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Engineering Failure Analysis 14 (2007) 408–416

Study of the fatigue failure of an anti-return valve of ahigh pressure machine

J.M. Alegre a,*, M. Preciado a, D. Ferreno b

a Department of Civil Engineering, University of Burgos, C/Villadiego s/n, 09001 Burgos, Spainb Laboratory of Science and Engineering of Materials, University of Cantabria, Av/Los Castos s/n, 30005 Santander, Spain

Available online 3 April 2006

Abstract

This paper presents a study of the fatigue failure of an anti-return valve, designed to work in the high pressure system(500 MPa) of a high pressure processing machine. To do this, the crack propagation has been simulated by means of thelinear-elastic fracture mechanics approach under mixed-mode loading conditions. From an initial crack, which size isrelated with the microstructure and superficial finish, the crack growth has been simulated using the stress intensity factorsKI and KII of the cracked valve axisymmetric geometry. The crack propagation path has been obtained step by step, apply-ing the criterion of the maximum circumferential stress at the crack tip. The experimental and simulated crack propagationpaths have been compared and, as a consequence of the reliable results obtained, the fatigue life of the valve has been cal-culated using the Paris law of the material with an effective stress intensity factor Keff. The good agreement with experi-mental fatigue life allows to perform new improved designs using the methodology presented.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Mixed model; Fatigue crack growth; High pressure processing

1. Introduction

The main objective of this work is to apply the linear-elastic fracture mechanics approach under mixed-mode conditions to analyse the fatigue failure of an anti-return valve working in the high pressure line(500 MPa) of a food processing machine. The fatigue failure of the valve has been detected after 6750 cyclesin operation. This unexpected failure requires an analysis of the failure origin to be done, in order to improvethe initial design.

The determination of the propagation path of a crack in a part is an essential aspect for the fatigue lifesimulation using fracture mechanics. In this sense, the stress intensity factors must be calculated for the crackevolution in order to determine the number of cycles to failure. Unfortunately, single-mode loadinginfrequently occurs in practice, and, usually, the cracks are subjected to mixed-mode loading conditions;

1350-6307/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfailanal.2006.02.014

* Corresponding author. Tel.: +34 947 259 420; fax: +34 947 259 485.E-mail address: jalegre@ubu.es (J.M. Alegre).

Fig. 1. Dimensions of the high pressure valve analysed.

J.M. Alegre et al. / Engineering Failure Analysis 14 (2007) 408–416 409

consequently, the crack deviates from its original direction. The crack propagation path should be correctlyestimated in order to correctly define the stress intensity factors.

Several methods have been proposed in the literature to model the crack propagation angle under mixed-mode loading conditions [1–7]. Some models are based on the local fields at the crack tip, as the maximumcircumferential stress criterion introduced by Erdogan and Sih [1] or the maximum strain criterion [2]. Othercriteria are based on the energy distribution throughout the cracked part, as the maximal strain energy releaserate criterion introduced by Hussain et al. [3]. Bittencourt et al. [5] have shown that if the crack orientation isallowed to change in automatic fracture simulation, the three criteria provide basically the same numericalresults. Only the mesh-dependence on the results for each criterion should be considered.

In this work the maximum circumferential stress criterion is used to simulate the fatigue crack growth of theanti-return valve, obtaining very good predictions of the fatigue path.

2. Valve description

The valve analysed form as a part of the high pressure piping system of a high pressure processing machine.Fig. 1 shows the general aspect of the valve and its dimensions.

The work cycle of the valve starts with the A and B zones without pressure. A low pressure intensifier pumpgets up the internal pressure of the valve up to 300 MPa (both zone A and zone B). After that, a high pressureintensifier injects water from zone B to zone A, gradually increasing the pressure in zone A. A small ball C,hold in place by a spring and a back support piece, not showed in Fig. 1, avoids the return of the fluid from thehigh pressure zone (A) to the low pressure zone (B).

The material of the valve was an 17-4 PH stainless steel, with a Young modulus of E = 198 GPa and a Pois-son ratio of m = 0.28. This steel has been selected due to its high yield stress, rY = 1200 MPa, combined with agood fracture toughness. The Paris law coefficients of this steel are C = 4.49 · 10�9 and m = 3.15 (da/dN = C Æ DKm) where da/dN is obtained in mm/cycle and DK must be included in MPa

ffiffiffiffimp

.

3. Experimental observations

Fig. 2 shows the fracture surface and the fracture path after 6750 operating cycles. The valve has been axi-ally sectioned and then polished in order to observe the internal fatigue path. The fatigue crack was completelyaxisymmetrical as can be observed in Fig. 2. Several initial cracks produced a circumferential crack thatgrowth due to the high pressure cycles. A detailed analysis of the initiation zone confirmed the presence of

Fig. 2. Fatigue crack evolution observed in the anti-return valve.

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corrosion evidences as can be observed in the micrograph of Fig. 3. In the microscopic analysis no evidences ofprecipitates or significant defects were observed; this fact suggests that the cracks initiation was a consequenceof the high stress intensity in the corner geometry. As the initial design of the valve was machined with a sim-ple drill, without any kind of fillet, the later hypothesis is strongly enforced. Future designs should consider anadequate change in geometry in order to reduce the stress concentrations in this zone.

Fig. 3. Detail of the crack initiation zone.

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4. Numerical computation of the crack propagation

The main characteristics of the finite element algorithms for fatigue life assessment were summarised in Sec-tion 1. In order to verify the numerical procedure developed in this paper, several experimental tests have beenundertaken. The crack propagation obtained from FE simulations have been compared with those experimen-tally obtained.

The crack propagation process can be separated into two basic steps, the crack initiation and the crackpropagation. Many initiation criteria have been proposed in the literature. However, the main goal of thispaper is to study the evolution of a pre-existing small crack, and as therefore, only the second step has beenconsidered in the simulations. Once the process initiates, the crack growth simulation must consider both acrack propagation rate criteria and a crack kinking criteria.

The stress intensity factors KI and KII, or their ranges, are usually used for description of the crack growthrate under mixed-mode cyclic loading. The Paris–Erdogan [8] model is commonly used to describe this fatigueprocess. They proposed the well-known equation in the form:

dadN¼ CDKm ð1Þ

where da/dN is the crack growth per cycle, DK the stress intensity factor range, C and m are material constantsto be found experimentally.

For crack growth under mixed modes I and II cyclic loading, and effective mixed-mode stress intensity fac-tor, DKeff is often used. Tanaka [9] proposed the relationship

DKeff ¼ DK4I þ 8 � DK4

II

� �1=4 ð2Þ

In this paper Eq. (2) has been used in Eq. (1) to evaluate the crack propagation rates.The displacement correlation technique (DCT) [10,11] has been used to compute the stress intensity factor

from the finite element analysis at specific locations. For quarter-point isoparametric plane elements the stressintensity factors can be evaluated by means of:

KI ¼l

jþ 1

ffiffiffiffiffiffi2pL

r4ðvb � vdÞ þ ve � vc½ � ð3Þ

KII ¼l

jþ 1

ffiffiffiffiffiffi2pL

r4ðub � udÞ þ ue � uc½ � ð4Þ

l ¼ E2ð1þ mÞ

j ¼3� 4m for plane strain

ð3� mÞ=ð1þ mÞ for plane stress

�

where E is the Young modulus, m is the Poisson’s ratio, L is the element length, and uj and vi are the nodaldisplacements in the x and y local crack directions, respectively, as presented in Fig. 4.

A concentric mesh around the crack tip together with isoparametric elements have been used to model thestress field singularity, as can be observed in the same Fig. 4. The rest of the part has been meshed with 8-node

θbd

ec

a

/

ac L

ab L 4

==

next crack direction

current crack direction

previous crack direction

Fig. 4. Quarter-points elements at the crack tip.

412 J.M. Alegre et al. / Engineering Failure Analysis 14 (2007) 408–416

quadrangular elements that provide better accuracy than triangular elements, typically used for free-meshareas. The effect of the number and size of elements on the simulation of the path crack has been analysedin order to obtain a result independent of the mesh. As the crack tip moves along the path, the areas thatdefine the crack tip also move changing its orientation but maintaining its initial size.

The crack path is modeled by a spline curve which is defined from the set of points followed by the crack tipas it advances. These points are sequentially stored in an array. The crack propagation is controlled by aparameter, named discrete crack advance Da(k), that control the number of steps. The effect of the discretecrack advance in the crack propagation has also been analysed.

The kinking angle of the propagating crack has been computed using the maximum circumferential stresscriterion, developed by Erdogan and Sih for elastic materials [1]. This model states that the crack extensionstarts at the crack tip and grows in the radial direction in the plane perpendicular to the direction of the max-imum circumferential tensile stress. Mathematically, the criterion leads to the next expression:

KI sinðhÞ þ KIIð3 cosðhÞ � 1Þ ¼ 0 ð5Þ

where KI and KII are the stress intensity factors in Mode I and II, respectively, and h is the crack propagationangle. This criterion also proves the existence of a limit angle corresponding to pure Mode II of h = ±70.54�.An exact solution for Eq. (5) resulting in:

h ¼ 2 tan�1 1

4

KI

KII

� 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKI

KII

� �2

þ 8

s24

35 for KII > 0

h ¼ 2 tan�1 1

4

KI

KII

þ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKI

KII

� �2

þ 8

s24

35 for KII < 0

ð6Þ

where the crack propagation angle h is defined as the angle between the line of the crack and the crack growthdirection with positive value defined in the anti-clockwise direction, Fig. 4.

The maximum circumferential stress criterion may be questionable for large cracks or high loads because inthese cases, the elastic Fracture Mechanics stress fields in the immediate vicinity of the crack tip are onlyapproximate. The existence of a plastic zone at the crack tip modifies the stress distributions, and, therefore,KI and KII are not representative fracture parameters. Only for small scale yielding conditions the crack prop-agation angle estimated by Eq. (6) can be assumed as correct.

The computational models described above have been implemented in the ANSYS finite element software[12]. Fig. 5 presents the algorithm flow-chart. In order to perform a simulation of the crack propagation under

, ,I IIK K θInitial crack size

Calculus of

Automatic geometry generation

(fatigue path : spline defined by i+1 points)

, , ( 3,4,6)I IIK K EqθSolve - Stress analysis

Calculus of

, ( 1),

,i i

Cycles for this step Eq and accumulated cycles

x yNew crack tip location ( )

i = 1

i = i+1

Automatic remesing

Fig. 5. Numerical procedure for the fatigue crack growth simulation.

Fig. 6. Maximum stressed zone (crack initiation).

Fig. 7. Initial angle effect in the crack propagation path (60�,45, y 0�).

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mixed-mode conditions an automatic re-meshing technique should be implemented [13–15]. Assuming aninitial crack position in the part, the procedure models a propagating crack as a spline line formed by the coor-dinates of the crack tip previous positions. For each step, the crack tip is simulated using a rosette of quarter-point elements. The rest of the part is then automatically re-meshed using 8-nodes quadrilateral elements.Then, the program solves the finite element equations, calculates the stress intensity factors using the Displace-ments Correlation Technique (3) and calculates the crack propagation angle (6). After that, a new crack tip isobtained at the calculated angle and at a user-defined crack increment length, Da(k). This cycle is repeated anumber of times specified by the user.

The initial crack is placed in the more stressed region obtained in the un-cracked geometry, as can beobserved in Fig. 6. The effect of the initial crack angle in the fatigue crack path evolution has been analysedconsidering different initial orientations. After a small number of steps the crack path appears independent ofthis initial angle, as can be observed in Fig. 7.

5. Results of the numerical simulations

Following the methodology previously presented, the fatigue crack evolution, from an initial crack size tothe final crack, has been simulated. It is worth noting that the pressure inside the crack faces must be consid-ered. Their effect in the stress intensity factor is essential being the main load in this fatigue process. In Fig. 8four steps of the numerical analysis performed are presented. Fig. 9 shows a macrograph of the valve with acomparison between the experimental fatigue crack and the numerical simulation result. A good agreementcan be clearly observed, that allows the Erdogan–Sih model to be used in the improvement of the fatiguedesign of this component.

Finally, the fatigue life of the specimen has been obtained by adding the number of cycles for all simulationssteps. Using an initial crack of 280 lm a total of 6880 cycles is obtained, that correctly reproduces the

Fig. 8. Fatigue crack simulation under mixed-mode conditions.

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experimental observation. This initial crack size can be related with the surface damage generated by the dril-ling process, as can be observed in Fig. 3. In Fig. 10 the stress intensity values obtained by the finite elementanalysis used for the fatigue life prediction, are presented.

Fig. 9. Simulation and experimental profiles of the fatigue crack path.

Fig. 10. Stress intensity factors versus the crack length.

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6. Conclusions

In the present paper the fatigue behaviour of an anti-return valve has been modelled using the linear-elasticfracture mechanics approach under mixed-mode load conditions. In order to predict the fatigue crack growththe maximum circumferential stress criterion developed by Erdogan–Sih has been used in an automatic pro-cedure implemented in the ANSYS finite element code. Extensive information about the crack growth proce-dure together with recommendations and details about the mesh have been provided.

The results obtained in the numerical simulations permit the Erdogan–Sih model to be used for predictingthe fatigue crack growth of this component. It has been observed that the high pressure actuating in the crackfaces is the main load for the stress intensity calculations. For the fatigue life an initial crack of 280 lm hasbeen obtained as the numerical value for the fatigue life simulations using the Paris law for the crackpropagation.

The steps presented here are essential to improve the initial geometry of the part in order to reduce the num-ber of prototypes proposed in the stage of design. The procedure presented here can be applied to other geom-etries and materials. However, plasticity considerations should be taken into account when the materialpresents ductile behaviour and high loads are applied. This plasticity can make the linear-elastic fracturemechanic predictions not adequate for fatigue crack path.

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Acknowledgements

This research has been supported by a collaboration project (BU-W07E-06) between NC-Hyperbaric andthe Structural Integrity Group of the University of Burgos (Spain).

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