Slippery Tracks: Wheel - Rail Contact

10
Slippery Tracks: Wheel - Rail Contact R.I. Popovici and D.J. Schipper Phone +31-(0)53-3892463, e-mail [email protected] University of Twente, Faculty of Engineering Technology, Laboratory for Surface Technology and Tribology, P.O. Box 217, NL 7500 AE Enschede Introduction Low friction due to a bio-film between wheel and rail is generating a security issue in which the braking manoeuvre is failing. Solving this is the biggest priority in railroad industry. Objective The objective of this research is to find a solution to remove the interfacial layer which causes the low friction between wheel and rail. A first step in the process is to predict the coefficient of friction as a function of velocity (Stribeck curve) and as a function of slip or slide to roll ratio (Traction curve) assuming that the interfacial layer behaviour is non-linear viscous. Model A fully deterministic microcontact and friction model was developed for the elliptical contact situation. The present friction model is based on the line contact model developed by Gelinck [1] and the hydrodynamic lubrication model. Inputs In order to get as close as possible to reality, rail and wheel samples were used for a first set of measurements (Figure 1): Surface roughness (Interferometer, IM); Coefficient of friction in BL regime (Surface Force Apparatus, SFA). IM SFA IM SFA Figure 1: Measurements of the rail samples. Outputs The main output of the model is the Stribeck curve which describes the variation of the coefficient of friction as a function of velocity. Stribeck curves for 3 types of interfacial layers (water, oil and grease) are presented in Figure 2. 0.23 0.135 0.00 0.05 0.10 0.15 0.20 0.25 0.01 0.1 1 10 100 log v [m/s] Coefficient of friction, f [-] Water Mineral Oil Grease Dry 0.06 20 Figure 2: Stribeck curve for wheel - rail contact. Second result of the model is represented by the evolution of the coefficient of friction with increasing the slip or slide to roll ratio. -0.15 -0.1 -0.05 0 0.05 0.1 0.15 -0.05 -0.03 -0.01 0.01 0.03 0.05 Slip [-] Coefficient of friction, f [-] v=0.1 m/s v=1 m/s v=2 m/s v=3 m/s v=10 m/s Sep Figure 3: Traction curve for wheel - rail contact. Based on these findings the effect of the interfacial layer on the friction level in these contacts is shown to be significant. Future work Introducing the side slip component into the model and viscoelastic behaviour of the interfacial layer. References [1] Gelinck, E.R.M. and Schipper, D.J. (2000), “Calculation of Stribeck Curves for line contacts”, Tribology International, 33, 175-181. Tenth Engineering Mechanics Symposium P-51

Transcript of Slippery Tracks: Wheel - Rail Contact

Page 1: Slippery Tracks: Wheel - Rail Contact

Slippery Tracks: Wheel - Rail Contact

R.I. Popovici and D.J. Schipper

Phone +31-(0)53-3892463, e-mail [email protected] of Twente, Faculty of Engineering Technology,

Laboratory for Surface Technology and Tribology, P.O. Box 217, NL 7500 AE Enschede

IntroductionLow friction due to a bio-film between wheeland rail is generating a security issue inwhich the braking manoeuvre is failing.Solving this is the biggest priority in railroadindustry.

ObjectiveThe objective of this research is to find asolution to remove the interfacial layerwhich causes the low friction betweenwheel and rail. A first step in the process isto predict the coefficient of friction as afunction of velocity (Stribeck curve) and asa function of slip or slide to roll ratio(Traction curve) assuming that theinterfacial layer behaviour is non-linearviscous.

ModelA fully deterministic microcontact andfriction model was developed for theelliptical contact situation. The presentfriction model is based on the line contactmodel developed by Gelinck [1] and thehydrodynamic lubrication model.

InputsIn order to get as close as possible toreality, rail and wheel samples were usedfor a first set of measurements (Figure 1):

• Surface roughness (Interferometer, IM);

• Coefficient of friction in BL regime(Surface Force Apparatus, SFA).

IM SFAIM SFA

Figure 1: Measurements of the railsamples.

OutputsThe main output of the model is the Stribeckcurve which describes the variation of thecoefficient of friction as a function ofvelocity. Stribeck curves for 3 types ofinterfacial layers (water, oil and grease) arepresented in Figure 2.

0.23

0.135

0.00

0.05

0.10

0.15

0.20

0.25

0.01 0.1 1 10 100log v [m/s]

Coeffic

ientoffr

iction,f[-

]

Water

Mineral

OilGrease

Dry

0.06 20

Figure 2: Stribeck curve for wheel - railcontact.

Second result of the model is representedby the evolution of the coefficient of frictionwith increasing the slip or slide to roll ratio.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.05 -0.03 -0.01 0.01 0.03 0.05

Slip [-]

Coeffic

ientof

fric

tion,f[-

]

v=0.1 m/sv=1 m/sv=2 m/sv=3 m/sv=10 m/s

Sep

Figure 3: Traction curve for wheel - railcontact.

Based on these findings the effect of theinterfacial layer on the friction level in thesecontacts is shown to be significant.

Future workIntroducing the side slip component into themodel and viscoelastic behaviour of theinterfacial layer.

References[1] Gelinck, E.R.M. and Schipper, D.J. (2000),“Calculation of Stribeck Curves for line contacts”,Tribology International, 33, 175-181.

Tenth Engineering Mechanics Symposium P-51

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A Computational Approach to Model

Fibre Reinforced Concrete

F.K.F. Radtke, A. Simone and L.J. Sluys

Delft University of Technology

Faculty of Civil Engineering and Geosciences

P.O. Box 5048, 2600 GA Delft

phone +31 15 278 3752, email: [email protected]

Introduction

Fibre reinforced cementitious composites (FRCCs)

are increasingly used in practice. However, there is

still a lack of adequate computational tools to simulate

the material behaviour with a proper inclusion of its

microstructure. For example, it is difficult to quantify

the interaction between specimen geometry, fibre dis-

tribution and mechanical properties. An FRCC speci-

men is shown in Figure 1.

Figure 1 : A Sample of Fibre Reinforced Concrete

Objective

The objective of this study is to develop a tool that

combines computational efficiency with the possibil-

ity of investigating the influence of the material mi-

crostructure on the global material characteristics.

MesoLevel

MicroLevel

FF

Fibre Matrix Crack

Fibre MatrixITZ

P

P

adapted from [Lin et al., 1999]

P

Figure 2 : Modelling Approach

Physical Problem

Depending on their microstructure, FRCCs show me-

chanical behaviour which is characterised either by

strain softening behaviour with localised cracks, sim-

ilar to that of normal concrete, or by strain hardening

behaviour combined with a distributed cracking pat-

tern.

Approach

Two main processes dominate the material be-

haviour: matrix cracking and fibre-matrix interaction.

The latter is characterised by the fibre pull-out be-

haviour. Both phenomena are modelled on different

levels as shown in Figure 2.

Results

The addition of fibres leads to a distributed damage

pattern and a more ductile force-displacement curve

as shown in Figure 3. Figure 4 shows the influence of

a fibre distribution on the material behaviour. A mesh

refinement study for a fibre distribution is depicted in

Figure 5.5

F[N

]

0 u [mm] 0.04

Fibre

Matrix Matrix

1 Fibre

Figure 3 : One Fibre in Tension

7

F[N

]

0 u [mm] 0.0012

Damage Plot

Fibre

Matrix

Figure 4 : Fibre Distribution in Tension

400 elements

1600 elements

6

F[N

]

0 u [mm] 0.0015

100 elements

200 elements

400 elements

800 elements1600 elements

Figure 5 : Mesh Refinement Study

Acknowledgements

Financial support from the Netherlands Technology Foundation

(STW) is gratefully acknowledged.

P-52 Tenth Engineering Mechanics Symposium

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Finite Element Based Reduced Complexity

Analysis of Shells of Revolution

T. Rahman and E.L. Jansen

Delft University of TechnologyFaculty of Aerospace EngineeringP.O. Box 5058, NL 2600 GB Delft

phone +31-(0)15-2782084, email [email protected]

Introduction and objectiveThe current simulation methods (Finite Element computermodels) for nonlinear structural analysis calculations re-quire a lot of effort and expertise from the user, and needa long computer run time. Therefore there is a strong needfirstly for Reduced Complexity intermediate tools that helpin providing a systematic approach to achieve reliable re-sults, and secondly for faster computational tools. Suchtools are not yet available in most of the commercial FEcodes while the research codes that have this capability donot include the important effect of pre-buckling nonlinear-ity. The current research aims at the FE implementationfor Reduced Complexity analysis of shells of revolution in-cluding the pre-buckling nonlinearity effect. Here a singlemode analysis is presented.

Perturbation Methods Reduced Basis Methods

FE based Reduced Complexity Methods

Full model linear analysis

Full model nonlinear analysis

Figure 1 : Reduced Complexity analysis

MethodsKoiter’s perturbation approach that describes the static ini-tial post-buckling path by an asymptotic expansion of thedisplacement field around the bifurcation point form the ba-sis of the present implementation. For a single mode anal-ysis the perturbation expansion of the displacement fieldcan be written as

u = u0 +u1ξ +u2ξ2 +u3ξ

3 + . . . (1)

where u0 is the nonlinear pre-buckling displacement mode,u1 and u2 are the buckling and second order modes, and ξ

is the modal amplitude. Perturbation expansion of the loadparameter λ gives

ξ (λ −λc) = aλcξ2 +bλcξ

3−αλcξ̄ −β (λ −λc)ξ̄ + . . . (2)

where λc is the buckling load, the post-buckling coefficientsa, b, α , and β correspond respectively to the post-bucklingslope, curvature, first, and second imperfection form fac-tors, and ξ̄ is the imperfection amplitude. Once the dis-placement modes and the post-buckling coefficients arecomputed the initial post-buckling path is characterized byequations 1 and 2. This approach has been implemented

for an existing class of curved shell elements in the gen-eral purpose FE code DIANA. The current implementationis largely based on the already existing DIANA implemen-tation and Tiso’s formulation [1].

Results and discussion

An isotropic conical shell representative of the conical in-terstage of the Vega launcher structure has been ana-lyzed. The resulting negative b-coefficient indicate anunstable post-buckling nature and imperfection sensitivity.We observe a fair agreement in the post-buckling coeffi-cients between DIANA and the semi-analytical tool BAAC[2]. A good agreement with full nonlinear analysis is alsoobtained (figure 2). This implementation will be extendedfor multi-mode analysis in near future.

Figure 2 : Reduced Complexity analysis of the Vega launcher:

(a) First buckling mode (b) Second order mode (c) Compari-

son between full and reduced analysis (with imperfection am-

plitude ξ̄ = 0.05t)

Buckling load [N] b α β

BAAC 9.1071×106 -0.44032 0.39031 -0.40697

DIANA 8.9895×106 -0.45524 0.38281 -0.46012

References1. P. Tiso, M. M. Abdalla, and E. L. Jansen. Koiter’s post-buckling anal-

ysis of general shell structures using the finite element method. Inproceedings of the 25th international congress of the aeronauticalsciences, Hamburg, Germany, 2006.

2. G. Zhang. Stability Analysis of Anisotropic Conical Shells. PhD the-sis, Delft University of Technology, 1993.

Tenth Engineering Mechanics Symposium P-53

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Strain Path Sensitivity of Mild Steel in

Experiment and Model

M. van Riel, A.H. van den Boogaard

University of Twente/ NIMR

P.O. Box 217, 7500 AE Enschede, The Netherlands

phone +31-(0)53-489 4567, email [email protected]

Material models

In finite element simulations the mechani-

cal behaviour of a material is captured in

a material model that is parame-

terized by some elementary uniax-

ial experiments. However, a deep

drawing process (figure) shows that

the deformation mode is not uniax-

ial and changes during the process.

Sheet metal forming processes are

strongly sensitive to the accuracy

of the material models and therefor

more complex tests and more ex-

tensive models are required.

Goal

The aim of this project is to investigate the mechanical

behaviour of sheet metal under different strain paths

and develop models that describe the observed phe-

nomena.

Experimental observations

During plastic deformation of the material the disloca-

tion structure changes (Figure 2) and causes harden-

ing. After some deformation the dislocation structure

is oriented according the deformation direction

Figure 2: Left: Biaxial tester. Right: Dislocation structure (Mc-

Cabe, Act.Mat. 52(705-714), 2004).

and acts as a barrier for deformations in other direc-

tions. Experiments with a biaxial tester have shown

that a strain path change requires large stresses,

hereby reorienting the dislocation structure with re-

spect to the new deformation direction.

Modelling the mechanical behaviour

The Teodosiu and Hu model is used to describe the

mechanical behaviour of the material. The model is

physically based and calculates the stresses using

the strength of the dislocation structure (S):

S = Sd N ⊗ N + SL

Dependent on the direction of deformation (N) a di-

rectional part (Sd) and a latent part (SL) contribute to

the strength of the dislocation structure.

Results

Results of the experiments and the model are dis-

played in Figure 3. The characteristic overshoot in

stress (σxy) can be qualitatively predicted with this

model.

0 0.1 0.2 0.3 0.4 0.5

0

100

200

300

400

εeq

p (−)

σ (

N/m

m2)

σyy

σxy

σxy

(ref)

0 0.1 0.2 0.3 0.40

50

100

150

200

250

300

350

εeq

p (−)

σ (

N/m

m2)

Sd

SL

σyy

σxy

σxy

(ref)

Figure 3: Experimental (left) and simulation (right) results.

Clearly, the evolution of the yield stress is strongly de-

pendent on Sd, but after the strain path change, Sd is

close to zero and the latent dislocation structure SL

initiates a overshoot in shear stress (σxy).

Future research

• Develop a strategy that determines the material

parameters from experiments.

• Strain rate sensitivity needs to be implemented

in the model, requiring experimental results.

P-54 Tenth Engineering Mechanics Symposium

Page 5: Slippery Tracks: Wheel - Rail Contact

distribution of dislocation walls in the pile-up. A

qualitative fit to a discrete analysis is shown in

Figure 2. Clearly the solution is inaccurate near

the obstacle.

On the relevance of discreteness in plasticityA. Roy , R.H.J. Peerlings, M.G.D. Geers

Eindhoven University of Technology

Faculty of Mechanical Engineering

P.O.Box 513, NL 5600MB Eindhovenphone +31-(0)40-2473048, email [email protected]

Motivation

Understanding dislocation interactions is

essential in a vast number of industrial

applications. For example: fatigue cracking in

polycrystalline/multiphase materials; micro

scale plasticity; dislocation patterning in metallic

crystals. Popular continuum plasticity theories

account for the long range stress fields of

dislocation distributions accurately but resort to

phenomenology to model the short range

interactions. Local gradients of dislocation

distributions are often used. The questions that

arise are: Is it possible to completely ignore the

underlying discrete nature of dislocation

distributions?

How accurate are the gradient approaches in

modeling short range interactions?

Directional Dependence

An idealized problem of a dislocation pile-up

against a hard obstacle (Figure 1) is rigorously

analyzed and it is seen that the discreteness in

the direction perpendicular to the slip direction

cannot be ignored. Interestingly, “smudging” of

dislocations is allowed in the slip direction [1].

.

Comparison with gradient theories

We compare a gradient dependent Statistical

Mechanics model as formulated in [2] applied to

our problem. The theory predicts an exponential

Reference[1] Roy et al. (2006) Poster EM 2006.

[2] Groma et al. (2003) Acta Mater, 51, 1271-81.

Figure 1. Discrete dislocation representation of a pile-up in an infinite medium. Slip plane separation is “h”. The

red circles represent the immobile wall at x=a.

−∞

2h

h

2h−

h−

0

x a=

y

x

σσσσ

σσσσ

Figure 2. Comparison of discrete analysis of 50 walls of edge dislocations to theory.

Observations and Comment

The theory assumes that characterizing the

nearest neighbor interaction is sufficient in

modeling short range interactions. Figure 4

demonstrates that though the distant walls

individually contribute less to the overall stress

at a dislocation position, the accumulation of the

contributions from these walls is substantial.

This kind of influences is currently lacking in

continuum theories and needs to be looked at.

Figure 4. Stress contributions from every dislocation

wall to the central wall

Tenth Engineering Mechanics Symposium P-55

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Towards enrichment of cohesive zone models for

numerical simulation of interfacial delamination

M. Samimi, B.A.E. van Hal, R.H.J. Peerlings,

J.A.W. van Dommelen, M.G.D. Geers

Department of Mechanical Engineering, Eindhoven University of Technology

e-mail: [email protected], Phone: (40) 247 5392, P.O. Box 513, 5600 MB, Eindhoven

Introduction

Interfacial failure, in the form of delamination, is one

of the most prevalent issues in microsystems which

consist of multiple thin and stacked layers (see figure

1). Here, a cohesive zone model (CZM) is employed

for the prediction of delamination.

Figure 1 : Microsystem.

Solution jump problem

Discontinuities in the numerical solutions of rate in-

dependent models are referred to as solution jumps.

This problem can be avoided by sufficiently refining

the mesh, as shown in figure 2.

However, for realistic interface parameters, i.e. small

size of the process zone in brittle interfaces in com-

parison with other dimensions, the element size has

to be extremely small, which results in high computa-

tional costs.

Figure 2 : Double Cantilever Beam (DCB) test [1].

Enriched cohesive zone model

The separation approximation of cohesive zone ele-

ments in the process zone is enriched by a piece-wise

linear function in order to reduce oscillations such as

those observed in the DCB test without need for fur-

ther mesh refinement or application of complicated

path-following approaches (see figures 3,4). The for-

mulation of bulk elements is enriched as well. The

values of crack tip position, a, and enrichment scaling

factors, ht/b, are driven by the deformation process.

Figure 3 : Enrichment of interpolation functions.

Figure 4 : Deformation of an enriched C.Z. element.

Results: mode I delamination

Considering one bulk and one C.Z. element with sep-

aration only in normal direction, the cohesive zone

opening profile is determined by prescribing the crack

tip degree of freedom at fixed positions (see figure 5).

Figure 5 : C.Z. element separation profile for prescribed

crack tip positions.

Conclusions and remarks

• Piece-wise linear enrichment can be used to im-

prove the efficiency and robustness of a CZM.

• Further steps should be taken to complete the

implementation of the proposed enrichment.

References1. B.A.E. van Hal, R.H.J. Peerlings, M.G.D. Geers (2006) Lo-

cal arc-length control method for cohesive zone modelling,

Submitted.

P-56 Tenth Engineering Mechanics Symposium

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A Qualitative Study of the Void Formation using Ultrasounds during VARTM

** Special thanks to the US Office of Naval Research and CCM, USA

M.K.Saraswat1,2

1Mekelweg 2, Technical University of Delft,2628 CD Delft, Netherlands.

Email: [email protected] of Composite Materials, University of Delaware,

Newark, DE-19716. USA.

INTRODUCTION

Dual scale flow process during a VARTM process results in the dry spots and void formation [Fig 1(a)] in the final composite part, severely affecting quality.

1(a) A composite part with high void content

1(b) An ultrasonic scan of a composite part with high void

content

ULTRASOUND NDE

An ultrasonic apparatus with a 5MHz central frequency transducer was used in the Through Transmission mode. Older sensing techniques such as LRDS, TIS fail to sense micro flow behavior and dual scale flow.

2(a) Sensing techniques like Long Range Distributed Flow Sensing, Tool Integrated Sensing are not useful for micro flow

sensing.

RESULTS1. The dual scale flow behavior is

independent of the choice of fiber preform, resin system etc.

2. The vacuum pressure employed during the infusion process directly influences the void formation with a critical pressure value Pc, below which no or minimal void formation takes place.

3. The physical variable (number of fiber preform layers in this case) do not influence the phenomenon of dual scale

flow.

CONCLUSION

Dual scale flow is the result of the vaporization effect of the resin system due to the drop in the partial pressure of the system during the infusion process.

4. The pressure regulatory experiments clearly indicate the influence of the critical vacuum pressure on the void

formation.

The classical explanation of dual scale flow behavior in terms of migration of the resin system from the inter tow spaces to the fiber mats, does not justify many observations like vacuum pressure dependence of void development.

Tenth Engineering Mechanics Symposium P-57

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Tyre/Road Noise

J.H. Schutte, Y.H. Wijnant, and A. de Boer

Institute of Mechanics, Processing and Control – Twente

Chair of Structural Dynamics and Acoustics, University of Twente

P.O. Box 217, 7500 AE Enschede, The Netherlands

phone +31 53 4893605, email [email protected]

Introduction

Traffic noise is considered to be an important

environmental problem, leads to annoyance and

even health damage. The interaction between the

tyre and road surface causes tyre vibrations and

air displacements, by which sound is generated

(figure 2). Today tyre/road noise is the dominant

noise source for driving speeds exceeding 40 km/h

(figure 1).

Figure 1 : Will tyre/road noise be dominant now?

The introduction of porous road surfaces has reduced

tyre/road noise significantly. After some years,

however, the sound reduction is decreased because

of pollution of the porous layer. Noise reduction

can also be accomplished by optimization of the

tyre. Simplified tyre models have been developed,

which suffer from limited accuracy. Finite element

models are more accurate, but computationally very

expensive at high frequencies.

air-pumping

impact

slip-stick

resonances

rotationradial and tangential vibrations

road

tyre

Figure 2 : Mechanisms of tyre/road noise generation.

Objective

The aim of this research is to develop a numerical

model, which is sufficiently fast to quantitatively

predict tyre/road noise by means of a direct contact

approach.

Methods

The tyre will be modelled by means of finite elements,

suitable for large deformations and anisotropic

material behaviour [1].

The contact will be modelled in the time domain,

because of the inherent non-linearity. A direct contact

approach [2] is used to model the contact between

tyre and road. The convergence is good and friction

between tyre and road is accounted for.

The boundary element method [3] will be used to

calculate the radiated sound from the tyre vibrations.

The road’s influence is taken into account. A

preliminary result is given in figure 3.

1050HzdB

60

70

80

90

100

110

120

Figure 3 : Sound pressure level at 1050 Hz of a slick

tyre on a flat, acoustic hard road (the tyre rolls to the

right) [2].

References1. Ten Thije, R.H.W. et al. (2007) Computer Methods in Applied

Mechanics and Engineering, 196, 3141–3150.

2. Wijnant, Y.H. and de Boer, A. (2006) Proceedings of

ISMA2006, Leuven, Belgium.

3. Visser, R. (2004) PhD thesis, University of Twente,

Enschede, The Netherlands.

The support of TNO and Vredestein within this CCAR project is

gratefully acknowledged. a

P-58 Tenth Engineering Mechanics Symposium

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Thermal effects on two-phase flow in porous media

Xuming Shan1, Tim C. Hooijkaas1, Cor van Kruijsdijk2 and Garth N. Wells3

1Delft University of Technology2Royal Dutch Shell

3University of Cambridgeemail: [email protected]

IntroductionIn thermal recovery processes, heated water or steam isinjected into a reservoir to stimulate oil production. Thisprocess has been modelled numerically in order to de-velop improved insights into the recovery of heavy oil. Thecomplexity of the problem presents numerous modellingand numerical challenges.

Here we simulate temperature-dependent two-phase flowin porous media and invoke the Boussinesq approxima-tion to model buoyancy effects. Using the FEniCS frame-work (www.fenics.org), the development of the compu-tational model has been automated.

ApproachTemperature-dependent two-phase immiscible flow, as-suming incompressibility of the fluids, is governed by:

• Darcy’s law for each phase

vα =krα k

µα

(−∇pα + ραg)

• Pressure equation

∇ ·

((krw

µw

+kro

µo(T )

)k ·∇p−

(krwρw

µw

+kroρo

µo(T )

)k ·g

)= f

• Saturation equation

∂S

∂ t+ v ·∇ f (S) = 0,

where v = vw + vo, and f (S) is the water fractionalflow function, given by

f (S) =krw/µw

krw/µw + kro/µo(T )

• Heat transfer

∂T

∂ t+ v ·∇T −∇ ·α∇T = 0

where S is water saturation, p is the pressure, T is temper-ature, k is the permeability, krα is the relative permeabilityof phase α, which is a funcition of water saturation. µα isthe viscosity of α phase, with µo being temperature depen-dent, since we are considering thermal recovery methodsof heavy oil. The capillary pressure has been neglected.

A finite element method is used, with upwinding in the formof the Streamline Upwind Petrov-Galerkin (SUPG) methodand a residual-based shock capturing operator.

Results

In the steam-assisted gravity drainage (SAGD) process,steam is injected, and rises and warms the surroundingoil. Gravity then causes the mobilised oil to flow downwardto the production well. This process is illustrated in Fig-ure 1, which shows a mixture of oil and water flowing tothe production well.

Figure 1: Total velocity field.

Figure 2 shows that the pressure field is smooth and nodiscernable pressure gradient appears between the injec-tion and production wells, avoiding the movement of in-jected steam directly to the production well.

Figure 2: Pressure distribution.

Figure 3 shows the extent to which oil has been recovered.From these results, we can see that the SAGD process canbe qualitatively simulated using our model.

Figure 3: Water saturation.

Future research

Future work includes two-phase flow in three dimensionsand three-phase flow coupled with poromechanical model.

Tenth Engineering Mechanics Symposium P-59

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The competition between solid-state phase

transformations and plastic deformation:

discrete interfaces and discrete dislocationsJ. Shi1, S. Turteltaub1, J. J. C. Remmers1, E. van der Giessen2

1 Delft University of Technology, Aerospace Engineering

Kluyverweg 1, 2629HS Delft, The Netherlands2 University of Groningen, Applied Physics, Materials Science Center

Nijenborgh 4, 9747AG Groningen, The Netherlands

Introduction

Steels assisted by transformation induced plasticity

(TRIP) are a class of high strength-high ductiltiy mate-

rials. During mechanical loading, TRIP steels gain ad-

ditional strength due to a martensitic transformation.

This phenomenon is strongly coupled to plastic defor-

mations in the austenitic matrix. In order to optimize

the mechanical properties of these materials, it is im-

portant to understand the coupling of the transforma-

tion and plasticity.

Method

The coupled transformation-plasticity problem is de-

composed into three sub-problems (see figure 1):

u 0

t0

boundaryconditions complementary

boundary conditions

= + +

Actual Field Martensitic Field Dislocation Field Complementary Field

Figure 1 : Decomposition of coupled problem

• A transformation problem for individual marten-

sitic plates that can be solved using Eshelby’s

inclusion theory and Mushkelishvili’s potentials.

• A plasticity problem for individual dislocations,

based on the discrete dislocation method [1].

• A complementary problem to account for polar-

ization stresses and actual boundary conditions.

This problem is solved using FEM.

Results

One polycrystal tensile case was simulated to illus-

trate the interaction between martensitic transforma-

tion and plastic deformations. The specimen has a

total of 37 grains with arbitrary polygonal shapes. Two

tensile cases were analyzed as follows:

• Case 1: a benchmark simulation where all

grains correspond to the ferrite phase

• Case 2: a multiphase simulation with four

austenitic grains embedded in a ferritic matrix.

+

+++

+ +

++++

+

+++

+ ++ ++

++++

+

++++

+

+

+

++

­

­­­

­

­

­ ­

­

­

­

­

­­

­

­

­

­­

­­­­

­­­­­

­

­­

­­

Case 1: Ferrite only

σxx (MPa)

+

++++

+

++

+

+++

+

+

+++ +

+

+++ ++++

+ ++++

+

++++

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500

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0

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Case 2: Ferrite and austenite

Figure 2 : Contour plots of axial stress σ11 when ǫ11 = 0.1%

Figure 2 shows the distribution of dislocations (cases

1 and 2) and distribution of martensitic plates (case 2)

for ǫ11 = 0.1%. Three out of the four austenitic grains

have partially transformed into martensite. For case

2, it can been seen that the martensitic transformation

that occurs in the austensitic grains is accommodated

by a significant plastic deformation in the surrounding

ferrite. In contrast, the benchmark simulation (case

1) shows a limited number of dislocations due to the

external loading.

Strain

Ma

rte

nsitic

vo

lum

efr

actio

n

0.0004 0.0006 0.0008 0.0010

0.02

0.04

0.06

0.08

0.1

Martensitic volume fraction (case 2)

Strain

Dis

loca

tio

nD

en

sity

0.0004 0.0006 0.0008 0.0010

10

20

30

40

50

Ferrite only (case 1)Austenite­Ferrite (case 2)

Figure 3 : Dislocation density and martensitic volume fraction

The evolution of the volume fraction of martensite for

case 2 and the change in dislocation density for cases

1 and 2 as functions of the axial strain are shown in

figure 3. The graphics indicate that sudden increases

in the number of dislocations tend to slow down the

transformation process. This is related to a reduction

in the transformation driving force due to plastic dissi-

pation in the surrounding ferrite.

References1. Van der Giessen, E., Needleman, A., (1995) Modelling Simul. Mater.

Sci. Eng., 3: 689-735.

P-60 Tenth Engineering Mechanics Symposium