Rate-dependent plasticity in Abaqus/Standard

Rate-dependent Mises plasticity

Elements tested

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

Rate dependence parameter, D=40.0

Rate dependence parameter, p=5.0

The rate dependence parameters are as follows for the test that verifies the temperature dependencies:

D=30.0, p=3.0 at 10.0°

D=50.0, p=7.0 at 20.0°

The power law is entered as a piecewise linear relationship for the cases in which rate-dependent test data are specified directly.

(The units are not important.)

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

mprooo3hut.inp: Uniaxial tension, power law, C3D8 elements.
mpryso3hut.inp: Uniaxial tension, yield ratios, C3D8 elements.
mproot3hut.inp: Uniaxial tension, temperature-dependent power law, C3D8 elements.
mpryst3hut.inp: Uniaxial tension, temperature-dependent yield ratios, C3D8 elements.
mprooo3vlp.inp: Linear perturbation uniaxial tension, power law, C3D8 elements.
mprpro3vlp.inp: Linear perturbation uniaxial tension, PLASTIC, RATE=option, C3D8 elements.

Adiabatic rate-dependent Mises plasticity

Elements tested

C3D8

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

Other properties

Density, $μ$ =1000.0

Specific heat, c=0.4

Inelastic heat fraction = 0.5

Rate dependence parameter, D=40.0

Rate dependence parameter, p=5.0

The power law is entered as a piecewise linear relationship for the cases in which rate-dependent test data are specified directly.

(The units are not important.)

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

mhriho3hut.inp: Uniaxial tension, power law, C3D8 elements.
mhrpro3hut.inp: Uniaxial tension, PLASTIC, RATE=option, C3D8 elements.
mhriho1hut.inp: Uniaxial tension, power law, T3D2 elements.
mhryso1hut.inp: Uniaxial tension, yield ratios, T3D2 elements.
mhriho3xmx.inp: Multiaxial, power law, C3D8 elements.
mhrpro3xmx.inp: Multiaxial, PLASTIC, RATE=option, C3D8 elements.

Rate-dependent Hill plasticity

Elements tested

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

Rate dependence parameter, D=40.0

Rate dependence parameter, p=5.0

The power law is entered as a piecewise linear relationship for the cases in which rate-dependent test data are specified directly.

(The units are not important.)

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

mpxooo3nt1.inp: Uniaxial tension in direction 1, power law, C3D8 elements.
mpxyso3nt1.inp: Uniaxial tension in direction 1, yield ratios, linear perturbation with LOAD CASE, C3D8 elements.
mpxooo3ot2.inp: Uniaxial tension in direction 2, power law, C3D8 elements.
mpxooo3pt3.inp: Uniaxial tension in direction 3, power law, C3D8 elements.
mpxpro3pt3.inp: Uniaxial tension in direction 3, PLASTIC, RATE=option, C3D8 elements.

Rate-dependent Drucker-Prager plasticity

Elements tested

C3D8

CPS4

Problem description

Material:

Elasticity

Young's modulus, E=300.0E3

Poisson's ratio, $ν$ =0.3

Plasticity

The linear Drucker-Prager model is used in each case.

Angle of friction, $β$ =40.0

Dilation angle, $ψ$ =40.0

Rate dependence parameter, D=10.0

Rate dependence parameter, p=1.0

For the test that verifies the temperature dependencies, the rate dependence parameters are as follows:

D=9.0, p=0.9 at 10.0°

D=11.0, p=1.1 at 20.0°

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 power law is entered as a piecewise linear relationship for the cases in which rate-dependent test data are specified directly.

(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. The constitutive path is integrated with 20 increments of fixed size.

Input files

mdrooo3euc.inp: Uniaxial compression, power law, C3D8 elements.
mdryso3euc.inp: Uniaxial compression, yield ratios, C3D8 elements.
mdroot3euc.inp: Uniaxial compression, temperature-dependent power law, C3D8 elements.
mdryst3euc.inp: Uniaxial compression, temperature-dependent yield ratios, C3D8 elements.
mdrooo2euc.inp: Uniaxial compression, power law, CPS4 elements.
mdryro2euc.inp: Uniaxial compression, linear perturbation with LOAD CASE, DRUCKER PRAGER HARDENING, RATE=option, CPS4 elements.
mdrooo3vlp.inp: Linear perturbation uniaxial compression, power law, C3D8 elements.
mdryso3vlp.inp: Linear perturbation uniaxial compression, yield ratios, C3D8 elements.

Rate-dependent crushable foam plasticity

Elements tested

C3D8

Problem description

Material:

Elasticity

Young's modulus, E=3.0E6

Poisson's ratio, $ν$ =0.2

Plasticity

Initial yield stress in hydrostatic compression, $p_{0}$ =2.0E5

Strength in hydrostatic tension, $p_{t}$ =2.0E4

Initial yield stress in uniaxial compression, $σ_{0}$ =2.2E5

Yield stress ratio, $k = σ_{0} / p_{0}$ =1.1

Yield stress ratio, $k_{t} = p_{t} / p_{0}$ =0.1

Rate dependence parameter, D=10.0

Rate dependence parameter, p=1.0

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

For the test that verifies the temperature dependencies, the rate dependence parameters are as follows:

D=9.0, p=0.9 at 10.0°

D=11.0, p=1.1 at 20.0°

The power law is entered as a piecewise linear relationship for the cases in which rate-dependent test data are specified directly.

(The units are not important.)

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

mfrooo3euc.inp: Uniaxial compression, power law, C3D8 elements.
mfryso3euc.inp: Uniaxial compression, yield ratios, C3D8 elements.
mfroot3euc.inp: Uniaxial compression, linear perturbation with LOAD CASE, temperature-dependent power law, C3D8 elements.
mfryst3euc.inp: Uniaxial compression, temperature-dependent yield ratios, elements.

Rate-dependent porous metal plasticity

Elements tested

C3D8

Problem description

Material:

Elasticity

Young's modulus, E=200.0E3

Poisson's ratio, $ν$ =0.3

Plasticity

Hardening curve:

Yield stress	Plastic strain
200.0	0.0000
220.0	0.0009
220.0	0.0029

Rate dependence parameter, D=40.0

Rate dependence parameter, p=5.0

Porous metal plasticity

$q_{1}$ = $q_{2}$ = $q_{3}$ =1.0

Initial relative density, $r_{0}$ =0.95 ( $f_{0}$ =0.05).

(The units are not important.)

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

mgrooo3vlp.inp: Uniaxial tension, power law, C3D8 elements.
mgrpro3vlp.inp: Uniaxial tension; PLASTIC, RATE=option; C3D8 elements; linear perturbation with LOAD CASE.
mgryso3hut.inp: Hydrostatic tension, yield ratios, C3D8 elements.