Equation of state material

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

where K is the bulk modulus of the material. The linear $U_s-U_p$ Hugoniot form is

where $\eta =1-{\rho }_0/\rho$ is the same as the nominal volumetric strain measure, ${\epsilon }_{vol}$. Thus, setting the parameters $s=$ 0.0 and ${\mathrm{\Gamma }}_0=$ 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, ${\rho }_0$, 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

where $\overline{\sigma }$ is the yield stress (in units of MPa) and ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain.

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 $f_1\left({\epsilon }_{\mathrm{vol}}\right)$ 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. Thus, setting the functions $f_1\left({\epsilon }_{\mathrm{vol}}\right)=-K{\epsilon }_{vol}$ and $f_2\left({\epsilon }_{\mathrm{vol}}\right)=$ 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, ${\rho }_0$, 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

where $\overline{\sigma }$ is the yield stress (in units of MPa) and ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain.

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 $P-\alpha$ 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:

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:

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.

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 ${\rho }_0$ is the reference density. The user subroutine needs to return the derivative of pressure with respect to density, $\partial p/\partial \rho$, 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

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, ${\rho }_0$, 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

where $\overline{\sigma }$ is the yield stress (in units of MPa) and ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain.