Equations of State

An equation of state (EOS) is a mathematical relationship describing how the thermodynamic properties of a material depend on one another. In shock-wave and detonation physics, these properties commonly include pressure, density, temperature, and specific internal energy.

A general form of an equation of state is:

where:

• P = pressure
• ρ = material density
• e = specific internal energy

The EOS establishes how pressure changes as a material is compressed, heated, or allowed to expand.

The conservation equations introduced in Lesson 02 describe how mass, momentum, and energy are conserved across a shock wave. However, conservation laws alone cannot completely determine the thermodynamic state of a material.

An EOS provides the additional relationship required to close the governing mathematical system.

For example, conservation of momentum establishes a relationship between pressure and material motion across a shock, but it does not independently determine how a particular material responds to compression. That response depends on its thermodynamic properties.

Figure 6.1 illustrates how three fundamental elements work together: conservation laws establish what must be conserved, equations of state describe the thermodynamic response of the material, and chemical kinetics describes how reactions progress and release energy.

Figure 6.1. The Role of Equations of State in Shock-Wave and Detonation Theory. Conceptual illustration showing how conservation laws, equations of state, and chemical kinetics contribute to the description of shock compression and detonation. The lower panels illustrate their application to shock Hugoniots, Chapman–Jouguet (CJ) theory, and the Zel’dovich–von Neumann–Döring (ZND) model. An EOS provides the thermodynamic relationships required to connect conservation equations with material behavior, while chemical equilibrium and finite-rate reaction models provide additional information for reactive systems. The diagrams are schematic and not based on material-specific calculations.

According to Equation (6.2), increasing temperature at constant density produces a corresponding increase in pressure. Similarly, compressing a gas at constant temperature increases its pressure.

These relationships make the ideal-gas EOS particularly useful for understanding fundamental concepts in compressible fluid dynamics and shock-wave propagation.

However, the ideal-gas assumptions become increasingly restrictive under extreme thermodynamic conditions.

In strongly shocked condensed materials and dense detonation products, intermolecular forces, nonlinear compressibility, phase transformations, and chemical composition may substantially influence the relationship between pressure, density, and energy.

Consequently, more sophisticated equations of state are generally required.

Many energetic materials, metals, and structural components initially exist as solids or liquids.

Unlike ideal gases, condensed materials resist compression strongly, and their thermodynamic response can be highly nonlinear.

When a strong shock propagates through a condensed material, several physical phenomena may occur:

• Elastic and plastic deformation: Solids may initially respond elastically before undergoing irreversible deformation.
• Nonlinear compression: Resistance to compression may change significantly as density increases.
• Shock heating: A portion of the mechanical work associated with compression increases internal energy and temperature.
• Phase transformations: At sufficiently high pressures and temperatures, materials may undergo structural changes or melting.
• Release and expansion: During unloading, a compressed material generally follows a thermodynamic path different from its original shock Hugoniot.

These effects require equations of state that account for the material’s compressibility, internal energy, and potentially its phase-dependent behavior.

Consider the inert shock Hugoniot of aluminum introduced in Lesson 03.

As a shock wave propagates through aluminum, pressure and density increase while specific volume decreases.

The Shock Hugoniot panel in Figure 6.1 illustrates the general relationship between pressure and specific volume during shock compression. The aluminum example presented in Figure 3.1 of Lesson 03 provides a material-specific conceptual illustration of this behavior.

An appropriate condensed-material EOS provides the thermodynamic relationships needed to determine these states when combined with the Rankine–Hugoniot conservation equations.

An important distinction: A Hugoniot is not itself a complete equation of state.

The Hugoniot represents a collection of states accessible through shock compression from a specified initial condition. An EOS, by comparison, describes a broader relationship among pressure, density, internal energy, and potentially other thermodynamic variables.

Two materials subjected to similar shock conditions may exhibit very different responses because their equations of state differ.

This is why material-specific thermodynamic characterization is important in shock physics.

For solid materials, an equation of state primarily describes the volumetric thermodynamic response to compression and changes in internal energy. A separate constitutive strength model may also be required to represent deviatoric stresses, plastic deformation, and other mechanical effects. Consequently, an EOS alone does not necessarily provide a complete description of a shocked solid.

When an energetic material detonates, its chemical composition changes as the original material transforms into reaction products.

These products may contain several molecular species whose relative concentrations and thermodynamic properties depend on pressure, temperature, and chemical equilibrium.

Unlike an inert material undergoing shock compression, a detonating energetic material experiences both mechanical compression and chemical transformation.

Consequently, an EOS describing the unreacted material is generally insufficient to represent its fully reacted products.

As discussed in Lessons 03–05, the leading shock compresses and heats the initially unreacted energetic material. Chemical reactions subsequently transform it into products while releasing energy.

This transformation changes the relationship among pressure, volume, and internal energy.

The distinction was illustrated previously in Lesson 03, Figure 3.2, where the unreacted explosive and equilibrium detonation products were represented by different Hugoniot curves.

The product EOS describes the thermodynamic behavior of the chemically transformed material.

As detonation products expand, their pressure, density, temperature, and internal energy evolve according to the applicable thermodynamic relationships and flow conditions.

In classical CJ theory, the equilibrium-products EOS contributes to defining the reacted Hugoniot.

The intersection and tangency relationships between the products Hugoniot and the appropriate Rayleigh line establish the classical CJ solution.

As shown in the lower-center panel of Figure 6.1, the CJ point is associated with tangency between the Rayleigh line and the equilibrium-products Hugoniot in pressure–specific-volume coordinates.

This condition also corresponds to sonic flow relative to the detonation wave at the ideal CJ state, as discussed in Lesson 04.

Therefore, the CJ solution depends not only on conservation laws but also on the thermodynamic properties of the reaction products and their equilibrium composition.

The ZND model extends classical CJ theory by describing the finite reaction zone behind the leading shock.

Within this zone, pressure, temperature, density, chemical composition, and flow velocity evolve as reactions proceed.

The lower-right panel of Figure 6.1 illustrates this internal structure, including the leading shock and finite chemical reaction zone.

To describe these changes, the conservation equations must be supplemented by thermodynamic relationships and a model of chemical reaction progress.

An EOS supplies the thermodynamic relationships, while reaction kinetics determines how the chemical transformations evolve through the reaction zone.

This distinction explains why the ZND model can describe the internal evolution of a detonation wave, whereas CJ theory primarily characterizes its ideal overall equilibrium state.

Figure 6.1 emphasizes three complementary elements:

Conservation laws establish the fundamental physical constraints governing mass, momentum, and energy.

Equations of state describe how materials respond thermodynamically to compression, heating, and expansion.

Chemical kinetics describes how chemical reactions proceed and release energy over time.

Together, these elements provide the theoretical foundation for understanding shock-wave propagation and detonation physics.

In summary, conservation laws tell us what must be conserved, equations of state tell us how materials respond, and chemical kinetics tells us how reactions evolve. Their combined application forms the basis of classical and modern detonation theory.

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