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                                                           R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46 
 

405 
 

 Focussed on Crack Paths 
 
 

 
  
Crack path simulation for cylindrical contact  
under fretting conditions  
 
 
R.A. Cardoso, J.A. Araújo, J.L.A. Ferreira, F.C. Castro  
University of Brasília, Brasil 
rapha_2213@hotmail.com  
 

 
ABSTRACT. In this work different strategies to estimate crack path for cylindrical contacts under fretting 
conditions are carried out. The main goal is to propose and to evaluate methodologies not only to estimate the 
direction of crack initiation but also the subsequent propagation in its earlier stages, where the stress field is 
multiaxial, non-proportional and decays very fast due to the proximity with the contact interface. Such complex 
conditions pose a substantial challenge to the modelling of crack path. The numerical simulations are provided 
by a 2D Finite Element Analysis taking into account interactions between the crack faces. The results show that, 
under fretting conditions, models based on the critical plane method are not effective to estimate the crack 
initiation orientation, while models based on a so called “critical direction” applied along a critical distance 
provide better results. Regarding the subsequent crack propagation orientation, it was possible to see that stress 
intensity factor based models where one considers an infinitesimal virtual crack emerging from an original pre-
existent crack are powerful mechanisms of crack orientation estimation.     
  
KEYWORDS. Fretting fatigue; Crack propagation; Critical distances. 
 
 
INTRODUCTION   
 

retting is the phenomenon that occur in mechanical couplings subjected to contact loads and small relative 
tangential displacement due to oscillatory loads. In association with remote fatigue loads, the process well known 
as fretting fatigue take place. Failure due to fretting fatigue usually is observed in engineering assemblies under 

vibration, such as riveted or bolted connections, dovetail joints in turbines and overhead conductors among, others. 
The fretting problem is characterized by a strong stress concentration in association with wear, which invariably leads to 
the nucleation of small cracks. Depending on the level of the stress gradient in such cracks they may arrest, since a 
threshold value at the crack tip be reached [1]. In order to conduct such analysis criteria able to precise the location, 
direction of initiation and the further crack path are essential.  In these aspects many challenges must be faced, since the 
fretting problem is characterized for high levels of multiaxility and non-proportionality  
The aim of this work is to estimate the crack path under fretting conditions in cylindrical contacts assessing some already 
used methodologies and propose a new methodology to estimate the direction of crack propagation in their earlier stages. 
Experimental data from literature will be used to confront the theory. 
 
 
CRACK ORIENTATION MODELING  
 

s extensively shown in the literature, crack initiation and further propagation is usually characterized by two 
distinct stages, Fig. 1. The Stage I describes the earlier stage of propagation, commonly governed by Mode II 
while the Stage II is characterized by a Mode I dominated crack. However, according to [2], Stage I may be 

F 

A 



 

R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46                                                             
 

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divided in two groups: type 1 when crack initiation is mainly influenced by the range of the shear stress Δτ, and type 2 
when the range of normal stress Δσn governs the process.  

 
 

Figure 1: Scheme of classical stages of crack initiation and short crack propagation. 
 

In [3], where tests were conducted in low carbon steel weakened by U-notches to study the high cycle fatigue cracking 
behaviour, was verified that Stage I, in this kind of stress raiser, is governed by a mixed mode, since the crack profile in 
Stage I was not parallel to the notch bisector line. It was also noted that the irregular mixed mode dominated path was 
confined within a distance from the notch tip of the order of L, the material characteristic length, Eq. (1). 
 

 

2

1

1













thK

L                                         (1) 

 
where Δσ-1 is the plain fatigue limit and ΔKth is the threshold value of stress intensity factor range, both for a load ratio of 
-1. Considering these assumptions, two multiaxial criteria based on the theory of critical distances will be applied to 
estimate the initial crack direction, Fig. 2.  
 
Early stage of crack propagation-The critical direction method combined with the TCD (the critical direction method) 
The theory of critical distances (TCD) [4] can be used in association with any fatigue criterion [5]. So based on the 
hypothesis that in stage I one may have cracks dominated by the maximum range of normal stress (crack type 2), the crack 
orientation will be defined here by a line of length 2L, inclined by an angle θσ with respect to the y axis, which maximizes 
θ in Eq. (2), Fig. 2a. Therefore, an important aspect that one should notice in Eq. (2) is that the maximum range of 
normal stress is always computed perpendicular to the line defined by angle θ, i.e, the line 2L (inclined by θ) is discretized 
in many material points and the maximum normal stress range is calculated for the same plane θ for all these points. This 
procedure provides the critical orientation θσ that a crack of length 2L would have in theory in its stages of initiation and 
early growth. A similar approach can be used to find the direction that minimizes the shear stress amplitude, Δτ, Eq. (3). 
The mechanical basis to support this hypothesis is the fact that, in these planes, less energy is wasted with friction and 
consequently more energy is available for crack propagation [7]. 
 

 








  drr

L

L

),(
2

1
max

2

0

                                        (2)  

 

 








  drr

L

L

r ),(
2

1
min

2

0

                            (3) 

                                
In Eqs. 2 and 3 the integrals are carried out over a constant length 2L, as defined by the so-called Line Method (LM). In a 
similar way one could assume the point or the area method (for 2D problems) [6], but these will not be assessed here due 
to space constraints.  
 
 



 

                                                           R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46 
 

407 
 

 
                                                 (a)                                                                                             (b) 
 

Figure 2: (a) Stress state in the vicinity of the critical zone by means of line method in cylindrical coordinates; (b) critical plane 
according to the MWCM. 
 
Early stage of crack propagation-The critical plane method combined with a multiaxial fatigue criterion applied in terms of the 
TCD (the critical plane method) 
In this case, one assumes that crack initiation is governed by the range of shear stress Δτ (crack type 1). Considering a 
critical plane approach, such as the Modified Wöhler Curve Method (MWCM), crack initiation is expected to take place 
on the material plane that experiences the highest shear stress amplitude. As the shear stress always has the same 
magnitude in two orthogonal planes, the critical plane between these two planes will be the one with the maximum 
normal stress along a cycle. Fig. 2b depicts the critical plane method in association with the point method (PM) to 
estimate the direction of crack propagation. Once again, in order of brevity only the PM will be assessed here. 
 
Crack propagation orientation under fretting conditions 
In fretting, the load conditions are complex and non-proportional, which makes the process to estimate the directions of 
propagation hard. Now a review of some relevant criteria already used to estimate the direction of crack propagation 
under fretting conditions will be presented. Notice that all these criteria are valid for pre-existing cracks (stage II). 
Classical crack path criteria, such as the maximum tangential stress (MTS) [8], the maximum strain energy density [9] or 
the maximum energy release rate [10] are not adequate for non-proportional multiaxial loading like in fretting conditions. 
Taking into account this non-proportionality [2] and [11] considered the following criteria based on [12]: 
(i) Crack propagates in the direction where k1(θ,t) is maximum during a cycle 
(ii) Crack propagates in the direction where Δk1(θ) is maximum during a cycle 
(iii) Crack propagates in the direction where da/dN(θ)is maximum during a cycle 
where k1 and k2 are the stress intensity factors in mode I and in mode II, respectively, of an infinitesimally small kinked 
crack emerging from the pre-existent crack with an angle θ, Fig. 3. The expression that relates k1 and k2 with the classical 
mode I and mode II stress intensities KI and KII is given by Eq. (4), where the angular functions Kij(θ) are reported in [11] 
and [13].  
 

 
 

Figure 3: Inclusion of a small virtual crack emerging from the crack tip of a pre-existing crack. 
 

 
)(

)(

)()(

)()(

),(

),(

2221

1211

2

1

tK

tK

KK

KK

tk

tk

II

I







                                                           (4) 

 
Based on the MTS criterion, Dubourg and Lamacq [2] proposed that the direction of crack propagation may be found 
searching for the direction that maximizes Δσθ along a cycle, considering that if σθ < 0 then σθ = 0, once that compressive 
stress does not encourage crack propagation. Recently Giner et al. [7], proposed the criterion of the minimum shear stress 
range, which consists in finding the plane that minimizes the shear stress range at the crack tip, Fig. 4. As the shear stress 



 

R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46                                                             
 

408 
 

always has the same magnitude in two orthogonal planes, the chosen plane for crack propagation direction is the one that 
experiences the maximum Δσn (less frictional energy is lost).  
 

 
Figure 4: Plane of the minimum shear stress at the crack tip. 

 
 
AVAILABLE DATA 
 

o evaluate the accuracy from the aforementioned methodologies experimental data available from tests under 
fretting conditions will be assessed. All the tests considered are from cylinder-on-plane configuration.  Tests were 
conducted on steel alloy 35NCD16 and AISI 1034 specimens. In both cases the pads were made from different 

materials, steel 100C6 and chromium 52100 steel, respectively. The mechanical properties are presented on Tab. 1, and the 
experiments have been reported and discussed in detail in [14] and [15], hence, only essential information necessary to 
carry out the analyses are briefly presented here.  
 

Material E (GPa) v σY (MPa) σu (MPa) σ-1 (MPa) τ-1 (MPa) ΔKth (MPa.m1/2) 

AISI 1034 200 0.3 350 600 270 - 7 

52100 210 0.3 1700 2000 - - - 

35NCD16 200 0.3 1127 1270 590 460 - 

100C6 195 0.3 1500 - - - - 
 

Table 1: Material properties. 
 
The load history applied in the tests are as follows. Firstly, a constant normal load, P, was applied to the fretting pad and 
held constant. Then an oscillatory shear force varying between ±Qa was applied. In the tests analysed here there is absence 
of a bulk load. All the tests were designed to run in partial slip regime, i.e., Qa/P<f where f is the coefficient of friction in 
the slip zones. One test will be analysed for AISI 1034 specimen and another for steel alloy 35NCD16. For AISI 1034 the 
tests analysed presented short crack arrest, hence, this data will be used only to assess the methodologies to estimate crack 
initiation path. The data from 35NCD16 will be used to assess methodologies in both the initiation and propagation 
stages. The most important parameters in the load history to carry out the analyses are reported in Tab. 2.  
 
 

Specimen P (N/mm) Qa (N/mm) R (mm) f 
AISI 1034 227 169 40 0.9 
35NCD16 1000 500 80 0.9 

 

Table 2: Parameters of the test; R is the pad radius. 
 
In both test programs after interrupting the experiment the crack path and its extension were observed in an optical 
microscope by cutting the samples in the centre of the contact zone [16]. In case of AISI 1034 the tests were conducted 
until 106 cycles, for 35NCD16 three tests were performed and each of them were interrupted after a different number of 
cycles and analysed. The crack extension and profile are thus obtained from information collected from different tests. 

T 



 

                                                           R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46 
 

409 
 

The results from these tests are reported in Tab. 3 and the crack path is described according to the scheme sketched in 
Fig. 5.  
 

 
 

Figure 5: Crack path scheme.  
 
 

AISI 1034   
1,000,000 cycles 1=68 μm θ1*=30° 
 
35NCD16 

  

100,000 cycles  1=105 μm  θ1=17° 
250,000 cycles  2=130 μm  θ2=13° 
500,000 cycles  3=248 μm  θ3=11° 

* The angle measured by [14] was established through a series of tests, where an angle of 30° ±3 was observed. 
 

Table 3: Crack path configuration. 
 
 
RESULTS FOR CRACK INITIATION ORIENTATION 
 
Crack initiation direction for AISI 1034 

n this case we have a material characteristic length given by 53.5 μm, Eq. (1). The results of the critical direction 
method are shown in Fig. 6. As can be seen, the estimated angle is very small when the maximum value of Δσn,eff  
(only positives values of normal stress are taken into account) is considered, whereas max(Δσn) and min(Δτ) provide 

similar results. The range of angles that were investigated were limited to -60° to 60°, as (i) it doesn’t make sense physically 
to assume crack propagation in directions larger than this (shallow angles practically parallel to the surface) (ii) it reduces 
the computational cost involved in the calculation when more complicated geometries that requires FEA analysis are 
assessed.  
 

 
 

Figure 6: The critical direction method for AISI 1034 data. 
 
Considering the complexity of the stress field and the dispersion involved in the calculation of material parameters as L, 
the critical direction criteria based on max(Δσn) and min(Δτ) provided quite reasonable results in the estimative of the 

I 



 

R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46                                                             
 

410 
 

crack initiation path. However, one should notice that the criterion based on Δσn is inconsistent, since compressive 
stresses (usually present during a large portion of the fretting load cycle) do not contribute to crack propagation.  
 

 
 

Figure 7: Critical plane model applied in the centre of the process zone for AISI 1034. 
 
Fig. 7 illustrates the results obtained by the critical plane model applied in the centre of the process zone (L/2). Note that 
the maximum shear stress amplitude is reached in two orthogonal planes (-31° and 59°), but the critical plane is the one at 
-31° once, between these two planes, it experiences the maximum normal stress during a cycle. Therefore, it is clear that 
the crack initiation path estimated by this method provides a direction, which is opposite to the actual one (30°). Similar 
kind of inaccuracy to determine crack initiation path from critical plane approach was obtained by Susmel and Taylor [3] 
for notched components under mixed mode loads. 
 

 
                                                                      (a)                                                                        (b) 

 

Figure 8: Results for 35NCD16 (a) the critical direction method; (b) the critical plane model applied in the centre of the process zone. 
 
Crack initiation direction for 35NCD16 
Unfortunately, the threshold value for the stress intensity factor range, ΔKth, is not available for steel 35NCD16, hence, an 
estimate based on the work of [17] was done, where the material characteristic length for a steel is given by: 
       

  

17.1

1

36200











 u

L                         (5) 

 



 

                                                           R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46 
 

411 
 

where L is given in mm, Δσ-1  is the plain fatigue limit (MPa) for a load ratio of -1, and σu is the ultimate tensile strength 
(MPa). The empirical equation has a correlation factor of 0.791, among a wide range of materials data for steels and cast 
irons. The results for the critical direction method and the critical plane model for the 35NCD16 are shown in Fig. 8. 
 

Material max(Δσn) max(Δσn,eff) min(Δτ) Critical Plane 
AISI 1034 20° 4° 19° -31° 
35NCD16 8° 4° 4° -40° 

 

Table 4: Results for crack initiation orientation. 
 
Note that Fig. 8b and Tab. 4 reinforce the previous result that in this kind of load conditions the critical plane model lead 
to severe mistakes in the estimation of the initial crack angle. Considering max(Δσn)  a better response was obtained again, 
however, as explained before this variable lead to the violation of a basic principle of the fracture mechanics (cracks 
should not propagate when they are closed). Therefore, considering this aspect and the results obtained for both materials 
the criterion chosen to predict the first orientation of the crack propagation hereinafter will be the critical direction 
method combined with the minimum shear stress range.  
 
 
RESULTS FOR CRACK PROPAGATION 
 
Contact modelling and stress intensity factor achievement 

or each plane fretting test configuration described in Tab. 2, a 2D-plain strain FEM was implemented in Abaqus. 
The models were composed from a cylinder submitted to normal and tangential loads pressed against a fixed 
plane, Fig. 9a. Contact interactions were described by Lagrange multiplier with a friction coefficient f. The crack tip 

zone was meshed in a round domain of 5 μm radius with a mesh of approximately 0.5 μm quadrilateral linear elements, 
Fig. 9b. The mesh in the contact zone was refined down to 20 μm by triangular linear elements. The coefficient of friction 
between crack faces was assumed to be the same one that occur in the slip zones for the cylindrical contacts in the fretting 
tests. The stress intensity factor was obtained by a contour integrals Abaqus routine.  
 

 
                                                                      (a)                                                                                            (b) 

 

Figure 9: FEM modelling (a) contact configuration; (b) illustration of the mesh refinement at the crick tip. 
 
Crack propagation modelling 
As described previously the crack propagation path was assessed only for 35NCD16 due to the fact that only in this case 
we have the experimental crack path profile for a long crack. Therefore, from the initial angle of propagation estimated, 
simulations were carried out with increments of crack length, Δb, where in each step the stress intensity factors mode I 
and II along a period were extracted as well as the stress field near to the crack tip. The direction of crack propagation in 
each step was defined applying two different methodologies, namely, the max(Δk1(θ)) in an infinitesimally kinked crack 
emerging from the original one or searching the material plane where the minimum shear stress range at the crack tip is 
reached[7], min(Δτ). The first step to apply this model is to define an initial crack (length and direction) in the specimen. 
In this setting, it was assumed that this crack had an initial length Δb=70µm and its initiation angle was given by the critical 
direction method in terms of min(Δτ), i.e., 4°. Fig. 10 shows the crack path simulation considering both criteria. As shown 
in [7] the increment size does not exert significant influence in global aspects, hence, a relatively high increment was 
considered to simplify the analysis. 

F 



 

R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46                                                             
 

412 
 

 

 
 

Figure 10: Results for crack path simulation, ∆b=70μm. 
 
Note that although the initial estimate of crack initiation path was 4° while the actual path was 17°, the direction of crack 
propagation obtained via max(Δk1) tends to the location of the real crack as it becomes larger. However, the min(Δτ) 
criterion leads to a complete different path that has no correlation at all with the experimental result. 
 
 
CONCLUSIONS 
 

ome methodologies to predict the direction of crack initiation and further propagation under fretting conditions 
were assessed, where available experimental data were used to evaluate the accuracy of the methodologies. Two 
methods based on critical distances to estimate the crack initiation planes were studied. The critical direction 

method, provided better results than the critical plane model, which was not able to produce any reasonable estimates of 
crack initiation direction. In order to predict the crack propagation path two distinct methodologies were applied, one 
based on the maximum SIF range mode I in an infinitesimal kinked crack with origin in the pre-existent crack and another 
based on the stress field near to the crack tip. The criteria based on stress field, min(Δτ), led to inconsistent predictions, 
whereas the criteria based on SIF, provided good agreement with the experimental response, even from a not so good 
choice for the direction of crack initiation.  Further work mainly for earlier stages of crack birth need to be conducted in 
order to establish more accurate estimates of crack path in fretting conditions. 
 
 
REFERENCES 
 
[1] Araújo, J.A., Nowell, D., Analysis of pad size effects in fretting fatigue using short crack arrest methodologies, Int J 

Fatigue, 21 (1999) 847-856. 
[2] Dubourg, M.C., Lamacq, V., Stage II crack propagation direction determination under fretting fatigue loading: a new 

approach in accordance with experimental observations. In: Hoeppner DW, et al., editors. Fretting fatigue: current 
technology and practices, ASTM STP 1367, West Conshohocken; (2000) 436-450. 

[3] Susmel, L., Taylor, D., Non-propagating cracks and high-cycle fatigue failures in sharply notched specimens under in 
phase Mode I and II loading, Eng F Analysis, 14 (2007) 861-876. 

[4] Taylor, D., Geometrical effects in fatigue: a unifying theoretical midel, Int J Fatigue, 21 (1999) 413-420. 
[5] Susmel, L., Taylor, D., Can the conventional high-Cycle multiaxial fatigue criteria be re-interpreted in terms of the 

theory of crical distancies: SDHM, 2 (2006) 91-180.  
[6] Taylor ,D., The Theory of Critical Distances. Oxford: Elsevier. 1st edition, (2007). 

S 



 

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[7] Giner, E., Sabsabi, M., Ródeans, J., Fuenmayor, J.F., Direction of crack propagation in a complete contact fretting-
fatigue, Int J Fatigue, 58 (2014) 172-180. 

[8] Erdogan, F., Sih, G.C., On the crack extension in plates under plane loading transverse shear, J Basic Eng, 85 (1963) 
519-27.  

[9] Sih, G.C., Strain-energy-density factor applied to mixed mode crack   problems, Int J Fract, 10(3) (1972) 305-321. 
[10] Palaniswamy, K., Knauss, W.G., Propagation of a crack under general, in-plane tension, In J Fract,8(1) (1972) 114-

117. 
[11] Ribeaucourt, R., Baietto-Dubourg, M.C., Gravouil, A., A new fa-tigue     frictional contact crack propagation model 

with the coupled X-FEM/LATIN method, Comput Methods Appl Mech Eng, 196 (2007) 3230–3247. 
[12] Hourlier, F., Pineau, A., Fatigue crack path behavior under complex mode loading. In: Francois D, editors. Advances 

in fracture research, Proc 5th Int Conference on Fracture. Pergamon, Oxford; (1981) 1841-1849. 
[13] Amestoy, M., Bui, H.-D., Dang, Van K., Déviation infinités d’une fissure dans une direction arbitraire. CR Acad Sci 

Paris, 289B (1979) 99-102.  
[14] Fouvry, S., Nowell, D., Nubiak, K., Hills, D.A., Prediction of fretting crack propagation based on a short crack 

methodology, Engineering Fracture Mechanics, 75 (2008) 1605-1622. 
[15] Baietto, M.C., Pierres, E., Gravouil, A., Berhtel, B., Fouvy, S., Trolle, B., Fretting fatigue crack growth simulation 

based on a combined experimental and XFEM strategy, Int J Fatigue, 47 (2013) 31-43. 
[16] Proudhon, H., Fouvy, S., Determination and prediction of the fretting crack initiation: introduction of the (P,Q,N) 

representation of a variation process volume, Int J Fatigue, 28 (2005) 707-713. 
[17] Susmel, L., Atzori, B., Meneghetti, G., Material fatigue properties for assessing mechanical components weakened by 

notches and defects, Fatigue Fract Engng Mater Struct, 28 (2005) 1-15.  
 
 
 
















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    /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke.  Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice