Application_of_user_defined_subroutine_U.pdf

Application of user defined subroutine UMESHMOTION in ABAQUS for simulating dry rolling/sliding wear

Master’s Thesis of  Basavaraj Basavaraj Kanavalli Kanavalli Master of Science (Scientific Computing) Royal Institute of Technology (KTH), Stockholm, Sweden. Supervised by: Dr. Vishwanath Vishwanath Hegadekatte Hegadekatte Institute for Reliability of Components and Systems University of Karlsruhe, Germany. Germany. Prof. Sören Andersson Andersson Department of Machine Design, Royal Institute of Technology (KTH), Stockholm, Sweden.

Abstract

Twin-disc rolling/sliding tribometer experiments try to mimic the rolling/sliding contact experienced by micro-machines, e.g., between the teeth of two mating micromicro-gear gears. s. In this this wo work, rk, a user defined subrouti subroutine ne UMESHM UMESHMOTIO OTION, N, in the commercial commercial finite element code, ABAQUS, has been applied to simulate simulate wear in a twin-disc twin-disc tribometer experiments, experiments, conducted conducted for defined slips. ABAQUS ABAQUS invokes invokes the adaptive meshing algorithm after the convergence of the equilibrium equations of the contact problem, which further, invokes the user defined sub-routine UMESHMOTION. UMESHMOTION is coded to compute the local wear using the generalized Archard’s wear model and it integrates the wear depth over the sliding sliding distance using Euler integration integration scheme. In the absence of wear coefficient coefficient data from such experiments, it is assumed to be the same as identified from pinon-disc experiments (Herz et al., 2004; J. Schneider & Herz, 2005). The computed wear is applied on each surface node as mesh-constraint by the adaptive meshing algorit algorithm. hm. The result resulting ing equilibri equilibrium um loss loss is correct corrected ed by solving solving the last time time increment. Thus the geometry and pressures are updated. Simulations were carried out with different applied loads (with constant slip) and different slips (with constant constant load) and the results obtained, obtained, are presented. presented. The results obtained from UMESHMOTION are discussed by comparing them with the Global Incremental Wear Model (GIWM) and the Wear-Processor (Hegadekatte, 2006a).

iii

Acknowledgement

Success of any work would not be completed unless we mention the names of  the people who contri contribut buted ed in the wo work rk directly directly or indire indirectl ctly y. While While I cannot cannot even begin to thank all those who have unknowingly helped me throughout my studies, I would like to express my deep felt gratitude to some people who have helped me with useful suggestions and guidance. First of all, I would like to express my deep felt gratitude for my adviser, Dr. Vishwanath Hegadekatte, without whose advises, encouragement and patience, this work would would not have been possible, even even to think of. The suggestions, suggestions, tips and advises given by Vishwanath were were immensly helpful for my thesis. I would like to thank Professor Oliver Kraft, Forschungszentrum Karlsruhe (FZK), for giving me this opportunity to work on my master thesis in an international research atmosphere. I would like to express my deep felt gratitude for Professor Norbert Huber for his advise advisess and support support during during the course of this this wo work. rk. I wo would uld also like to thank Yixiang Gan, at IMF II, FZK, for his useful tips and suggestions. I would like to express my deep felt gratitude for Swedish Government, Dr Lennart Edsberg and Dr Katarina Gustavsson, for keeping faith in me and giving me an opportunity to study at Royal Institute of Technology (KTH). I am greatly indebted for the scholarship awarded by STINT for my studies at KTH. I would like to express my sincere thanks to Professor Sören Andersson for accepting me as his student and guiding me, from Sweden, during my master thesis, amidst his busy schedule schedule.. I wo would uld also like like to thank DAAD DAAD and IAESTE IAESTE for helpin helpingg me in getting getting my visa for my thesis at FZK, Germany Germany. Especially Especially, I am greatly indebted to the cooperation that I received from Barhammar Corolla, Jasmine Agassi, Gerd Aye and Maike Weisskopf. v

Last but not the least, I want to acknowledge my family members for their continuous support and encouragement. This work is dedicated to my mother Mrs. Kamaladevi Kanavalli and my father Mr. Chudamani Kanavalli. Basavaraj Kanavalli Karlsruhe, 13 June 2006

vi

Contents Abstract

iii

Acknowledgement

v

1 Introduction 1.1 Tribometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of chapters . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3

2 Wear Simulation of Twin Disc Rolling/Sliding Tribometer 2.1 Formulation of contact problem using FEM . . . . . . . . . . 2.2 Wear simulation strategy . . . . . . . . . . . . . . . . . . . . . 2.2.1 Wear model for twin-disc rolling/sliding tribometer . . 2.2.1.1 Generalized Archard’s wear model . . . . . . 2.3 Wear simulation of twin-disc rolling/sliding tribometer. . . . . 2.3.1 Adaptive meshing . . . . . . . . . . . . . . . . . . . . . 2.3.1.1 Mesh sweeping . . . . . . . . . . . . . . . . . 2.3.1.2 Advection . . . . . . . . . . . . . . . . . . . . 2.3.2 Wear-Simulator . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Computation of inward surface normals . . . 3 Wear Simulation Results 3.1 FE model of the twin-disc rolling/sliding tribometer . 3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Wear simulation with different loads and slips . . . . 3.3.1 Wear Simulations with different loads . . . . . 3.3.2 Wear Simulations with different slips . . . . . 3.4 Comparison with GIWM . . . . . . . . . . . . . . . . vii

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5 5 8 11 12 14 15 16 16 17 19

23 . . . . 23 . . . . 26 . . . . 28 . . . . 28 . . . . 29 . . . . 30

4 Discussion and conclusion 4.1 Numerical integration method . . . . . . . . . . . . . . . . . . . . 4.2 Pressure field updated by ABAQUS . . . . . . . . . . . . . . . . . 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 35 38 41

5 Summary

43

List of Figures

45

List of Tables

46

Bibliography

49

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Chapter 1

Introduction

Micro-machines are part of a rapidly emerging technology, finding wide variety of  applications in our day today life. The reliability of micro-machines is, therefore, a critical factor. The primary factors determining the performance and reliability of micro-machines are adhesion, friction, wear, film stress, fracture strength and fatigue (Romig et al., 2003). Out of these modes of failure, excessive wear is the most critical parameter, reducing the life span of micro-machines substantially, owing to their high operating speeds and high surface to volume ratio. For example, Williams (2001) observed that 1 minute of life for a micro-machine represents a degree of wear and degradation equal to more than 10 years as in a well designed watch bearing. In Figure 1.1 (a) shows a linear rack micro-motor. We can see that, between the two mating gear teeth, there will be rolling/sliding contact and between the gear and shaft, on which it is mounted, there will be sliding contact. Figure 1.1 (b) shows how a gap is formed between the center of  the micro gear and the shaft, because of wear. Compared to the research work done on modeling and simulations of wear in the macro world, less effort has been made to study wear in micro-machines, due to obvious difficulties at micro dimensions. For example, one of the difficulties is that the surface tension of  fluids used in the final rinse can glue the parts of a micro-machine together.

1.1. Tribometers 

2 (a)

(b)

Figure 1.1: (a) Picture showing linear rack micro-motor, (b) Picture showing gap formation on the gear, because of wear. Courtesy: Dannelle M. Tanner (Sandia National Laboratories).

1.1

Tribometers

At present, in-situ wear measurements are the most realistic methods to predict useful life-span of micro-machines. But manufacturing of prototypes and measuring of wear to estimate the useful life span of micro-machines involves extremely high costs, both in terms of money and time. Because of this, wear rate is usually determined experimentally using a tribometer like the twin-disc rolling/sliding tribometer (see Figure 1.2). These tribometers try to mimic the contact conditions experienced by micro-machines. e. g., the twin-disc rolling/sliding tribometer tries to mimic the rolling/sliding contact experienced by teeth of two mating micro-gears. The specimens used in tribometers are manufactured exactly by the same production/manufacturing processes as that of the micro-machines and therefore are assumed to posses same microstructure as that of the micromachines. In twin-disc rolling/sliding tribometer, the two discs are loaded against each other and are rotated at different speeds. The weight loss from either of the discs, is measured after certain number of rotations. Wear is expressed as worn volume per unit load and per unit sliding distance which is called as the dimensional wear coefficient, kD . Dimensional wear coefficient is the most important wear parameter determined from a tribometer. Tribometer gives a qualitative picture of the suitability of a material combination

1. Introduction

3

Figure 1.2: Twin disc rolling/sliding tribometer. for a given application and does not predict the useful life span of micro-machines. A wear simulation strategy that uses the finite element method (FEM), has been proposed by Hegadekatte et al. (2005) and the working of the wear simulation tool, Wear-Processor is explained in brief, in the second chapter.

1.2

Overview of chapters

In this chapter we discussed about the importance of wear in micro-machines and the role of tribometers in wear measurement. In chapter two, formulation of contact problem using FEM will be discussed. Later in this chapter wear simulation strategy adopted by Hegadekatte et al. (2005) for simulating dry sliding wear in pin-on-disc tribometer is discussed. This will be followed by discussion about modification of Archard’s wear model to simulate wear in twin-disc rolling/sliding tribometer. In the same chapter, adaptive meshing and working of UMESHMOTION, which is a user defined subroutine in commercial FE package, ABAQUS, to simulate dry sliding wear in twin-disc rolling/sliding tribometer, is explained in detail. In chapter three, the finite element model used in this work is presented. In this chapter, the results obtained from the FE based wear simulation using UMESHMOTION are presented. in chapter four, the results obtained from UMESHMOTION are compared with two different simulation tools, namely, the

4

1.2. Overview of chapters 

Global Incremental Wear Model (GIWM) and the Wear-Processor. Finally, a summary of this work is given in chapter five.

Chapter 2 Wear Simulation of Twin Disc Rolling/Sliding Tribometer

The whole process of simulating wear on any general tribosystem consists of two main tasks, namely, solution of the contact problem and using this solution to compute wear. In the first part of this chapter the formulation of contact problem using FEM is explained in brief. In the second part, a brief description about the wear simulation strategy adopted by Hegadekatte et al. (2005), for simulating wear in pin-on-disc tribometer and modification of the Archard’s wear model to compute wear in twin-disc tribometer is given. In the third part, detailed description of adoptive meshing and the working of user defined subroutine UMESHMOTION and its application for wear simulation is given.

2.1

Formulation of contact problem using FEM

In FE analysis, the contact conditions are applied as constraints. Unlike boundary conditions which remain fixed throughout the analysis, the constraints are discontinuous. Two methods for applying constraints are very popular, namely the Lagrange multiplier method, and the penalty method. In the following lines, the formulation of the contact problem in FEM using the Lagrange multiplier

2.1. Formulation of contact problem using FEM 

6

method for application of contact constraints will be discussed in brief (Bathe, 1996). The assumptions made are: 1. arbitrarily shaped body, 2. small deformation, 3. linear elastic material property, 4. static problem and 5. external forces include volume forces, surface forces and concentrated forces (which include the contact forces). Assuming a linear elastic continuum, the total potential is given by:

                     

1 Π(u, λ) = 2

εT DεdV 

V  

          

Strain Energy uT f V  dV 

−

V  

Work done by body forces T 

uS  f S dS 

−

S 

Work done by surface forces T 

uC  FC 

−

C 

Work done by concentrated forces T 

T 

λk (Qk uk − ∆k ),

+

k

 

(2.1)

Work done by contact forces (Lagrange multiplier method),

where, ε is the strain tensor, D  is the stress-strain matrix for the linear elastic material, u is the displacement vector, f V  is the body force vector, f S is the vector of surface forces, FC  is the vector of concentrated forces, λk is the vector of  T 

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer 

7

contact forces for the contact node k, Qk is the contact matrix for contact node k, ∆k is the penetration of contact node k, V  is the volume, S  is the surface, () C is the superscript for concentrated forces and the respective displacements and ()k is the superscript for contact node k. By applying the principle of minimum potential energy, i.e, δ Π = 0, we can formulate the governing equations for the body. δ Π =

 

T 

δε DεdV  −

V  

−



T  V  

δ u f  dV  −

V  

C T  C 

δ u

 

f  +

C 



T 

 

T 

δ uS  f S dS 

S 

T 

T 

T 

[δλ k (Qk uk − ∆k ) + λk Qk δ uk ].

 

(2.2)

k

In finite element analysis the domain is approximated as a set of discrete subdomains called as finite elements with elements being interconnected at nodal points on the element boundaries. Therefore the equilibrium equations that correspond to the nodal point displacements of the assemblage of finite elements is written as a sum of integrations over the volume and the areas of all finite elements. Then applying the principle of stationarity or the principle of minimum potential energy, we get the following governing equations for the system, (for more details refer Hegadekatte (2006b)),

ˆ T  δ U

   M 



T 

ˆ B(M ) D(M ) B(M ) dV  ( M ) U

V  (M )

= +∂  ˆ U

T 

           T 

HB(M ) f B(M ) dV  ( M )

M 

V  (M )

T 

ˆ T  +∂ U

HS (M ) f S (M ) dS (M )

S (M )

M 

 ˆ ˆ Qλ + ∂  UT 

ˆ T  −∂ U

k(M )

FC 

C (M )

ˆ T  QT U ˆ − ∆k , δ λ

−

 

(2.3)

k(M )

where the superscript M   stands for elements in the domain, B(M ) is the strain ˆ is the vector displacement matrix, H(M ) is displacement interpolation matrix, U of global displacements at all nodal points. F is the vector of all the concentrated

2.2. Wear simulation strategy 

8

ˆ the vector of all the contact forces, Q is the complete contact matrix forces, λ is of the assemblage and ∆   is the vector for all penetrations. Since, for a linear elastic continuum, the principle of minimum potential energy is identical to the principle of virtual displacement, the unknown nodal point displacement can be obtained from Equation (2.3) by imposing unit virtual displacement in turn at ˆ = I. all displacement components. In this way we have δ U The equilibrium equations of the element assemblage, corresponding to the nodal point displacements are given by:



K

Q

QT 

0

      ˆ U ˆ λ

 =

R

0

Rk

 +

∆

,

 

(2.4)

where,

     

K=

M 

V  (M )

T 

B(M ) D(M ) B(M ) dV  ( M )

R = RB  + RS 

=

M 

HB V  (M )

ˆ T . Rk = − Qλ

(M )T 

f B

(M )

dV  ( M ) +

   M 

S (M )

T 

HS (M ) f S (M ) dS (M )

For nonlinear problems the system of equations is formulated in an appropriate way and is solved using the Newton Raphson Method (Deuflhard, 2004; Ypma, 1995).

2.2

Wear simulation strategy

In this section the wear simulation strategy adopted by Hegadekatte et al. (2005) for simulating dry sliding wear in pin-on-disc tribometer will be explained, in brief. A wear simulation tool, Wear-Processor, has been developed, which works in conjunction with a commercial FE package ABAQUS. In the following lines the working of Wear-Processor, to simulate wear on the pin and the disc, is explained in brief. The contact simulation of the FE model is solved using the commercial FE package, ABAQUS. The contact pressures on the surface nodes are calculated using the stress tensors and surface normals and are used to compute wear using

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer 

9

the Archard’s wear law. Archard’s wear law is the simplest wear model available in the literature and can be expressed as,

V  F N  = k s H  = kD F N ,

 

(2.5)

where V  is the volume of material removed, s is the sliding distance, F N  is the applied normal load, k  is the dimensionless wear coefficient and k D [mm3 /Nmm] is the dimensional wear coefficient. The hardness (ratio of load over projected area), H , is that of the softer material. By dividing both sides of Equation (2.5) by real area of contact we get the expression for wear depth, h = k D p, s

(2.6)

where p is the pressure. This wear model is applicable on the global scale and in this work it is assumed that it is also applicable on the local scale. That is, contact pressures at each node on the surface can be used to compute wear on these nodes. Therefore, Equation (2.6) is discretized with respect to sliding distance as, dh = k D p. ds

 

(2.7)

As the disc wears out, the contact area increases and hence the contact pressure decreases. Therefore, Equation (2.7) is integrated over the sliding distance by using Euler integration scheme, to obtain expression for wear depth, as shown below, hi+1  = hi  + kD pi ds,

 

(2.8)

where hi+1  is the total wear up to the (i+1)th wear increment, and hi  is total wear up to the i th wear increment, p i  is the contact pressure during i th wear increment and kD pi ds   is the wear depth for the (i + 1)th (current) wear increment. The Equation (2.8) computes wear on a node which is in continuous contact with its mating surface. e.g., for a node on the surface of the pin. In pin-on-disc tribometer, any point on the disc comes in contact with the pin surface, only once in one complete revolution of the pin over disc. It is during this period, that

2.2. Wear simulation strategy 

10

the wear takes place on each of these disc surface nodes, depending on the contact pressure that the considered node experiences. As the pin moves over the disc, the contact pressure on the considered node, increases from zero to maximum and then decreases to zero. Therefore, to compute wear on the disc surface nodes, contact pressure is integrated over the circumferential length at a given radius, R, of the disc, as,

 

2π

hi+1  = h i + kD

 pi Rdθ,

 

(2.9)

θ =0

where θ  is the circumferential coordinate. In order to compute the integral in Equation (2.9) it is necessary that the finite element surface nodes are located exactly along the circumference of the various streamlines on the disc. This restriction on the mesh is over come by assuming an imaginary grid of points on the disc such that these grid points are located at definite pre-determined radii of the circumferential streamlines on the disc. The grid points are first identified with the three dimensional surface element faces that they are located on. The contact pressures at these grid points are obtained by interpolating the contact pressures calculated at the surface nodes. With the help of the computed contact pressures at grid points, the integral in the Equation (2.9) is calculated. Then, wear depth is computed using Equation (2.9). This wear depth is applied to all surface nodeson the pertcular streamline, in the inward normal direction. If the wear depth after certain amount of sliding is more than the product of surface element height and a pre-set factor, then the domain is re-meshed. The process is repeated until the desired amount of sliding distance is reached. This work is based on the wear simulation strategy adopted by Hegadekatte et al. (2005), but for twin-disc rolling/sliding tribometer. The computation of  wear on the disc in Hegadekatte et al. (2005) is similar to the computation of  wear in twin-disc rolling/sliding tribometer. In both the cases, the point under consideration comes in contact with its mating surface once per rotation of the disc. The considered point enters the contact interface and then leaves the contact interface. Therefore, Equation (2.9) is modified suitably to compute wear in twindisc rolling/sliding tribometer. In the next sub-section the modification of the Equation (2.9) to compute wear in twin-disc rolling/sliding tribometer along with the assumptions made in this work are explained in detail.

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer  (a)

11

(b)

Figure 2.1: (a) Schematic view of the twin-disc tribometer; (b) Figure showing the contact simulation geometric model for twin-disc tribometer.

2.2.1

Wear model for twin-disc rolling/sliding tribometer

When two circular rotating bodies come in contact with each other and they have same tangential velocity at all points of contact, then they are said to exhibit pure rolling contact. In pure rolling there will be no slip at the contact interface. However in reality it is very difficult to find a contact situation which exhibits pure rolling. Local sliding takes place in a part of the contact. Most of the rolling contacts are, therefore, actually rolling and sliding contacts (Spiegelberg, 2005). Therefore, in this work, twin-disc rolling/sliding tribometer with definite slip, between the two discs, is considered for wear simulation and also the experiments are conducted by (Herz et al., 2004; J. Schneider & Herz, 2005) for the same reason. The existence of slip between the discs together with normal load acting on the them, results in sliding wear. Figure 2.1a, shows the twin-disc rolling/sliding tribometer. The two circles on top disc indicate that the top disc has radius of curvature whereas the bottom disc has a flat surface. This will help for better alignment between the two discs while conducting the experiments. This type of surface results in an elliptical contact area. The two discs rotate with velocities V 1 , and V 2 , respectively, such that V 1  = V 2 . Such a system, from mechanics point of view, can be assumed to be like the one shown on the right hand side of Figure 2.1b, in which the bottom disc is fixed and the top disc rotates with the slip velocity. With this

2.2. Wear simulation strategy 

12

assumption, the problem will be reduced from rolling/sliding contact to time (pseudo) dependent sliding contact. Equation (2.9), computes wear depth for nodes on the disc in a pin-on-disc tribometer. Therefore Equation (2.9) needs to be modified, to compute wear in twin-disc rolling/sliding tribometer. The modified wear model used in this work is explained in the following sub-section.

2.2.1.1

Generalized Archard’s wear model

Figure 2.2: Figure showing different positions assumed by reference node.

Consider a reference node, A, on the surface of the top disc, which comes in contact with the bottom disc. As the disc rotates, the contact pressure on this node increases from zero to maximum and then decreases gradually to zero (see Figure 2.2). The node wears only when it experiences the pressure, i.e., when it moves through the contact interface. Therefore, pressure acting on this point, is integrated along the sliding distance which is the circumference of the disc. For one rotation of the disc, the wear taking place on this node, on the top disc, can

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer 

13

be written using the Equation (2.8) as,

 

Θ=2π

hi+1  = hi  + kD

 

 p j R1i dΘ,

(2.10)

Θ=0

where θ  is the circumferential coordinate, the subscript j   corresponds to the positions that the point occupies along its circumference while the disc rotates through an angle of  dθ , R1i  is the radius of the top disc at the ith wear increV  1 −V  2 )| ment. For a given increment of time, ∆ti , the top disc makes ∆t |(2πR rotations. 1 Where V 1 and V 2  are the velocities of the top disc and the bottom disc, respectively. Therefore the amount of wear depth, for a time increment of ∆ti , is, i

i

∆ti | (V 1 − V 2 ) | hi+1  = hi  + kD 2πR 1i

 

Θ=2π

 p j R1i dΘ.

 

(2.11)

Θ=0

This is the wear model used in this work. Equation (2.11) is called as generalized Archard’s wear model. The wear depth is computed using Equation (2.11) for all surface nodes.

In order to compute the integral in Equation (2.11), it is necessary that the nodes on the surface of the FE model are located exactly along the circumference of  the various streamlines on the disc. A streamline is a linear path on the surface of the top disc, traced by the surface nodes as the disc rotates. The width of the disc can be assumed to be made of several streamlines. Therefore, the FE model has been modeled in such a way that all the nodes, on the surface of the disc, appear along various streamlines. Figure 2.3 shows the bottom view of the top disc. This figure represents only a quarter of the top disc (read Section 3.1) as this will save a lot of computation time and memory. We can see that various nodes on the surface, which come in contact with the bottom disc, are located along the streamlines. Figure 2.3 also shows the edge nodes and the corner nodes. In this work, for the edge nodes and corner nodes the inward surface normal is calculated in the user subroutine UMESHMOTION. More details of computation of normals is given in Sub-section 2.3.2.1.

14

2.3. Wear simulation of twin-disc rolling/sliding tribometer.

Figure 2.3: Schematic representation, showing bottom view of the top disc.

2.3

Wear simulation of twin-disc rolling/sliding tribometer.

The wear simulation of twin-disc tribometer using generalized Archard’s wear model is accomplished using commercial FE code, ABAQUS 6.5, together with user defined subroutine, UMESHMOTION (see Figure 2.4). Using adaptive meshing algorithm, it is possible, in ABAQUS 6.5, to specify the mesh constraints through a user defined subroutine, UMESHMOTION. In this work, wear depth computed using Equation (2.11), by UMESHMOTION is applied as mesh constraint on all surface nodes. The user subroutine, UMESHMOTION coded for simulating wear in twin-disc rolling/sliding tribometer is named as   Wear-

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer 

15

Contact Pressure, Normals

A B A

User Subroutine, UMESHMOTION

Q U S Wear increment

Figure 2.4: Figure showing connection between ABAQUS and the user defined subroutine, UMESHMOTION. Simulator. In the following lines a detailed description of adaptive meshing and working of Wear-Simulator is given

2.3.1

Adaptive meshing

In structural analysis, sometimes, as the deformation takes place, some of the elements are overly distorted and may not produce good results. So it is advisable that those elements are re-meshed near those regions. This can be accomplished by using adaptive meshing which re-locates the mesh, without creating new elements in the domain. This property of the adaptive meshing algorithms is used for wear simulation to re-locate the mesh by an amount equal to the computed wear depth, h. In ABAQUS 6.5, adaptive meshing combines the features of  pure Langrangian analysis and Eulerian analysis (ABAQUS, 2004). That is, the structural analysis is carried out using pure Lagrangian approach and the relocation (also called as mesh smoothing) of the distorted elements is carried out using Eulerian approach. Thus adaptive meshing maintains high quality mesh throughout the analysis, even when large deformation or loss of material occurs, by allowing the mesh to move independently of the material. Adaptive meshing consists of two fundamental tasks, (1) creating new mesh, through a process called as  sweeping, (2) remapping the solution variables from the old mesh to the new mesh through a process called as advection.

2.3. Wear simulation of twin-disc rolling/sliding tribometer.

16 2.3.1.1

Mesh sweeping

Adaptive mesh smoothing is performed after the structural equilibrium equations have converged. The mesh smoothing equations are solved explicitly by sweeping, iteratively, over the adaptive mesh domain. During each mesh sweep, nodes in the domain are re-located, based on the positions of the neighboring nodes obtained during the previous mesh sweep, to reduce element distortion. The new position, x j +1 , of a node is obtained as, x j +1 = X + u j +1  = NN x jN ,

where X is the original position of the node, u j +1  is the displacement, x jN  are the neighboring nodal positions obtained during the previous mesh sweep, and NN  are the weight functions obtained from a least squares minimization procedure that minimizes displacement in the original configuration (ABAQUS, 2004) .

2.3.1.2

Advection

During advection, the solution variables are re-mapped from the old mesh to the new mesh by integrating the advection equations using the Lax-Wendroff method (Leveque, 2002). The scheme is explicit, second order accurate and provides some upwinding. Upwinding is a behavior of the solution in which it remains constant along the characteristic curves. For a linear advection equation as shown below, q t + aq x = 0, = a, and these lines are called as the solution is constant along the lines dx dt characteristic curves (for more details, refer chapter 3 and 4 of Leveque (2002)). The Lax-Wendroff scheme for linear advection equation is, n+1  j

q 

a∆t n  (a∆t)2 n n = q  − (q  j +1 − q  j −1) + (q  j +1 − 2q  jn + q  jn−1 ), 2 2∆x 2∆x n  j

where the subscript n  stands for the time increment and j  stands for the node number in the adaptive mesh domain, q   is the solution variable, being advected e.g., contact pressure, displacement etc., a is advection velocity, ∆t and ∆x are the increments in time and space respectively. The advection of material quantities can lead into loss of equilibrium, for two main reasons. The first reason

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer 

17

is, the error in the advection process itself. To minimize the errors in advection, ABAQUS restricts the magnitude of the advection velocity by taking Courant number ( for more details about Courant number and numerical stability please refer Gustavsson (2004)) to be less than one. The second reason for loss of equilibrium is the changes in the representation of the underlying material quantities by the changed mesh. This effect is more severe in case of wear simulation. Because, as the disc wears, the contact area increases and hence more number of  elements come in contact. This will lead to slightly different volume of integration while computing the internal forces. Therefore ABAQUS solves the last time increment, to correct the equilibrium.

2.3.2

Wear-Simulator

It is to be noted that in wear simulation of twin-disc tribometer, the top disc, which is defined as deformable body, is not rotated physically, but it is assumed to be rotated for certain time increment in the user defined subroutine UMESHMOTION which we call the Wear-Simulator. During this time increment, it is assumed that the configuration changes are negligible and have minor effect on the contact solution. Bottom disc, which is defined as rigid body, is rotated by a small angle, physically, to include the frictional effects as shown in Figure ??. First, the three dimensional, deformable-rigid contact problem is solved in ABAQUS. After the equilibrium equations have converged, ABAQUS invokes the adaptive meshing algorithm. Adaptive meshing algorithm, further, invokes the user defined subroutine UMESHMOTION. The working of Wear-Simulator will be explained with the help of a flow-chart, shown in Figure 2.5. The Wear-Simulator accesses the nodes one-by-one and stores them in a global matrix in the order of their appearance on the bottom surface of the top disc. Then, nodal coordinates and respective contact pressures are accessed. The pressure experienced by the node changes as it moves through the contact interface. Therefore pressure is integrated over the sliding distance for each streamline. To compute the number of rotations in Equation (2.11) the circumference of the disc, for that increment, is calculated . Wear depth in this time increment is calculated using Equation (2.11) for all surface nodes. The inward surface nor-

2.3. Wear simulation of twin-disc rolling/sliding tribometer.

18

•Geometry •Contact Definition •Material Model •Load •Boundary Condition

Finite Element Simulation (ABAQUS)

Wear-Simulator 

•Surface Nodes •Coordinates •Contact Pressure

 Yes

Inc=1

Surface Node Map

No

•Integrate Pressure •Circumference •Local Wear Model (Generalised Archard’s Wear Model) •Wear Depth

Node on Edge

 Yes

Inward Surface Normal

No

Wear in the Direction of  Inward Surface Normal

•Sweep the Mesh •Advect

Adaptive Meshing

t   t 

 Yes

max

No

END

Figure 2.5: Flowchart for wear simulation in a twin-disc tribometer.

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer 

19

mals for all surface nodes are supplied by ABAQUS, whereas for the nodes which come on edges and corners (see Figure 2.3), the inward surface normals have to be computed. Computing the inward normals for edge nodes and corner nodes is discussed in the Subsection 2.3.2.1. The wear depth for all surface nodes are returned to the adaptive meshing algorithm. Wear depth, received by adaptive meshing algorithm (see Sub-section 2.3.1) for all surface no des, are applied in two steps. First, the surface nodes are swept in the inward normal direction. Sweeping of the nodes is carried out purely as Eulerian analysis. Thus the geometry is updated. Second, the material quantities are re-mapped to the new positions. This is accomplished by advecting the material quantities from the old location to the new location by solving advection equations using a second order numerical method, called as Lax-Wendroff method as explained above. The advection of  material quantities can cause equilibrium loss, as explained in Subsection 2.3.1.2. The equilibrium loss is corrected by solving the last time increment. Thus the contact pressure is updated. The process from invoking the adaptive meshing algorithm to advecting the material point information is called one wear increment. It is to be noted that, in this work, the complete contact problem is not solved at the end of each wear increment. Instead, only the last time increment is solved to correct the loss of equilibrium. This will save a lot of computation time compared to wear simulation tool, developed in Hegadekatte et al. (2005).

2.3.2.1

Computation of inward surface normals

For nodes which are on the surface the inward surface, normals are supplied by ABAQUS. But for nodes which are on the edges as well as those which are on the corners (Figure 2.3) the normals have to be computed. In this section a brief  description of computing inward surface normals is given (for more details refer Hegadekatte et al. (2005)). The Figure (2.6) shows the node number 1, which is on one of the edges of the FE model.

2.3. Wear simulation of twin-disc rolling/sliding tribometer.

20

4 s1 1 2 y

z

n 1 1

2

n 1 2

r 1

r 2

1

3

n 2 2

n 2 1

Figure 2.6: Computation of inward surface normals for edge nodes.

The node number 2 and 3 are to its left and right respectively, and node number 4 is a subsurface node. All these nodes lie on y − z  plane (see Figure 2.6) and that is why the x coordinate for these nodes can be neglected while calculating the inward normal. The inward surface normal at node 1 can be calculated using the following condition, ri  · nki = 0,

 

(2.12)

where the subscript i stands for the element number and superscript, k, stands for the normal number of these elements at node 1. Applying this condition we get four normals at node 1, out of which two will be acting outwards and the remaining two will be acting inwards. Identification of the two normals which are acting inwards is accomplished by imposing the condition that the dot product of the inward normals and the vector formed by the subsurface node and node under consideration, is always greater than or equal to zero. i.e., nki · si  ≥ 0,

 

(2.13)

where si  is the subsurface vector formed between node 1 and node 4. With the help of this condition we get two normals which are acting inwards. The unit inward normal from these inward normals is calculated as, (n1i + n2i ) ni  = .  n 1i + n2i 

 

(2.14)

2. Wear Simulation of Twin Disc Rolling/Sliding Tribometer 

21

For the nodes which come on the corner, the inward normal is approximated by but the unit normal in the direction of subsurface node. For example, in Figure 2.7, for nodes 1 and 2 the inward surface normal is in the direction of  the respective subsurface nodes. A vector from the corner node to its subsurface node is computed and is divided by its magnitude to get the unit vector in that direction. Wear is applied in the direction of this unit vector. 2

1 n1

n2

y

z

Figure 2.7: Figure showing direction of wear for corner nodes.

In this chapter we discussed about the formulation of contact problem using FEM and wear model used for simulating wear in twin-disc rolling/sliding tribometer. A detailed description of wear model used in this work, adaptive meshing and working of Wear-Simulator was also given. In the next chapter, the FE model used for wear simulation and the results obtained from Wear-Simulator are presented.

Chapter 3 Wear Simulation Results In this chapter, the results obtained using the Wear-Simulator to simulate wear in a twin-disc rolling/sliding tribometer will be presented. First, the FE model used for this purpose and the parameters used for the wear simulation will be presented. Later the results of wear simulation will be presented.

3.1

FE model of the twin-disc rolling/sliding tribometer

The geometric model used for wear simulation is shown in Figure 3.2. In this work a three dimensional FE model has been used for two reasons: 1. The top disc has a curved surface and the bottom disc has a flat surface, as shown in Figure 3.2. Therefore the contact area will be an ellipse. Axisymmetric FE models cannot be used to represent an elliptical contact. 2. In (Hegadekatte, 2006b), it was shown that, for sliding contact, the asymmetric effects resulting from friction can be considerable. Only a quarter of the top disc has been modeled (the shaded region in Figure 3.2), since the disc is symmetric about z-axis. Very fine mesh has been generated near the contact region and the mesh is coarser as we go away from the contact

24

3.1. FE model of the twin-disc rolling/sliding tribometer 

Figure 3.1: Geometric model of the twin-disc rolling/sliding tribometer.

region (see Figure ?? (a) and (b)). This will not only reduce, substantially, the computation time and memory required, but will also result in more data points for calculating the integral in the Equation (2.11). The aspect ratio of elements near the contact region is very close to one. The FE model has been built in such a way that each surface node appears along a streamline (Figure 2.3). This will help in integrating pressure with respect to the sliding distance. First, the mesh is created in 2-D in x-y plane. This mesh is then swept about the center of the disc, to obtain a 3-D mesh. The bottom disc has been defined as a rigid surface (analytical rigid surface in ABABQUS 6.5). The wear results are obtained from the deformable top disc. The bottom disc is rotated by an angle of 0.01 radians to include the frictional effects. The parameters used for simulation are listed in the Table 3.1. In the absence of wear coefficient data from twin-disc rolling/sliding tribometer experiments, it is assumed to be the same as identified from pin-on-disc experiments (Herz et al., 2004; J. Schneider & Herz, 2005). Therefore dimensional wear coefficient, kD , and the frictional coefficient, µ, are taken from pin-on-disc experimental data.

3. Wear Simulation Results 

25

Table 3.1: Wear parameters for the twin-disc rolling/sliding tribometer (see Figure 3.2). Parameter Normal Load Velocity of Wheel 1 Velocity of Wheel 2 Young’s Modulus Poisson’s ratio Frictional Coefficient Dimensional Wear Coefficient

Value F N  = 300, 600, 900 mN  V 1  = 800 mm/s V 1  = 880 mm/s E 1 = E 2  = 152 GP a ν 1 = ν 2 = 0.32 µ = 0.6 kD = 1 × 10−10 mm3 /Nmm

Figure 3.2: Three dimensional picture of FE model of the top disc with zoomed view showing the fine mesh near the contact region.

3.2. Results 

26 0.004 Wear-Simulator

0.003

     ]     m     m 0.002      [

Elastic + Wear

  w

   h

0.001

0 0

6e+06

1.2e+07

 N

1.8e+07

2.4e+07

[-]

Figure 3.3: Graph of wear depth versus number of rotations.

3.2

Results

The wear simulations were carried out until 3 µm of wear was obtained on the top disc. Figure 3.3 shows the graph of wear depth, h, versus the number of  rotations, N . As the disc surface wears, the contact area, A r , increases and hence the pressure, p, decreases. Therefore the slope of the h versus N  continuously decreases as seen in Figure 3.3. This is called as ”running in”, in the literature. Unlike for wear depth curve, the curve for the elastic displacement and wear depth starts at a non-zero value equal to the elastic displacement. It can be see that this curve approaches the wear depth curve with increasing number of rotations. The curve for the elastic displacement plus wear depth, is shown in Figure 3.3. Figure 3.4 (a) and (b) shows how the contact pressure changes, when plotted on surface nodes along x and z directions respectively. We can see that the pressure drops along x and z directions as the disc wears out. However, as the disc wears out, because of the geometry of the contacting surfaces, the semi-major axis, a, of the elliptical contact area increases and the semi-minor axis, b, of the contact area decreases (see Figure 3.5 (a) and (b)). With the increase in wear on the disc, the elliptical contact slowly approaches towards a line contact. But the contact area, on the whole, increases continuously as shown in t Figure 3.6 (a). Figure 3.6 (b) shows the decrease in pressure as the wear progresses on the disc.

3. Wear Simulation Results 

27

(a) [mm]

 x

0 0

0.05

   ]-150   m   m    / -300    N    [

(b) 0.1

 z

0.15

-0.05

0.2

   2

[mm]

-0.025

0

0.025

0.05

   ]-150   m   m    / -300    N    [

   2

After 0.0E+00 Rotations After 1.5E+05 Rotations After 4.7E+05 Rotations After 20.5E+05 Rotations After 203.8E+05 Rotations

N

  p-450

  p

-450

-600

After 0.0E+00 Rotations After 1.5E+05 Rotations After 4.7E+05 Rotations After 20.5E+05 Rotations After 203.8E+05 Rotations

N

-600

Figure 3.4: Graph showing decay in contact pressure as wear progresses: (a) for nodes located along x-axis; (b) for nodes located along z-axis. (a)

(b)

0.2

0.024

     ]     m     m0.1      [

0.15

     ]     m 0.012     m      [

0.05

0.006

0.018

   b

  a

0 0

 

6e+06

0 0

1.2e+07 1.8e+07 2.4e+07

 N

 

6e+06

1.2e+07 1.8e+07 2.4e+07

 N

[-]

[-]

Figure 3.5: (a) Graph showing increase in length of semi-major axis of the contact area; (b) Graph showing decrease in length of semi-minor axis of the contact area. (a)

(b)

0.002

600

 ] 450    m    m 300      /      N      [

   ] 0.0015   m   m 0.001    [

     2

   2

  r

   A0.0005 0 0

  p150

 

6e+06 1.2e+07 1.8e+07 2.4e+07

 N

[-]

0 0

 

6e+06

1.2e+07   1.8e+07 2.4e+07

 N

[-]

Figure 3.6: (a) Graph showing increase in contact area; (b) Graph showing decrease in contact pressure as the disc wears.

3.3. Wear simulation with different loads and slips 

28

3.3

Wear simulation with different loads and slips

As mentioned earlier, the experimental results for twin-disc rolling/sliding tribometer were not available. Therefore, in this work, two sets of simulations were carried out, i.e., by varying the applied load with a constant slip and by varying the slip with a constant load. With the help of these simulations, the time taken by the twin-disc tribometer in the laboratory, to obtain a wear of 3 µm, has been approximated.

3.3.1

Wear Simulations with different loads

0.004

0.003

     ]     m     m 0.002      [

F   = 0.3 N 

  w

 N 

   h

F   = 0.6 N   N 

0.001

F   = 0.9 N   N 

0 0

 

6e+06

1.2e+07

1.8e+07

2.4e+07

 N [-]

Figure 3.7: Graph showing wear depth versus number of rotations for different applied loads (slip = 10%).

Figure 3.7 shows the graph of wear depth versus number of rotations. The simulations are carried out for three different applied loads with a slip of 10%. As the load increases the number of rotations required for 3 µm of wear on the disc decreases. The number of rotations shown on the x-axis of the Figure 3.7 are for the system shown in Figure 2.1b, in which the top disc rotates with the slip

3. Wear Simulation Results 

29

velocity and the bottom disc is rotated through a small angle (Sub-section 2.2.1). The duration for which the experiments with the twin-disc tribometer need to be conducted with different loads, for obtaining 3 µm of wear on the disc, is listed in Table 3.2. Table 3.2: Time required for 3 µm of wear in twin-disc tribometer for different loads (see Figure 3.7). Load (N) 0.3 0.6 0.9

3.3.2

Time (hours) 174 91 61

Wear Simulations with different slips

0.004

     ]0.003     m     m 0.002      [   w

Slip = 5 % Slip = 10 % Slip = 20 %

   h 0.001 0 0

4e+06

8e+06 t

[s]

1.2e+07

1.6e+07

Figure 3.8: Graph showing wear depth versus number of rotations for different slips (load = 0.3N).

Figure 3.8 shows the graph of wear depth versus number of rotations. The simulations are carried out for three different slips with a load of 0.3N . The time

3.4. Comparison with GIWM 

30

shown on the x-axis of the Figure 3.8, are for the system shown in Figure 2.1b, in which the top disc rotates with the slip velocity and the bottom disc is rotated through a small angle (Sub-section 2.2.1). The duration for which the experiments with the twin-disc tribometer need to be conducted for obtaining 3 µm of  wear on the disc, is listed in Table 3.3.

Table 3.3: Time required for 3 µm of wear in twin-disc tribometer for different slips (see Figure 3.8). Slip 5% 10% 20%

3.4

Time (hours) 182 174 172

Comparison with GIWM 0.006 GIWM UMESHMOTION

     ]0.0045     m     m0.003      [   w

   h

0.0015 0 0

 

6e+06

1.2e+07

 N

1.8e+07

2.4e+07

[-]

Figure 3.9: Graph showing the comparison of wear results between GIWM and the Wear-Simulator.

3. Wear Simulation Results 

31

The results obtained from the Wear-Simulator are compared with Global Incremental Wear Model (GIWM) (Hegadekatte, 2006b). In GIWM, while calculating the contact area for a simple geometry as in the case of the twin disc tribometer, the elastic displacements are taken into consideration by using the Oliver & Pharr (1992) relation. The average contact pressure (global quantity) is calculated since the applied load is known. Then, the global incremental wear is calculated and integrated over sliding distance. Figure 3.9 shows the graph of wear depth obtained from both GIWM and the Wear-Simulator. It can be seen from the graph that the wear calculated by Wear-Simulator is less than that of GIWM. The difference in results between the two, is 35.63%. To look further into this difference the results obtained from Wear-Simulator are discussed by comparing them with another simulation tool, Wear-Processor, in the next chapter.

Chapter 4 Discussion and conclusion

In this chapter wear results obtained from the Wear-Simulator are discussed in comparison with GIWM and Wear-Processor Hegadekatte et al. (2005). To evaluate more closely, a test model with a coarse mesh, to save computational time, was used, in the same way as the main model. Since the test model has coarse mesh, in order to obtain reasonable number of nodes in contact, a higher load was applied. Also, the dimensional wear coefficient used, was hundred times that of the value used for FE model discussed in previous chapter (main model). The remaining parameters like the velocities of the two discs, frictional coefficient, Young’s modulus and Poisson’s ratio remain the same as those used for main model. The wear parameters used for test model are listed in the Table 4.1. Table 4.1: Wear parameters for test model. Parameter Normal Load Velocity of Disc 1 Velocity of Disc 2 Young’s Modulus Poisson’s ratio Frictional Coefficient Dimensional Wear Coefficient

Value F N  = 300 N  V 1  = 800 mm/s V 1  = 880 mm/s E 1 = E 2  = 152 GP a ν  = 0.32 µ = 0.6 kD = 1 × 10−8 mm3 /Nmm

34

0.0300 GIWM Wear-Processor

     ]0.0225     m     m 0.0150      [

Wear-Simulator

  w

   h

0.0075

0.0000 0

 

1500

 

3000

 N

[-]

 

4500

 

6000

Figure 4.1: Graph showing wear depth obtained from three different tools: GIWM, Wear-Processor and Wear-Simulator. Figure 4.1 shows the linear wear obtained from three different tools, namely, Wear-Simulator, Wear-Processor and GIWM. The working of Wear-Simulator and GIWM has been discussed in chapter 2 (Sub-section 2.3.2) and chapter 3 (Section 3.4) respectively. And working of Wear-Processor is explained in brief  in Section 2.2. In Figure 4.1 we can see that the results from GIWM and WearProcessor match each other, favorably. The wear calculated by Wear-Simulator is less than that of Wear-Processor. The difference in wear depth between WearProcessor and Wear-Simulator is 17.46 %. In Figure 4.2 variation of pressure for the two simulation tools is shown. One of the differences between the two simulation tools, namely the WearProcessor and the Wear-Simulator, is in calculating the contact pressures. In the Wear-Processor, the contact pressures are calculated, for each surface node, by using the stress tensors and normals. Whereas in the Wear-Simulator contact pressures are accessed, directly, from the main code of ABAQUS. There are two principal differences between the two simulation tools. First one, is in updating the pressure field at the end of every wear increment. As discussed earlier, in the Wear-Processor the complete contact problem is solved, whereas, in the Wear-Simulator, pressure field is updated by correcting the equilibrium loss. The second one, is in the numerical integration method used in these two simulation

4. Discussion and conclusion

35

4500 4000

 

]    m 3500    m      /      N      [ 3000

Wear-Simulator Wear-Processor

     2

  p

2500 2000 0

 

1500

 

3000

 N

[-]

 

4500

6000

Figure 4.2: Graph showing decrease of pressure obtained by two different simulation tools: the Wear-Processor and the Wear-Simulator. tools. In Wear-Processor, the numerical integration method used for integrating pressure, p, with respect to sliding distance along the streamlines, was Simpson’s 1/3rd rule and in Wear-Simulator it was mid-point rectangular rule. In the next section of this chapter, the results from two simulation tools: WearSimulator and Wear-Processor, are discussed with respect to numerical integration method used. In the later section, updating of the pressure field by ABAQUS, is discussed. In the last section a conclusion is given.

4.1

Numerical integration method

In Wear-Processor, the numerical integration method used for integrating pressure, p, with respect to s, along each the streamlines, was Simpson’s 1/3rd rule and in Wear-Simulator it was mid-point rectangular rule. When the difference between these two methods was studied, it was seen that for the first streamline, before wear, there is a difference of 16.58 %, as shown in Table 4.2.

4.1. Numerical integration method 

36

Table 4.2: Comparison of integration methods used. Integration Method   Simpson’s 1/3rd rule (Wear-Processor) Mid-point rectangular rule (Wear-Simulator)

Value 1170 N/mm 980 N/mm

The wear depth calculated and applied by Wear-Processor is more than that of Wear-Simulator as the integration of pressure lies in the numerator of the Equation (2.11). This could be one of the reasons why the pressure (Figure 4.2), as a function of number of rotations, drops faster for Wear-Processor than that of the Wear-Simulator.

0.0300 GIWM Wear-Processor

     ]0.0225     m     m 0.0150      [

Wear-Simulator (Simpson’s rule)

  w

   h 0.0075

0.0000 0

Wear-Simulator (Mid-point rectangular rule)

 

1500

 

3000    N [-]

4500

 

6000

Figure 4.3: Graph showing improved wear results on the test model from WearSimulator by using Simpson’s 1/3rd rule in comparison to Wear-Processor and GIWM.

4. Discussion and conclusion

37

4500 Wear-Simulator Wear-Processor

4000

]    m 3500    m      /      N      [ 3000

     2

Using Simpson’s rule for integration

  p

2500 2000 0

 

1500

 

3000    N [-]

4500

6000

Figure 4.4: Graph showing the pressure drop obtained from Wear-Simulator on test model by using Simpson’s 1/3rd rule in comparison to that obtained from Wear-Processor.

In order to compare the results obtained from the Wear-Processor and the WearSimulator, Simpson’s 1/3rd rule was implemented in Wear-Simulator as well. It can be seen from Figure 4.3 that using Simpson’s 1/3rd rule the wear depth obtained from Wear-Simulator are higher than that of wear depth obtained from mid-point rectangular rule. Also, Figure 4.4 shows that the pressure drop obtained from Simpson’s 1/3rd rule is faster than the pressure drop obtained from mid-point rectangular rule. The difference in wear depth between the WearProcessor and the Wear-Simulator is 8.26 % and the pressure difference between the two is 3.9 %. From Figure 4.3 and 4.4 we can see that by using a better integration scheme the wear results have improved. However, for the FE model considered in the previous chapter, there are sufficient number of data points to integrate pressure over a streamline. Therefore, there is a negligible difference between the two different integration methods, as seen in Figure 4.5.

4.2. Pressure field updated by ABAQUS 

38

0.002  

GIWM Wear-Simulator

     ]0.0015     m     m0.001      [

Using Rectangular rule

Using Simpson’s rule

  w

   h0.0005 0 0   1.5e+06   3e+06   4.5e+06   6e+06

 N

[-]

Figure 4.5: Graph showing wear depth obtained on main model using two different integration methods.

4.2

Pressure field updated by ABAQUS 4500 4000

]      2    m 3500    m      /      N      [ 3000

With pressure correction Without pressure correction

  p

2500 2000 0

 

1500

 

3000

 N

[-]

 

4500

 

6000

Figure 4.6: Graph showing pressure updated by ABAQUS 6.5 using test model.

4. Discussion and conclusion

39

As discussed earlier ABAQUS does not solve the complete contact problem at the end of a wear increment, but only the last increment is solved to correct the equilibrium loss. Therefore, at the end of certain number of rotations the pressure updated by ABAQUS were verified. Simulation on the test model using Simpson’s 1/3rd rule, was terminated after 2000 rotations, and the pressure field was updated by solving the complete contact problem. Figure 4.6 shows the updated contact pressure as a function of number of rotations. It can be seen that, the pressure updated by ABAQUS does not agree, completely, with the pressure obtained by solving the complete contact problem. The difference of  7.54 % is present between the two after 2000 rotations. However, this difference is negligible compared to the saving in the computation time. Wear simulation was continued with the Wear-Simulator using the corrected pressure field, for another 1300 rotations. In this simulation, pressure decreased faster. This is because, with the corrected pressure field, the calculated wear was more and hence there was increase in contact area. Similarly, pressure was corrected after 3300 rotations and the wear simulation was continued with this new pressure field. Figure 4.7 shows wear depth results obtained after the pressure is corrected. We can see that there is a marginal increase in the wear obtained after pressure correction. Similarly pressure field was corrected at different intervals using main model. From Figure 4.8 and Figure 4.9, we can see that the corrected pressure field has produced negligible amount of increase in wear depth. 0.04

0.03

With pressure correction

     ]     m     m 0.02      [

Without pressure correction

  w

   h 0.01

0.00 0

 

1500

 

3000

 N

[-]

 

4500

6000

Figure 4.7: Graph showing change in wear depth after pressure correction, using test model

4.2. Pressure field updated by ABAQUS 

40

500 Wear-Simulator

]400    m    m 300      /      N      [

     2

Without Pressure Correction With Pressure Correction

  p

200

100 0

2.5e+05

5e+05

 N

[-]

7.5e+05

1e+06

Figure 4.8: Graph showing pressure updated by ABAQUS 6.5 using main model.

0.0005

     ]     m 0.0003     m      [ 0.0004

  w

 

GIWM

With Pressure Correction

Without Pressure Correction

0.0002

   h

0.0001 0 0

2.5e+05

5e+05

 N

[-]

7.5e+05   1e+06

Figure 4.9: Graph showing change in wear depth after pressure correction, using main model

4. Discussion and conclusion

4.3

41

Conclusion

On the basis of the results presented so far, the following conclusions are made: 1. It was shown that the user defined subroutine UMESHMOTION (WearSimulator) in ABAQUS can be coded to simulate rolling/sliding wear in a twin-disc tribometer. 2. The Wear-Simulator is faster than that of the Wear-Processor. It was observed that, for the test model, the time taken by Wear-Simulator was 40 minutes, whereas the time taken by Wear-Processor for the same amount of  wear, on the same computer, was 600 minutes. However, the computation time required by GIWM is negligible. 3. The wear results obtained from Wear-Simulator are compared with GIWM and a difference of 35.63 % was observed. 4. A better numerical integration method, like Simpson’s 1/3rd rule, was used and it was seen that this does not produce significant improvement on the wear results. 5. Pressure field updated by ABAQUS at the end of every wear increment is not very accurate. A difference of 7.54 % was observed between the pressure updated by ABAQUS by correcting only the equilibrium loss and pressure obtained by solving the complete contact simulation. However, this difference is difference is negligible when the saving in the computation time is considered. 6. A difference of 8.26 % was observed, in the wear depth between the WearProcessor and the Wear-Simulator using Simpson’s 1/3rd rule, which is negligible. Since Wear-Simulator is computationally less expensive compared to the Wear-Processor, a best compromise between computation time and accuracy has to be made in using the two FE based wear simulation tools.

Chapter 5 Summary In this work, an attempt has been made to simulate wear in a twin-disc rolling/sliding tribometer with defined slip. A wear simulation strategy in association with FEM (Hegadekatte et al., 2005) has been used. User defined subroutine UMESHMOTION (Wear-Simulator) in ABAQUS has been applied, as an alternative tool, to simulates dry sliding wear between the two discs. Wear simulations were carried out for different applied loads and different slips and the results obtained from these simulations were presented. Wear depth obtained from Wear-Simulator were compared with GIWM and a difference of 35.63 % was found. Results obtained from Wear-Processor, GIWM and Wear-Simulator, on a test model, were discussed with respect to their numerical integration methods. It was shown that, use of a better numerical integration scheme, like Simpson’s 1/3rd rule, improves the results on the test model. However, use of Simpson’s 1/3rd rule does not produce significant improvement on the main model as the main model has sufficient number of data points for integrating pressure with respect to sliding distance for each streamline. The pressure updated by ABAQUS was verified and a difference of 7.54 % was observed which is negligible. Unlike in WearProcessor, where the complete contact problem is solved at the end of every wear increment, in this work the adopted method (UMESHMOTION/Wear-Simulator) updates the contact pressure by correcting the equibrium loss by solving the last time increment of the contact simulation. This saves a lot of computation time. Therefore Wear-Simulator is computationally less expensive compared to WearProcessor. However, a difference of 8.26 % in the wear results from the two tools, 43

using test model with Simpson’s 1/3rd rule, is present. A best compromise between computation time and accuracy has to be made, in using the two FE based wear simulation tools. This work forms the basis of the future work, which aims to simulate wear in micro-machines and thus predict useful life-span.

List of Figures 1.1 (a) Picture showing linear rack micro-motor, (b) Picture showing gap formation on the gear, because of wear. Courtesy: Dannelle M. Tanner (Sandia National Laboratories). . . . . . . . . . . . . .

2

1.2

Twin disc rolling/sliding tribometer. . . . . . . . . . . . . . . . . .

3

2.1 (a) Schematic view of the twin-disc tribometer; (b) Figure showing the contact simulation geometric model for twin-disc tribometer. .

11

2.2

Figure showing different positions assumed by reference node. . .

12

2.3

Schematic representation, showing bottom view of the top disc. .

14

2.4 Figure showing connection between ABAQUS and the user defined subroutine, UMESHMOTION. . . . . . . . . . . . . . . . . . . . .

15

2.5

Flowchart for wear simulation in a twin-disc tribometer. . . . . .

18

2.6

Computation of inward surface normals for edge nodes. . . . . . .

20

2.7

Figure showing direction of wear for corner nodes. . . . . . . . . .

21

3.1

Geometric model of the twin-disc rolling/sliding tribometer. . . .

24

3.2 Three dimensional picture of FE model of the top disc with zoomed view showing the fine mesh near the contact region. . . . . . . . .

25

3.3

Graph of wear depth versus number of rotations. . . . . . . . . . .

26

3.4 Graph showing decay in contact pressure as wear progresses: (a) for nodes located along x-axis; (b) for nodes located along z-axis.

27

3.5

(a) Graph showing increase in length of semi-major axis of the contact area; (b) Graph showing decrease in length of semi-minor axis of the contact area. . . . . . . . . . . . . . . . . . . . . . . .

27

3.6 (a) Graph showing increase in contact area; (b) Graph showing decrease in contact pressure as the disc wears. . . . . . . . . . . .

27

45

LIST OF FIGURES 

46

3.7 Graph showing wear depth versus number of rotations for different applied loads (slip = 10%). . . . . . . . . . . . . . . . . . . . . . 3.8 Graph showing wear depth versus number of rotations for different slips (load = 0.3N). . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Graph showing the comparison of wear results between GIWM and the Wear-Simulator. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3

4.4

4.5 4.6 4.7 4.8 4.9

Graph showing wear depth obtained from three different tools: GIWM, Wear-Processor and Wear-Simulator. . . . . . . . . . . . Graph showing decrease of pressure obtained by two different simulation tools: the Wear-Processor and the Wear-Simulator. . . . . Graph showing improved wear results on the test model from Wear-Simulator by using Simpson’s 1/3rd rule in comparison to Wear-Processor and GIWM. . . . . . . . . . . . . . . . . . . . . . Graph showing the pressure drop obtained from Wear-Simulator on test model by using Simpson’s 1/3rd rule in comparison to that obtained from Wear-Processor. . . . . . . . . . . . . . . . . . . . . Graph showing wear depth obtained on main model using two different integration methods. . . . . . . . . . . . . . . . . . . . . Graph showing pressure updated by ABAQUS 6.5 using test model. Graph showing change in wear depth after pressure correction, using test model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graph showing pressure updated by ABAQUS 6.5 using main model. Graph showing change in wear depth after pressure correction, using main model . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 29 30

34 35

36

37 38 38 39 40 40

List of Tables 3.1 Wear parameters for the twin-disc rolling/sliding tribometer (see Figure 3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Time required for 3 µm of wear in twin-disc tribometer for different loads (see Figure 3.7). . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Time required for 3 µm of wear in twin-disc tribometer for different slips (see Figure 3.8). . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2

Wear parameters for test model. . . . . . . . . . . . . . . . . . . . Comparison of integration methods used. . . . . . . . . . . . . . .

47

25 29 30 33 36

48

LIST OF TABLES 

Bibliography ABAQUS (2004).   V 6.5 . Hibbit, Karlsson and Soresen Inc., Providence, RI, USA. Bathe, K. J. (1996).  Finite element procedures In engineering Analysis . Prentice Hall. Deuflhard, P. (2004). Newton Methods for Non-linear Problems, Affine Invariance  and Adaptive Algorithms , volume 35. Springer,Berlin. Gustavsson, K. (2004). Computational fluid dynamics. Lecture Notes. Hegadekatte, V. (2006a). Global incremental wear model for twin disc rolling/sliding tribometer. Personal Communication. Hegadekatte, V. (2006b). Modelling and simulation of dry sliding wear for micromachine applications. Doctoral Thesis . Hegadekatte, V., Huber, N., & Kraft, O. (2005). Finite element based simulation of dry sliding wear.  Modelling Simul. Mater. Sci. Eng., 13, 57–75. Herz, J., Schneider, J., & Zum-Gahr, K. H. (2004). Tribologische charakterisierung von werkstoffen für mikrotechnische anwendungen. In R. W. Schmitt (Ed.), GFT Tribologie-Fachtagung 2004 Göttingen, Germany. on CD. J. Schneider, K. H. Z.-G. & Herz, J. (2005). Tribological characterization of mold inserts and materials. In D. Löhe & J. H. Haußelt (Eds.),  Micro-Engineering  of Metals and Ceramics, part I and part II (pp. 579-603). Wiley-VCH Verlag GmBH, Weinheim, Germany. Leveque, R. (2002). Finite Volume Methods for Hyperbolic Problems . Cambridge University Press. 49