulations, mesoscopic dislocation dynamics and rate-dependent,
    anisotropic continuum-scale plasticity models.

Comments similar to those immediately above for damage apply more
generally to any material in which localization occurs due to (any kind
of) material heterogeneity on a sub-grid spatial scale. Although the
details will surely differ from application to application, this type of
connection among disparate scales and dominant physical processes,
involving one or more homogenizations to yield a constitutive model
suitable for macroscale simulation, is needed whether the specific
target is a single- or poly-crystal metal, or a complicated,
multi-component organic material. However, in order to be reliable and
of real predictive power, each of these up-scalings and homogenization
steps needs to be performed in the most rigorous way possible, taking
care on a case-by-case basis to capture the essential physics or
chemistry that drives the observed phenomenology on a given
spatio-temporal scale.

Prediction of strength behavior that is dependent on 'weakest link'
failure requires characterizing and representing variability of strength
in a material and sample size effects. Both micro-scale and macro-scale
simulations can contribute to determining good representations of
strength distributions.

-   **Relevant research needs:** Development of stochastic modeling
    methodsto predict strength behavior and provide reliable
    representations of strength distributions .

**Acknowledgement**

This work was, in part, performed by the Lawrence Livermore National
Laboratory, Los Alamos National Laboratory, and Sandia National
Laboratories under auspices of the U.S. Department of Energy.

This document was prepared as an account of work sponsored by an agency
of the United States Government. Neither the United States Government
nor the University of California nor any of their employees, makes any
warranty, express or implied, or assumes any legal liability or
responsibility for the accuracy, completeness, or usefulness of any
information, apparatus, product, or process disclosed, or represents
that its use would not infringe privately owned rights. Reference herein
to any specific commercial product, process, or service by trade name,
trademark, manufacturer, or otherwise, does not necessarily constitute
or imply its endorsement, recommendation, or favoring by the United
States Government or the University of California. The views and
opinions of authors expressed herein do not necessarily state or reflect
those of the United States Government or the University of California,
and shall not be used for advertising or product endorsement purposes.

This work was performed under the auspices of the U.S. Department of
Energy by University of California, Lawrence Livermore National
Laboratory under Contract W-7405-Eng-48.