### 17.9.1 Equation of state

Products: ABAQUS/Explicit  ABAQUS/CAE

### Overview

Equations of state:

• provide a hydrodynamic material model in which the material's volumetric strength is determined by an equation of state;

• determine the pressure (positive in compression) as a function of the density, , and the specific energy (the internal energy per unit mass), : ;

• are available as Mie-Grüneisen equations of state (thus providing the linear Hugoniot form);

• are available as equations of state for the compaction of ductile porous materials and must be used in conjunction with the Mie-Grüneisen equation of state for the solid phase;

• are available as JWL high explosive equations of state;

• are available in the form of an ideal gas;

• assume an adiabatic condition unless a dynamic fully coupled temperature-displacement analysis is used;

• can be used to model a material that has only volumetric strength (the material is assumed to have no shear strength) or a material that also has isotropic elastic or Newtonian viscous deviatoric behavior;

• can be used with the Mises (Classical metal plasticity, Section 18.2.1) or the Johnson-Cook (Johnson-Cook plasticity, Section 18.2.7) plasticity models; and

• can be used with the tensile failure model (Dynamic failure models, Section 18.2.8) to model dynamic spall or a pressure cutoff.

### Energy equation and Hugoniot curve

The equation for conservation of energy equates the increase in internal energy per unit mass, , to the rate at which work is being done by the stresses and the rate at which heat is being added. In the absence of heat conduction the energy equation can be written as

where p is the pressure stress defined as positive in compression, is the pressure stress due to the bulk viscosity, is the deviatoric stress tensor, is the deviatoric part of strain rate, and is the heat rate per unit mass.

The equation of state is assumed for the pressure as a function of the current density, , and the internal energy per unit mass, :

which defines all the equilibrium states that can exist in a material. The internal energy can be eliminated from the above equation to obtain a p versus V relationship (where V is the current volume) or, equivalently, a p versus relationship that is unique to the material described by the equation of state model. This unique relationship is called the Hugoniot curve and is the locus of pV states achievable behind a shock (see Figure 17.9.1–1). The Hugoniot pressure, , is a function of density only and can be defined, in general, from fitting experimental data.

### Mie-Grüneisen equations of state

A Mie-Grüneisen equation of state is linear in energy. The most common form is

where and are the Hugoniot pressure and specific energy (per unit mass) and are functions of density only, and is the Grüneisen ratio defined as

where is a material constant and is the reference density.

The Hugoniot energy, , is related to the Hugoniot pressure by

where is the nominal volumetric compressive strain. Elimination of and from the above equations yields

The equation of state and the energy equation represent coupled equations for pressure and internal energy. ABAQUS/Explicit solves these equations simultaneously at each material point.

Linear equations of state can always be written in the form

where and are functions of density only and depend on the particular equation of state model.

#### Linear Us – Up Hugoniot form

A common fit to the Hugoniot data is given by

where and s define the linear relationship between the linear shock velocity, , and the particle velocity, , as follows:

With the above assumptions the linear Hugoniot form is written as

where is equivalent to the elastic bulk modulus at small nominal strains.

There is a limiting compression given by the denominator of this form of the equation of state

or

At this limit there is a tensile minimum; thereafter, negative sound speeds are calculated for the material.

 Input File Usage: Use both of the following options: ```*DENSITY (to specify the reference density ) *EOS, TYPE=USUP (to specify the variables , s, and )```

 ABAQUS/CAE Usage: Property module: material editor: GeneralDensity (to specify the reference density )OtherEos: Type: Us - Up (to specify the variables , s, and )

##### Initial state

The initial state of the material is determined by the initial values of specific energy, , and pressure stress, p. ABAQUS/Explicit will automatically compute the initial density, , that satisfies the equation of state, . You can define the initial specific energy and initial stress state (see Initial conditions, Section 27.2.1). The initial pressure used by the equation of state is inferred from the specified stress states. If no initial conditions are specified, ABAQUS/Explicit will assume that the material is at its reference state:

 Input File Usage: Use either or both of the following options, as required: ```*INITIAL CONDITIONS, TYPE=SPECIFIC ENERGY *INITIAL CONDITIONS, TYPE=STRESS```

 ABAQUS/CAE Usage: Initial specific energy and initial stress are not supported in ABAQUS/CAE.

### P–α equation of state

The equation of state is designed for modeling the compaction of ductile porous materials. The implementation in ABAQUS/Explicit is based on the model proposed by Hermann (1968) and Carroll and Holt (1972). The constitutive model provides a detailed description of the irreversible compaction behavior at low stresses and predicts the correct thermodynamic behavior at high pressures for the fully compacted solid material. In ABAQUS/Explicit the solid phase is assumed to be governed by a Mie-Grüneisen equation of state. The relevant properties of the porous material in the virgin state, to be discussed later, and the material properties of the solid phase are specified separately.

The porosity of the material, n, is defined as the ratio of pore volume, , to total volume, , where is the solid volume. The porosity remains in the range , with 0 indicating full compaction. It is convenient to introduce a scalar variable , sometimes referred to as “distension,” defined as the ratio of the density of the solid material, , to the density of the porous material, , both evaluated at the same temperature and pressure:

For a fully compacted material ; otherwise, is greater than 1. Assuming that the density of the pores is negligible compared to that of the solid phase, can be expressed in terms of the porosity n as

An equation of state is assumed for the pressure of the porous material as a function of ; current density, ; and internal energy per unit mass, , in the form

Assuming that the pores carry no pressure, it follows from equilibrium considerations that when a pressure p is applied to the porous material, it gives rise to a volume-average pressure in the solid phase equal to . Assuming that the specific internal energies of the porous material and the solid matrix are the same (i.e., neglecting the surface energy of the pores), the equation of state of the porous material can be expressed as

where is the equation of state of the solid material. For the fully compacted material (that is, when ), the equation of state reduces to that of the solid phase, therefore predicting the correct thermodynamic behavior at high pressures.

The equation of state must be supplemented by an equation that describes the behavior of as a function of the thermodynamic state. This equation takes the form

where is a state variable corresponding to the minimum value attained by during plastic (irreversible) compaction of the material. The state variable is initialized to the elastic limit for a material that is at its virgin state. The specific form of the function used by ABAQUS/Explicit is illustrated in Figure 17.9.1–2 and is discussed next.

The function captures the general behavior to be expected in a ductile porous material. The unloaded virgin state corresponds to the value , where is the reference porosity of the material. Initial compression of the porous material is assumed to be elastic. Recall that decreasing porosity corresponds to a reduction in . As the pressure increases beyond the elastic limit, , the pores in the material start to crush, leading to irreversible compaction and permanent (plastic) volume change. Unloading from a partially compacted state follows a new elastic curve that depends on the maximum compaction (or, alternatively, the minimum value of ) ever attained during the deformation history of the material. The absolute value of the slope of the elastic curve decreases as decreases, as will be quantified later. The material becomes fully compacted when the pressure reaches the compaction pressure ; at that point , a value that is retained forever. The function therefore has multiple branches: a plastic branch, , and multiple elastic branches, , corresponding to elastic unloading from partially compacted states. The appropriate branch of A is selected according to the following rule:

These expressions can be inverted to solve for p:

The equation for the plastic curve takes the form

or, alternatively,

The elastic curve originally proposed by Hermann (1968) is given by the differential equation

where is the elastic bulk modulus of the solid material at small nominal strains; is the reference density of the solid; and and are the reference sound speeds in the solid and virgin (porous) materials, respectively. (Here is equivalent to the reference sound speed, , of the Mie-Grüneisen equation of state for the solid phase.) Following Wardlaw et al. (1996), the above equation for the elastic curve in ABAQUS/Explicit is simplified and replaced by the linear relations

and

 Input File Usage: Use the following options to specify the properties of the solid phase (the reference density, ; the reference sound speed, ; the slope of the curve, s; and the Grüneisen ratio, ): ```*DENSITY *EOS, TYPE=USUP```Use the following option to specify the properties of the porous material (the reference sound speed, ; the reference porosity, ; the elastic limit, ; and the compaction pressure, ):`*EOS COMPACTION`

 ABAQUS/CAE Usage: Property module: material editor: GeneralDensity (to specify the reference density )OtherEos: Type: Us - Up (to specify the variables , s, and )OtherEos: Suboptions Eos Compaction (to specify the reference sound speed, ; the porosity of the unloaded material, ; the pressure required to initialize plastic behavior, ; and the pressure at which all pores are crushed, )

#### Initial state

The initial state of the porous material is determined from the initial values of porosity, ; specific energy, ; and pressure stress, p. ABAQUS/Explicit will automatically compute the initial density, , that satisfies the equation of state, . You can define the initial porosity, initial specific energy, and initial stress state (see Initial conditions, Section 27.2.1). If no initial conditions are given, ABAQUS/Explicit will assume that the material is at its virgin state:

ABAQUS/Explicit will issue an error message if the initial state lies outside the region of allowed states (see Figure 17.9.1–2). When initial conditions are specified only for p (or for ), ABAQUS/Explicit will compute (or p) assuming that the state lies on the primary (monotonic loading) curve.

 Input File Usage: Use some or all of the following options, as required: ```*INITIAL CONDITIONS, TYPE=SPECIFIC ENERGY *INITIAL CONDITIONS, TYPE=STRESS *INITIAL CONDITIONS, TYPE=POROSITY```

 ABAQUS/CAE Usage: Initial specific energy, initial stress, and initial porosity are not supported in ABAQUS/CAE.

### JWL high explosive equation of state

The Jones-Wilkens-Lee (or JWL) equation of state models the pressure generated by the release of chemical energy in an explosive. This model is implemented in a form referred to as a programmed burn, which means that the reaction and initiation of the explosive is not determined by shock in the material. Instead, the initiation time is determined by a geometric construction using the detonation wave speed and the distance of the material point from the detonation points.

The JWL equation of state can be written in terms of the initial energy per unit mass, , as

where and are user-defined material constants; is the user-defined density of the explosive; and is the density of the detonation products.

 Input File Usage: Use both of the following options: ```*DENSITY (to specify the density of the explosive ) *EOS, TYPE=JWL (to specify the material constants and )```

 ABAQUS/CAE Usage: Property module: material editor: GeneralDensity (to specify the density of the explosive )OtherEos: Type: JWL (to specify the material constants and )

#### Arrival time of detonation wave

ABAQUS/Explicit calculates the arrival time of the detonation wave at a material point as the distance from the material point to the nearest detonation point divided by the detonation wave speed:

where is the position of the material point, is the position of the Nth detonation point, is the detonation delay time of the Nth detonation point, and is the detonation wave speed of the explosive material. The minimum in the above formula is over the N detonation points that apply to the material point.

#### Burn fraction

To spread the burn wave over several elements, a burn fraction, , is computed as

where is a constant that controls the width of the burn wave (set to a value of 2.5) and is the characteristic length of the element. If the time is less than , the pressure is zero in the explosive; otherwise, the pressure is given by the product of and the pressure determined from the JWL equation above.

#### Defining detonation points

You can define any number of detonation points for the explosive material. Coordinates of the points must be defined along with a detonation delay time. Each material point responds to the first detonation point that it sees. The detonation arrival time at a material point is based upon the time that it takes a detonation wave (traveling at the detonation wave speed ) to reach the material point plus the detonation delay time for the detonation point. If there are multiple detonation points, the arrival time is based on the minimum arrival time for all the detonation points. In a body with curved surfaces care should be taken that the detonation arrival times are meaningful. The detonation arrival times are based on the straight line of sight from the material point to the detonation point. In a curved body the line of sight may pass outside of the body.

 Input File Usage: Use both of the following options to define the detonation points: ```*EOS, TYPE=JWL *DETONATION POINT```

 ABAQUS/CAE Usage: Property module: material editor: OtherEos: Type: JWL: SuboptionsDetonation Point

#### Initial state

Explosive materials generally have some nominal volumetric stiffness before detonation. It may be useful to incorporate this stiffness when elements modeled with a JWL equation of state are subjected to stress before initiation of detonation by the arriving detonation wave. You can define the pre-detonation bulk modulus, . The pressure will be computed from the volumetric strain and until detonation, at which time the pressure will be determined by the procedure outlined above.

If you specify a nonzero value of , you can also define an initial stress state for the explosive materials.

 Input File Usage: *INITIAL CONDITIONS, TYPE=STRESS

 ABAQUS/CAE Usage: Initial stress is not supported in ABAQUS/CAE.

The initial relative density () used in the JWL equation is assumed to be unity. You can specify nonzero values of initial specific energy, .

 Input File Usage: `*INITIAL CONDITIONS, TYPE=SPECIFIC ENERGY`

 ABAQUS/CAE Usage: Initial specific energy is not supported in ABAQUS/CAE.

### Ideal gas equation of state

An ideal gas equation of state can be written in the form of

where is the ambient pressure, R is the gas constant, is the current temperature, and is the absolute zero on the temperature scale being used. It is an idealization to real gas behavior and can be used to model any gases approximately under appropriate conditions (e.g., low pressure and high temperature).

One of the important features of an ideal gas is that its specific energy depends only upon its temperature; therefore, the specific energy can be integrated numerically as

where is the initial specific energy at the initial temperature and is the specific heat at constant volume (or the constant volume heat capacity), which depends only upon temperature for an ideal gas.

Modeling with an ideal gas equation of state is typically performed adiabatically; the temperature increase is calculated directly at the material integration points according to the adiabatic thermal energy increase caused by the work , where v is the specific volume (the volume per unit mass, ). Therefore, unless a fully coupled temperature-displacement analysis is performed, an adiabatic condition is always assumed in ABAQUS/Explicit.

When performing a fully coupled temperature-displacement analysis, the pressure stress and specific energy are updated based on the evolving temperature field. The energy increase due to the change in state will be accounted for in the heat equation and will be subject to heat conduction.

For the ideal gas model in ABAQUS/Explicit you define the gas constant, R, and the ambient pressure, . For an ideal gas R can be determined from the universal gas constant, , and the molecular weight, , as follows:

In general, the value R for any gas can be estimated by plotting as a function of state (e.g., pressure or temperature). The ideal gas approximation is adequate in any region where this value is constant. You must specify the specific heat at constant volume, . For an ideal gas is related to the specific heat at constant pressure, , by

 Input File Usage: Use both of the following options: ```*EOS, TYPE=IDEAL GAS *SPECIFIC HEAT, DEPENDENCIES=n```

 ABAQUS/CAE Usage: Property module: material editor: OtherEos: Type: Ideal GasThermalSpecific Heat

#### Initial state

The initial state of the gas is determined from the user-specified initial density, , and either the initial pressure stress, , or the initial temperature, . You can choose between specifying initial pressure or temperature. If both are specified, the value for stress will be used to initialize the state of the gas. For those material points that have zero or no pressure stress specified, the initial state is determined from the initial density and the initial temperature.

You can define the initial specific energy, . The initial specific energy for an ideal gas equation of state affects only the output of the total specific energy and should be mathematically equivalent to

 Input File Usage: Use some or all of the following options, as required: ```*DENSITY, DEPENDENCIES=n *INITIAL CONDITIONS, TYPE=SPECIFIC ENERGY *INITIAL CONDITIONS, TYPE=STRESS *INITIAL CONDITIONS, TYPE=TEMPERATURE```

 ABAQUS/CAE Usage: Property module: material editor: GeneralDensity Load module: Create Predefined Field: Step: Initial: choose Other for the Category and Temperature for the Types for Selected StepInitial specific energy and initial stress are not supported in ABAQUS/CAE.

#### The value of absolute zero

When a non-absolute temperature scale is used, you must specify the value of absolute zero temperature.

 Input File Usage: `*PHYSICAL CONSTANTS, ABSOLUTE ZERO=`

 ABAQUS/CAE Usage: Any module: ModelEdit Attributesmodel_name: Absolute zero temperature

#### A special case

In the case of an adiabatic analysis with constant specific heat (both and are constant), the specific energy is linear in temperature

The pressure stress can, therefore, be recast in the common form of

where is the ratio of specific heats and can be defined as

where

for a monatomic;

for a diatomic; and

for a polyatomic gas.

#### Comparison with the hydrostatic fluid model

The ideal gas equation of state can be used to model wave propagation effects and the dynamics of a spatially varying state of a gaseous region. For cases in which the inertial effects of the gas are not important and the state of the gas can be assumed to be uniform throughout a region, the hydrostatic fluid model (Hydrostatic fluid models, Section 20.4.1) is a simpler, more computationally efficient alternative.

### Deviatoric behavior

The equation of state defines only the material's hydrostatic behavior. It can be used by itself, in which case the material has only volumetric strength (the material is assumed to have no shear strength). Alternatively, ABAQUS/Explicit allows you to define deviatoric behavior, assuming that the deviatoric and volumetric responses are uncoupled. Two models, a linear isotropic deviatoric model or a Newtonian viscous deviatoric model, are available. The material's volumetric response is governed then by the equation of state model, while its deviatoric response is governed by either the linear isotropic elastic model or the Newtonian viscous fluid model.

You must provide the elastic shear modulus, , or the viscosity, , when you define the deviatoric behavior. For the elastic shear behavior the increment in the deviatoric stress is defined from the deviatoric elastic strain increment as

where is the deviatoric stress and is the deviatoric elastic strain; for the Newtonian viscous shear behavior the deviatoric stress is related to the deviatoric strain rate by the Navier-Poisson law of a Newtonian fluid

where is the deviatoric stress and is the deviatoric part of the strain rate.

 Input File Usage: Use both of the following options in the case of elastic shear behavior: ```*EOS *EOS SHEAR, TYPE=ELASTIC```or use both of the following options in the case of viscous shear behavior:```*EOS *EOS SHEAR, TYPE=VISCOUS```

 ABAQUS/CAE Usage: In the case of elastic shear behavior: Property module: material editor: OtherEos: SuboptionsEos Shear: Type: ElasticIn the case of viscous shear behavior:Property module: material editor: OtherEos: SuboptionsEos Shear: Type: Viscous

### Use with the Mises or the Johnson-Cook plasticity models

An equation of state model can be used with the Mises (Classical metal plasticity, Section 18.2.1) or the Johnson-Cook (Johnson-Cook plasticity, Section 18.2.7) plasticity models to model elastic-plastic behavior. In this case you must define the elastic part of the shear behavior. The material's volumetric response is governed by the equation of state model, while the deviatoric response is governed by the linear elastic shear and the plasticity model.

 Input File Usage: Use the following options: ```*EOS *EOS SHEAR, TYPE=ELASTIC *PLASTIC```

 ABAQUS/CAE Usage: Property module: material editor: OtherEos: SuboptionsEos Shear: Type: ElasticMechanicalPlasticityPlastic

#### Initial conditions

You can specify initial conditions for the equivalent plastic strain, (Initial conditions, Section 27.2.1).

 Input File Usage: `*INITIAL CONDITIONS, TYPE=HARDENING`

 ABAQUS/CAE Usage: Initial equivalent plastic strain is not supported in ABAQUS/CAE.

### Use with the tensile failure model

An equation of state model (except the ideal gas equation of state) can also be used with the tensile failure model (Dynamic failure models, Section 18.2.8) to model dynamic spall or a pressure cutoff. The tensile failure model uses the hydrostatic pressure stress as a failure measure and offers a number of failure choices. You must provide the hydrostatic cutoff stress.

You can specify that the deviatoric stresses should fail when the tensile failure criterion is met. In the case where the material's deviatoric behavior is not defined, this specification is meaningless and is, therefore, ignored.

The tensile failure model in ABAQUS/Explicit is designed for high-strain-rate dynamic problems in which inertia effects are important. Therefore, it should be used only for such situations. Improper use of the tensile failure model may result in an incorrect simulation.

 Input File Usage: Use the following options: ```*EOS *EOS SHEAR *TENSILE FAILURE```

 ABAQUS/CAE Usage: The tensile failure model is not supported in ABAQUS/CAE.

An adiabatic condition is always assumed for materials modeled with an equation of state unless a dynamic coupled temperature-displacement procedure is used. In other words, specifying an adiabatic dynamic stress analysis step or specifying temperature as a predefined field has no effect on the behavior of this model.

### Modeling fluids

A linear equation of state model can be used to model incompressible viscous and inviscid laminar flow governed by the Navier-Stokes equation of motion. The volumetric response is governed by the equations of state, where the bulk modulus acts as a penalty parameter for the incompressible constraint.

To model a viscous laminar flow that follows the Navier-Poisson law of a Newtonian fluid, use the Newtonian viscous deviatoric model and define the viscosity as the real linear viscosity of the fluid. Appropriate initial conditions for velocity and stress are essential to get an accurate solution for this class of problem.

To model an incompressible inviscid fluid such as water in ABAQUS/Explicit, it is useful to define a small amount of shear resistance to suppress shear modes that can otherwise tangle the mesh. Here the shear stiffness or shear viscosity acts as a penalty parameter. The shear modulus or viscosity should be small because flow is inviscid; a high shear modulus or viscosity will result in an overly stiff response. To avoid an overly stiff response, the internal forces arising due to the deviatoric response of the material should be kept several orders of magnitude below the forces arising due to the volumetric response. This can be done by choosing an elastic shear modulus that is several orders of magnitude lower than the bulk modulus. If the viscous model is used, the shear viscosity specified should be on the order of the shear modulus, calculated as above, scaled by the stable time increment. The expected stable time increment can be obtained from a data check analysis of the model. This method is a convenient way to approximate a shear resistance that will not introduce excessive viscosity in the material.

If a shear model is defined, the hourglass control forces are calculated based on the shear resistance of the material. Thus, in materials with extremely low or zero shear strengths such as inviscid fluids, the hourglass forces calculated based on the default parameters are insufficient to prevent spurious hourglass modes. Therefore, a sufficiently high hourglass scaling factor is recommended to increase the resistance to such modes.

### Elements

Equations of state can be used with any solid (continuum) elements in ABAQUS/Explicit except plane stress elements.

### Output

In addition to the standard output identifiers available in ABAQUS (ABAQUS/Explicit output variable identifiers, Section 4.2.2), the following variables have special meaning for the equation of state models:

 PALPH Distension, , of the porous material. The current porosity is equal to one minus the inverse of :
 PALPHMIN Minimum value, , of the distension attained during plastic compaction of the porous material.
 PEEQ Equivalent plastic strain, where is the initial equivalent plastic strain (zero or user-specified; see “Initial conditions”). This is relevant only if the equation of state model is used in combination with the Mises or Johnson-Cook plasticity models.