# 25.2.1 EQUATION OF STATE Products: Abaqus/Explicit Abaqus/CAE # References • “Hydrodynamic behavior: overview,” Section 25.1.1 • “Material library: overview,” Section 21.1.1 • “VUEOS,” Section 1.2.15 of the Abaqus User Subroutines Reference Guide • \*EOS • \*EOS COMPACTION • \*ELASTIC • \*VISCOSITY • \*DETONATION POINT • \*GAS SPECIFIC HEAT • \*REACTION RATE • \*TENSILE FAILURE • “Defining equations of state” in “Defining other mechanical models,” Section 12.9.4 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # 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, $\rho ,$ and the specific energy (the internal energy per unit mass), $E _ { m } \colon p = f ( \rho , E _ { m } )$ ; • are available as Mie-Grüneisen equations of state (thus providing the linear $U _ { s } - U _ { p }$ Hugoniot form); • are available as tabulated equations of state linear in energy; • are available as $P - \alpha$ equations of state for the compaction of ductile porous materials and must be used in conjunction with either the Mie-Grüneisen or the tabulated equation of state for the solid phase; • are available as JWL high explosive equations of state; • are available as ignition and growth equations of state; • are available in the form of an ideal gas; • are available in the form of user-defined equations of state (VUEOS); • 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 viscous deviatoric behavior; • can be used with the Mises (“Classical metal plasticity,” Section 23.2.1) or the Johnson-Cook (“Johnson-Cook plasticity,” Section 23.2.7) plasticity models; • can be used with the extended Drucker-Prager (“Extended Drucker-Prager models,” Section 23.3.1) plasticity models (without plastic dilation); and • can be used with the tensile failure model (“Dynamic failure models,” Section 23.2.8) 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, $E _ { m }$ , 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 $$ \rho \frac {\partial E _ {m}}{\partial t} = (p - p _ {b v}) \frac {1}{\rho} \frac {\partial \rho}{\partial t} + \mathbf {S}: \dot {\mathbf {e}} + \rho \dot {Q}, $$ where p is the pressure stress defined as positive in compression, $p _ { b v }$ is the pressure stress due to the bulk viscosity, is the deviatoric stress tensor, 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, $E _ { m }$ : $$ p = f (\rho , E _ {m}), $$ 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 ${ \pmb { p } } { - } V$ states achievable behind a shock (see Figure 25.2.1–1). 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 $$ p = f + g E _ {m}, $$ where $f ( \rho )$ and $g ( \rho )$ are functions of density only and depend on the particular equation of state model. # Mie-Grüneisen equations of state A Mie-Grüneisen equation of state is linear in energy. The most common form is $$ p - p _ {H} = \Gamma \rho (E _ {m} - E _ {H}), $$  Figure 25.2.1–1 A schematic representation of a Hugoniot curve. where $p _ { H }$ and $E _ { H }$ are the Hugoniot pressure and specific energy (per unit mass) and are functions of density only, and is the Grüneisen ratio defined as $$ \Gamma = \Gamma_ {0} \frac {\rho_ {0}}{\rho}, $$ where $\Gamma _ { 0 }$ is a material constant and $\rho _ { 0 }$ is the reference density. The Hugoniot energy, $E _ { H }$ , is related to the Hugoniot pressure by $$ E _ {H} = \frac {p _ {H} \eta}{2 \rho_ {0}}, $$ where $\eta = 1 - \rho _ { 0 } / \rho$ is the nominal volumetric compressive strain. Elimination of and $E _ { H }$ from the above equations yields $$ p = p _ {H} \left(1 - \frac {\Gamma_ {0} \eta}{2}\right) + \Gamma_ {0} \rho_ {0} E _ {m}. $$ 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 $U _ { s } \mathrm { ~ - ~ } U _ { p }$ Hugoniot form A common fit to the Hugoniot data is given by $$ p _ {H} = \frac {\rho_ {0} c _ {0} ^ {2} \eta}{(1 - s \eta) ^ {2}}, $$ where $c _ { 0 }$ and s define the linear relationship between the shock velocity, $U _ { s }$ , and the particle velocity, $U _ { p }$ , as follows: $$ U _ {s} = c _ {0} + s U _ {p}. $$ With the above assumptions the linear $U _ { s } - U _ { p }$ Hugoniot form is written as $$ p = \frac {\rho_ {0} c _ {0} ^ {2} \eta}{(1 - s \eta) ^ {2}} (1 - \frac {\Gamma_ {0} \eta}{2}) + \Gamma_ {0} \rho_ {0} E _ {m}, $$ where $\rho _ { 0 } c _ { 0 } ^ { 2 }$ 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 $$ \eta_ {l i m} = \frac {1}{s} $$ or $$ \rho_ {l i m} = \frac {s \rho_ {0}}{s - 1}. $$ 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 $\rho _ { 0 } )$ \*EOS, TYPE=USUP (to specify the variables , s, and $\Gamma _ { 0 } )$ Abaqus/CAE Usage: Property module: material editor: General→Density (to specify the reference density $\rho _ { 0 } )$ Mechanical→Eos: Type: Us - Up (to specify the variables $c _ { 0 } ,$ s, and $\Gamma _ { 0 } )$ # Initial state The initial state of the material is determined by the initial values of specific energy, $E _ { m }$ , and pressure stress, p. Abaqus/Explicit will automatically compute the initial density, $\rho ,$ that satisfies the equation of state, $p = f ( \rho , E _ { m } )$ . You can define the initial specific energy and initial stress state (see “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). 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: $$ E _ {m} = 0, $$ $$ p = 0, $$ $$ \rho = \rho_ {0}. $$ 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: Load module: Create Predefined Field: Step: Initial: choose Mechanical for the Category and Stress for the Types for Selected Step Initial specific energy is not supported in Abaqus/CAE. 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 $$ p = f _ {1} (\varepsilon_ {\mathrm{vol}}) + \rho_ {0} f _ {2} (\varepsilon_ {\mathrm{vol}}) E _ {m}, $$ where $f _ { 1 } ( \varepsilon _ { \mathrm { v o l } } )$ and $f _ { 2 } ( \varepsilon _ { \mathrm { v o l } } )$ are functions of the logarithmic volumetric strain $\varepsilon _ { \mathrm { v o l } }$ only, with $\varepsilon _ { \mathrm { v o l } } =$ $\ln ( \rho _ { 0 } / \rho )$ , and $\rho _ { 0 }$ is the reference density. You can specify the functions $f _ { 1 } ( \varepsilon _ { \mathrm { v o l } } )$ and $f _ { 2 } ( \varepsilon _ { \mathrm { v o l } } )$ 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. Input File Usage: Use both of the following options: \*DENSITY (to specify the reference density $\rho _ { 0 } )$ \*EOS, TYPE=TABULAR (to specify and $f _ { 2 }$ as functions $o f \varepsilon _ { \mathrm { v o l } } )$ Abaqus/CAE Usage: Property module: material editor: General→Density (to specify the reference density ) Mechanical→Eos: Type: Tabular (to specify and $f _ { 2 }$ as functions $o f \varepsilon _ { \mathrm { v o l } } )$ # Initial state The initial state of the material is determined by the initial values of specific energy, $E _ { m }$ , and pressure stress, p. Abaqus/Explicit automatically computes the initial density, $\rho ,$ that satisfies the equation of state. You can define the initial specific energy and initial stress state (see “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). The initial pressure used by the equation of state is inferred from the specified stress states. If no initial conditions are specified, Abaqus/Explicit assumes that the material is at its reference state: $$ E _ {m} = 0, $$ $$ p = 0, $$ $$ \rho = \rho_ {0} (\varepsilon_ {\mathrm{vol}} = 0). $$ 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: Load module: Create Predefined Field: Step: Initial: choose Mechanical for the Category and Stress for the Types for Selected Step Initial specific energy is not supported in Abaqus/CAE. The user-defined equation of state provides a general capability for modeling the volumetric response of materials through user subroutine VUEOS (see “VUEOS,” Section 1.2.15 of the Abaqus User Subroutines Reference Guide). The equation of state defines the pressure as a function of the current density, $\rho ,$ and the internal energy per unit mass, $E _ { m } \colon p = f ( \rho , E _ { m } )$ . 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 “User subroutines: overview,” Section 18.1.1). Input File Usage: Use the following option: $\ast \mathrm { E O S , ~ T Y P E = U S E R , ~ P R O P E R T I E S } = n$ Abaqus/CAE Usage: The user-defined equation of state is not supported in Abaqus/CAE. # Initial state You need to make sure that the initial specific energy, the initial stress, and the initial density satisfy the equation of state. If you do not specify the initial conditions, Abaqus/Explicit assumes that the material is at its reference state: $$ E _ {m} = 0, $$ $$ p = 0, $$ $$ \rho = \rho_ {0}. $$ Input File Usage: Use either or both of the following options to define the initial specific energy and/or initial pressure stress: \*INITIAL CONDITIONS, TYPE=SPECIFIC ENERGY \*INITIAL CONDITIONS, TYPE=STRESS Use the following option to define the initial density: \*DENSITY Abaqus/CAE Usage: Load module: Create Predefined Field: Step: Initial: choose Mechanical for the Category and Stress for the Types for Selected Step Initial specific energy is not supported in Abaqus/CAE. # P – α equation of state 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, $\pmb { n } ,$ 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 $0 \leq n < 1$ , 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: $$ \alpha = \frac {\rho_ {s}}{\rho} \geq 1. $$ For a fully compacted material $\alpha = 1$ ; 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 $$ \alpha = \frac {\rho_ {s}}{\rho} = \frac {V}{V _ {s}} = \frac {V}{V - V _ {p}} = \frac {1}{1 - V _ {p} / V} = \frac {1}{1 - n}. $$ 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 $$ p = p (\alpha , \rho , E _ {m}). $$ Assuming that the pores carry no pressure, it follows from equilibrium considerations that when a pressure $\pmb { p }$ 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 $$ p (\alpha , \rho , E _ {m}) = \frac {1}{\alpha} p _ {s} (\alpha \rho , E _ {m}) = \frac {1}{\alpha} p _ {s} (\rho_ {s}, E _ {m}), $$ where $p _ { s } ( \rho _ { s } , E _ { m } )$ is the equation of state of the solid material. For the fully compacted material (that is, when $\alpha = 1 \gamma$ , 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 as a function of the thermodynamic state. This equation takes the form $$ \alpha = A (p, \alpha_ {m i n}), $$ where $\alpha _ { m i n }$ 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 $A ( p , \alpha _ { m i n } )$ used by Abaqus/Explicit is illustrated in Figure 25.2.1–2 and is discussed next. 

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| p | α | Label | |-------|-------|--------------| | 0 | α₀ | α₀ | | pₑ | αₑ | αₑ | | pₛ | 1 | Aₑₗ(p, αₑ) | | pₛ | 1 | Aₑₗ(p, α₁ₘ) | | pₛ | 1 | Aₑₗ(p, α₂ₘ) | | pₛ | 1 | Aₑₗ(p, α₂ₘ) |

Figure 25.2.1–2 $P - \alpha$ elastic and plastic curves for the description of compaction of ductile porous materials. The function $A ( p , \alpha _ { m i n } )$ captures the general behavior to be expected in a ductile porous material. The unloaded virgin state corresponds to the value $\alpha _ { 0 } = 1 / ( 1 - n _ { 0 } )$ , 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 . 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 $( \mathrm { o r } ,$ 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 $\alpha _ { m i n }$ decreases, as will be quantified later. The material becomes fully compacted when the pressure reaches the compaction pressure $p _ { S } \mathrm { ; }$ ; at that point $\alpha = \alpha _ { m i n } = 1$ , a value that is retained forever. The function $A ( p , \alpha _ { m i n } )$ therefore has multiple branches: a plastic branch, $A _ { p l } ( p )$ , and multiple elastic branches, $A _ { e l } ( p , \alpha _ { m i n } )$ , corresponding to elastic unloading from partially compacted states. The appropriate branch of A is selected according to the following rule: $$ \alpha = A (p, \alpha_ {m i n}) = \left\{ \begin{array}{c c c} A _ {p l} (p) & \text {if} & A _ {p l} (p) \leq \alpha_ {m i n} \\ A _ {e l} (p, \alpha_ {m i n}) & \text {if} & A _ {p l} (p) > \alpha_ {m i n} \end{array} \right. $$ These expressions can be inverted to solve for ${ \pmb p } \mathrm { . }$ $$ p = P (\alpha , \alpha_ {m i n}) = \left\{ \begin{array}{c c c} P _ {p l} (\alpha) & \text {if} & \alpha \leq \alpha_ {m i n} \\ P _ {e l} (\alpha , \alpha_ {m i n}) & \text {if} & \alpha > \alpha_ {m i n} \end{array} \right. $$ The equation for the plastic curve takes the form $$ A _ {p l} (p) = 1 + (\alpha_ {e} - 1) \left(\frac {p _ {S} - p}{p _ {S} - p _ {e}}\right) ^ {2} $$ or, alternatively, $$ P _ {p l} (\alpha) = p _ {S} - (p _ {S} - p _ {e}) \left(\frac {\alpha - 1}{\alpha_ {e} - 1}\right) ^ {\frac {1}{2}}. $$ The elastic curve originally proposed by Hermann (1968) is given by the differential equation $$ \frac {d \bar {A} _ {e l}}{d p} (\alpha) = \frac {\alpha^ {2}}{K _ {0}} \left(1 - \frac {1}{h ^ {2} (\alpha)}\right), \quad h (\alpha) = 1 + \frac {(c _ {e} - c _ {s}) (\alpha - 1)}{c _ {s} (\alpha_ {0} - 1)}, $$ where $K _ { 0 } ~ = ~ \rho _ { s 0 } c _ { s } ^ { 2 }$ is the elastic bulk modulus of the solid material at small nominal strains; $\rho _ { s 0 }$ 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, $c _ { s } = \sqrt { K _ { 0 } / \rho _ { s 0 } }$ . 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 $$ A _ {e l} (p, \alpha_ {m i n}) = \alpha_ {m i n} + (p - P _ {p l} (\alpha_ {m i n})) \left. \frac {d \bar {A} _ {e l}}{d p} \right| _ {\alpha = \alpha_ {m i n}} $$ and $$ P _ {e l} (\alpha , \alpha_ {m i n}) = P _ {p l} (\alpha_ {m i n}) + \frac {(\alpha - \alpha_ {m i n})}{\left. \frac {d \bar {A} _ {e l}}{d p} \right| _ {\alpha = \alpha_ {m i n}}}. $$

Input File Usage:

Use the following option to specify the reference density of the solid phase, $\rho_{s0}$ :*DENSITYUse one of the following options to specify additional material properties for the solid phase:*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)Use the following option to specify the properties of the porous material (the reference sound speed, $c_e$ ; the reference porosity, $n_0$ ; the elastic limit, $p_e$ ; and the compaction pressure, $p_S$ ):*EOS COMPACTION

Abaqus/CAE Usage:

Property module: material editor:General→Density (to specify the reference density $\rho_0$ )Use one of the following options to specify additional material properties for the solid phase:Mechanical→Eos: Type: Us - Up (if the solid phase is modeled using the Mie-Grüneisen equation of state)Mechanical→Eos: Type: Tabular (if the solid phase is modeled using the tabulated equation of state)Use the following option to specify the properties of the porous material:Mechanical→Eos: 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, $p_S$ )

# Initial state The initial state of the porous material is determined from the initial values of porosity, $n = ( \alpha - 1 ) / \alpha ;$ specific energy, $E _ { m }$ ; and pressure stress, p. Abaqus/Explicit automatically computes the initial density, $\rho ,$ that satisfies the equation of state, $p ~ = ~ f ( \alpha , \rho , E _ { m } )$ . You can define the initial porosity, initial specific energy, and initial stress state (see “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). If no initial conditions are given, Abaqus/Explicit assumes that the material is at its virgin state: $$ \begin{array}{l} E _ {m} = 0, \\ p = 0, \\ \alpha = \alpha_ {0}, (n = n _ {0}), \\ \rho = \rho_ {s 0} / \alpha_ {0}. \\ \end{array} $$