^{1,a)}

### Abstract

A general formulation was developed to represent material models for applications in dynamic loading. Numerical methods were devised to calculate response to shock and ramp compression and ramp decompression, generalizing previous solutions for scalar equations of state. The numerical methods were found to be flexible and robust, and matched analytic results to a high accuracy. The basic ramp and shock solution methods were coupled to solve for compositedeformation paths, such as shock-induced impacts and shock interactions with a planar interface between different materials. These calculations capture much of the physics of typical material dynamics experiments, without requiring spatially resolving simulations. Example calculations were made of loading histories in metals, illustrating the effects of plastic work on the temperatures induced in quasi-isentropic and shock-release experiments and the effect of a phase transition.

Ian Gray introduced the author to the concept of multimodel material properties software. Lee Markland developed a prototype Hugoniot-calculating computer program for equations of state while working for the author as an undergraduate summer student.

Evolutionary work on material property libraries was supported by the UK Atomic Weapons Establishment, Fluid Gravity Engineering Ltd., and Wessex Scientific and Technical Services Ltd. Refinements to the technique and applications to the problems described were undertaken at Los Alamos National Laboratory (LANL) and Lawrence Livermore National Laboratory (LLNL).

The work was performed partially in support of, and funded by, the National Nuclear Security Agency’s Inertial Confinement Fusion program at LANL (managed by Steven Batha), and LLNL’s Laboratory-Directed Research and Development Project No. 06-SI-004 (Principal Investigator: Hector Lorenzana). The work was performed under the auspices of the U.S. Department of Energy under Contracts Nos. W-7405-ENG-36, DE-AC52-06NA25396, and DE-AC52-07NA27344.

I. INTRODUCTION

II. CONCEPTUAL STRUCTURE FOR MATERIAL PROPERTIES

III. IDEALIZED ONE-DIMENSIONAL LOADING

A. Ramp compression

B. Shock compression

C. Accuracy: Application to air

IV. COMPLEX BEHAVIOR OF CONDENSED MATTER

A. Temperature

1. Density-temperature EOS

2. Temperature model for mechanical EOS

B. Strength

1. Preferred representation of isotropic strength

2. Beryllium

C. Phase changes

V. COMPOSITE LOADING PATHS

VI. CONCLUSIONS

### Key Topics

- Materials properties
- 45.0
- Equations of state
- 42.0
- Stress strain relations
- 15.0
- Elasticity
- 11.0
- Spatial analysis
- 11.0

## Figures

Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations for general material models, compared to analytic solutions.

Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations for general material models, compared to analytic solutions.

Shock Hugoniot for Al in pressure-temperature space, for different representations of the EOS.

Shock Hugoniot for Al in pressure-temperature space, for different representations of the EOS.

Principal adiabat and shock Hugoniot for Be in normal stress-compression space, neglecting strength (dashed), for Steinberg–Guinan strength (solid), and for elastic-perfectly plastic with (dotted).

Principal adiabat and shock Hugoniot for Be in normal stress-compression space, neglecting strength (dashed), for Steinberg–Guinan strength (solid), and for elastic-perfectly plastic with (dotted).

Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space, neglecting strength (dashed), for Steinberg–Guinan strength (solid), and for elastic-perfectly plastic with (dotted).

Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space, neglecting strength (dashed), for Steinberg–Guinan strength (solid), and for elastic-perfectly plastic with (dotted).

Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress-temperature space, neglecting strength (dashed), for Steinberg–Guinan strength (solid), and for elastic-perfectly plastic with (dotted).

Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress-temperature space, neglecting strength (dashed), for Steinberg–Guinan strength (solid), and for elastic-perfectly plastic with (dotted).

Demonstration of shock Hugoniot solution across a phase boundary: shock melting of Al, for different initial porosities.

Demonstration of shock Hugoniot solution across a phase boundary: shock melting of Al, for different initial porosities.

Wave interactions for the impact of a flat projectile moving from left to right with a stationary target. The dashed arrows are a guide to the sequence of states. For a projectile moving from right to left, the construction is the mirror image reflected in the normal stress axis.

Wave interactions for the impact of a flat projectile moving from left to right with a stationary target. The dashed arrows are a guide to the sequence of states. For a projectile moving from right to left, the construction is the mirror image reflected in the normal stress axis.

Wave interactions for the release of a shocked state (shock moving from left to right) into a stationary “window” material to its right. The release state depends whether the window has a higher or lower shock impedance than the shocked material. The dashed arrows are a guide to the sequence of states. For a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis.

Wave interactions for the release of a shocked state (shock moving from left to right) into a stationary “window” material to its right. The release state depends whether the window has a higher or lower shock impedance than the shocked material. The dashed arrows are a guide to the sequence of states. For a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis.

Wave interactions for the release of a shocked state by tension induced as materials try to separate in opposite directions when joined by a bonded interface. Material damage, spall, and separation are neglected: the construction shows the maximum tensile stress possible. For general material properties, e.g., if plastic flow is included, the stress maximum tensile stress is not just the negative of the initial shock stress. The dashed arrows are a guide to the sequence of states. The graph shows the initial state after an impact by a projectile moving from right to left; for a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis.

Wave interactions for the release of a shocked state by tension induced as materials try to separate in opposite directions when joined by a bonded interface. Material damage, spall, and separation are neglected: the construction shows the maximum tensile stress possible. For general material properties, e.g., if plastic flow is included, the stress maximum tensile stress is not just the negative of the initial shock stress. The dashed arrows are a guide to the sequence of states. The graph shows the initial state after an impact by a projectile moving from right to left; for a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis.

Schematic of uniaxial wave interactions induced by the impact of a flat projectile with a composite target.

Schematic of uniaxial wave interactions induced by the impact of a flat projectile with a composite target.

Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF release window, after impact. Structures on the waves are elastic precursors.

Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF release window, after impact. Structures on the waves are elastic precursors.

## Tables

Interface to material models required for explicit forward-time continuum dynamics simulations. Parentheses in the interface calls denote functions, e.g., “stress (state)” for “stress as a function of the instantaneous, local state.” The evolution functions are shown in the operator-split structure that is most robust for explicit, forward-time numerical solutions, and can also be used for calculations of the shock Hugoniot and ramp compression. Checks for self-consistency include that mass density is positive, volume or mass fractions of components of a mixture add up to one, etc.

Interface to material models required for explicit forward-time continuum dynamics simulations. Parentheses in the interface calls denote functions, e.g., “stress (state)” for “stress as a function of the instantaneous, local state.” The evolution functions are shown in the operator-split structure that is most robust for explicit, forward-time numerical solutions, and can also be used for calculations of the shock Hugoniot and ramp compression. Checks for self-consistency include that mass density is positive, volume or mass fractions of components of a mixture add up to one, etc.

Examples of types of material model, distinguished by different structures in the state vector. The symbols are : mass density; : specific internal energy, : temperature, : volume fraction, : mass fraction, : stress deviator, : fraction of plastic work converted to heat, : shear modulus, : elastic and plastic parts of strain rate deviator, : scalar equivalent plastic strain, : factor in effective strain magnitude. Reacting solid explosives can be represented as heterogeneous mixtures, one component being the reacted products; reaction, a process of internal evolution, transfers material from unreacted to reacted components. Gas-phase reaction can be represented as a homogeneous mixture, reactions transferring mass between components representing different types of molecule. Symmetric tensors such as the stress deviator are represented more compactly by their six unique upper triangular components, e.g., using Voigt notation.

Examples of types of material model, distinguished by different structures in the state vector. The symbols are : mass density; : specific internal energy, : temperature, : volume fraction, : mass fraction, : stress deviator, : fraction of plastic work converted to heat, : shear modulus, : elastic and plastic parts of strain rate deviator, : scalar equivalent plastic strain, : factor in effective strain magnitude. Reacting solid explosives can be represented as heterogeneous mixtures, one component being the reacted products; reaction, a process of internal evolution, transfers material from unreacted to reacted components. Gas-phase reaction can be represented as a homogeneous mixture, reactions transferring mass between components representing different types of molecule. Symmetric tensors such as the stress deviator are represented more compactly by their six unique upper triangular components, e.g., using Voigt notation.

Outline hierarchy of material models, illustrating the use of polymorphism (in the object-oriented programming sense). Continuum dynamics programs can refer to material properties as an abstract “material type” with an abstract material state. The actual type of a material (e.g., mechanical EOS), the specific model type (e.g., polytropic), and the state of material of that type are all handled transparently by the object-oriented software structure. The reactive EOS has an additional state parameter , and the software operations are defined by extending those of the mechanical EOS. Spalling materials can be represented by a solid state plus a void fraction , with operations defined by extending those of the solid material. Homogeneous mixtures are defined as a set of thermal EOS, and the state is the set of states and mass fractions for each. Heterogeneous mixtures are defined as a set of “pure” material properties of any type, and the state is the set of states for each component plus its volume fraction.

Outline hierarchy of material models, illustrating the use of polymorphism (in the object-oriented programming sense). Continuum dynamics programs can refer to material properties as an abstract “material type” with an abstract material state. The actual type of a material (e.g., mechanical EOS), the specific model type (e.g., polytropic), and the state of material of that type are all handled transparently by the object-oriented software structure. The reactive EOS has an additional state parameter , and the software operations are defined by extending those of the mechanical EOS. Spalling materials can be represented by a solid state plus a void fraction , with operations defined by extending those of the solid material. Homogeneous mixtures are defined as a set of thermal EOS, and the state is the set of states and mass fractions for each. Heterogeneous mixtures are defined as a set of “pure” material properties of any type, and the state is the set of states for each component plus its volume fraction.

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