Equation of state material

Linear U_s − U_p Hugoniot equation of state

Elements tested

C3D8R

CPE4R

Features tested

Linear $U_{s} - U_{p}$ equation of state (EOS) material model with plasticity.

Problem description

This verification test consists of a list of single-element models that use either C3D8R or CPE4R elements and are run under simple loading conditions (uniaxial tension, uniaxial compression, and simple shear). The purpose of this example is to test the equation of state material model and its combination with the Mises and Johnson-Cook plasticity models. Two parallel sets of models are studied. The first set uses the linear elastic, linear elastic with Mises plastic, and linear elastic with Johnson-Cook plastic materials. The second set uses the linear $U_{s} - U_{p}$ type of EOS, linear $U_{s} - U_{p}$ type of EOS with Mises plastic, and linear $U_{s} - U_{p}$ type of EOS with Johnson-Cook plastic materials.

For linear elasticity the volumetric response is defined by

p = K ε_{v o l},

where K is the bulk modulus of the material. The linear $U_{s} - U_{p}$ Hugoniot form is

p = \frac{ρ_{0} c_{0}^{2} η}{{(1 - s η)}^{2}} (1 - \frac{Γ_{0} η}{2}) + Γ_{0} ρ_{0} E_{m},

where $η = 1 - ρ_{0} / ρ$ is the same as the nominal volumetric strain measure, $ε_{v o l}$ . Thus, setting the parameters $s = $ 0.0 and $Γ_{0} = $ 0.0 gives the simple hydrostatic bulk response, which is identical to the elastic volumetric response. The elastic deviatoric response of the material is defined by the shear modulus.

The elastic material properties are Young's modulus = 207 GPa and Poisson's ratio = 0.29. The initial material density, $ρ_{0}$ , is 7890 kg/m³. The equivalent properties for the linear $U_{s} - U_{p}$ type of equation of state material model are $c_{0}$ = 4563.115 m/s and shear modulus = 80.233 GPa. For models in which plasticity (including both Mises and Johnson-Cook plasticity models) is used, the plastic hardening is chosen to be

\bar{σ} = 175 + 380 {({\bar{ε}}^{p l})}^{0.32},

where $\bar{σ}$ is the yield stress (in units of MPa) and ${\bar{ε}}^{p l}$ is the equivalent plastic strain.

Results and discussion

The results obtained from the analyses that use the EOS material model match the corresponding results obtained from the analyses that use the linear elasticity model. The comparison of the pressure and Mises stresses obtained with the EOS material model (with Johnson-Cook plastic shear response) and the linear elasticity model (with the same Johnson-Cook plastic shear response) using the C3D8R element under uniaxial tension loading are shown in Figure 1 and Figure 2, respectively. The uniaxial compression comparisons are shown in Figure 3 and Figure 4.

Input files

eosshrela.inp: Uniaxial tension test.
eosshrela_pre.inp: Uniaxial compression test.
eosshrela_shr.inp: Simple shear test.
eosshrelainit_shr.inp: Simple shear test with nonzero initial conditions for ${\bar{ε}}^{p l}$ .

Figures

Figure 1. Pressure stress in uniaxial tension: elastic response versus linear

U_{s} - U_{p}

type of equation of state response.

Figure 2. Mises stress in uniaxial tension: elastic response versus linear

U_{s} - U_{p}

type of equation of state response.

Figure 3. Pressure stress in uniaxial compression: elastic response versus linear

U_{s} - U_{p}

type of equation of state response.

Figure 4. Mises stress in uniaxial compression: elastic response versus linear

U_{s} - U_{p}

type of equation of state response.

Tabulated equation of state

Elements tested

C3D8R

CPE4R

Features tested

Tabulated equation of state (EOS) material model with plasticity.

Problem description

This verification test consists of single-element models that use either C3D8R or CPE4R elements and are run under simple loading conditions (uniaxial tension, uniaxial compression, and simple shear). The purpose of this example is to test the tabulated EOS material model and its combination with the Mises and Johnson-Cook plasticity models. Two parallel sets of models are studied. The first set uses the linear elasticity, linear elasticity with Mises plasticity, and linear elasticity with Johnson-Cook plasticity materials. The second set uses the tabulated EOS, tabulated EOS with Mises plasticity, and tabulated EOS with Johnson-Cook plasticity materials.

For linear elasticity the volumetric response is defined by

p = - K ε_{v o l},

where K is the bulk modulus of the material. The tabulated EOS is linear in energy and assumes the form

p = f_{1} (ε_{vol}) + ρ_{0} f_{2} (ε_{vol}) E_{m},

where $f_{1} (ε_{vol})$ and $f_{2} (ε_{vol})$ are functions of the logarithmic volumetric strain $ε_{vol}$ only, with $ε_{vol} = \ln (ρ_{0} / ρ)$ , and $ρ_{0}$ is the reference density. Thus, setting the functions $f_{2} (ε_{vol}) = - K ε_{v o l}$ and $f_{2} (ε_{vol}) = $ 0.0 gives the simple hydrostatic bulk response, which is identical to the elastic volumetric response. The elastic deviatoric response of the material is defined by the shear modulus.

The elastic material properties are Young's modulus = 207 GPa and Poisson's ratio = 0.29. The initial material density, $ρ_{0}$ , is 7890 kg/m³. The properties for the tabular EOS material model are computed using $K$ = 164.286 GPa and shear modulus = 80.233 GPa. For models in which plasticity (including both Mises and Johnson-Cook plasticity models) is used, the plastic hardening is chosen to be

\bar{σ} = 175 + 380 {({\bar{ε}}^{p l})}^{0.32},

where $\bar{σ}$ is the yield stress (in units of MPa) and ${\bar{ε}}^{p l}$ is the equivalent plastic strain.

Results and discussion

The results obtained from the analyses that use the EOS material model match the corresponding results obtained from the analyses that use the linear elasticity model.

Input files

eostabshrela.inp: Uniaxial tension test.
eostabshrela_pre.inp: Uniaxial compression test.
eostabshrela_shr.inp: Simple shear test.
eostabshrelainit_shr.inp: Simple shear test with nonzero initial conditions for ${\bar{ε}}^{p l}$ .

P–α equation of state

Elements tested

C3D8R

CPE4R

Features tested

$P - α$ equation of state (EOS) material model.

Problem description

This verification test consists of single-element models that use either C3D8R or CPE4R elements and are run under simple loading conditions (uniaxial, hydrostatic, and simple shear). The purpose of this example is to test the $P - α$ equation of state material model and its combination with different models for the deviatoric behavior: linear elastic, Newtonian viscous shear, and Mises and Johnson-Cook plasticity; as well as itscombination with different models for the hydrodynamic response of the solid phase: Mie-Grüneisen and tabulated equations of state.

The material properties used for the tests are representative of partially saturated sand. They are summarized below:

Material:

Solid phase

The solid phase is described by a Mie-Grüneisen equation of state:

$ρ_{s 0}$	2070 kg/m³
$C_{s}$	1480 m/sec
s	1.93
$Γ_{0}$	0.880

For models using the tabulated equation of state, the functions $f_{1} (ε_{vol})$ and $f_{2} (ε_{vol})$ are defined such as to provide similar hydrodynamic behavior as the above Mie-Grüneisen equation of estate.

Compaction properties

$C_{e}$	600 m/sec
$n_{0}$ ( $α_{0}$ )	0.049758 (1.052364)
$p_{e}$	0.0 MPa
$p_{S}$	6.5 MPa

Viscous shear behavior

μ

5.0E+4

Elastic shear behavior

E	124 MPa
$ν$	0.3

Plasticity

For models with plastic shear behavior (either Mises or Johnson-Cook plasticity), the plastic hardening is chosen to be

\bar{σ} = 1.97 + 4.28 {({\bar{ε}}^{p l})}^{0.32},

where $\bar{σ}$ is the yield stress (in units of MPa) and ${\bar{ε}}^{p l}$ is the equivalent plastic strain. The plasticity models are used in combination with linear elastic shear behavior.

Results and discussion

The results obtained from the analyses agree well with exact analytical or approximate solutions. The evolution of the distension $α$ with hydrostatic pressure during a cyclic volumetric test is shown in Figure 5.

Input files

eospalpha_uni.inp: Uniaxial test.
eospalpha_vol.inp: Cyclic hydrostatic test.
eospalpha_shr.inp: Simple shear test.
eospalphainit_shr.inp: Simple shear test with nonzero initial conditions for ${\bar{ε}}^{p l}$ .

Figures

Figure 5.

P - α

elastic and plastic curves during the cyclic volumetric test.

Viscous shear behavior

Elements tested

C3D8R

CPE4R

Features tested

Viscosity models for equation of state materials with viscous shear behavior.

Problem description

This verification test consists of single-element models that use either C3D8R or CPE4R elements and are run under simple shear loading conditions. The purpose of this example is to test the different viscosity models for both Newtonian and non-Newtonian fluids. The hydrodynamic response of the material is described by the Mie-Grüneisen equation of state in all cases. Some tests include thermorheologically simple temperature-dependent viscosity using the Arrhenius form.

The material properties used for the tests are summarized below:

Material:

Hydrodynamic properties

The hydrodynamic response described by a Mie-Grüneisen equation of state:

$ρ_{s 0}$	2070 kg/m³
$C_{s}$	1480 m/sec
s	1.93
$Γ_{0}$	0.880

Viscous properties

The properties for each of the tested viscosity models are given below:

Mat1:

Newtonian viscosity:

η

1 MPa sec

Mat2:

Power Law viscosity:

$k$	2.173 MPa (sec)ⁿ
$n$	0.392
$η_{max}$	1 MPa sec
$η_{min}$	0.1 MPa sec

Mat3:

Carreau-Yasuda viscosity:

$η_{0}$	1 MPa sec
$η_{\infty}$	0.1 MPa sec
$λ$	0.11 sec
$n$	0.392
$a$	0.644

Mat4:

Cross viscosity:

$η_{0}$	1 MPa sec
$η_{\infty}$	0.1 MPa sec
$λ$	0.11 sec
$n$	0.392

Mat5:

Herschel-Bulkley viscosity:

$η_{0}$	1 MPa sec
$τ_{0}$	3.59 MPa
$k$	2.173 MPa (sec)ⁿ
$n$	0.392

Mat6:

Ellis-Meter viscosity:

$η_{0}$	1 MPa sec
$η_{\infty}$	0.1 MPa sec
$τ_{1 / 2}$	5.665 MPa
$n$	0.392

Mat7:

Powell-Eyring viscosity:

$η_{0}$	1 MPa sec
$η_{\infty}$	0.1 MPa sec
$λ$	0.11 sec

Mat8:

Tabular viscosity:

$η$ (MPa sec)	$\dot{γ}$ (sec^-1)
1.00000	0.0
0.83383	1.0
0.76532	2.0
0.71776	3.0
0.68112	4.0
0.65134	5.0
0.62631	6.0
0.60477	7.0
0.58593	8.0
0.56921	9.0
0.55422	10.0
0.54066	11.0
0.52830	12.0
0.51697	13.0
0.50652	14.0
0.49684	15.0

Mat9:

User-defined Cross viscosity. The viscosity is expressed as

η = \frac{η_{0}}{1 + {(λ \dot{γ})}^{1 - n}},

$η_{0}$	1 MPa sec
$λ$	0.11 sec
$n$	0.392

TRS properties

Arrhenius form:

$E_{0}$	109100 joule/mole
$θ_{0}$	308 kelvin
$θ^{Z}$	0 kelvin
$R$	8.31434 joule/(mole kelvin)

Results and discussion

The results obtained from the analyses agree well with exact analytical or approximate solutions.

Input files

eosshrvisc.inp: Simple shear test.
eosshrvisctrs.inp: Material with Arrhenius TRS properties. Simple shear test.
eosshrvisc.f: User subroutine VUVISCOSITY for the user-defined Cross viscosity model used in eosshrvisc.inp and eosshrvisctrs.inp.

Pressure-dependent shear plasticity

Elements tested

C3D8R

CPE4R

CAX4R

Features tested

Equation of state (EOS) material model with pressure-dependent (Drucker-Prager) shear plasticity.

Problem description

This verification test consists of single-element models that use either C3D8R, CPE4R, or CAX4R elements and are run under simple loading conditions (uniaxial tension, uniaxial compression, and simple shear). The purpose of this example is to test the combination of EOS models for the volumetric response of the material with the extended Drucker-Prager pressure-dependent plasticity models for the shear response. Some of the models also include Johnson-Cook strain-rate dependence in the plasticity definition.

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

eosjcratedpexpuni3d.inp: Uniaxial tension test, Johnson-Cook strain-rate dependence, Drucker-Prager plasticity with exponent form shear criterion, C3D8R element.
eosjcratedpexpunicpe.inp: Uniaxial tension test, Johnson-Cook strain-rate dependence, Drucker-Prager plasticity with exponent form shear criterion, CPE4R element.
eosjcratedpexpuniaxi.inp: Uniaxial tension test, Johnson-Cook strain-rate dependence, Drucker-Prager plasticity with exponent form shear criterion, CAX4R element.
eosjcratedphypuni3d.inp: Uniaxial tension test, Johnson-Cook strain-rate dependence, Drucker-Prager plasticity with hyperbolic shear criterion, C3D8R element.
eosjcratedphypunicpe.inp: Uniaxial tension test, Johnson-Cook strain-rate dependence, Drucker-Prager plasticity with hyperbolic shear criterion, CPE4R element.
eosjcratedphypuniaxi.inp: Uniaxial tension test, Johnson-Cook strain-rate dependence, Drucker-Prager plasticity with hyperbolic shear criterion, CAX4R element.
eosdruckerprager.inp: Uniaxial tension test, C3D8R and CPE4R elements.
eosdruckerprager_pre.inp: Uniaxial compression test, C3D8R and CPE4R elements.
eosdruckerprager_shr.inp: Simple shear test, C3D8R and CPE4R elements.

User-defined equation of state

Elements tested

C3D8R

CPE4R

Features tested

User-defined equation of state (EOS) material model with plasticity.

Problem description

This verification test consists of single-element models that use either C3D8R or CPE4R elements and are run under simple loading conditions (uniaxial tension, uniaxial compression, and simple shear). The purpose of this example is to test the user-defined EOS material model (user subroutine VUEOS) and its combination with the Mises and Johnson-Cook plasticity models. Two parallel sets of models are studied. The first set uses the linear elasticity, linear elasticity with Mises plasticity, and linear elasticity with Johnson-Cook plasticity materials. The second set uses the user-defined EOS, user-defined EOS with Mises plasticity, and user-defined EOS with Johnson-Cook plasticity materials.

For linear elasticity the volumetric response is defined by

p = - K ε_{v o l},

where K is the bulk modulus of the material. To obtain the same elastic volumetric response with the user-defined EOS, the pressure update inside user subroutine VUEOS is

p = - K ε_{v o l} = - K \ln (ρ_{0} / ρ),

where $ρ_{0}$ is the reference density. The user subroutine needs to return the derivative of pressure with respect to density, $\partial p / \partial ρ$ , which is needed for the the evaluation of the effective moduli of the material that enters the stable time calculation. User subroutine VUEOS also returns the derivative of pressure with respect to the energy, $\partial p / \partial E_{m}$ , which is usually needed to solve the nonlinear pressure-energy dependency using the Newton method. In the case considered here, these quantities are

\partial p / \partial ρ = K / ρ, \partial p / \partial E_{m} = 0.

The elastic deviatoric response of an equation of state material can be defined by using the ELASTIC, TYPE=SHEAR option.

The elastic material properties are Young's modulus = 207 GPa and Poisson's ratio = 0.29. The initial material density, $ρ_{0}$ , is 7890 kg/m³. The properties for the tabular EOS material model are computed using $K$ = 164.286 GPa and shear modulus = 80.233 GPa. For models in which plasticity is used (including both Mises and Johnson-Cook plasticity models), the plastic hardening is chosen to be

\bar{σ} = 175 + 380 {({\bar{ε}}^{p l})}^{0.32},

where $\bar{σ}$ is the yield stress (in units of MPa) and ${\bar{ε}}^{p l}$ is the equivalent plastic strain.

Results and discussion

The results obtained from the analyses that use the EOS material model match the corresponding results obtained from the analyses that use the linear elasticity model.

Input files

eosusershrela.inp: Uniaxial tension test.
eosusershrela_pre.inp: Uniaxial compression test.
eosusershrela_shr.inp: Simple shear test.
eosusershrelainit_shr.inp: Simple shear test with nonzero initial conditions for ${\bar{ε}}^{p l}$ .
vueos_tab.f: User subroutine VUEOS used in the input files listed above.

Equation of state material

Linear Us − Up Hugoniot equation of state

Elements tested

Features tested

Problem description

Results and discussion

Input files

Figures

Tabulated equation of state

Elements tested

Features tested

Problem description

Results and discussion

Input files

P–α equation of state

Elements tested

Features tested

Problem description

Material:

Results and discussion

Input files

Figures

Viscous shear behavior

Elements tested

Features tested

Problem description

Material:

Results and discussion

Input files

Pressure-dependent shear plasticity

Elements tested

Features tested

Problem description

Results and discussion

Input files

User-defined equation of state

Elements tested

Features tested

Problem description

Results and discussion

Input files

Linear U_s − U_p Hugoniot equation of state