COMSOL 6.4 - Hysteresis in Soil Using the Small-Strain Overlay Model

Hysteresis in Soil Using the Small-Strain Overlay Model

The majority of soil types have a high stiffness at small strains in the elastic regime while it decreases monotonically with increasing strain. Soils also displays a hysteresis effect when subjected to cyclic loads. The Small-Strain Overlay model presented in Ref. 1 captures the effect of high stiffness at low strain as well as the hysteric behavior under cyclic loading. The formulation allows stiffness degradation with an increase in shear strain, and the full recovery of stiffness at load reversal.

In this example, cyclic tensile and shear tests show the stiffness degradation and the hysteresis effect with the small-strain overlay model. The cyclic tensile test mimic the cyclic triaxial loading without any isotropic compression step.

Model Definition

A rectangular soil specimen of 5 cm in width and 10 cm in height is used for both tests. The specimen is represented by a 2D geometry.

Soil Properties

Properties for dense sand are presented in Table 1 as taken from Ref. 1.

Table 1: Material properties for The soil model.
Property	Variable	Value
Poisson’s ratio	ν	0.2
Density	ρ	2400 kg/m3
Initial shear modulus	G0	185 MPa
Reference shear strain	γs	5.1948E-4

Constraints and Loads

•

For the tensile test, the vertical left and bottom boundaries are constrained in the normal direction. On the top boundary, a prescribed displacement is applied in the normal direction.

•

For the shear test, the vertical left and bottom boundaries are constrained in the tangential direction. On the vertical right and top boundary, a prescribed displacement is applied in the tangential direction.

Results and Discussion

Note that for consistency with the geomechanics sign convention, compressive stress and strain are plotted along the positive axis in all figures, while tensile stress and strain are plotted along the negative axis.

For the cyclic triaxial test, the axial stress versus axial strain is shown in Figure 1, while the variation of Young’s modulus versus axial strain is shown in Figure 2. The small-strain overlay model captures two important aspects of the behavior of soils:

•

Monotonic degradation of stiffness with increasing strain.

•

Hysteresis in the cyclic loading.

From Figure 1 it is clear that the model satisfies two basic rules of Masing:

•

The shear modulus at the start of unloading is equal to the initial modulus of the primary loading curve.

•

The shape of the unloading and reloading curve should be the same as the initial loading curve except the scale must be enlarged by the factor of two.

Note that a log scale is used for the axial strain in the Figure 2, which is why the first parametric steps corresponding to zero strain are excluded. The stiffness at the beginning of the unloading curve is the same as the initial stiffness at the beginning of the primary loading.

The hysteresis effect in the cyclic shear test is shown in Figure 3. Figure 4 shows a reduction in the shear stiffness with increasing shear strain. From both figures it can be observed that the stiffness at the start of the unloading cycle is regained and that both Massing rules are satisfied.

Figure 1: Axial stress versus axial strain for the cyclic tensile test.

Figure 2: Young’s modulus versus axial strain for the cyclic tensile test.

Figure 3: Shear stress versus shear strain in the cyclic shear test.

Figure 4: Shear modulus versus shear strain in the cyclic shear test.

Notes About the COMSOL Implementation

For cyclic loading, the load reversal points in the small-strain overlay model are found automatically based on the increments of the deviatoric strain tensor. However, in some load cases, the automatic algorithm is not effective. In such scenarios the user can set to . With this option, a set of load reversal points can be specified in terms of the parameter for a parametric study or the time variable for a transient study.

Reference

1. T. Benz, P. A. Vermeer, and R. Schwab, “A small-strain overlay model,” Int. J. Numer. Anal. Methods Geomech., vol. 33, pp. 25–44, 2009.

Geomechanics_Module/Verification_Examples/hysteresis_small_strain_overlay

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
para	0	0	Parameter

Definitions

Interpolation 1 (int1)

In the toolbar, click

In the window for , locate the section.

In the text field, type appliedDisp.

In the table, enter the following settings:


t	f(t)
0	0
1	5e-3
2	0
3	-5e-3
4	0
5	5e-3

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


Function	Unit
appliedDisp	cm

In the table, enter the following settings:


Argument	Unit
t	1

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 5[cm].

In the text field, type 10[cm].

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Solid Mechanics (solid)

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From the list, choose .

Nonlinear Elastic Material 1

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and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

Roller 1

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and choose .

Select Boundaries 1 and 2 only.

Prescribed Displacement 1

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and choose .

Select Boundary 3 only.

In the window for , locate the section.

From the list, choose .

In the u 0 y text field, type -appliedDisp(para).

Prescribed Displacement 2

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and choose .

Select Boundary 4 only.

In the window for , locate the section.

From the list, choose .

In the u 0 y text field, type appliedDisp(para).

Prescribed Displacement 3

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In the window for , locate the section.

Click

Select Boundary 1 only.

Locate the section. In the u 0 y text field, type 0.

Prescribed Displacement 4

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and choose .

Select Boundary 3 only.

In the window for , locate the section.

From the list, choose .

In the u 0 x text field, type appliedDisp(para).

Prescribed Displacement 5

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Click

Select Boundary 2 only.

Locate the section. In the u 0 x text field, type 0.

Prescribed Displacement 1, Roller 1

In the window, under > , Ctrl-click to select and .

Right-click and choose .

Cyclic Triaxial Loading

In the window for , type Cyclic Triaxial Loading in the text field.

Prescribed Displacement 2, Prescribed Displacement 3, Prescribed Displacement 4, Prescribed Displacement 5

In the window, under > , Ctrl-click to select , , , and .

Right-click and choose .

Cyclic Shear Loading

In the window for , type Cyclic Shear Loading in the text field.

Materials

Soil Material

In the window, under right-click and choose .

In the window for , type Soil Material in the text field.

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


Property	Variable	Value	Unit	Property group
Reference shear strain	gammaRef	5.1948E-4	1	Nonlinear elastic material
Poisson’s ratio	nu	0.2	1	Young’s modulus and Poisson’s ratio
Initial shear modulus	G0	185[MPa]	N/m²	Nonlinear elastic material
Density	rho	2400[kg/m^3]	kg/m³	Basic