Brian Whitney
FWM Project
Introduction
When performing complex finite element analysis in Abaqus, there are often problems that we
assign a friction coefficient between two bodies that come into contact. While often a book
value for the friction coefficient is used, there has not been verification that the friction value is
mimicking real life correctly.
The standard friction formulation used in Abaqus is the classical isotropic Coulomb friction
model. This model assumes that no motion occurs unless the equivalent frictional stress is
greater than the critical stress. The critical stress is defined as the friction coefficient multiplied
by the contact pressure normal to the surface. This friction model is expected to perform very
closely to a real life state.
The purpose of this project is to mimic real life 'friction tests' using finite elements to see if the
results would match the theoretical solutions. Three different friction scenarios are considered.
These three scenarios are delrin on steel (friction coefficient = 0.27), delrin on delrin (friction
coefficient = 0.38), and steel on steel (friction coefficient = 0.8). These materials were chosen
because of their high applicability to marine ship environments. These three friction scenarios
occur many times in different areas of ships. Also, a full range of friction coefficients is
evaluated with this selection; a small, a medium, and a large.
Theory and Methodology
Two different tests will be examined for this project. The first is an inclined plane friction test
and the second is a dragging friction test. Both will compare finite elements (modeled after
real life tests) to the theoretical solution to determine the accuracy of using finite elements for
different friction values and mesh densities.
Inclined Plane Friction Test
The first example is to have a block on a flat surface that is tilted to an incline. The friction
value is measured based on the angle at which the block begins to slide. This analysis would be
very similar to representing the ‘inclined plane test’ from www.worldoftest.com
Real Life Inclined Plane Friction Tested
The “Coefficient of Friction Tester - COF-P01 - COF Tester Series” can be seen in the figure below.
This machine has a range of 0 to 85 degrees with an accuracy of .01 degrees. It has an angular
velocity range of 0.1 to 10.0 degrees per second. This machine is considered state of the art in
friction testing and meets ASTM friction testing standards.
Inclined Plane Friction Theory
This machine can be modeled as a 2-d system with a block on a tilted plane. The angle at which
the block begins to slip can be calculated by creating a system of equations using a free body
diagram. A diagram, along with an example MathCAD calculation for delrin on delrin, can be
seen in the figure below.
Inclined Plane Friction Finite Element Model
This test was simulated using a 2-D finite element model. A square block was modeled sitting
on a flat plate. The block is 10 inches wide and 10 inches tall. It sits approximately half way on
a 50 inch long plate. This can be seen in the figure below.
The plate was either given steel (for steel on steel) or delrin properties (for delrin on steel and
delrin on delrin). Friction values were also assigned for each case. Example material and
friction values (for delrin) are shown below.
Three different mesh densities were used. The coarse mesh density was 2.5 inch element
length. The normal mesh density was 1.0 inch element length. The fine mesh density was 0.5
inch element length. These three element densities can be seen in the figure below.
The analysis was run as a nonlinear elastic quasi-static analysis using the Abaqus explicit solver.
The left end of the flat plate was rotated at an angular velocity of 0.25 rad/s for a total of 3
seconds. This angular velocity is slightly over the machine specifications that it is compared to,
but was needed to solve the analysis in a reasonable amount of time. The analysis still behaved
quasi-statically, so a slower angular velocity should produce the same result.
The velocity of the block was measured at the lower left corner. When this velocity reached
0.001 in/s, to ensure that the velocity was not just noise, the block was considered ‘sliding’.
The time at which the block started sliding multiplied by the angular velocity was used to
determine the angle at which slip occurred. This was then used to calculate the coefficient of
friction and compare it to the theoretical value. The figure shows the block just as it starts
sliding.
Dragging Friction Test
The second example is to have a block on a flat surface that is dragged with the reaction force
being used to measure the coefficient of friction. The friction value is measured based on the
reaction force resulting when the block begins to slide. This is based off of ‘the Coefficient of
Friction Machine’ created by Chemsultants.
Real Life Inclined Plane Friction Tested
The figure below shows ‘the Coefficient of Friction machine’.
This machine meets the friction testing standard ASTM D1894. This machine has four dragging
rates (6, 12, and 24 inches per minute). Test samples can be from 0.5 to 6 inches long with a
maximum weight of 10 pounds. The accuracy of this test is within 0.1%. This is a state of the
art machine and very accurate.
Dragging Friction Theory
This machine can be modeled as a 2-d system with a block on a flat plane. A diagram, along
with an example MathCAD calculation for delrin on delrin, can be seen in the figure below.
Dragging Friction Test Finite Element Model
This test was simulated using a 2-D finite element model. A square block was modeled sitting
on a flat plate. The block is 10 inches wide and 10 inches tall. It sits approximately half way on
a 50 inch long plate. This block is constrained in the x-direction to RP-1. RP-1 is a fixed point
and will be used to measure the reaction force created by the dragging friction test. This can be
seen in the figure below.
The plate was either given steel (for steel on steel) or delrin properties (for delrin on steel and
delrin on delrin). Friction values were also assigned for each case. These materials are the
same as the inclined plane test.
Three different mesh densities were used. The coarse mesh density was 2.5 inch element
length. The normal mesh density was 1.0 inch element length. The fine mesh density was 0.5
inch element length. These densities are identical to the inclined plane test and are repeated in
the figure below.
The analysis was run as a nonlinear elastic quasi-static analysis using the Abaqus standard
solver. The left end of the flat plate was translated in the negative x direction at a rate of 1 in/s
for a duration of four seconds. This velocity is slightly over the machine specifications that it is
compared to, but was needed to solve the analysis in a reasonable amount of time. The
analysis behaved quasi-statically, so a slower velocity will produce the same result.
The reaction force of the block was measured at RP-1. The reaction force was divided by the
mass of the block times the acceleration of gravity to determine the coefficient of friction value
measured. The figure below shows the state at the end of the sliding friction test for a delrin
on delrin test.
Results and Discussion
This section will discuss results from both tests as well as the conclusions drawn from them.
Results from Inclined Plane Test
The following three tables present the results from the inclined plane friction test. The first line
in each table presents what the theoretical value of friction should be given the variables used
in the test. The next three lines in each table present the results from Abaqus Explicit for a
course, normal, and fine mesh respectively.
Delrin on Delrin
Scenario
Time
Angle
Friction Value
% Error
Theoretical
n/a
0.363
0.38
0.00%
Coarse Mesh
1.47502
0.368755
0.386431557
1.69%
Normal Mesh
1.47486
0.368715
0.386385585
1.68%
Fine Mesh
1.47412
0.36853
0.38617298
1.62%
Delrin on Steel
Scenario
Time
Angle
Friction Value
% Error
Theoretical
n/a
0.264
0.27
0.00%
Coarse Mesh
1.07763
0.2694075
0.276120358
2.27%
Normal Mesh
1.0789
0.269725
0.276462095
2.39%
Fine Mesh
1.07937
0.2698425
0.27658858
2.44%
Steel on Steel
Scenario
Time
Angle
Friction Value
% Error
Theoretical
n/a
0.675
0.8
0.00%
Coarse Mesh
2.71299
0.6782475
0.805766956
0.72%
Normal Mesh
2.71586
0.678965
0.806950985
0.87%
Fine Mesh
2.71303
0.6782575
0.805783449
0.72%
The results overall are reasonably accurate especially since an explicit code is being used. The
steel on steel test proved to be the most accurate of all. The accuracy also seems to be
independent of mesh size for this test. In the delrin on delrin test, the results became slightly
better for a finer mesh. In the delrin on steel test, the results surprisingly became slightly less
accurate for a finer mesh.
Overall it seems that in an Abaqus Explicit analysis that higher values of coefficient of friction
seem to produce more accurate results. A trend can be seen that the highest coefficient of
friction value, 0.8, is most accurate, while the lowest coefficient of friction, 0.27, is least
accurate. Caution and other side studies should be performed when trying to analyze
coefficients of friction that are even less than this.
Results from Dragging Friction Test
The following three tables present the results from the dragging friction test. The first line in
each table presents what the theoretical value of friction should be given the variables used in
the test. The next three lines in each table present the results from Abaqus Standard for a
course, normal, and fine mesh respectively.
Delrin on Delrin
Scenario
Theoretical
Coarse Mesh
Normal Mesh
Fine Mesh
Block
Mass
0.0133
0.0133
0.0133
0.0133
Reaction Force Friction Value
1.949
1.94943
1.94943
1.94943
0.380
0.380
0.380
0.380
%
Error
0.00%
0.02%
0.02%
0.02%
Delrin on Steel
Scenario
Theoretical
Coarse Mesh
Normal Mesh
Fine Mesh
Block
Mass
0.0133
0.0133
0.0133
0.0133
Reaction Force Friction Value
1.385
1.38512
1.38512
1.38512
0.270
0.270
0.270
0.270
%
Error
0.00%
0.01%
0.01%
0.01%
Steel on Steel
Scenario
Theoretical
Coarse Mesh
Normal Mesh
Fine Mesh
Block
Mass
0.0732
0.0732
0.0732
0.0732
Reaction Force Friction Value
22.628
22.6276
22.6276
22.6276
0.800
0.800
0.800
0.800
%
Error
0.00%
0.00%
0.00%
0.00%
The results overall are extremely accurate. In all three cases, it seems as the answer is
independent of mesh size. This is most likely due to the use of the Abaqus Standard solver,
which solves for static equilibrium in each step. The Abaqus Explicit solver will solve for
dynamic equilibrium in each step. While the answers are not 100% matching to the theoretical
value, these small variances are most likely due to slight rounding errors.
Conclusions
Overall, Abaqus is very accurate in the modeling of friction. Two simple tests, which were
designed to mimic real world friction tests, were performed to validate this accuracy and both
were successful. The first test was the inclined plane friction test. This test explored the use of
friction in the Abaqus Explicit solver. The results from this test were overall harder to interpret
due to the inherent noise in explicit solvers. It was found that while all friction values tested
were reasonably accurate, the higher coefficients of friction produced better results. There
were also slight differences based on the mesh density as well. The second test was the
dragging friction test. This test explored the use of friction in the Abaqus Standard solver. The
results from this test were extremely accurate as well as consistent. The values were in all
cases within 0.02% of the theoretical value and independent of mesh size. If possible and
convenient, the Abaqus Standard solver should be used for a more accurate result if friction is a
primary concern.
References:
Abaqus 6.12 Theory Manual – 5.2.3 Coulomb Friction
Abaqus 6.12 User’s Manual – 36.1.5 Frictional Behavior
http://www.worldoftest.com/pdf/Qualitest_COF_PO1.pdf - Inclined plane tester
http://www.chemsultants.com/testing-equipment-products/testing-devices/coefficient-offriction.aspx - Dragging friction tester
Friction testing standard - ASTM D1894
http://plastics.dupont.com/plastics/pdflit/europe/delrin/DELDGe.pdf - delrin information and
friction information
http://rustam.uwp.edu/202/individual/friction.pdf -friction formulas
http://www.engineeringtoolbox.com/friction-coefficients-d_778.html - steel on steel friction
0.8