# Rate-independent plasticity

 This problem contains basic test cases for one or more Abaqus elements and features. The following topics are discussed:

ProductsAbaqus/StandardAbaqus/Explicit

## Mises plasticity with isotropic hardening

C3D8

CPS4

T3D2

### Problem description

#### Material:

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Hardening:

Yield stress Plastic strain
200. 0.0000
220. 0.0009
220. 0.0029

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mpliho3hut.inp

Uniaxial tension, C3D8 elements.

mpliho2hut.inp

Uniaxial tension, CPS4 elements.

mpliho1hut.inp

Uniaxial tension, T3D2 elements.

mpliho3gsh.inp

Shear, C3D8 elements.

mpliho2gsh.inp

Shear, CPS4 elements.

mpliho1mcy.inp

mpliho3vlp.inp

Linear perturbation steps containing LOAD CASE, uniaxial tension, C3D8 elements.

mplihi3hut.inp

Uniaxial tension with nonzero initial condition for $ε¯p⁢l$, C3D8 elements.

## Mises plasticity with linear kinematic hardening

T3D2

### Problem description

#### Material:

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Hardening:

Yield stress Plastic strain
200. 0.0000
220. 0.0009

The linear kinematic hardening model is defined by the slope of the stress-strain data given earlier. (The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

#### Abaqus/Standard input files

mplkho1mcy.inp

mplkhi1hut.inp

Uniaxial tension with nonzero initial condition for $α11$, load control, T3D2 elements.

#### Abaqus/Explicit input files

mplkho1mcy_xpl.inp

mplkhi1hut_xpl.inp

Uniaxial tension with nonzero initial condition for $α11$, load control, T3D2 elements.

## Mises plasticity with multilinear kinematic hardening

C3D8

### Problem description

#### Material:

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Hardening:

Yield stress Plastic strain Temperature Field variable
2000. 0.00 0. 0.
3000. 0.25 0. 0.
5500. 1.0 0. 0.
800. 0. 100. 0.
2000. 0.25 100. 0.
4000. 0.8 100. 0.
400. 0. 0. 200.
800. 0.1 0. 200.
1600. 0.4 0. 200.
200. 0. 100. 200.
500. 0.5 100. 200.
750. 1. 100. 200.
Thermal properties

Coefficient of expansion, $α$ = 1.E-4

### Results and discussion

The results agree well with the analytical solution.

### Input files

#### Abaqus/Standard input file

Uniaxial test, C3D8 element.

## Mises plasticity with combined isotropic/kinematic hardening

B21

C3D8

C3D8R

CPE4

CPS4

M3D4

SAX1

T3D2

### Problem description

#### Material 1

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Initial yield stress: $σ0$ = 200.0

Isotropic hardening parameter, $Q∞$ = 2000

Isotropic hardening parameter, b = 0.25

Kinematic hardening parameter, C = 2.222 × 104

Kinematic hardening parameter, $γ$ = 34.65

The parameters given above are used to generate data for some of the input files that use tabular data. (The units are not important.)

#### Material 2

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Initial yield stress: $σ0$ = 200.0

Kinematic hardening parameter, C = 2.222 × 104

Kinematic hardening parameter, $γ$ = 0.0

The parameters given above are used to generate data for some of the input files that use tabular data. (The units are not important.)

#### Material 3

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Initial yield stress: $σ0$ = 200.0

Isotropic hardening parameter, $Q∞$ = 0.0

Isotropic hardening parameter, b = 0.0

Kinematic hardening parameter, C = 2.222 × 104

Kinematic hardening parameter, $γ$ = 34.65

(The units are not important.)

#### Material 4

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Initial yield stress: $σ0$ = 200.0

Isotropic hardening parameter, $Q∞$ = 2000

Isotropic hardening parameter, b = 0.25

Kinematic hardening parameter, $C1$ = 1.111 × 104

Kinematic hardening parameter, $γ1$ = 34.65

Kinematic hardening parameter, $C2$ = 5.555 × 103

Kinematic hardening parameter, $γ2$ = 34.65

Kinematic hardening parameter, $C3$ = 5.555 × 103

Kinematic hardening parameter, $γ3$ = 0.0

(The units are not important.)

#### Material 5

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Initial yield stress: $σ0$ = 200.0

Isotropic hardening parameter, $Q∞$ = 0.0

Isotropic hardening parameter, b = 0.0

Kinematic hardening parameter, $C1$ = 1.111 × 104

Kinematic hardening parameter, $γ1$ = 34.65

Kinematic hardening parameter, $C2$ = 5.555 × 103

Kinematic hardening parameter, $γ2$ = 34.65

Kinematic hardening parameter, $C3$ = 5.555 × 103

Kinematic hardening parameter, $γ3$ = 0.0

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

#### Abaqus/Standard input files

mplchi3nt1mb.inp

Uniaxial tension, nonzero initial conditions for $α11$, $ε¯p⁢l$, and $σ11$; displacement control; M3D4 elements with rebar; number backstresses = 3.

Material 1:
mplchb2hut.inp

Uniaxial tension with temperature and field variable dependence, displacement control, SAX1 elements.

mplchb3hut.inp

Uniaxial tension with temperature-dependent $γ$, displacement control, C3D8R elements.

mplcho1hut.inp

Uniaxial tension, tabulated data, load control, B21 elements.

mplcho1hutmb.inp

Uniaxial tension, tabulated data, load control, B21 elements, number backstresses = 3.

mplcho3nt1.inp

Uniaxial tension, load control, C3D8 elements.

mplchi3nt1.inp

Uniaxial tension, nonzero initial conditions for $α11$, $ε¯p⁢l$, and $σ11$; displacement control; M3D4 elements with rebar.

mplchi2hut.inp

Uniaxial tension with orientation and nonzero initial conditions for $α11$ and $ε¯p⁢l$, displacement control, CPE4 elements.

Material 2:
mplcho1mcy.inp

mplcho1mcymb.inp

Cyclic loading, no isotropic hardening, displacement control, T3D2 elements, number backstresses = 3.

Material 3:
mplcho2gsh.inp

Simple shear including perturbation step, CPS4 elements.

Material 4:
mplchb2hutmb.inp

Uniaxial tension with temperature and field variable dependence, displacement control, SAX1 elements, number backstresses = 3.

mplchi2hutmb.inp

Uniaxial tension with orientation and nonzero initial conditions for $α11$ and $ε¯p⁢l$, displacement control, CPE4 elements, number backstresses = 3.

mplcho3nt1mb.inp

Uniaxial tension, load control, C3D8 elements, number backstresses = 3.

Material 5:
mplcho2gshmb.inp

Simple shear including perturbation step, CPS4 elements, number backstresses = 3.

#### Abaqus/Explicit input files

mplchi3nt1mb_xpl.inp

Uniaxial tension, nonzero initial conditions for $α11$, $ε¯p⁢l$, and $σ11$; displacement control; M3D4R elements with rebar; number backstresses = 3.

Material 1:
mplchb2hut_xpl.inp

Uniaxial tension with temperature and field variable dependence, displacement control, SAX1 elements.

mplchb3hut_xpl.inp

Uniaxial tension with temperature-dependent $γ$, displacement control, C3D8R elements.

mplcho1hut_xpl.inp

Uniaxial tension, tabulated data, load control, B21 elements.

mplcho1hutmb_xpl.inp

Uniaxial tension, tabulated data, load control, B21 elements, number backstresses = 3.

mplcho3nt1_xpl.inp

Uniaxial tension, load control, C3D8R elements.

mplchi3nt1_xpl.inp

Uniaxial tension, nonzero initial conditions for $α11$, $ε¯p⁢l$, and $σ11$; displacement control; M3D4R elements with rebar.

mplchi2hut_xpl.inp

Uniaxial tension with orientation and nonzero initial conditions for $α11$ and $ε¯p⁢l$, displacement control, CPE4R elements.

Material 2:
mplcho1mcy_xpl.inp

mplcho1mcymb_xpl.inp

Cyclic loading, no isotropic hardening, displacement control, T3D2 elements, number backstresses = 3.

Material 3:
mplcho2gsh_xpl.inp

Simple shear including perturbation step, CPS4R elements.

Material 4:
mplchb2hutmb_xpl.inp

Uniaxial tension with temperature and field variable dependence, displacement control, SAX1 elements, number backstresses = 3.

mplchi2hutmb_xpl.inp

Uniaxial tension with orientation and nonzero initial conditions for $α11$ and $ε¯p⁢l$, displacement control, CPE4R elements, number backstresses = 3.

mplcho3nt1mb_xpl.inp

Uniaxial tension, load control, C3D8R elements, number backstresses = 3.

Material 5:
mplcho2gshmb_xpl.inp

Simple shear including perturbation step, CPS4R elements, number backstresses = 3.

C3D8

CPS4

T3D2

### Problem description

#### Material:

Elasticity

Young's modulus, E = 30.0E6

Poisson's ratio, $ν$ = 0.3

Plasticity

Hardening:

Yield stress Plastic strain Temperature
30.0E3 0.000 0.0
50.0E3 0.200 0.0
50.0E3 2.000 0.0
3.0E3 0.000 100.0
5.0E3 0.200 100.0
5.0E3 2.000 100.0
Other properties

Density, $μ$ = 1000.0

Specific heat, c = 0.4

Inelastic heat fraction = 0.5

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mhliho3hut.inp

Uniaxial tension, C3D8 elements.

mhliho1hut.inp

Uniaxial tension, T3D2 elements.

mhliho3gsh.inp

Shear, C3D8 elements.

mhliho2gsh.inp

Shear, CPS4 elements.

mhliho3ltr.inp

Triaxial, C3D8 elements.

mhliht3hut.inp

Uniaxial tension, C3D8 elements.

mhliht3xmx.inp

Multiaxial, C3D8 elements.

## Hill plasticity

C3D8

### Problem description

#### Material:

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Hardening:

Yield stress Plastic strain
200. 0.0000
220. 0.0009
220. 0.0029

Anisotropic yield ratios: 1.5, 1.2, 1.0, 1.0, 1.0, 1.0

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mppiho3nt1.inp

Uniaxial tension in direction 1, C3D8 elements.

mppiho3ot2.inp

Uniaxial tension in direction 2, C3D8 elements.

mppiho3pt3.inp

Uniaxial tension in direction 3, C3D8 elements.

mppiho3vlp.inp

Linear perturbation steps containing LOAD CASE, uniaxial tension in direction 1, C3D8 elements.

## Deformation plasticity

C3D8

CPS4

T3D2

### Problem description

#### Material:

Elasticity

Young's modulus, E = 200.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Yield stress, $σ0$ = 200.0

Exponent, n = 21.315

Yield offset, $α$ = 0.11802

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mdfooo3hut.inp

Uniaxial tension, C3D8 elements.

mdfooo3huti.inp

Uniaxial tension with initial stress, C3D8 elements.

mdfooo2hut.inp

Uniaxial tension, CPS4 elements.

mdfooo2huti.inp

Uniaxial tension with initial stress, CPS4 elements.

mdfooo1hut.inp

Uniaxial tension, T3D2 elements.

mdfooo1huti.inp

Uniaxial tension with initial stress, T3D2 elements.

## Drucker-Prager plasticity with linear elasticity

C3D8

C3D8R

CAX4

CPE4

CPS4

### Problem description

#### Material:

Elasticity

Young's modulus, E = 300.0E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Angle of friction, $β$ = 40.0

Dilation angle, $ψ$ = 40.0

Third invariant ratio, K = 0.78 (when included; otherwise, 1.0)

Hardening curve:

Yield stress Plastic strain
6.0E3 0.000000
9.0E3 0.020000
11.0E3 0.063333
12.0E3 0.110000
12.0E3 1.000000

(The units are not important.)

The hyperbolic and exponent forms of the yield criteria are verified by using parameters that reduce them into equivalent linear forms. Reducing the hyperbolic yield function into a linear form requires that $pt|0=d/tan⁡β$. Reducing the exponent yield function into a linear form requires that b = 1.0 and that a = ($tan⁡β$)−1.

### Results and discussion

Most tests in this section are set up as cases of the homogeneous deformation of a single element of unit dimensions. Consequently, the results are identical for all integration points within the element. To test certain conditions, however, it is necessary to set up inhomogeneous deformation problems. In each case the constitutive path is integrated with 20 increments of fixed size.

### Input files

#### Shear criterion: linear Drucker-Prager

mdeooo3euc.inp

Uniaxial compression, C3D8 elements.

mdeooo2euc.inp

Uniaxial compression, CPS4 elements.

mdeooo3ctc.inp

Triaxial compression, CAX4 elements.

mdeooo3dte.inp

Triaxial extension, CAX4 elements.

mdekoo3dte.inp

K = 0.78, triaxial extension, CAX4 elements.

mdeooo3gsh.inp

Shear, C3D8 elements.

mdeooo2gsh.inp

Shear, CPS4 elements.

mdekoo3gsh.inp

K = 0.78, shear, C3D8 elements.

mdekoo2gsh.inp

K = 0.78; shear, CPS4 elements.

mdeooo3hut.inp

Uniaxial tension, C3D8 elements.

mdeooo2hut.inp

Uniaxial tension, CPS4 elements.

mdekoo3hut.inp

K = 0.78, uniaxial tension, C3D8 elements.

mdekoo2hut.inp

K = 0.78, uniaxial tension, CPS4 elements.

mdekot3hut.inp

K = 0.78, uniaxial tension with temperature dependence, C3D8R elements.

mdeooo3jht.inp

Hydrostatic tension, C3D8 elements.

mdeooo3ltr.inp

Triaxial stress, CPE4 elements (inhomogeneous).

mdeooo2ltr.inp

Triaxial stress, CPS4 elements (inhomogeneous).

mdekoo3ltr.inp

K = 0.78, triaxial stress, CPE4 elements (inhomogeneous).

mdekoo2ltr.inp

K = 0.78, triaxial stress, CPS4 elements (inhomogeneous).

mdeoot3euc.inp

Uniaxial compression with temperature dependence, C3D8 elements.

mdeooo3vlp.inp

Linear perturbation uniaxial compression, C3D8 elements.

mdedos3euc.inp

Uniaxial compression with rate dependence, C3D8 elements.

mdeooi3euc.inp

Uniaxial compression with nonzero initial condition for $ε¯p⁢l$, C3D8 elements.

mdedoo2euc.inp

Uniaxial tension, perfect plasticity, CPS4 elements.

#### Shear criterion: exponent

mdeeoo3jht.inp

Hydrostatic tension, C3D8 elements.

mdeeoo3ltr.inp

Triaxial stress, CPE4 elements (inhomogeneous).

mdeeoo3dte.inp

Triaxial extension, CAX4 elements.

mdeeoo3hut.inp

Uniaxial tension, C3D8 elements.

mdeeoo2hut.inp

Uniaxial tension, CPS4 elements.

mdeeoo3gsh.inp

Shear, C3D8 elements.

mdeeoo2gsh.inp

Shear, CPS4 elements.

mdeeoo3ctc.inp

Triaxial compression, CAX4 elements.

mdeeoo3euc.inp

Uniaxial compression, C3D8 elements.

mdeeoo2euc.inp

Uniaxial compression, CPS4 elements.

mdeeos3euc.inp

Uniaxial compression with rate dependence, C3D8 elements.

mdeeot3euc.inp

Uniaxial compression with temperature dependence, C3D8 elements.

mdeeoo3vlp.inp

Linear perturbation uniaxial compression, C3D8 elements.

Abaqus/Standard input files
mdeeoo3jht_xpl.inp

Hydrostatic tension, C3D8R elements.

mdeeoo3ltr_xpl.inp

Triaxial stress, CPE4R elements (inhomogeneous).

mdeeoo3dte_xpl.inp

Triaxial extension, CAX4R elements.

mdeeoo3hut_xpl.inp

Uniaxial tension, C3D8R elements.

mdeeoo2hut_xpl.inp

Uniaxial tension, CPS4R elements.

mdeeoo3gsh_xpl.inp

Shear, C3D8R elements.

mdeeoo2gsh_xpl.inp

Shear, CPS4R elements.

mdeeoo3ctc_xpl.inp

Triaxial compression, CAX4R elements.

mdeeoo3euc_xpl.inp

Uniaxial compression, C3D8R elements.

mdeeoo2euc_xpl.inp

Uniaxial compression, CPS4R elements.

mdeeos3euc_xpl.inp

Uniaxial compression with rate dependence, C3D8R elements.

mdeeot3euc_xpl.inp

Uniaxial compression with temperature dependence, C3D8R elements.

Abaqus/Explicit input files

#### Shear criterion: exponent with test data

mdeeto3jht.inp

Hydrostatic tension, C3D8 elements.

mdeeto3ltr.inp

Triaxial stress, CPE4 elements (inhomogeneous).

mdeeto3dte.inp

Triaxial extension, CAX4 elements.

mdeeto3hut.inp

Uniaxial tension, C3D8 elements.

mdeeto2hut.inp

Uniaxial tension, CPS4 elements.

mdeeto3gsh.inp

Shear, C3D8 elements.

mdeeto2gsh.inp

Shear, CPS4 elements.

mdeeto3ctc.inp

Triaxial compression, CAX4 elements.

mdeeto3euc.inp

Uniaxial compression, C3D8 elements.

mdeeto2euc.inp

Uniaxial compression, CPS4 elements.

mdeets3euc.inp

Uniaxial compression with rate dependence, C3D8 elements.

mdeeto3vlp.inp

Linear perturbation uniaxial compression, C3D8 elements.

mdeeto3jht_xpl.inp

Hydrostatic tension, C3D8R elements.

mdeeto3ltr_xpl.inp

Triaxial stress, CPE4R elements (inhomogeneous).

mdeeto3dte_xpl.inp

Triaxial extension, CAX4R elements.

mdeeto3hut_xpl.inp

Uniaxial tension, C3D8R elements.

mdeeto2hut_xpl.inp

Uniaxial tension, CPS4R elements.

mdeeto3gsh_xpl.inp

Shear, C3D8R elements.

mdeeto2gsh_xpl.inp

Shear, CPS4R elements.

mdeeto3ctc_xpl.inp

Triaxial compression, CAX4R elements.

mdeeto3euc_xpl.inp

Uniaxial compression, C3D8R elements.

mdeeto2euc_xpl.inp

Uniaxial compression, CPS4R elements.

mdeets3euc_xpl.inp

Uniaxial compression with rate dependence, C3D8R elements.

Abaqus/Standard input filesAbaqus/Explicit input files

#### Shear criterion: hyperbolic

mdehoo3jht.inp

Hydrostatic tension, C3D8 elements.

mdehoo3ltr.inp

Triaxial stress, CPE4 elements (inhomogeneous).

mdehoo3dte.inp

Triaxial extension, CAX4 elements.

mdehoo3hut.inp

Uniaxial tension, C3D8 elements.

mdehoo2hut.inp

Uniaxial tension, CPS4 elements.

mdehoo3gsh.inp

Shear, C3D8 elements.

mdehoo2gsh.inp

Shear, CPS4 elements.

mdehoo3ctc.inp

Triaxial compression, CAX4 elements.

mdehoo3euc.inp

Uniaxial compression, C3D8 elements.

mdehoo2euc.inp

Uniaxial compression, CPS4 elements.

mdehos3euc.inp

Uniaxial compression with rate dependence, C3D8 elements.

mdehot3euc.inp

Uniaxial compression with temperature dependence, C3D8 elements.

mdehoo3vlp.inp

Linear perturbation uniaxial compression, C3D8 elements.

mdehoo3jht_xpl.inp

Hydrostatic tension, C3D8R elements.

mdehoo3ltr_xpl.inp

Triaxial stress, CPE4R elements (inhomogeneous).

mdehoo3dte_xpl.inp

Triaxial extension, CAX4R elements.

mdehoo3hut_xpl.inp

Uniaxial tension, C3D8R elements.

mdehoo2hut_xpl.inp

Uniaxial tension, CPS4R elements.

mdehoo3gsh_xpl.inp

Shear, C3D8R elements.

mdehoo2gsh_xpl.inp

Shear, CPS4R elements.

mdehoo3ctc_xpl.inp

Triaxial compression, CAX4R elements.

mdehoo3euc_xpl.inp

Uniaxial compression, C3D8R elements.

mdehoo2euc_xpl.inp

Uniaxial compression, CPS4R elements.

mdehos3euc_xpl.inp

Uniaxial compression with rate dependence, C3D8R elements.

mdehot3euc_xpl.inp

Uniaxial compression with temperature dependence, C3D8R elements.

Abaqus/Standard input filesAbaqus/Explicit input files

#### Transferring results between Abaqus/Standard and Abaqus/Explicit

sx_s_druckerprager.inp

Base problem for carrying out import from Abaqus/Standard to Abaqus/Explicit, C3D8R elements, uniaxial tension.

sx_x_druckerprager_y_y.inp

Explicit dynamic continuation of sx_s_druckerprager.inp with both the reference configuration and the state imported, C3D8R elements, uniaxial tension.

sx_x_druckerprager_n_y.inp

Explicit dynamic continuation of sx_s_druckerprager.inp with only the state imported, C3D8R elements, uniaxial tension.

sx_x_druckerprager_n_n.inp

Explicit dynamic continuation of sx_s_druckerprager.inp without importing the state or the reference configuration, C3D8R elements, uniaxial tension.

xs_s_druckerprager_y_y.inp

Import into Abaqus/Standard from sx_x_druckerprager_y_y.inp with both the reference configuration and the state imported, C3D8R elements, uniaxial tension.

xs_s_druckerprager_n_y.inp

Import into Abaqus/Standard from sx_x_druckerprager_n_y.inp with only the state imported, C3D8R elements, uniaxial tension.

xs_s_druckerprager_n_n.inp

Import into Abaqus/Standard from sx_x_druckerprager_n_n.inp without importing the state or the reference configuration, C3D8R elements, uniaxial tension.

## Drucker-Prager plasticity with porous elasticity

CAX4

### Problem description

#### Material:

Elasticity

Logarithmic bulk modulus, $κ$ = 1.49

Poisson's ratio, $ν$ = 0.1

Plasticity

Angle of friction, $β$ = 10.0

Dilation angle, $ψ$ = 10.0

Hardening curve:

Yield stress Plastic strain
100.0 0.0
500.0 0.5
Initial conditions

Initial void ratio, $e0$ = 4.1

The hyperbolic and exponent forms of the yield criteria are verified by using parameters that reduce them into equivalent linear forms. Reducing the hyperbolic yield function into a linear form requires that $pt|0=d/tan⁡β$. Reducing the exponent yield function into a linear form requires that b = 1.0 and that a = ($tan⁡β$)−1.

(The units are not important.)

### Results and discussion

The tests in this section are set up as cases of homogeneous deformation of a single element of unit dimensions. Consequently, the results are identical for all integration points within the element. In each case the constitutive path is integrated with 20 increments of fixed size.

### Input files

#### Shear criterion: linear Drucker-Prager

mdpdoo3bus.inp

Uniaxial strain, CAX4 elements.

mdpdoo3ctc.inp

Triaxial compression, CAX4 elements.

#### Shear criterion: exponent

mdpeoo3bus.inp

Uniaxial strain, CAX4 elements.

mdpeoo3ctc.inp

Triaxial compression, CAX4 elements.

#### Shear criterion: exponent with test data

mdpeto3bus.inp

Uniaxial strain, CAX4 elements.

mdpeto3ctc.inp

Triaxial compression, CAX4 elements.

#### Shear criterion: hyperbolic

mdphoo3bus.inp

Uniaxial strain, CAX4 elements.

mdphoo3ctc.inp

Triaxial compression, CAX4 elements.

## Cap plasticity

C3D8R

CAX4

CPE4

### Problem description

#### Material:

In the tests described in this section, the following data for linear elasticity, cap plasticity I, cap hardening I, and K = 1.0 are used unless otherwise specified. With this data, the elastic shear modulus is 5000.0 and the bulk modulus is 10000.0. First yield in pure shear occurs at S12 = 100.0, first yield in pure hydrostatic compression occurs at PRESS = 270.0, first yield in pure hydrostatic tension occurs at PRESS = 300.0, and first yield with PRESS = $pa$ occurs at PRESS = 120.0 and S12 = 125.0. C3D8 elements are used unless otherwise specified.

Linear elasticity (used in nearly all tests)

Young's modulus, E = 12857.1429

Poisson's ratio, $ν$ = 0.28571429 (= 1/7)

Cap plasticity I (used in nearly all tests)

Cohesion, d = 173.20508 (= 100$3$)

Slope of Drucker-Prager failure surface, $β$ = 30.0

Cap ellipticity, R = 0.61858957

Initial volumetric plastic strain, $εv⁢o⁢lp⁢l⁢(0)$ = 0.027

Transition parameter, $α$ = 0.69258232

Third invariant factor, K = 1.0 or 0.8, depending on the test.

Cap hardening I (used in nearly all tests)

Position of the yield surface in pure hydrostatic compression, $pb$

Volumetric compressive plastic strain, $εv⁢o⁢lp⁢l$

$pb$ $εv⁢o⁢lp⁢l$
213.0 0.00
222.0 0.01
242.0 0.02
282.0 0.03
362.0 0.04
522.0 0.05
842.0 0.06
1482.0 0.07
2762.0 0.08
Cap plasticity II

d = 0.2286E6

$β$ = 85.0

R = 0.0875

$εv⁢o⁢lp⁢l⁢(0)$ = 1.22

$α$ = 0.07877

K = 1.0

Cap hardening II

Position of the yield surface in pure hydrostatic compression, $pb$

Volumetric compressive plastic strain, $εv⁢o⁢lp⁢l$

$pb$ $εv⁢o⁢lp⁢l$
0.03E6 0.0
0.20E6 1.22
2.00E6 2.44
2.00E7 3.66
Porous elasticity I

Logarithmic bulk modulus, $κ$ = 20.0

Poisson's ratio, $ν$ = 0.28571429

Tensile strength limit, $pt$ = 1.0E5

Porous elasticity II

$κ$ = 0.09

$ν$ = 0.0

$pt$ = 0.02E6

Initial conditions

Initial void ratio, $e0$ = 1.0

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mcaooo3mcy.inp

Hydrostatic cyclic test, displacement control.

The following six steps are executed:

1. Load, yielding in hydrostatic compression

mcaooo3euc.inp

Uniaxial compressive stress test; displacement control.

Step 2 reverses the displacement causing yielding in tension.

mcaooo3gsh.inp

Shear test; load control (S22 = −S11); overlaid soft linear element.

There will be some hydrostatic stress due to transverse restraint.

mcaooo3ucs.inp

Cyclical shear test; displacement control; S12 dominant.

mcaoot3ctc.inp

Hydrostatic compression to $p=pa$, then pure shear; displacement control; temperature dependence.

Yielding should be volume preserving.

mcakoo3gsh.inp

Shear test; load control; two primary elements and two overlaid soft elements.

One set loaded with principal stresses $∝$ (1, 1, −2), the other with $∝$ (−1, −1, 2).

The ratio of yield stresses should be K = 0.8.

mcaoob3bus.inp

Uniaxial compressive strain (odometer) test; CPE4 element; load control; with temperature and field variable dependence of the CAP PLASTICITY and CAP HARDENING data.

The temperatures and field variables are specified to give CAP PLASTICITY and CAP HARDENING data exactly the same as cap plasticity I and cap hardening I data.

mcaooo3bus.inp

Uniaxial compressive strain (odometer) test; load control; NLGEOM and porous elasticity I.

The tangent bulk and shear moduli of porous elasticity I differ from that of the linear elasticity by about 1% over the strain range of the test.

mcaooo3ctc.inp

Triaxial test. Hydrostatic loading to $p=pa$, then increase S11 only.

mcaooo3vlp.inp

Uniaxial compressive strain (odometer) test; load control; the nonlinear analysis is split into two steps, each of which is preceded by a linear perturbation step.

The results of the nonlinear steps should correspond to those of mca0003bus.inp.

The results of the two linear perturbation steps (STATIC) should be identical because small displacements are assumed and the elasticity is linear.

mcakoo3ltr.inp

A displacement pattern designed to produce different stress states at the 8 Gauss points but dominated by shear.

The aim is to test the robustness of the Newton loops, so very large strain increments are taken.

Displacement control. K = 0.8.

mcaooo3ltr.inp

Another test of the robustness of the algorithm.

CAX4 element, porous elasticity II, cap plasticity II, and cap hardening II is used.

mcaooo3xmx.inp

Tests adjustment of the initial position of the cap.

Two C3D8R elements with different initial stress states.

The initial stress in element 1 will cause an adjustment that will make the stress point lie on the cap yield surface.

The initial stress in element 2 will cause an adjustment that will make the stress point lie on the transition yield surface.

## Clay plasticity with porous elasticity

C3D8

CAX8R

### Problem description

#### Material:

Porous elasticity

Logarithmic bulk modulus, $κ$ = 0.026

Poisson's ratio, $ν$ = 0.3

Plasticity

Logarithmic plastic bulk modulus, $λ$ = 0.174

Critical state slope, M = 1.0

Initial yield surface size, $a0$ = 58.3

(except in tests mclxxxxahc.inp where we use $a0$ = 130.9 or $e1$ = 1.904)

Cap parameter, $β$ = 0.5 (when included; otherwise, 1.0)

Third invariant ratio, K = 0.78 (when included; otherwise, 1.0)

Initial conditions

Initial void ratio, $e0$ = 1.08

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mclooo3ahc.inp

Hydrostatic compression, C3D8 elements.

mcloio3ahc.inp

Hydrostatic compression with intercept option, C3D8 elements.

mclooo3ctc.inp

Triaxial compression, CAX8R elements.

mclott3ctc.inp

Triaxial compression, temperature dependence, CAX8R elements.

mclobo3ctc.inp

$β$ = 0.5, triaxial compression, CAX8R elements.

mclooo3dte.inp

Triaxial extension, CAX8R elements.

mclkoo3dte.inp

K = 0.78, triaxial extension, CAX8R elements.

mclktd3dte.inp

K = 0.78, triaxial extension, field variable dependence, CAX8R elements.

mclkbo3dte.inp

$β$ = 0.5, K = 0.78, triaxial extension, CAX8R elements.

mcloto3euc.inp

Uniaxial compression, CAX8R elements.

mclooo3gsh.inp

Shear, C3D8 elements.

mcloto3gsh.inp

Shear, tabulated hardening, C3D8 elements.

mclooo3vlp.inp

Linear perturbation hydrostatic compression, C3D8 elements.

## Crushable foam plasticity

C3D8

CPE4

### Problem description

#### Material:

Elasticity

Young's modulus, E = 3.0E6

Poisson's ratio, $ν$ = 0.2

Plasticity

Initial yield stress in hydrostatic compression, $p0$ = 2.0E5

Strength in hydrostatic tension, $pt$ = 2.0E4

Initial yield stress in uniaxial compression, $σ0$ = 2.2E5

Yield stress ratio, $k=σ0/p0$ = 1.1

Yield stress ratio, $kt=pt/p0$ = 0.1

Hardening curve (from uniaxial compression):

Yield stress Plastic strain
2.200E5 0.0
2.465E5 0.1
2.729E5 0.2
2.990E5 0.3
3.245E5 0.4
3.493E5 0.5
3.733E5 0.6
3.962E5 0.7
4.180E5 0.8
4.387E5 0.9
4.583E5 1.0
4.938E5 1.2
5.248E5 1.4
5.515E5 1.6
5.743E5 1.8
5.936E5 2.0
6.294E5 2.5
6.520E5 3.0
6.833E5 5.0
6.883E5 10.0
Initial conditions

Initial volumetric compacting plastic strain, $-εvolp⁢l$, is set to 0.02 for the cases in which specifying an initial equivalent plastic strain is tested.

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mfeoto3ahc.inp

Hydrostatic compression, C3D8 elements.

mfeoto3euc.inp

Uniaxial compression, C3D8 elements.

mfeoto3gsh.inp

Shear, C3D8 elements.

mfeoto3hut.inp

Uniaxial tension, C3D8 elements.

mfeoti3euc.inp

Uniaxial compression, CPE4 elements.

mfeoto3ltr.inp

Triaxial stress, CPE4 elements (inhomogeneous).

mfeoto3vlp.inp

Linear perturbation with LOAD CASE and hydrostatic compression, C3D8 elements.

## Clay plasticity with linear elasticity

C3D8

C3D8R

CAX4R

CAX8R

CPE4R

### Problem description

#### Material 1

Elasticity

The Young's modulus used in each test is given in the input file description. The modulus of each test is based on the average elastic stiffness of the equivalent test with porous elasticity at increments 10 and 20. A direct comparison with the results documented in Drucker-Prager plasticity with linear elasticity is, therefore, possible.

Poisson's ratio, $ν$ = 0.3

Plasticity

Critical state slope, M = 1.0

Initial volumetric plastic strain, $εvolp⁢l|0$ = 0.4

Cap parameter, $β$ = 0.5 (when included; otherwise, 1.0)

Third invariant ratio, K = 0.78 (when included; otherwise, 1.0)

The exponential hardening curve used in Drucker-Prager plasticity with linear elasticity is entered in tabulated form with an initial volumetric plastic strain that corresponds to a yield surface size of either $a0$ = 58.3 or $a0$ = 130.9.

(The units are not important.)

#### Material 2

Elasticity

Young's modulus, $E$ = 18820

Poisson's ratio, $ν$ = 0.3

Plasticity

Critical state slope, M = 1.0

Initial volumetric plastic strain, $εvolp⁢l|0$ = 0.0

Cap parameter, $β$ = 1.0

Third invariant ratio, K = 1.0

Tabulated curves are used for defining the compressive and tensile hardening.

Softening regularization

$lc(m)$ = 0.8

$nr$ = 2.0

$fm⁢a⁢x$ = 2.5

(The units are not important.)

#### Material 3

Elasticity

Young's modulus, $E$ = 18820

Poisson's ratio, $ν$ = 0.3

Plasticity

Critical state slope, M = 1.0

Initial volumetric plastic strain, $εvolp⁢l|0$ = 0.0

Cap parameter, $β$ = 1.0

Third invariant ratio, K = 1.0

Tabulated curves are used for defining the compressive and tensile hardening.

Softening regularization

$lc(m)$ = 0.5

$nr$ = 1.0

$fm⁢a⁢x$ = 2.5

(The units are not important.)

#### Material 4 [Crook et al. (2002)]

Elasticity
Engineering constants
$E1$ 200000.0
$E2$ 342000.0
$E3$ 342000.0
$ν12$ 0.32
$ν13$ 0.32
$ν23$ 0.32
$G12$ 89900.0
$G13$ 89900.0
$G23$ 129545.5
Plasticity

Critical state slope, M = 1.0

Initial volumetric plastic strain, $εvolp⁢l|0$ = 0.0

Cap parameter, $β$ = 1.0

Third invariant ratio, K = 1.0

Tabulated curves are used for defining the compressive and tensile hardening.

Anisotropic yield ratios: 1.2, 1.0, 1.0, 0.71, 0.71, 0.99

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

#### Abaqus/Standard input files

Material 1:
mceooo3ahc.inp

Hydrostatic compression, C3D8 elements, E = 18820.

mceoot3ctc.inp

Triaxial compression, CAX8R elements, E = 30732.

mceobo3ctc.inp

$β$ = 0.5, triaxial compression, CAX8R elements, E = 29556.

mceooo3dte.inp

Triaxial extension, CAX8R elements, E = 21114.

mcekod3dte.inp

K = 0.78, triaxial extension, CAX8R elements, E = 28140.

mcekbo3dte.inp

$β$ = 0.5, K = 0.78, triaxial extension, CAX8R elements, E = 27580.

mceooo3euc.inp

Uniaxial compression, CAX8R elements, E = 30000.

mceooo3gsh.inp

Shear, C3D8 elements, E = 2798.

mceooo3vlp.inp

Linear perturbation with LOAD CASE and hydrostatic compression, C3D8 elements, E = 18820.

Material 2:
clayteninipssoft3d.inp

Hydrostatic compression, C3D8 and C3D8R elements.

clayteninipssoftgpe.inp

Hydrostatic compression, CAX4R elements.

Material 4:
clayorthelaspoten3d.inp

Uniaxial compression and shear, C3D8R elements.

clayorthelaspotengpe.inp

Shear, CPE4R elements.

#### Abaqus/Explicit input files

Material 1:
mceooo3ahc_xpl.inp

Hydrostatic compression, C3D8 elements, E = 18820.

mceoot3ctc_xpl.inp

Triaxial compression, CAX4R elements, E = 30732.

mceobo3ctc_xpl.inp

$β$ = 0.5, triaxial compression, CAX4R elements, E = 29556.

mceooo3dte_xpl.inp

Triaxial extension, CAX4R elements, E = 21114.

mcekod3dte_xpl.inp

K = 0.78, triaxial extension, CAX4R elements, E = 28140.

mcekbo3dte_xpl.inp

$β$ = 0.5, K = 0.78, triaxial extension, CAX4R elements, E = 27580.

mceooo3euc_xpl.inp

Uniaxial compression, CAX4R elements, E = 30000.

mceooo3gsh_xpl.inp

Shear, C3D8 elements, E = 2798.

Material 2:
clayteninipssoft3d_xpl.inp

Hydrostatic compression, C3D8 and C3D8R elements.

Material 3:
clayteninipssoftgpe_xpl.inp

Hydrostatic compression, CAX4R elements.

Material 4:
clayorthelaspoten3d_xpl.inp

Uniaxial compression and shear, C3D8R elements.

clayorthelaspotengpe_xpl.inp

Shear, CPE4R elements.

#### Transferring results from Abaqus/Standard to Abaqus/Explicit

Material 1:
mceooo3ahc_sx_s.inp

Abaqus/Standard analysis, hydrostatic compression, C3D8 elements, E = 18820.

mceooo3ahc_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mceoot3ctc_sx_s.inp

Abaqus/Standard analysis, triaxial compression, CAX4R elements, E =27580.

mceoot3ctc_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mceobo3ctc_sx_s.inp

Abaqus/Standard analysis, $β$ = 0.5, triaxial compression, CAX4R elements, E =29556.

mceobo3ctc_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mceooo3dte_sx_s.inp

Abaqus/Standard analysis, triaxial extension, CAX4R elements, E =21114.

mceooo3dte_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mcekod3dte_sx_s.inp

Abaqus/Standard analysis, K = 0.78, triaxial extension, CAX4R elements, E =28140.

mcekod3dte_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mcekbo3dte_sx_s.inp

Abaqus/Standard analysis, $β$ = 0.5, K = 0.78, triaxial extension, CAX4R elements, E =27580.

mcekbo3dte_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mceooo3euc_sx_s.inp

Abaqus/Standard analysis, uniaxial compression, CAX4R elements, E =30000.

mceooo3euc_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mceooo3gsh_sx_s.inp

Abaqus/Standard analysis, shear, C3D8 elements, E =2798.

mceooo3gsh_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

#### Transferring results from Abaqus/Explicit to Abaqus/Standard

Material 1:
mceoot3ctc_xs_x.inp

Abaqus/Explicit analysis, triaxial compression, CAX4R elements, E =27580.

mceoot3ctc_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mceooo3dte_xs_x.inp

Abaqus/Explicit analysis, triaxial extension, CAX4R elements, E =21114.

mceooo3dte_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mceooo3gsh_xs_x.inp

Abaqus/Explicit analysis, shear, C3D8 elements, E =2798.

mceooo3gsh_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mceobo3ctc_xs_x.inp

Abaqus/Explicit analysis, $β$ = 0.5, triaxial compression, CAX4R elements, E =29556.

mceobo3ctc_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mcekod3dte_xs_x.inp

Abaqus/Explicit analysis, K = 0.78, triaxial extension, CAX4R elements, E =28140.

mcekod3dte_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mceooo3ahc_xs_x.inp

Abaqus/Explicit analysis, hydrostatic compression, C3D8 elements, E = 18820.

mceooo3ahc_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

### References

1. Crook A. J. L.J. GYu, and S. MWillson, Development of an Orthotropic 3D Elastoplastic Material Model for Shale, SPE/ISRM Paper SPE 78238, 2002.

## Porous metal plasticity

C3D8

CAX4

CAX4T

CPE4

### Problem description

#### Material:

Elasticity

Young's modulus, E = 300.0

Poisson's ratio, $ν$ = 0.3

Plasticity

Hardening curve:

Yield stress Plastic strain
1.0000000 0.00
1.7411011 0.05
2.7276924 0.50
2.9950454 0.80
Porous metal plasticity

Modified Gurson's model: $q1$ = 1.5, $q2$ = 1.0, $q3$ = 2.25

(otherwise, $q1$ = $q2$ = $q3$ = 1.0)

Void nucleation parameters (when included): $ϵN$ = 0.3, $sN$ = 0.1, $fN$ = 0.04

Initial relative density, $r0$ = 0.95 ($f0$ = 0.05).

#### Material properties used in coupled temperature-displacement analysis

Elasticity

Young's modulus, E = 200.0E9

Poisson's ratio, $ν$ = 0.3

Plasticity

Hardening curve:

Yield stress Plastic strain
7.0E8 0.00
3.7E9 10.0
Porous metal plasticity

$q1$ = $q2$ = $q3$ = 1.0

Initial relative density, $r0$ = 0.95 ($f0$ = 0.05).

Thermal properties

Specific heat, $cp$ = 586.0

Density, $ρ$ = 7833.0

Conductivity, k = 52.0

Coefficient of expansion, $α$ = 1.2E−5

#### General:

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

mgrono2xmx.inp

Inhomogeneous deformation, displacement control, CPE4 elements.

mgrono2xmx1.inp

Same as mgrono2xmx.inp except that the initial relative density is specified using the INITIAL CONDITIONS, TYPE = RELATIVE DENSITY option.

mgrono3hut.inp

Uniaxial tension, traction control, nucleation of voids, C3D8 elements.

mgrono3jht.inp

Hydrostatic tension, displacement control, nucleation of voids, C3D8 elements.

mgrooo2bus.inp

Uniaxial strain (confined compression), traction control, CAX4 elements.

mgrooo2euc.inp

Uniaxial compression, traction control, CAX4 elements.

mgrooo2gsh.inp

Shear, CPE4 elements.

mgrooo2hut.inp

Uniaxial tension, displacement control, CAX4 elements.

mgrooo2jht.inp

Hydrostatic tension, displacement control, CAX4 elements.

mgrooo3gsh.inp

Shear, C3D8 elements.

mgrooo3jht.inp

Hydrostatic tension, displacement control, C3D8 elements.

mgrqno2hut.inp

Modified Gurson's model, uniaxial tension, displacement control, nucleation of voids, CAX4 elements.

mgrqoo2ahc.inp

Modified Gurson's model, hydrostatic compression, displacement control, CAX4 elements.

mgtooo2hut.inp

Uniaxial tension, coupled temperature-displacement, CAX4T elements.

mgroob2hut.inp

Uniaxial tension, displacement control, CAX4 elements, temperature and field variable dependencies.

mgrqnt2hut.inp

Modified Gurson's model, uniaxial tension, nucleation of voids, temperature dependencies.

## Mohr-Coulomb plasticity

C3D8

C3D8R

CAX4

CAX4R

CPE4

CPE4R

### Problem description

#### Material 1

Elasticity

Young's modulus, E = 300.E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Angle of friction, $ϕ$ = 40°

Dilation angle, $ψ$ = 40°

Cohesion hardening curve:

Yield stress Plastic strain
6.0E3 0.000000
9.0E3 0.020000
11.0E3 0.063333
12.0E3 0.110000
12.0E3 1.000000
Tension cutoff

Perfectly plastic, yield stress = 600.0

(The units are not important.)

#### Material 2

Elasticity

Young's modulus, E = 300.E3

Poisson's ratio, $ν$ = 0.3

Plasticity

Angle of friction, $ϕ$ = 30°

Dilation angle, $ψ$ = 20°

Cohesion hardening curve:

Yield stress Plastic strain
866.025 0.0
1732.05 1.0
Tension cutoff

Softening response:

Yield stress Plastic strain
1000.0 0.0
100.0 1.0

(The units are not important.)

#### Material 3

Elasticity

Young's modulus, E = 2 .E7

Poisson's ratio, $ν$ = 0.3

Plasticity

Angle of friction, $ϕ$ = 30°

Dilation angle, $ψ$ = 20°

Perfectly plastic cohesion:

Yield stress Plastic strain
1000.0 0.0
1000.0 1.0
Tension cutoff

Perfectly plastic:

Yield stress Plastic strain
1000.0 0.0
1000.0 1.0

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

#### Abaqus/Standard input files

Material 1:
mmoooo3jht.inp

Hydrostatic tension, C3D8 elements.

mmoooo3ltr.inp

Triaxial stress, CPE4 elements (inhomogeneous).

mmoooo3dte.inp

Triaxial extension, CAX4 elements.

mmoooo3hut.inp

Uniaxial tension, C3D8 elements.

mmoooo3gsh.inp

Shear, C3D8 elements.

mmoooo3ctc.inp

Triaxial compression, CAX4 elements.

mmoooo3euc.inp

Uniaxial compression, C3D8 elements.

mmooot3euc.inp

Uniaxial compression with temperature dependence, C3D8 elements.

mmoooo3vlp.inp

Linear perturbation steps containing LOAD CASE, uniaxial compression, C3D8 elements.

mctc_trxs.inp

Triaxial extension with tension cutoff, CAX4 elements.

Material 2:
mctc_ucut.inp

Tension cutoff, uniaxial compression followed by uniaxial tension, C3D8R and CAX4R elements.

mctc_psss.inp

Tension cutoff, plane strain compression/tension and simple shear, CPE4R elements.

Material 3:
mctc_btbc.inp

Tension cutoff, biaxial tension followed by biaxial compression, C3D8R element.

mctc_ptpc.inp

Tension cutoff, hydrostatic tension followed by hydrostatic compression, C3D8R element.

#### Abaqus/Explicit input files

Material 1:
mmoooo3jht_xpl.inp

Hydrostatic tension, C3D8 elements.

mmoooo3ltr_xpl.inp

Triaxial stress, CPE4R elements (inhomogeneous).

mmoooo3dte_xpl.inp

Triaxial extension, CAX4R elements.

mmoooo3hut_xpl.inp

Uniaxial tension, C3D8 elements.

mmoooo3gsh_xpl.inp

Shear, C3D8 elements.

mmoooo3ctc_xpl.inp

Triaxial compression, CAX4R elements.

mmoooo3euc_xpl.inp

Uniaxial compression, C3D8 elements.

mmooot3euc_xpl.inp

Uniaxial compression with temperature dependence, C3D8 elements.

mctc_trxs_xpl.inp

Triaxial extension with tension cutoff, CAX4R elements.

Material 2:
mctc_ucut_xpl.inp

Tension cutoff, uniaxial compression followed by uniaxial tension, C3D8R and CAX4R elements.

mctc_psss_xpl.inp

Tension cutoff, plane strain compression/tension and simple shear, CPE4R elements.

#### Transferring results from Abaqus/Standard to Abaqus/Explicit

Material 1:
mmoooo3jht_sx_s.inp

Abaqus/Standard analysis, hydrostatic tension, C3D8 elements.

mmoooo3jht_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mmoooo3ltr_sx_s.inp

Abaqus/Standard analysis, triaxial stress, CPE4R elements.

mmoooo3ltr_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mmoooo3dte_sx_s.inp

Abaqus/Standard analysis, triaxial extension, CAX4R elements.

mmoooo3dte_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mmoooo3hut_sx_s.inp

Abaqus/Standard analysis, uniaxial tension, C3D8 elements.

mmoooo3hut_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mmoooo3gsh_sx_s.inp

Abaqus/Standard analysis, shear, C3D8 elements.

mmoooo3gsh_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mmoooo3ctc_sx_s.inp

Abaqus/Standard analysis, triaxial compression, CAX4R elements.

mmoooo3ctc_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

mmoooo3euc_sx_s.inp

Abaqus/Standard analysis, uniaxial compression, C3D8 elements.

mmoooo3euc_sx_x.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

Material 3:
sx_s_mctc.inp

Abaqus/Standard analysis, uniaxial tension followed by compression, C3D8R element.

sx_x_mctc_n_y.inp

Abaqus/Explicit import analysis, UPDATE=NO, STATE=YES.

#### Transferring results from Abaqus/Explicit to Abaqus/Standard

Material 1:
mmoooo3dte_xs_x.inp

Abaqus/Explicit analysis, triaxial extension, CAX4R elements.

mmoooo3dte_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mmoooo3gsh_xs_x.inp

Abaqus/Explicit analysis, shear, C3D8 elements.

mmoooo3gsh_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mmoooo3ctc_xs_x.inp

Abaqus/Explicit analysis, triaxial compression, CAX4R elements.

mmoooo3ctc_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mmoooo3ltr_xs_x.inp

Abaqus/Explicit analysis, triaxial stress, CPE4R elements.

mmoooo3ltr_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mmoooo3jht_xs_x.inp

Abaqus/Explicit analysis, hydrostatic tension, C3D8 elements.

mmoooo3jht_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

mmoooo3hut_xs_x.inp

Abaqus/Explicit analysis, uniaxial tension, C3D8 elements.

mmoooo3hut_xs_s.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

Material 3:
xs_x_mctc.inp

Abaqus/Explicit analysis, uniaxial tension, C3D8R element.

xs_s_mctc_n_y.inp

Abaqus/Standard import analysis, UPDATE=NO, STATE=YES.

## Cast iron plasticity

C3D8

CAX4

CAX4T

CPE4

T3D2

### Problem description

#### Material:

Elasticity

Young's modulus, E = 14.773E6

Poisson's ratio, $ν$ = 0.2273

Plasticity

Plastic “Poisson's ratio,” $νp⁢l$ = 0.039

Hardening curves: The hardening curves in tension and compression are illustrated in Figure 1.

Thermal properties

Specific heat, $cp$ = 47.52

Density, $ρ$ = 439.92

Conductivity, k = 9.4

Coefficient of expansion, $α$ = 11.0E−6

Figure 1. Stress versus plastic strain under uniaxial tension and uniaxial compression.

(The units are not important.)

### Results and discussion

Most tests in this section are set up as cases of the homogeneous deformation of a single element of unit dimensions. Consequently, the results are identical for all integration points within the element.

### Input files

#### Abaqus/Standard input files

mciooo3jht.inp

Hydrostatic tension, C3D8 elements.

mciooo3gsh.inp

Shear, C3D8 elements.

mciooo3hut.inp

Uniaxial tension, CAX4 elements.

mcioot3euc.inp

Uniaxial compression with temperature dependence, C3D8 elements.

mctoot3hut.inp

Uniaxial tension, coupled temperature-displacement, CAX4T elements.

mciooo3xmx.inp

Inhomogeneous deformation, CPE4 elements.

mciooo1hut.inp

Uniaxial tension and linear perturbation steps containing LOAD CASE, T3D2 elements.

#### Abaqus/Explicit input files

mciooo3jht_xpl.inp

Hydrostatic tension, C3D8 elements.

mciooo3gsh_xpl.inp

Shear, C3D8 elements.

mciooo3hut_xpl.inp

Uniaxial tension, CAX4R elements.

mcioot3euc_xpl.inp

Uniaxial compression with temperature dependence, C3D8 elements.

mciooo3xmx_xpl.inp

Inhomogeneous deformation, CPE4 elements.

mciooo1hut_xpl.inp

Uniaxial tension, T3D2 elements.

#### Transferring results from Abaqus/Standard to Abaqus/Explicit

mciooo3gsh_sx_s.inp

Abaqus/Standard analysis, shear, C3D8 elements.

mciooo3gsh_sx_x.inp

Abaqus/Explicit import analysis from mciooo3gsh_sx_s.inp.

mciooo3gsh_xs_s.inp

Abaqus/Standard import analysis from mciooo3gsh_sx_x.inp.

mciooo3gsh_xx_x2.inp

Abaqus/Explicit import analysis from mciooo3gsh_sx_x.inp.

## Soft rock plasticity with porous elasticity

C3D8R, CPE4R

### Problem description

#### Material 1

Power law–based porous elasticity
 $Eref$ = 75000.0 $pref$ = 1.0 $p0$ = 1.0 $n$ = 0. $ν∞$ = 0.2 $ν0$ = 0.2 $m$ = 1.0
Plasticity
 $β$ = 60.0 $ψ$ = 50.0 $ny$ = 1.6 $f0$ = 0.001 $f1$ = 0.0007 $α$ = 0.25 $εvolp⁢l|0$ = 0.05

Tabulated curves are used for defining the compressive and tensile hardening.

(The units are not important.)

#### Material 2

Power law–based porous elasticity
 $Eref$ = 75000.0 $pref$ = 1.0 $p0$ = 1.0 $n$ = 0. $ν∞$ = 0.2 $ν0$ = 0.2 $m$ = 1.0
Plasticity
 $β$ = 45.0 $ψ$ = 45.0 $ny$ = 1. $f0$ = 0.001 $f1$ = 0.0007 $α$ = 0.0 $εvolp⁢l|0$ = 0.0

Tabulated curves are used for defining the compressive and tensile hardening.

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

softrockporelatests.inp

Hydrostatic compression, uniaxial compression, and shear using materials 1 and 2.

## Soft rock plasticity with linear elasticity

C3D8R, CPE4R

### Problem description

#### Material 1

Linear elasticity

Young's modulus, E = 75.E3

Poisson's ratio, $ν$ = 0.2

Plasticity
 $β$ = 60.0 $ψ$ = 50.0 $ny$ = 1.6 $f0$ = 0.001 $f1$ = 0.0007 $α$ = 0.25 $εvolp⁢l|0$ = 0.05

Tabulated curves are used for defining the compressive and tensile hardening.

(The units are not important.)

#### Material 2

Linear elasticity

Young's modulus, E = 75.E3

Poisson's ratio, $ν$ = 0.2

Plasticity
 $β$ = 45.0 $ψ$ = 45.0 $ny$ = 1. $f0$ = 0.001 $f1$ = 0.0007 $α$ = 0.0 $εvolp⁢l|0$ = 0.0

Tabulated curves are used for defining the compressive and tensile hardening.

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

#### Abaqus/Standard input files

softrocklineelatests.inp

Hydrostatic compression, uniaxial compression, and shear using materials 1 and 2.

#### Abaqus/Explicit input files

softrocklineelatests_xpl.inp

Hydrostatic compression, uniaxial compression, and shear using materials 1 and 2.

## Plasticity model for superelastic materials

### Elements tested

C3D8, CPS4, CPS4R

### Problem description

#### Material

Elasticity
 $EA$ = 40000.0 $νA$ = 0.33 $EM$ = 32000.0 $νM$ = 0.33 $εL$ = 0.041 $σtLS$ = 440.0 $σtLE$ = 540.0 $σtUS$ = 250.0 $σtUE$ = 140.0 $σcLS$ = 440.0 $T0$ = 22.0 $(δσδT)L$ = 6.7 $(δσδT)U$ = 6.7
Plasticity

Tabulated curves are used for defining the yield stress as a function of total strain.

(The units are not important.)

### Results and discussion

The results agree well with exact analytical or approximate solutions.

### Input files

#### Abaqus/Standard input files

superelas_std.inp

Tension test.

#### Abaqus/Explicit input files

superelas_xpl.inp

Tension test.