COMSOL 6.4 - Stress Corrosion

Steel pipelines are often subjected to complex stress/strain conditions in oil and gas industry. In addition to stress from internal pressure, pipelines are subjected to significant longitudinal strain due to surrounding soil movement. As a result of the changed surface energies at the pipe surface, the resulting stresses may have an effect on the pipe corrosion rate. The intercoupled effects of elastic and plastic deformation on pipeline corrosion is demonstrated in this model example.

The elasto-plastic stress simulations are performed here using a small strain plasticity model and von-Mises yielding criterion. Iron dissolution (anodic) and hydrogen evolution (cathodic) are considered as electrochemical reactions, using kinetic expressions that account for the effect of elasto-plastic deformations.

The example is based on a paper by L. Y. Xu and Y. F. Cheng (Ref. 1). The model is built in two parts. In the first part, the stationary effect of the elasto-plastic stress on corrosion is investigated by varying the longitudinal strain using a parametric study. In the second part, a transient simulation is performed to investigate the intercoupled effects of a dissolving, and hence dynamically varying, corrosion defect depth on a pipe, subject to a constant strain.

This model requires the Structural Mechanics, or the MEMS Module, with the addition of either the Nonlinear Structural Materials or the Geomechanics Module.

Model Definition

The model geometry is shown in Figure 1.

Figure 1: The model geometry consists of a pipeline with corrosion defect and surrounding soil domain.

The model geometry consists of high strength alloy steel pipeline and surrounding soil domain. The pipeline length is 2 m and wall thickness is 19.1 mm. The corrosion defect on the exterior side of the pipeline is elliptical in shape with a length of 200 mm and a depth of 11.46 mm. The electrolyte conductivity of soil domain in 0.096 S/m.

Elastoplastic Stress

An elastoplastic stress simulation is performed over pipeline domain using the small strain plasticity model. The user defined isotropic hardening model is used where the hardening function, σyhard, is defined as:

where σexp is the experimental stress-strain curve, εp is the plastic deformation, σe is the von Mises stress, E is the Young’s modulus (207·109 Pa), and σys is the yield strength of high strength alloy steel(806·106 Pa).

The experimental stress-strain curve used in the model is prescribed in terms of a piecewise cubic interpolation function and is taken from Ref. 2.

Electrochemical Reactions

The iron dissolution (anodic) and hydrogen evolution (cathodic) reactions are the two electrochemical reactions that occur at the corrosion defect surface of pipelines. The rest of pipeline surfaces are assumed to be electrochemically inactive.

An anodic Tafel expression is used to model the iron dissolution reaction, with a local anodic current density defined as

where i0,a is the exchange current density (2.353·10−3 A/m2), Aa is the Tafel slope (0.118 V) and the overpotential ηa for the anodic reaction is calculated from

The equilibrium potential for the anodic reaction is calculated from

where Eeq0,a is the standard equilibrium potential for the anodic reaction (−0.859 V), ΔPm is the excess pressure to elastic deformation (2.687·108 Pa), Vm is the molar volume of steel (7.13·10−6 m3/mol), z is the charge number for steel (2), F is the Faraday’s constant, T is the absolute temperature (298.15 K), R is the ideal gas constant, νis an orientation dependent factor (0.45), α is a coefficient (1.67·1015 m−2) and N0 is the initial dislocation density (1·1012 m−2).

A cathodic Tafel expression is used to model the hydrogen evolution reaction, this sets the local cathodic current density to

where i0,c is the exchange current density, Ac is the Tafel slope (−0.207 V) and the overpotential ηc (SI unit: V) for the cathodic reaction is calculated from

where Eeq0,c is the standard equilibrium potential for the cathodic reaction (−0.644 V)

The exchange current density for the cathodic reaction is calculated from

where i0,c,ref is the reference exchange current density for the cathodic reaction in the absence of external stress/strain (1.457·10−2 A/m2).

Deformed geometry

In the second part of the model, dissolution of iron from the corrosion defect is modeled using deformed geometry. The dissolution of iron causes the electrode boundary to move, with a velocity in the normal direction, v (SI unit: m/s), according to

where ia (SI unit: A/m2) is the anodic current density, M is the molar mass (55.845 g/mol) and ρ the density (7870 kg/m3) of the iron.

Solve the model in a time-dependent study, simulating the corrosion for 20 years of exposure.

Results and Discussion

Figure 2 shows the electrolyte potential distribution (V) over the soil domain and the von Mises stress distribution (MPa) over the pipe domain, as indicated by the color bars for a prescribed displacement of 4 mm in the x direction. It can be seen that the local stresses are significantly higher near the corrosion defect than that at the rest of the pipeline. A nonuniform electrolyte potential distribution near the corrosion defect is also evident in Figure 2, as indicated by a semi-circular area.

Figure 2: The electrolyte potential distribution over the soil domain and the von Mises stress distribution over the pipeline domain for a prescribed displacement of 4 mm.

Figure 3 shows the von Mises stress distribution along the length of the corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm, respectively. The von Mises stress increases with an increase in the tensile strain and it is found to be maximum at the center of corrosion defect. For the tensile strain of 2 mm, 3 mm and 4 mm, it is observed that the local stress, particularly at the center of corrosion defect, exceeds the yield strength of high strength alloy steel (806·106 Pa). This results in the plastic deformation at the center of corrosion defect while deformation in the remaining area of corrosion defect remains in the elastic range. For the lower tensile strain of 1 mm, the entire corrosion defect is observed to be in the elastic deformation range.

Figure 3: The von Mises stress distribution along the length of the corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm.

Figure 4 shows the corrosion potential distribution along the length of the corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm. For lower tensile strain of 1 mm, the variation in the corrosion potential is found to be uniform along the length of the corrosion defect. However, for higher tensile strains of 2 mm, 3 mm and 4 mm, the variation in the corrosion potential is nonuniform with the more negative corrosion potential at the center of the corrosion defect than that at both the sides of the corrosion defect.

Figure 4: The corrosion potential distribution along the length of the corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm.

Figure 5 shows the anodic current density distribution along the length of corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm. For lower tensile strain of 1 mm, the variation in the anodic current density is found to be uniform along the length of the corrosion defect, similar to the corrosion potential behavior. However, for higher tensile strains of 2 mm, 3 mm and 4 mm, the variation in the anodic current density is significantly nonuniform, particularly at the center of the corrosion defect. It can be seen that the anodic current density increases significantly at the center of the corrosion defect whereas it decreases slightly at both the sides of the corrosion defect for higher tensile strains. The increase in the anodic current density for tensile strains of 2 mm, 3 mm and 4 mm is attributed to the plastic deformation observed at the center of the corrosion defect (see Figure 3).

Figure 5: The anodic current density distribution along the length of the corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm.

Figure 6 shows the cathodic current density distribution along the length of the corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm. It can be seen that the cathodic current density increases negatively with an increase in the tensile strain and it is found to be the most negative at the center of the corrosion defect. The nonuniformity in the cathodic current density is also found to increase with an increase in the tensile strain. Thus, the cathodic current density distribution is found to be the most nonuniform for a tensile strain of 4 mm.

Figure 6: The cathodic current density distribution along the length of the corrosion defect for prescribed displacements of 1 mm, 2 mm, 3 mm, and 4 mm.

Figure 7 shows the corrosion defect profile at time t = 0 a and 20 a. It can be seen that after 20 years, the depth of the corrosion defect is increased particularly at the center of the corrosion defect.

Figure 7: The corrosion defect profile at time t = 0 a and 20 a.

Figure 8 shows the von Mises stress distribution along the length of the corrosion defect at time t = 0 a and 20 a. It can be seen that after 20 years, the von Mises stress is higher at the center of the corrosion defect than that at both the sides of the corrosion defect.

Figure 8: The von Mises stress distribution along the length of the corrosion defect at time t = 0 a and 20 a.

Figure 9 shows the corrosion potential distribution along the length of the corrosion defect at time t = 0 a and 20 a. It can be seen that after 20 years, the corrosion potential is more negative throughout the length of the corrosion defect.

Figure 9: The corrosion potential distribution along the length of the corrosion defect at time t = 0 a and 20 a.

Figure 10 shows the anodic current density distribution along the length of corrosion defect at time t = 0 a and 20 a. It can be seen that after 20 years, the anodic current density is marginally higher at the center of the corrosion defect when compared with that at time t = 0 a.

Figure 10: The anodic current density distribution along the length of the corrosion defect at time t = 0 a and 20 a.

Figure 11 shows the cathodic current density distribution along the length of the corrosion defect at time t = 0 a and 20 a. It can be seen that after 20 years, the cathodic current density is more negative at the center of the corrosion defect than that at both the sides of the corrosion defect. The cathodic current density is also found to be marginally more negative at the center of the corrosion defect when compared with that at time t = 0 a.

Figure 11: The cathodic current density distribution along the length of the corrosion defect at time t = 0 a and 20 a.

Notes About the COMSOL Implementation

The model is implemented using the Solid Mechanics interface and the Secondary Current Distribution interface. In the first study, the Solid Mechanics model does not depend upon the results of the Secondary Current Distribution model, and we use a sequential solver set up with a Parametric Sweep to study the impact of elastoplastic deformations on electrochemical reactions. In the second part of the model, a transient analysis is performed using Deformed Geometry to analyze the effect of varying depth, due to dissolution, on stress as well as corrosion.

References

1. L.Y. Xu and Y.F. Cheng, “Development of a finite element model for simulation and prediction of mechanoelectrochemical effect of pipeline corrosion,” Corrosion Science, vol. 73, pp. 150–160, 2013.

2. L. Xu, Assessment of corrosion defects on high-strength steel pipelines, PhD thesis, Department of mechanical and manufacturing engineering, University of Calgary, Alberta, August 2013.

Corrosion_Module/General_Corrosion/stress_corrosion

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Geometry 1

Draw the pipe geometry with a corrosion defect and a surrounding soil domain.

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.

In the text field, type 19.1[mm].

Ellipse 1 (e1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 100[mm].

In the text field, type 11.46[mm].

Locate the section. In the text field, type 1.

In the text field, type 19.1[mm].

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Click to select the

toggle button for .

Select the object only.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.

Click

Click the

button in the toolbar.

Your geometry should look like that in Figure 1.

Global Definitions

Parameters 1

Load the model parameters.

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file stress_corrosion_parameters.txt.

Definitions

Load the model variables.

Variables 1

In the window, expand the > node.

Right-click and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file stress_corrosion_variables.txt.

Interpolation 1 (int1)

Load the stress strain interpolation data from a text file.

In the toolbar, click

In the window for , locate the section.

In the text field, type stress_strain_curve.

Click

Browse to the model’s Application Libraries folder and double-click the file stress_corrosion_stress_strain_curve_interpolation.txt.

Locate the section. From the list, choose .

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


Argument	Unit
t	1

In the table, enter the following settings:


Function	Unit
stress_strain_curve	MPa

Solid Mechanics (solid)

Start setting up the physics. First, set the elastoplastic deformation at the Linear Elastic Material node by adding a Plasticity node.

In the window, under click .

In the window for , locate the section.

In the list box, select .

Click

Select Domain 1 only.

Linear Elastic Material 1

In the window, under > click .

Plasticity 1

In the toolbar, click

and choose .

In the window for , locate the section.

Find the subsection. From the list, choose .

Materials

Now, add a high-strength alloy steel material for the pipe and set the values for initial yield stress, hardening function, Young’s modulus, and Poisson’s ratio.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

High-strength alloy steel (mat1)

In the window for , locate the section.

In the list box, select .

Click

Select Domain 1 only.

Locate the section. In the table, set the following property values:


Property	Variable	Value	Unit	Property group
Initial yield stress	sigmags	806[MPa]	Pa	Elastoplastic material model
Hardening function	sigmagh	hardening	Pa	Elastoplastic material model
Young’s modulus	E	207[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.33	1	Young’s modulus and Poisson’s ratio