Equation of state

The equation for conservation of energy equates the increase in internal energy per unit mass, $E_m$, 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, $p_{bv}$ is the pressure stress due to the bulk viscosity, $\mathbf{S}$ is the deviatoric stress tensor, $\dot{\mathbf{e}}$ is the deviatoric part of strain rate, and $\dot{Q}$ is the heat rate per unit mass.

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

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 $1/\rho$ 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 p–V states achievable behind a shock (see Figure 1).

Figure 1. A schematic representation of a Hugoniot curve.

The Hugoniot pressure, $p_H$, is a function of density only and can be defined, in general, from fitting experimental data.

An equation of state is said to be linear in energy when it can be written in the form

where $f\left(\rho \right)$ and $g\left(\rho \right)$ are functions of density only and depend on the particular equation of state model.

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

where $p_H$ and $E_H$ are the Hugoniot pressure and specific energy (per unit mass) and are functions of density only, and $\mathrm{\Gamma }$ is the Grüneisen ratio defined as

where ${\mathrm{\Gamma }}_0$ is a material constant and ${\rho }_0$ is the reference density.

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

where $\eta =1-{\rho }_0/\rho$ is the nominal volumetric compressive strain. Elimination of $\mathrm{\Gamma }$ and $E_H$ 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.

The tabulated equation of state provides flexibility in modeling the hydrodynamic response of materials that exhibit sharp transitions in the pressure-density relationship, such as those induced by phase transformations. The tabulated equation of state is linear in energy and assumes the form

where $f_1\left({\epsilon }_{\mathrm{vol}}\right)$ and $f_2\left({\epsilon }_{\mathrm{vol}}\right)$ are functions of the logarithmic volumetric strain ${\epsilon }_{\mathrm{vol}}$ only, with ${\epsilon }_{\mathrm{vol}}=\ln \left({\rho }_0/\rho \right)$, and ${\rho }_0$ is the reference density.

You can specify the functions $f_1\left({\epsilon }_{\mathrm{vol}}\right)$ and $f_2\left({\epsilon }_{\mathrm{vol}}\right)$ directly in tabular form. The tabular entries must be given in descending values of the volumetric strain (that is, from the most tensile to the most compressive states). Abaqus/Explicit will use a piecewise linear relationship between data points. Outside the range of specified values of volumetric strains, the functions are extrapolated based on the last slope computed from the data.

DENSITY (to specify the reference density ${\rho }_0$)
EOS, TYPE=TABULAR (to specify $f_1$ and $f_2$ as functions of ${\epsilon }_{\mathrm{vol}}$)

Property module: material editor:
GeneralDensity (to specify the reference density ${\rho }_0$)
MechanicalEos: Type: Tabular (to specify $f_1$ and $f_2$ as functions of ${\epsilon }_{\mathrm{vol}}$)

The user-defined equation of state provides a general capability for modeling the volumetric response of materials through user subroutine VUEOS (see VUEOS). The equation of state defines the pressure as a function of the current density, $\rho$, and the internal energy per unit mass, $E_m$: $p=f\left(\rho ,E_m\right)$. Abaqus/Explicit solves the energy equation together with the equation of state using an iterative method. The pressure stress, $p$, and the derivatives of the pressure with respect to the internal energy and to the density, $\partial p/\partial E_m$ and $\partial p/\partial \rho$, must be provided by user subroutine VUEOS. The latter is needed for the evaluation of the effective bulk modulus of the material, which is necessary for the stable time increment calculation.

Optionally, you can also specify the number of property values needed as data in the user subroutine as well as the number of solution-dependent variables (see About user subroutines and utilities).

EOS, TYPE=USER, PROPERTIES=n

The $P-\alpha$ 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 either the Mie-Grüneisen equation of state or the tabulated 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, $V_p$, to total volume, $V=V_s+V_p$, where $V_s$ is the solid volume. The porosity remains in the range $0\le n<1$, with 0 indicating full compaction. It is convenient to introduce a scalar variable $\alpha$, sometimes referred to as “distension,” defined as the ratio of the density of the solid material, ${\rho }_s$, to the density of the porous material, $\rho$, both evaluated at the same temperature and pressure:

For a fully compacted material $\alpha =1$; otherwise, $\alpha$ is greater than 1. Assuming that the density of the pores is negligible compared to that of the solid phase, $\alpha$ 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 $\alpha$; current density, $\rho$; and internal energy per unit mass, $E_m$, 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 $p_s=\alpha p$. 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 $p_s\left({\rho }_s,E_m\right)$ is the equation of state of the solid material. For the fully compacted material (that is, when $\alpha =1$), the $P-\alpha$ equation of state reduces to that of the solid phase, therefore predicting the correct thermodynamic behavior at high pressures.

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

where ${\alpha }_{min}$ is a state variable corresponding to the minimum value attained by $\alpha$ during plastic (irreversible) compaction of the material. The state variable is initialized to the elastic limit ${\alpha }_e$ for a material that is at its virgin state. The specific form of the function $A\left(p,{\alpha }_{min}\right)$ used by Abaqus/Explicit is illustrated in Figure 2 and is discussed next.

Figure 2. $P-\alpha$ elastic and plastic curves for the description of compaction of ductile porous materials.

The function $A\left(p,{\alpha }_{min}\right)$ captures the general behavior to be expected in a ductile porous material. The unloaded virgin state corresponds to the value ${\alpha }_0=1/\left(1-n_0\right)$, where $n_0$ 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 $\alpha$. As the pressure increases beyond the elastic limit, $p_e$, 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 $\alpha$) ever attained during the deformation history of the material. The absolute value of the slope of the elastic curve decreases as ${\alpha }_{min}$ decreases, as will be quantified later. The material becomes fully compacted when the pressure reaches the compaction pressure $p_S$; at that point $\alpha ={\alpha }_{min}=1$, a value that is retained forever. The function $A\left(p,{\alpha }_{min}\right)$ therefore has multiple branches: a plastic branch, $A_{pl}\left(p\right)$, and multiple elastic branches, $A_{el}\left(p,{\alpha }_{min}\right)$, 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 $K_0={\rho }_{s0}{c}_s^2$ is the elastic bulk modulus of the solid material at small nominal strains; ${\rho }_{s0}$ is the reference density of the solid; and $c_s$ and $c_e$ are the reference sound speeds in the solid and virgin (porous) materials, respectively.

If the solid phase is modeled using the Mie-Grüneisen equation of state, $c_s$ is given directly by the reference sound speed, $c_0$. On the other hand, if the solid phase is modeled using the tabulated equation of state, $c_s$ is computed from the initial bulk modulus and reference density of the solid material, $c_s=\sqrt{K_0/{\rho }_{s0}}$. In this case the reference density is required to be constant; it cannot be a function of temperature or field variables.

Following Wardlaw et al. (1996), the above equation for the elastic curve in Abaqus/Explicit is simplified and replaced by the linear relations

and

DENSITY

EOS, TYPE=USUP (if the solid phase is modeled using the Mie-Grüneisen equation of state)
EOS, TYPE=TABULAR (if the solid phase is modeled using the tabulated equation of state)

EOS COMPACTION

Property module: material editor:
GeneralDensity (to specify the reference density ${\rho }_0$)

MechanicalEos: Type: Us - Up (if the solid phase is modeled using the Mie-Grüneisen equation of state)
MechanicalEos: Type: Tabular (if the solid phase is modeled using the tabulated equation of state)

MechanicalEos: Suboptions Eos Compaction (to specify the reference sound speed, $c_e$; the porosity of the unloaded material, $n_0$; the pressure required to initialize plastic behavior, $p_e$; and the pressure at which all pores are crushed, $p_S$)

The Jones-Wilkins-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 internal energy per unit mass, $E_m$, as

where $A,B,R_1,R_2,$ and $\omega$ are user-defined material constants; ${\rho }_0$ is the user-defined density of the explosive; and $\rho$ is the density of the detonation products.

DENSITY (to specify the density of the explosive ${\rho }_0$)
EOS, TYPE=JWL (to specify the material constants $A,B,R_1,R_2,$ and $\omega$)

Property module: material editor:
GeneralDensity (to specify the density of the explosive ${\rho }_0$)
MechanicalEos: Type: JWL (to specify the material constants $A,B,R_1,$
$R_2,$ and $\omega$)

The ignition and growth equation of state models shock initiation and detonation wave propagation of solid high explosives. The heterogeneous explosive is modeled as a homogeneous mixture of two phases: the unreacted solid explosive and the reacted gas products. Separate JWL equations of state are prescribed for each phase:

where

and

The subscript s refers to the unreacted solid explosive, and g refers to the reacted gas products. $A_i,B_i,R_{1i},R_{2i},$ and ${\omega }_i$ are user-defined material constants used in the JWL equations; $E_d$ is the detonation energy; ${\rho }_0$ is the user-defined reference density of the explosive, and ${\rho }_i$ is the density of the unreacted explosive or the reacted products.

DENSITY(to specify the density of the explosive ${\rho }_0$)
EOS, TYPE=IGNITION AND GROWTH, DETONATION ENERGY=$E_d$
(to specify the material constants $A,B,R_1,$ $R_2,$ and $\omega$
 of the unreacted solid explosive and the reacted gas product)

Property module: material editor:
GeneralDensity (to specify the density of the explosive ${\rho }_0$)
MechanicalEos: Type: Ignition and growth: Detonation energy: $E_d$;
Solid Phase tabbed page and Gas Phase tabbed page
(to specify the material constants $A,B,R_1,$ $R_2,$ and $\omega$
 of the unreacted solid explosive and the reacted gas product)

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

where $p_A$ is the ambient pressure, R is the gas constant, $\theta$ is the current temperature, and ${\theta }^Z$ 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 $E_{m_0}$ is the initial specific energy at the initial temperature ${\theta }_0$ and $c_v$ 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 $pdv$, where v is the specific volume (the volume per unit mass, $v=1/\rho$). 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, $p_A$. For an ideal gas R can be determined from the universal gas constant, $\tilde{R}$, and the molecular weight, $MW$, as follows:

In general, the value R for any gas can be estimated by plotting $pv/\left(\theta -{\theta }^Z\right)$ 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, $c_v$. For an ideal gas $c_v$ is related to the specific heat at constant pressure, $c_p$, by

EOS, TYPE=IDEAL GAS
SPECIFIC HEAT, DEPENDENCIES=n

Property module: material editor:
MechanicalEos: Type: Ideal Gas
ThermalSpecific Heat

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 are available for the deviatoric response: a linear isotropic elastic model and a viscous model. 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 viscous fluid model.

An equation of state model, with the exception of the ignition and growth model, can be used with the Mises (Classical metal plasticity) or the Johnson-Cook (Johnson-Cook plasticity) 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.

EOS
ELASTIC, TYPE=SHEAR
PLASTIC

Property module: material editor:
MechanicalElasticityElastic; Type: Shear
MechanicalPlasticityPlastic

An equation of state model, with the exception of the ignition and growth model, can be used in conjunction with the extended Drucker-Prager (Extended Drucker-Prager models) plasticity models to model pressure-dependent plasticity behavior. This approach can be appropriate for modeling the response of ceramics and other brittle materials under high velocity impact conditions. In this case you must define the elastic part of the shear behavior. The material's deviatoric response is governed by the linear elastic shear and the pressure-dependent plasticity model, while the volumetric response is governed by the equation of state model. In particular, no plastic dilation effects are taken into account (if you specify a dilation angle other than zero, the value is ignored and Abaqus/Explicit issues a warning message).

High-velocity impact of a ceramic target illustrates the use of an equation of state model with the extended Drucker-Prager plasticity models.

EOS
ELASTIC, TYPE=SHEAR
DRUCKER PRAGER
DRUCKER PRAGER HARDENING

Property module: material editor:
MechanicalElasticityElastic; Type: Shear
MechanicalPlasticityDrucker Prager: SuboptionsDrucker Prager Hardening

An equation of state model (except the ideal gas equation of state) can also be used with the tensile failure model (Dynamic failure models) 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.

EOS
TENSILE FAILURE

An adiabatic condition is always assumed for materials modeled with an equation of state unless a dynamic coupled temperature-displacement procedure is used. The adiabatic condition is assumed irrespective of whether an adiabatic dynamic stress analysis step has been specified. The temperature increase is calculated directly at the material integration points according to the adiabatic thermal energy increase caused by the mechanical work

where $c_v$ is the specific heat at constant volume. Specifying temperature as a predefined field has no effect on the behavior of this model.

When performing a fully coupled temperature-displacement analysis, the specific energy is updated based on the evolving temperature field using

A linear $U_s-U_p$ 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. To model non-Newtonian viscous flow, use one of the nonlinear viscosity models available in Abaqus/Explicit. Appropriate initial conditions for velocity and stress are essential to get an accurate solution for this class of problems.

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.