Linear 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 type of EOS, linear type of EOS with Mises plastic, and
linear type of EOS with Johnson-Cook plastic
materials.
For linear elasticity the volumetric response is defined by
where K is the bulk modulus of the material. The linear Hugoniot form is
where is the same as the nominal volumetric strain measure, . Thus, setting the parameters 0.0 and 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, , is 7890 kg/m3. The equivalent properties for the linear type of equation of state material model are = 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
where is the yield stress (in units of MPa) and 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.
Simple shear test with nonzero initial conditions for .
Figures
Figure 1. Pressure stress in uniaxial tension: elastic response versus linear type of equation of state response. Figure 2. Mises stress in uniaxial tension: elastic response versus linear type of equation of state response. Figure 3. Pressure stress in uniaxial compression: elastic response versus linear type of equation of state response. Figure 4. Mises stress in uniaxial compression: elastic response versus linear 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
where K is the bulk modulus of the material. The tabulated
EOS is linear in energy and assumes the form
where and are functions of the logarithmic volumetric strain only, with , and is the reference density. Thus, setting the functions and 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, , is 7890 kg/m3. The properties for the tabular
EOS material model are computed using = 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
where is the yield stress (in units of MPa) and 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.
Simple shear test with nonzero initial conditions for .
P–α equation of state
Elements tested
C3D8R
CPE4R
Features tested
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 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:
2070 kg/m3
1480 m/sec
s
1.93
0.880
For models using the tabulated equation of state, the functions and are defined such as to provide similar hydrodynamic behavior as
the above Mie-Grüneisen equation of estate.
Compaction properties
600 m/sec
()
0.049758 (1.052364)
0.0 MPa
6.5 MPa
Viscous shear behavior
5.0E+4
Elastic shear behavior
E
124 MPa
0.3
Plasticity
For models with plastic shear behavior (either Mises or Johnson-Cook plasticity),
the plastic hardening is chosen to be
where is the yield stress (in units of MPa) and 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 with hydrostatic pressure during a cyclic volumetric test is shown in
Figure 5.
Simple shear test with nonzero initial conditions for .
Figures
Figure 5. 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:
2070 kg/m3
1480 m/sec
s
1.93
0.880
Viscous properties
The properties for each of the tested viscosity models are given below:
Mat1:
Newtonian viscosity:
1 MPa sec
Mat2:
Power Law viscosity:
2.173 MPa (sec)n
0.392
1 MPa sec
0.1 MPa sec
Mat3:
Carreau-Yasuda viscosity:
1 MPa sec
0.1 MPa sec
0.11 sec
0.392
0.644
Mat4:
Cross viscosity:
1 MPa sec
0.1 MPa sec
0.11 sec
0.392
Mat5:
Herschel-Bulkley viscosity:
1 MPa sec
3.59 MPa
2.173 MPa (sec)n
0.392
Mat6:
Ellis-Meter viscosity:
1 MPa sec
0.1 MPa sec
5.665 MPa
0.392
Mat7:
Powell-Eyring viscosity:
1 MPa sec
0.1 MPa sec
0.11 sec
Mat8:
Tabular viscosity:
(MPa sec)
(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
1 MPa sec
0.11 sec
0.392
TRS properties
Arrhenius form:
109100 joule/mole
308 kelvin
0 kelvin
8.31434 joule/(mole kelvin)
Results and discussion
The results obtained from the analyses agree well with exact analytical or approximate
solutions.
User subroutine VUVISCOSITY for the
user-defined Cross viscosity model used in eosshrvisc.inp and eosshrvisctrs.inp.
Nonlinear viscoelastic shear behavior
Elements tested
C3D8R
CPE4R
Features tested
Nonlinear viscoelastic models for equation of state materials using a combination of
elastic and 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 family of nonlinear
viscoelastic shear models defined using a combination of elastic and viscous shear
behaviors with different Newtonian and non-Newtonian shear viscosities. 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 the same as in Viscous shear behavior except that an elastic shear modulus is defined: = 1000Pa.
Results and discussion
The results obtained from the analyses agree well with exact analytical or approximate
solutions.
User subroutine VUVISCOSITY for the
user-defined Cross viscosity model used in eosshrelasvisc.inp
and eosshrelasvisctrs.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.
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
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
where is the reference density. The user subroutine needs to return the
derivative of pressure with respect to density, , 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, , which is usually needed to solve the nonlinear pressure-energy
dependency using the Newton method. In the case considered here, these quantities are
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, , is 7890 kg/m3. The properties for the tabular
EOS material model are computed using = 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
where is the yield stress (in units of MPa) and 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.