Lemaitre–Chaboche Viscoplastic Model

Most metals and alloys undergo viscoplastic deformation at high temperatures. In case of cyclic loading, a constitutive law with both isotropic and kinematic hardening is necessary to describe effects such as ratcheting, cyclic softening/hardening, and stress relaxation. The Lemaitre–Chaboche viscoplastic model combines isotropic hardening with nonlinear kinematic hardening to model these effects.

This tutorial model demonstrates the uniaxial deformation of an indium IN 100 test specimen submitted to cyclic tension-compression loading at high temperature, as described in Ref. 1

Figure 1: Specimen used for the uniaxial tension-compression tests.

Model Definition

The test specimen consists of a small cylinder with a thin central section to ensure uniform stress and strain distributions. Thick parts are added to the ends of the specimen to ease the mounting in a universal testing machine. The loading for the tension-compression cycle is directed in the axial direction. For shortening the computation time, half of the specimen is modeled in a 2D axisymmetric geometry.

Governing equations

The strain tensor consists of the sum of the elastic strain tensor εel and the viscoelastic strain tensor εvp:

The constitutive law between stress and strain is given by Hooke’s law

The viscoplastic strain tensor is defined by Lemaitre–Chaboche viscoplastic rule

where σb is the back-stress tensor, derived by the nonlinear kinematic hardening rule; and σ' and σb' are the deviatoric parts of the stress and back-stress tensors σ and σb.

The equivalent viscoplastic strain rate is defined by the expression

where A is the viscoplastic rate coefficient, σref is a reference stress, n is the stress exponent, and σy is the yield stress given by the nonlinear isotropic hardening rule.

Mixed Hardening

The isotropic hardening model represents the change in yield stress as a function of the equivalent viscoplastic strain. Lemaitre and Chaboche (Ref. 1) derived a nonlinear isotropic hardening relation of the type

where σys0 is the initial yield stress, and σsat and β are material parameters.

The kinematic hardening rule represents the translation of the yield surface as a function of the back-stress σb and the equivalent viscoplastic strain εvp. Lemaitre and Chaboche (Ref. 1) derived a nonlinear kinematic hardening relation where the back-stress tensor is calculated from the ordinary differential equation

where Ck and γk are material parameters. The kinematic hardening parameter γk also depends exponentially on the equivalent viscoplastic strain:

where γs, γ0, and βk are material parameters.

Material data

The material parameters of the indium alloy IN 100 at a temperature of 900 K are given in Ref. 1.

Table 1: Material parameters, IN 100 at 900 K.
Parameter	Value
K	490 MPa.s1/n
n	9
σy0	60 MPa
σy0+σsat	25 MPa
b	200
Ck	362,500 MPa
γ0	1200
γs	1540
βk	1000

In Ref. 1, the parameter K is given in units of MPa·s1/n. The parameters K, A, and σref are related by the expression

Use the values A = 1/s and σref = 490 MPa to obtain similar results as in given in Ref. 1.

Results and Discussion

Stress vs. Time

After the first tensile cycle the axial stress (blue line in Figure 2) increases linearly, exceeding the yield strength (cyan line). Because of viscous effects, the onset of plasticity starts once the viscous stress (green line) reaches the yield strength. At this point, the relation between stress and strain is no longer linear, kinematic/isotropic hardening evolves, and the back-stress (red line) increases.

When the maximum allowed tensile strain is reached, the prescribed axial velocity turns negative to prescribe axial unloading. The axial stress then decreases linearly, and the back stress remains constant to account for kinematic hardening and the Bauschinger effect. The yield strength in compression after a tensile cycle is lower than the initial yield strength.

In the compressive cycle, the viscoplastic flow starts earlier and lasts for a longer period. When the maximum allowed compressive strain is reached, the prescribed axial velocity becomes positive to prescribe tensile unloading. This procedure is repeated for each load cycle.

Figure 2: Evolution of the axial stress variables with time.

Four studies are performed to prescribe four different types of loading cycles: prescribed symmetric strain, prescribed symmetric stress, prescribed nonsymmetric strain, and prescribed nonsymmetric stress. The stress-strain graph for each type of loading cycle highlights material properties of the Lemaitre–Chaboche viscoplastic model.

symmetric Cycles

Symmetric loading aims to show the stabilization effect of the hysteresis cycle. This emphasizes the softening or hardening effect during cyclic loadings for a given hardening rule.

For the symmetric prescribed axial strain cycle shown in Figure 3, there is a periodic response to the periodic load and a stabilized state is reached after few cycles. A closer view shows that the stress amplitude decreases during the first cycles due to material softening.

A similar effect can be seen in for the symmetric prescribed stress cycle in Figure 4, but more cycles are needed to reach a stabilized state when compared to the prescribed strain loading. Lastly, the axial strain amplitude increases with the number of cycles. This is an effect of the isotropic hardening rule. With kinematic hardening only, the stabilized state is reached after the first cycle of prescribed stress. Thus, a mixed hardening formulation is needed to model both cyclic softening/hardening and Bauschinger effects.

Figure 3: Axial stress versus axial strain for a symmetric prescribed strain loading.

Figure 4: Axial stress versus axial strain for a symmetric prescribed stress loading.

Nonsymmetric Strain Cycles

When the average strain is not zero in a prescribed strain cycle as in Figure 5, the initial asymmetry in the axial stress gradually disappears over the cycles: This happens because of the stress relaxation effect, which is observed in many alloys.

Once a stabilized cycle is reached, the tensile and compressive stresses are equal in absolute value. This is an effect of the nonlinear kinematic hardening rule. Applying linear kinematic hardening only (or isotropic hardening) would not allow to observe the relaxation of the mean stress, so a nonlinear kinematic hardening rule is needed to model stress relaxation.

Figure 5: Axial stress versus axial strain for a nonsymmetric prescribed strain loading.

Nonsymmetric Stress Cycles

The gradual deformation that occurs in tension-compression cycles when the mean stress is nonzero is called the ratcheting effect. The ratcheting effect is most pronounced when the lower stress limit is −σys0. This behavior is possible to observe thanks to the nonlinear kinematic hardening rule.

Figure 6: Axial stress versus axial strain for a nonsymmetric prescribed stress loading.

Notes About the COMSOL Implementation

Several yield functions and plastic potentials are available in the node. Use von Mises equivalent stress to reproduce the results described in Ref. 1.

Solve four studies to account for the different types of load cycles: the load cycle can be symmetric or nonsymmetric, and the control can be done by prescribing either the axial strain or the axial stress.

Apply a constant axial velocity on one end of the test specimen to avoid using complicate loading functions. Multiply this axial velocity by 1 to represent axial tension, or by −1 to represent axial compression.

To achieve this, define a discrete state called LoadingType in an interface. Add in the Events interface to define tension and compression limits based on the stress or strain state. For instance, for a symmetric loading cycle with a prescribed strain of 0.4 % the following are used:


Tension	intop1(solid.el33)-0.004
Compression	intop1(solid.el33)+0.004

Then add two nodes to define the discrete state LoadingType: when Tension > 0, then LoadingType is set to −1, and when Compression < 0, LoadingType is set to 1.

Reference

1. J. Lemaitre and J.-L. Chaboche, Mécanique des matériaux solides, 2nd ed., Dunod, 2001 (in French).

Nonlinear_Structural_Materials_Module/Viscoplasticity/lemaitre_chaboche_viscoplastic_model

Modeling Instructions

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Global Definitions

Parameters 1

Define the parameters that will be needed in the model.

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In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
e0t	1e-3[1/s]	0.001 1/s	Prescribed strain rate
L0	(35/2+14+10)[mm]	0.0415 m	Half length
gamma0	1200	1200	Initial kinematic hardening parameter
gammas	1540	1540	Saturation kinematic hardening parameter
beta	1000	1000	Kinematic hardening parameter exponent

Add a step function to apply the load smoothly.

Step 1 (step1)

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In the text field, type 5[ms].

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Geometry 1

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Rectangle 1 (r1)

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Rectangle 2 (r2)

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Rectangle 3 (r3)

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Circular Arc 1 (ca1)

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Convert to Solid 1 (csol1)

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Select the objects and only.

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Add conditions to toggle the boundary conditions between tension and compression. This is controlled by the interface. First create a nonlocal integration coupling to get variable from a point.

Definitions

Integration 1 (intop1)

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From the list, choose .

Select Point 6 only.

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Events (ev)

Discrete States 1

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In the window for , locate the section.

In the table, enter the following settings:


Name	Initial value (u0)	Description
LoadingType	1

The LoadingType variable is 1 when tension is applied, and -1 when compression is applied.

The first indicator state is used to control the loading with a strain measure. The tension and compression limits are symmetric.

Indicator States: Strain, Symmetric

In the toolbar, click

and choose .

In the window for , type Indicator States: Strain, Symmetric in the text field.

Locate the section. In the table, enter the following settings:


Name	g(v,vt,vtt,t)
Tension	intop1(solid.el33)-0.004
Compression	intop1(solid.el33)+0.004

The second indicator state is used to control the loading with a stress measure. The tension and compression limits are symmetric.

Indicator States: Stress, Symmetric

In the toolbar, click

and choose .

In the window for , type Indicator States: Stress, Symmetric in the text field.

Locate the section. In the table, enter the following settings:


Name	g(v,vt,vtt,t)
Tension	intop1(solid.sl33)-500[MPa]
Compression	intop1(solid.sl33)+500[MPa]

The third indicator state is used to control the loading with a strain measure. The tension and compression limits are not symmetric.

Indicator States: Strain, Nonsymmetric

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and choose .

In the window for , type Indicator States: Strain, Nonsymmetric in the text field.

Locate the section. In the table, enter the following settings:


Name	g(v,vt,vtt,t)
Tension	intop1(solid.el33)-0.006
Compression	intop1(solid.el33)+0.002

The fourth indicator state is used to control the loading with a stress measure. The tension and compression limits are not symmetric.

Indicator States: Stress, Nonsymmetric

In the toolbar, click

and choose .

In the window for , type Indicator States: Stress, Nonsymmetric in the text field.

Locate the section. In the table, enter the following settings:


Name	g(v,vt,vtt,t)
Tension	intop1(solid.sl33)-500[MPa]
Compression	intop1(solid.sl33)+100[MPa]

Implicit Event 1

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In the window for , locate the section.

In the text field, type Tension>0.

Locate the section. In the table, enter the following settings:


Variable	Expression
LoadingType	-1

Implicit Event 2

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type Compression<0.

Locate the section. In the table, enter the following settings:


Variable	Expression
LoadingType	1

Solid Mechanics (solid)

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From the list, choose .

Symmetry Plane 1

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Select Boundary 2 only.

Prescribed Velocity 1

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Select Boundary 9 only.

In the window for , locate the section.

Select the check box.

In the vz text field, type e0t*L0*step1(t)*LoadingType.

Linear Elastic Material 1

Add viscoplasticity with combined isotropic and kinematic hardening.

In the window, click .

Viscoplasticity 1

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Select Domains 1–3 only.

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From the list, choose .

Find the subsection. From the list, choose .

Set the material properties.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	200[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.3	1	Young’s modulus and Poisson’s ratio
Density	rho	7500	kg/m³	Basic
Viscoplastic rate coefficient	A_cha	1	1/s	Chaboche viscoplasticity
Reference stress	sigRef_cha	490[MPa]	N/m²	Chaboche viscoplasticity
Stress exponent	n_cha	9	1	Chaboche viscoplasticity
Initial yield stress	sigmags	60[MPa]	Pa	Elastoplastic material model
Saturation flow stress	sigma_voc	-35[MPa]	Pa	Voce
Saturation exponent	beta_voc	200	1	Voce
Kinematic hardening modulus	Ck	362.5[GPa]	Pa	Armstrong-Frederick

The kinematic hardening parameter gammak is nonlinear and a function of the equivalent viscoplastic strain. Add the latter from the model input list to make it available.