The separation of EOS (volumetric) and strength (deviatoric) is a pragmatic convenience, not a physical reality. At high pressure, both derive from the same interatomic potential. Selected materials reveal that:
A next-generation “strength-aware EOS” must embed dislocation dynamics or phase-field damage directly into the free energy. Until then, users of Hugoniot databases should treat tabulated “pressure” as the longitudinal stress, subtract ( \frac23Y ) to recover hydrostatic pressure, and always cite the strain rate.
Acknowledgments
This article synthesizes work from the shock physics community, including decades of data from LLNL, LANL, Sandia, CEA, and the Institute for Shock Physics (WSU).
Correspondence
For access to the Jupyter Notebook that generates the figures for Cu and Ta strength scaling, see the author’s GitHub repository (link in published version).
References (abbreviated)
If you intended a different completion of the prompt (e.g., “selected high explosives,” “selected planetary ices,” or “selected materials for additive manufacturing”), please clarify, and I will rewrite the article accordingly.
The study of materials under extreme conditions relies on two pillars of constitutive modeling: the Equation of State (EOS) , which governs how a material compresses, and strength models
, which define how it resists shear deformation and eventually yields. A seminal reference in this field is Daniel J. Steinberg’s 1991 report, equation of state and strength properties of selected
Equation of State and Strength Properties of Selected Materials
, which provides critical data for approximately 50 materials often used in high-velocity impact and shock wave analysis. AIP Publishing 1. Theoretical Framework of the Equation of State
An EOS represents a macroscopic relationship between thermodynamic variables—typically pressure ( ), volume ( ), and temperature (
). In shock physics, the material response is often decomposed into a "cold" compression part and a thermal contribution. Springer Nature Link Mie-Gruneisen EOS
: This is one of the most widely used models for solids. It relates the thermal pressure to the internal energy through the Grüneisen parameter
), which describes how vibrational frequencies change with volume. Birch-Murnaghan & Vinet EOS
: These are empirical forms used to fit isothermal compression data. The The separation of EOS (volumetric) and strength (deviatoric)
is often called the "universal" equation of state because it remains valid even at ultra-high compressions where other models might diverge. The Shock Hugoniot
: In dynamic experiments, the "Hugoniot" represents the locus of end states reached by a shock wave, serving as a primary calibration for pressure in high-energy physics. OSTI (.gov) 2. Strength Properties and Constitutive Modeling
While the EOS handles the "fluid-like" response of materials at extreme pressures, the strength model characterizes the yield surface
—the threshold where a material stops behaving elastically and begins permanent, plastic deformation. AIP Publishing
It sounds like you are looking for a technical guide on the Equation of State (EOS) and Strength Properties of selected materials (likely metals, ceramics, polymers, or geomaterials) under high-pressure and high-strain-rate conditions. This is a common need in fields like shock physics, planetary science, ballistic impact modeling, and materials engineering.
Below is a structured guide covering the key concepts, common models, and how to select/apply them for a given material.
| Material | Density (g/cm³) | Bulk Modulus (GPa) | Shear Modulus (GPa) | HEL (GPa) | Spall Strength (GPa) | Dominant Failure Mode | |----------|----------------|--------------------|---------------------|-----------|----------------------|----------------------| | Copper | 8.93 | 140 | 48 | 0.2 | 1.8–2.5 | Ductile void growth | | Tantalum | 16.65 | 200 | 69 | 1.2 | 4.0–6.0 | Adiabatic shear bands | | SiC | 3.21 | 220 | 193 | 14.5 | 1.5–2.0 | Brittle fracture / comminution | | Quartzite | 2.65 | 37 (low-P) → 100 (high-P) | 44 | ~6.0 | 0.3–0.5 | Phase transition + fragmentation | | Dry sand | 1.6 (loose) / 1.8 (dense) | ~0.1–0.3 (bulk) | N/A | N/A | ~0 | Compaction + shear localization | Acknowledgments This article synthesizes work from the shock
Though not in our "selected" list exhaustively, Fe is the ultimate test case for EOS and strength under extreme conditions (Earth’s inner core: 330 GPa, 6000 K).
This demonstrates that high-pressure strength properties of selected materials often diverge from ideal EOS predictions due to microstructural evolution (grain growth, recrystallization).
For decades, the equation of state—a thermodynamic relation between pressure, volume, and temperature (P-V-T)—was treated separately from strength properties (resistance to plastic deformation, fracture, and shear). However, under dynamic loading (e.g., ballistic impact, planetary accretion, or explosive forming), these properties are intimately coupled. A material's compressive response influences its shear strength, and its strength affects the onset of melting and phase transitions.
This article focuses on selected materials that serve as:
We review their EOS parameters (bulk modulus K₀, its pressure derivative K₀', Grüneisen parameter γ₀) and strength metrics (Hugoniot elastic limit HEL, shear strength G, spall strength).
The most widely used form for solids:
[ P(V, T) = P_\textcold(V) + \frac\gamma(V)V [E_\textth(T) - E_0] ]
where ( \gamma(V) = V \left(\frac\partial P\partial E\right)_V ) is the Grüneisen parameter, often assumed ( \gamma(V) = \gamma_0 (V/V_0)^q ). For metals, ( q \approx 1 ) (Slater model). Limitations: fails near melt or phase transitions.