20 Material properties. Stochastic fatigue limits and fatigue life variability in oriented PVC pipe.pdf

Polymer Testing 31 (2012) 304

–

311

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Polymer Testing

journal homepage: www.elsevier.com/locate/polytest

Material properties

Stochastic fatigue limits and fatigue life variability in oriented

–

PVC pipe

D.B. West 1 , R.W. Truss *

School of Mechanical and Mining Engineering, The University of Queensland, St Lucia, Brisbane Qld 4072, Australia

article info

abstract

Article history:

Received 4 November 2011

Accepted 22 December 2011

Fatigue data has been collected on oriented polyvinyl chloride (PVC-O) pipe samples up to

fatigue lifetimes of 10 7 cycles. A random fatigue limit model based on the work of Pascual

and Meeker has been used to establish the fatigue limit in this material. This model

assumes that the fatigue limit is a stochastic quantity and that the fatigue life is conditional

on the fatigue limit. Moreover, it allows the use of censored data in establishing the fatigue

limit magnitude. Although the most appropriate distributions for the fatigue life and the

fatigue limit were dif

Keywords:

Oriented PVC pipes

Fatigue

Random fatigue limit

cult to determine unambiguously, the presence of a fatigue limit in

PVC-O was established. This statistical analysis of the fatigue data allowed failure proba-

bility quantiles to be calculated for PVC-O pipe materials.

1. Introduction

limit is a stochastic quantity and that fatigue life vari-

ability is affected by the variability of a material

sfatigue

limit. Fatigue limits for PVC-U and chlorinated poly-

ethylene modi

’

In many modern pipeline designs, unplasticised poly-

vinyl chloride (PVC-U) has been superseded by other PVC

based materials. This work is concerned with one of these,

oriented PVC (PVC-O). Molecular orientation is achieved in

PVC pipe by increasing the diameter of conventionally

extruded pipe at elevated temperature and cooling rapidly

[1] . This has been shown to signi

ed PVC (PVC-M) have been proposed

previously [10

12] but without consideration of fatigue

limit variability. If the fatigue limit is a stochastic quantity,

as is the case with all other material properties, then

designing on the mean or median fatigue limit assumes an

approximate 0.50 probability (depending on the distri-

butions used) that an individual sample will fail. An

important feature of the RFL model is that it allows

conservative design based on a selected failure probability

at high stress amplitudes and stress amplitudes in the

region of the fatigue limit.

–

cantly increase strength

in the direction of orientation [2

4] and to provide superior

fatigue performance compared with PVC-U [5,6] . Although

PVC-O pipes have been in use for over 20 years, there is

limited fatigue life data, especially for lifetimes over 10 6

cycles. This paper presents fatigue life data for PVC-O using

the hoop and D-block geometry and a novel method for

increasing the data production rate.

The fatigue data will be analysed using the Random

Fatigue limit (RFL) model introduced by Pascual and

Meeker [7] . In a previous paper [8] , the RFL model was

shown to work well for the PVC-U pipe fatigue data of

Joseph and Leevers [9] . The RFL model assumes the fatigue

–

2. Experimental methods

2.1. Materials

The PVC-O pipe used was Vinidex Supermain PVC-O 500

Series 2 DN150 PN16 made in accordance with the

Australian/New Zealand Standard AS/NZS4441. Sections of

the PVC-O pipe were mounted in a lathe and surface

damage removed using wet 400, 800 and 1200 grit silicon

carbide papers. Hoops were parted at nominally 12.5 mm

widths from the polished pipe.

61 7 33653729.

E-mail address: r.truss@uq.edu.au (R.W. Truss).

1 Present address: Origin Energy Australia, Level 2, 12 Cribb St, Milton

Qld 4064, Australia.

Corresponding author. Tel.:

0142-9418/$

doi: 10.1016/j.polymertesting.2011.12.011

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see front matter

D.B. West, R.W. Truss / Polymer Testing 31 (2012) 304

311

305

–

Commercially produced pipes display dimensional

variability and tolerances for out-of-roundness and wall

thickness are given in appropriate standards. For the pipes

used here, the wall thickness had a mean of 4.76 mm with

a standard deviation of 0.131 mm. The outside diameter

had a mean of 177.60 mm (s.d. 1.00 mm) while the inner

diameter had a mean of 172.84 mm (s.d. 0.99 mm).

maximum load speci

ed for the fatigue cycle. The hoops-

in-series

jig was used for

stress amplitudes below

9.5 MPa at R

0.1. Up to four hoops were tested concur-

rently using the jig. The hoop samples were tested on

a servo-controlled INSTRON 8031 testing machine at room

temperature (22

1 C) at 5 Hz.

The stress in the hoops, s, was calculated as the simple

section stress in the region between the D-blocks i.e.

2.2. Fatigue testing methods

2tw

(1)

Fatigue testing of PVC-O pipes presents an experimental

challenge. Ideally, direct pressure testing of full pipes

should be used but this requires signicant investment in

time and equipment. Fabrication of

where P is the applied load, w is the width of the pipe ring

and t is the wall thickness. This approach ignores the small

bending moment generated in the gap between the D-

blocks which would impart a slight tensile stress on the

inside surface, but this is generally considered to be less

than 10% of the section stress. To avoid overestimation of

the stress amplitude, an upper bound of the wall thickness

was used equal to the mean plus 2 standard deviations.

Hence, in this work, the stress amplitude data should be

interpreted as a lower bound. Since the wall thickness was

found to be normally distributed, the actual stress at the

point of failure should be above the calculated lower bound

stress for 95% of fatigue samples.

at standard test

geometries requires heating or deformation of the pipe

which will alter the properties of the pipe material.

Moreover, the direction of crack propagation in oriented

plastics often aligns with the molecular orientation. Thus

cracks that start on the inside or outside surface of an

oriented pipe deviate from a purely radial direction and

tend to propagate in a partially circumferential direction.

This behavior makes crack growth rate experiments prob-

lematic both experimentally and because the crack growth

direction results in mixed mode stress intensity factors at

the crack tip. Consequently, in this work, conventional

stress amplitude cycles to failure (S-N) curves were

generated using the hoop and D-block geometry used by

Moore et al. [13] which allows a tensile

2.3. The random fatigue limit model

The fatigue life data generated for the PVC-O pipes was

analysed using the Random Fatigue Limit (RFL) model

developed by Pascaul and Meeker [7] and applied in

a previous paper to fatigue data for unoriented unplasti-

cised PVC pipe [8] . Details of the RFL model can be found in

these two references.

The RFL model is a failure-time regression model. The

data is assumed to have a particular statistical distribution

and the model parameters are

tensile fatigue load

cycle to be applied to a full hoop specimen cut from a pipe.

In the hoop and D-block method, the cross-section of

the hoop is generally reduced at the D-block gap to elevate

stress and initiate a fatigue crack at a controlled location.

For PVC-O, however, crack growth is de

–

ected towards the

direction of molecular orientation (hoop direction) result-

ing in the crack tip propagating away from the minimum

section, which decreases the stress amplitude experienced

by the crack tip. In this work, hoops with constant section

were used. While this resolved the problem of non-

constant stress amplitude due to non-radial crack growth,

the location of crack initiation could not be determined

prior to the experiment.

The

tted to the data using

maximum likelihood estimation (MLE). The best

t model

is established by examining the likelihood value,

the

distribution

t via probability (P-P) plots and the deviation

of the model from the data via standardized residual plots.

3. Results and discussion

nal experimental challenge is that, due to the low

thermal conductivity of polymer samples, low frequencies

must be used to avoid adiabatic heating. This is the primary

reason for the lack of long term data for plastic pipe

materials beyond 10 6 -10 7 cycles. The RFL model assists here

in that it allows the use of censored (no failure) data to

establish the fatigue limit. In addition, a novel method for

increasing the data production rate was developed and

used in this work. The method consisted of testing

a number of D-block sub-assemblies in series using

a specially designed slotted connector between the top and

bottom D-block in each station, as shown in Fig. 1 . This

allowed for de

The majority of samples failed from the inside surface at

the gap between the D

blocks while approximately 30% of

samples failed from the outside surface. Crack initiation

was observed on the outside surface of the hoops between

–

45 from the horizontal and on the inside

surface in the gap between the D-blocks. The cracks did not

propagate in the radial direction but at

15 and

80 to the radial

direction. These cracks have been termed delamination

cracks as the cracks were diverted towards the direction of

molecular orientation in the pipes.

ection of the pipe sample during testing but

provided a rigid connection when the sample failed. The

program Wavemaker Pro was used to control the servo-

hydraulic machine. On detection of a hoop failure, the

program caused the hydraulic actuator to reduce the load in

a controlled fashion before bringing the load back to the

desired level. Monitoring of the load showed that the load

on the remaining hoop samples never exceeded the

3.1. Failure position

The inside surface between the D-blocks was the ex-

pected failure position because of the slight bending

moment due to the straightening of the hoop sample on

loading. However,

30% of the samples did not fail at this

position but rather from the outside of the hoop sample.

306

D.B. West, R.W. Truss / Polymer Testing 31 (2012) 304

–

311

Fig. 1. D block assembly

tted to the Instron. Inserrt shows the slotted connected that allowed deformation of the pipe ring during testing but provided a rigid

connector on failure of the hoop.

This may have resulted from a number of factors. It is

assumed that the failure point will be a function of both the

3.1.1. Wall thickness and ovality

Although the dimensions of the pipe were within the

limits set by the standard, small variations in wall

thickness and ovality are expected in commercial

production and will produce variation in the stress

around the hoop sample. Since the D-blocks have

aw distribution and the stress amplitude at any point. The

aw distribution was dif

cult to quantify but the most

severe

aws were likely to be on the surface of the pipe due

to handling damage. Although the outside surfaces of the

pipes were polished before testing to remove surface

xed

radius, the ovality introduces bending into the pipe as the

pipe assumes the D-block shape under load. To quantify

this effect, strain gauges were mounted on the outside of

three pipe samples, R1, R2, R3, at the points of maximum,

D max , and minimum, D min , pipe diameter. The rings were

loaded in the D-blocks and the stress recorded. The rings

were then rotated with respect to the D-block and the

loading repeated to assess the effect of position on the

stress in the ring. Fig. 2 shows the measured load as

a function of angle measured from the horizontal position

aws,

some

aws may have remained undetected and caused the

failures. An alternative explanation is that there may have

been variation in the stress amplitude around the hoop

sample resulting from either residual stresses or variations

in wall thickness and ovality. In addition, it is possible that

the orientation of the pipe may have varied slightly with

changes in pipe dimensions. This would affect the

mechanical properties of the pipe material, which in turn

may have in

uenced the position of failure.

D.B. West, R.W. Truss / Polymer Testing 31 (2012) 304

311

307

–

To assess the variation in elastic constants, and thus

orientation, samples were cut from the axial, radial and

hoop planes and at 45 between these planes at position of

maximum and minimum diameter. A pulse-echo technique

using a piezo-electric transducer that acts as both trans-

mitter and receiver was used to measure the elastic

constants of these specimens and to establish the elastic

constants matrix for the oriented pipes. The results are

shown in Table 1 . The variation in elastic constants between

points of maximum and minimum diameter was found to

be within the experimental errors of these measurements

so it is considered unlikely that variations in orientation

were the cause of different failure positions.

3.1.3. Residual stress

Due to different cooling rates on the inside and outside

of pipes, most commercial pipes retain a level of residual

stress and, in the case of PVC-O, there may be residual

stress both in the hoop and axial directions. Residual stress

was estimated in two ways. Clutton and Williams [14]

described a method of measuring the circumferential

ection of axially slit pipe hoops of different lengths.

They gave the circumferential de

ection, d , for a pipe of

radius, r, and wall thickness, t, as

d ¼

24 p r 2

n M a F

b L

Et 3

M h

ÞÞ =

(2)

s modulus, n is Poisson ’ s ratio, M h is the

hoop bending moment and M a is the axial bending

moment. F( b L) is a function of the hoop specimen length,

pipe dimensions and n . A series of PVC-O pipe hoops of

different lengths were slit in the axial direction and the

circumferential de

where E is Young

’

Fig. 2. Outside surface stress calculated from strain gauge measurements on

PVC-O hoops at applied loads of a) 100 N; b) 4500N. (R1, R2, R3 designates

data from three separate pipes).

ection measured. Using equation (2) ,

the hoop residual stress was found to be 1.41 MPa and the

axial residual stress was - 4.33 MPa. The Clutton and Wil-

liams analysis assumes a linear stress pro

(gap point in the D-blocks). Included on the graphs are

the values of nominal stress applied using equation (1)

and the upper bound to the wall thickness discussed in

section 2.2 above. At low loads (100N), position had

negligible effect on the measured stress. Two of the three

pipes tested showed a small compressive stress at the

point of maximum pipe diameter which is consistent

with the pipe bending slightly to put the outside surface

into compression. At higher loads (4500N) more typical

of the maximum load on the hoop during the fatigue

testing, the measured stresses were within the experi-

mental scatter in two of the three pipes, while in the

third pipe, variations in the measured stress of up to 10%

were recorded. This may have been suf

le through the

wall of the pipe. To check the validity of these values for the

residual stresses, direct measurement was also made using

strain gauges attached to the inside and the outside of the

pipes before slitting of the hoops. The measured hoop

residual stress was adjusted for the axial residual stress and

resulted in a small tensile stress of 0.6

1.1 MPa on the

inside of the pipe and a compressive stress of 3.4

–

4.2 MPa

on the outside of the pipe. These direct measurements

suggest that the assumption of a symmetric residual stress

distribution was not correct and that the compressive

residual stress existed to a shallow depth from the outside

surface. The residual stress distributionwas consistent with

the fatigue failures occurring on the inside of the pipe rings

as the compressive residual stress on the outside of the

pipe should have increased the applied stress required to

initiate fatigue cracks.

–

cient to alter the

point of failure.

3.1.2. Variations in the level of orientation

Oriented pipe is made by extruding a conventional pipe

and then expanding its diameter. This can be done in an off-

line process inwhich the ends of a cut section of pipe are held

and the pipe heated and expanded or in an in-line process in

which the pipe is expanded by drawing over a mandrill

immediately after extrusion [1] . In both cases, the orientation

is not strictly uniaxial in the hoop direction but there is

a degree of orientation also in the axis of the pipe. There is

also the possibility that variations in thewall thickness in the

conventional precursor pipemayalso lead to variations in the

degree of orientation in the oriented pipe.

Table 1

Elastic constants (GPa) for PVC-O pipe at 25 C. Direction 1, 2, 3 are the

axial, radial and hoop directions respectively.

C 11

C 22

C 33

C 12

C 13

C 23

C 44

C 55

C 66

D max

7.61

9.14

3.42

4.77

3.32

1.87

1.81

1.71

D min

7.38

6.43

9.06

2.90

4.96

3.66

1.99

1.73

308

D.B. West, R.W. Truss / Polymer Testing 31 (2012) 304

–

311

The above observations suggested that the reason for

failures occurring from the outside of the hoop samples

was most probably a result of a combination of the pres-

ence of

aws on the outside of the pipe and higher

stresses as a result of variations in wall thickness and pipe

ovality.

3.2. Fatigue data analysis

The S-N data for the PVC-O pipes is shown in Figs. 3 and

4 with data

tted to two RFL models discussed below. The

data generated in this work were consistent with previous

data reported for PVC-O pipes [5] but extended the range

and quantity of data. It also contained censored data at 10 7

cycles.

Four RFL models were

tted to the data. These assumed

either a normal (norm) or a smallest extreme value (sev)

distribution of the fatigue life data and also for the fatigue

limit. A model labeled as norm-sev assumes a normal

distribution for the fatigue life and a smallest extreme value

distribution for the fatigue limit. In addition, two linear

models were

tted to the data assuming either a normal

(norm) or smallest extreme value (sev) distribution to the

fatigue life. The goodness of

t to the data was assessed

using a maximum likelihood estimation with more positive

values of log(L) indicating a better

t. The values of log(L)

are shown in Table 2 . Log(L) values of the linear models

Fig. 4. Fit of the RFL norm-sev model to the PVC-O fatigue data showing the

0.01, 0.05 and the 0.95 quantiles. a)

t over lifetime of data; b)

t extended

to show the fatigue limit.

were noticeably more negative than the RFL models, indi-

cating that the RFL models give a superior

t and that

a fatigue limit is probable for PVC-O materials. This is

similar to the behaviour found for PVC-U [8] . Of the RFL

models, the sev

sev and norm-sev models possessed log(L)

values indicating slightly better

–

ts than for norm

–

norm

and sev-norm models.

Probability (P-P) plots should display a linear diagonal

when Expected Values are plotted against Observed

Values. P-P plots for the norm

norm and sev-norm

models, Fig. 5 , show some deviation from the diagonal

and do not possess the quality found in the coherent

PVC-U data set of Joseph and Leevers [9] modeled

previously [8] . The P-P plots suggest that the RFL models

–

Table 2

Maximum likelihood estimation (MLE) parameter results for the PVC-O

fatigue data.

RFL

linear

norm

–

norm sev

–

sev sev

–

norm norm

–

sev norm

sev

Log(L)

25.316

24.469

26.133

24.795

30.623

29.784

b 0

24.147

20.668

20.361

25.569

30.551

33.417

b 1

4.737

3.657

3.492

5.294

6.853

7.970

0.160

0.300

0.100

0.505

0.403

Fig. 3. Fit of the RFL sev

sev model to the PVC-O fatigue data showing the

0.01, 0.05 and the 0.95 quantiles. a)

–

m g

0.950

1.668

1.631

0.918

t over lifetime of data; b)

t extended

s g

0.294

0.080

0.111

0.265

to show the fatigue limit.

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