Analysis of accuracy of Williams series approximation of stress field in cracked body – influence of area of interest around crack-tip on multi-parameter regression performance
Abstract
A study on the accuracy of an approximation of the stress field in a cracked body is presented. Crack-tip stress tensor is expressed using the linear elastic fracture mechanics (LEFM) theory in this work, more precisely via its multi-parameter formulation, i.e. by Williams power series (WPS). Determination of coefficients of terms of this series is performed using a least squares-based regression technique known as over-deterministic method (ODM) for which results from finite element (FE) method computation are usually taken as inputs. Main attention is paid to a detailed analysis of a suitable selection of FE nodes whose results serve as the inputs to the employed method. Two different ways of FE nodal selection are compared – nodes selected from the crack tip vicinity lying at a ring of a certain radius versus nodes selected more or less uniformly from a specified part of the test specimen body. Comparison of these approaches is made with the help of procedures developed by the authors which enable both the determination of the coefficients of terms of the analytical WPS approximation of the stress field based on the FE results and the backward reconstruction of the field (again using WPS) from those determined terms’ coefficients/functions. The wedge-splitting test (WST) specimen with a crack is taken as example for the study
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J. Sobek e alii, F a u a ed In eg i à S u u ale, 39 (2017) 129-142; DOI: 10.3221/IGF-ESIS.39.14
129
Focussed on Modelling in Mechanics
Analysis o accu acy o Williams se ies app oxima ion o s ess
ield in c acked body – in luence o a ea o in e es a ound
c ack- ip on mul i-pa ame e eg ession pe o mance
J. Sobek, P. F an ík, V. Veselý
B no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ins i u e o S uc u al Mechanics, B no, Czech Republic
sobek.j@ ce. u b .cz, h p://o cid.o g/0000-0003-4215-1029
[email p o ec ed]
esely. 1@ ce. u b .cz, h p://o cid.o g/0000-0002-7723-971X
ABSTRACT. A s udy on he accu acy o an app oxima ion o he s ess ield in
a c acked body is p esen ed. C ack- ip s ess enso is exp essed using he
linea elas ic ac u e mechanics (LEFM) heo y in his wo k, mo e p ecisely
ia i s mul i-pa ame e o mula ion, i.e. by Williams powe se ies (WPS).
De e mina ion o coe icien s o e ms o his se ies is pe o med using a leas
squa es-based eg ession echnique known as o e -de e minis ic me hod
(ODM) o which esul s om ini e elemen (FE) me hod compu a ion a e
usually aken as inpu s. Main a en ion is paid o a de ailed analysis o a
sui able selec ion o FE nodes whose esul s se e as he inpu s o he
employed me hod. Two di e en ways o FE nodal selec ion a e compa ed –
nodes selec ed om he c ack ip icini y lying a a ing o a ce ain adius
e sus nodes selec ed mo e o less uni o mly om a speci ied pa o he es
specimen body. Compa ison o hese app oaches is made wi h he help o
p ocedu es de eloped by he au ho s which enable bo h he de e mina ion o
he coe icien s o e ms o he analy ical WPS app oxima ion o he s ess
ield based on he FE esul s and he backwa d econs uc ion o he ield
(again using WPS) om hose de e mined e ms’ coe icien s/ unc ions. The
wedge-spli ing es (WST) specimen wi h a c ack is aken as example o he
s udy.
KEYWORDS. Mul i-pa ame e ac u e mechanics; Williams powe se ies;
C ack- ip ields; O e -de e minis ic me hod; Highe o de e ms.
Ci a ion: Sobek, J., F an ík, P., Veselý, V.,
Analysis o accu acy o Williams se ies
app oxima ion o s ess ield in c acked body
– in luence o a ea o in e es a ound c ack-
ip on mul i-pa ame e eg ession
pe o mance, F a u a ed In eg i à S u u ale,
39 (2017) 129-142.
Recei ed: 20.07.2016
Accep ed: 19.09.2016
Published: 01.01.2017
Copy igh : © 2017 This is an open access
a icle unde he e ms o he CC-BY 4.0,
which pe mi s un es ic ed use, dis ibu ion,
and ep oduc ion in any medium, p o ided
he o iginal au ho and sou ce a e c edi ed.
INTRODUCTION
e e mina ion o he highe o de e ms o Williams powe se ies [1] app oxima ing he ields o s ess and
displacemen s in a c acked body is o subs an ial in e es o he p esen ed wo k. Pa icula mo i a ion o i is
ep esen ed by he need o cap u ing he c ack- ip ields in a a he dis ance om he c ack ip.
D
J. Sobek e alii, F a u a ed In eg i à S u u ale, 39 (2017) 129-142; DOI: 10.3221/IGF-ESIS.39.14
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Tensile ailu e o he quasi-b i le ma e ials is accompanied wi h a c ack p opaga ion oge he wi h a nonlinea zone [2]
( ac u e p ocess zone – FPZ) de elopmen , whe e he dec ease o ma e ial in eg i y akes place. The size o he FPZ is
no negligible in compa ison wi h he es o he body. P ocesses o ailu e he e o e occu in a wide a ea a ound he
c ack ip. The desc ip ion o ailu e mechanism mus be in ag eemen wi h he s ess and displacemen ield a ound he
c ack ip in an ex ended a ea. Recen wo ks o many au ho s show he ele ance o he opic o he c ack ip ields
accu a e desc ip ion [3–6], mo eo e , ex ending i in o 3D and aking in o accoun he e ec s o a ious loading modes
[7–12]. Special ocus on b i le and quasi-b i le ac u e is summa ized in [13]. Wo ks [3,14,15] epo ed he ac , ha o
he desc ip ion o he s ess/displacemen ield in a c acked body in a mo e dis an su oundings om he c ack ip he
necessi y o usage o he se e al e ms o Williams expansion (WE), no only he i s o he i s wo e ms, is c ucial.
P ocedu es enabling he mul i-pa ame e desc ip ion o he nea c ack- ip ields (using e.g. hyb id c ack elemen s [16]
o mula ion, o e -de e minis ic me hod [17] based on s anda d FE compu a ion, o o he echniques based e.g. on
ex apola ion o displacemen s o selec ed nodes o FE mesh) usually p ocess esul s om FE nodes selec ed om he
close icini y o he c ack ip. This is adequa e o de e mina ion o he classical/ wo-pa ame e LEFM cha ac e is ics
(SIF, T-s ess). Howe e , se e al ques ions a ise i he a ea o accu a e enough desc ip ion o he s ess and displacemen
ield ex ends o a g ea e dis ance om he c ack ip, whe e K + T dominance anishes? How many e ms o he se ies
should be aken in o accoun ? How o selec he FE nodes conside ed o he eg ession echnique? And how o op imize
he mu ual ela ionship be ween he a ea om whe e he nodes a e conside ed o he eg ession (and how a e hey
loca ed/dis ibu ed in ha a ea) and he ex en whe e he app oxima ion o he ields is o ele an accu acy? Answe ing
hese ques ions p esen s he ac ual mo i a ion o his wo k.
This pape in es iga es he abo e-desc ibed issue ia a pa ame ic s udy e alua ing he in luence o he nodal selec ion on
he quali y o he ob ained app oxima ion o he ield. The wo k p esen ed he e u he builds on p e ious s udies. A
classical (common) way o nodal selec ion, whe e he nodes a e selec ed om a ing in he icini y o he c ack ip was
used in ecen pape s. In luences o se e al pa ame e s on he desc ip ion o s ess/displacemen ields in c acked bodies
by Williams se ies we e in es iga ed. Recons uc ion o he s ess ield wi h he help o a so wa e ool de eloped by he
au ho ’s eam was shown in [14,15] whe e he accu acy o he app oxima ion by WE was e i ied by a isual compa ison
(which was ega ded as su icien o i s in ended pu pose) wi h he FE solu ion (which was ega ded as he exac
solu ion). Howe e , he nodal selec ion ha was used o ob aining he WE e ms was pe o med om he ing a ound
he c ack ip (wi h he dis ance gi en by ecommenda ion om [17,18]).
Published s udy [19] on WPS app oxima ion wi h nodal selec ion om mo e han one ing a ound he c ack ip esul ed
in o nex s udies. Ano he ype o nodal selec ion was conside ed in subsequen wo ks by he au ho s [20,21,22]; he
nodes we e selec ed om speci ic pa s o he es specimen body wi h speci ic dis ibu ion unc ions o hei dis ance and
angula posi ion om he c ack ip. This way was employed wi h expec a ions ha he ields will be be e (mo e
accu a ely/e icien ly) app oxima ed.
An au oma ic u ili y o de e mine he alues o coe icien s o he highe o de e ms o WPS using he o e -de e minis ic
me hod was de eloped [20] o enable mul i-pa ame e desc ip ion o s ess ield, whe e he coe icien s o WPS e ms a e
calcula ed om se e al laye s and angula sec ions. And he dis ance dis ibu ion o he nodal selec ion is go e ned by
a ious unc ions. Compa ison be ween hose used a ian s was gi en by isual echnique again [21].
Hence, a new de ailed way o e alua ion o he accu acy o he econs uc ion was used in [22]. Me hod based on he plo
o he ela i e de ia ion (pe cen di e ence) o he s ess ield be ween he co ec solu ion ( he FE solu ion) and he
solu ion gi en by he app oxima ion using WPS wi h a ce ain a ian o nodal selec ion was in oduced.
Main mo i a ion o his s udy is o ind he easies way o he mul i-pa ame e s ess ield desc ip ion o ob ain he
su icien ly accu a e solu ion alid o he as a as possible a ea om he c ack ip.
METHODS
Mul i-pa ame e linea elas ic ac u e mechanics (MP-LEFM)
he s ess and displacemen ields in a plana homogeneous iso opic c acked body can be o mula ed as an in ini e
powe expansion – Williams se ies [1] by Eq. (1) and (2), espec i ely – o mo e de ails see [14,21,22,23].
1
2,
1
,,, ,
2
n
ij n ij
n
n
A n ij xy
(1)
T
J. Sobek e alii, F a u a ed In eg i à S u u ale, 39 (2017) 129-142; DOI: 10.3221/IGF-ESIS.39.14
131
2,u
1
,,, , ,
n
ini
n
uA nE ixy
(2)
In his s udy, he a en ion is paid o he mode I c ack p oblem. MP-LEFM akes in o accoun se e al ini ial e ms o WE,
i.e. n anges om 1 o N (no ∞); coe icien s o hese e ms a e o en exp essed as dimensionless shape unc ions gn
( unc ions o he ela i e c ack leng h
= a/W ).
O e -de e minis ic me hod
Fo de e mina ion o coe icien s o he Williams se ies e ms he so-called O e -de e minis ic me hod (ODM, [17]) is
used. Based on he linea leas -squa es o mula ion, i sol es a sys em o 2k equa ions, whe e k ep esen s he numbe o
selec ed nodes (in he o iginal pape , hey we e selec ed om a nodal ing a ound he c ack ip), o N chosen e ms o
he powe se ies. De ailed analyses o his me hod can be ound in wo ks [18,19,24]. Displacemen s o selec ed nodes,
oge he wi h hei coo dina es, se e as he inpu o ha me hod. This issue was s udied in de ail in p e ious wo ks which
p esen ed, among o he s, an implemen a ion o his echnique in o an au oma ic nume ical ool called ODeMApp [20] –
he ODM analysis based on an a bi a y nodal selec ion and es specimen geome y a ian is enabled.
ReF aP o app oach
The ReF aP o (Recons uc ion o F ac u e P ocess [25]) is a Ja a applica ion which allows an ad anced de e mina ion o
ac u e cha ac e is ics o ma e ials ailing in a quasi-b i le manne (silica e-based ma e ials). Es ima ion o he FPZ (i s
shape and size) is implemen ed by a echnique de eloped by he au ho s combining MP-LEFM, classical non-linea
models and plas ici y app oach. Recons uc ion o he ac u e p ocess is made by his applica ion gene ally based on he
measu ed loading cu es and basic mechanical p ope ies o he ma e ial.
Fo he pu pose o his s udy, a pa o his p og am is used which p o ides he econs uc ion o s ess ield om he
a ailable shape unc ions (co esponding o alues o coe icien s o e ms o he WE) o he gi en es geome y. This
pa is accompanied wi h a special ool (a class called pe cen di e ence) which allows he display o he de iance (pe cen
di e ence) be ween he app oxima ed s ess ield and he exac solu ion ( o which he FE solu ion is ega ded) o es he
accu acy o he solu ion wi h nodal selec ion om an a ea o in e es a ound he c ack ip. The de ia ion i sel is
exp essed ia pixmap g id as he ela i e di e ence [22].
NUMERICAL STUDY
o he analysis o he accu acy o he WPS app oxima ion, he wedge-spli ing es specimen (WST) is used.
Specimen loaded by eccen ic ension h ough wo s eel pla ens wi h pins among which he s eel wedge is
imp essed was de eloped by Linsbaue and Tscheg in [26]. Schema o he analysed es con igu a ion is displayed
in Fig. 1 le accompanied wi h d awings o i s geome y in Fig. 1 igh .
Compu a ions o he s ess and displacemen ields we e ealized in he ANSYS ini e elemen so wa e [27]. C ack
elemen s (PLANE82) we e u ilized o he FE solu ion. An au oma ic in e connec ion p ocedu e has been de eloped
be ween he compu a ional ool and he ODeMApp echnique o he shape unc ions de e mina ion. A WST a ian wi h
wo suppo s and specimens wid h W = 100 mm (Fig. 1 igh ) was conside ed. The employed FE mesh is shown in
Fig. 2a wi h he c ack- ip de ails, whe e wo basic nodal selec ions (se ing as he e e ence nodal selec ions) a e depic ed.
The i s is aken om he ing a he dis ance o 5 mm om he c ack ip (Fig. 2b) and he second selec ion is aken om
he ing a he dis ance o 0.5 mm om he c ack ip (see Fig. 2c). These a ian s a e labelled as ing 5 mm and ing 0.5 mm
u he in he ex .
Fig. 3a shows he FE mesh in ended o models wi h a mo e gene al nodal selec ion. Va ian om Fig. 3b labelled as
con 180° (0°) ep esen s a nodal selec ion go e ned by a cons an dis ibu ion unc ion (in he dis ance selec ion) aken
om he whole a ea o es specimen (excep o s eel pla ens); qua 90° (45°) – a nodal selec ion wi h a quad a ic dis ance
dis ibu ion unc ion om a ea o 90° angula sec ion wi h he o igin o a ed by 45° angle ( om he c ack p opaga ion
di ec ion) displayed in Fig. 3c; exp 90° (80°) – a nodal selec ion ( om Fig. 3d wi h an exponen ial dis ibu ion unc ion
om he 90° sec ion wi h ini ial o a ion angle o 80°. De ailed desc ip ion o he me hod o he conduc ed nodal
selec ion, including se e al o he a ian s, is ca ied ou in [20,21,22]. The numbe o selec ed nodes is kep he same o
each selec ion ype, k = 49.
F
J. Sobek e alii, F a u a ed In eg i à S u u ale, 39 (2017) 129-142; DOI: 10.3221/IGF-ESIS.39.14
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Figu e 1: Wedge spli ing es wi h he schema o loading imposi ion (le ); geome y o he analysed wedge spli ing es specimen
( igh )[15].
Shape unc ions, ep esen ing he non-dimensional exp essions o he coe icien s o he highe o de e ms o he WE as
unc ions o he ela i e c ack leng h
, co esponding o he men ioned nodal selec ion a ian s we e de e mined by he
ODeMApp p ocedu e and hen hey we e used as inpu s o ReF aP o applica ion (which allows displaying he ela i e
de ia ion om he “exac ” solu ion o he s ess ield dis ibu ion). Subsequen ly, a pos -p ocessing ega ding he s ess
ields econs uc ions based on one, wo and e en highe o de e ms o he WPS was conduc ed.
Figu e 2a: Nume ical model o he
wedge spli ing es , ini e elemen
mesh.
Figu e 2b: De ail o he FE mesh o he
ing 5 mm a ian wi h nodal selec ion ( ed
do s a he dis ance 5 mm om he c ack ip).
Figu e 2c: De ail o he FE mesh o
ing 0.5 a ian wi h nodal selec ion ( ed do s
a he dis ance 0.5 mm om he c ack ip).
J. Sobek e alii, F a u a ed In eg i à S u u ale, 39 (2017) 129-142; DOI: 10.3221/IGF-ESIS.39.14
133
Figu e 3a: Nume ical model o he
WST specimen, used ini e elemen
mesh [22].
Figu e 3b: Dis ibu ion o
FE
nodes o con 180° (0°) a ian
o nodal selec ion [22].
Figu e 3c: Dis ibu ion o FE
nodes o qua 90° (45°)
a ian o nodal selec ion
[22].
Figu e 3d: Dis ibu ion o FE
nodes o exp 90° (80°)
a ian o nodal
selec ion [22].
RESULTS AND DISCUSSION
he econs uc ion o he ields o he p incipal ensile s ess
1 and he c ack opening s ess
yy was pe o med
and compa ed o he exac solu ion. Figs. 4 and 5 display he ela i e di e ences be ween he app oxima ion o
he ield co esponding o chosen a ian o nodal selec ion and he FEM solu ion. Only ew examples o he
econs uc ion (based on he numbe o used Williams se ies’ e ms equal o 1, 2, 4, 7 and 11 – posi ioned
ho izon ally) a e gi en o illus a ion. Cases wi h he ela i e c ack leng h
= 0.5 a e shown he e as an example.
I can be seen ha low numbe o e ms o he Williams se ies used p o ides e y inaccu a e app oxima ion o he s ess
ield (mainly o a wide a ea a ound he c ack ip). Usage o only i s wo e ms leads o a su icien accu acy only in he
e y icini y o he c ack ip (whe e he classical and wo-pa ame e LEFM holds). A su icien ly accu a e solu ion can be
p o ided by a usage o a leas ou e ms o Williams expansion i an a ea o abou 2 cm om he c ack ip is eques ed.
Compa ison be ween he used a ian s o nodal selec ion shows one impo an ac – in gene al [21,22], a ian
con 180° (0°) (uni o m selec ion om he whole body o he es specimen wi h he cons an dis ibu ion unc ion o
dis ance selec ion) comes ou as he bes a ian o nodal selec ion (in Fig. 4 and 5 i is emphasized by a g een colou ed
ame). As i can be seen, he g ea e is he sha e o he ed colou in he pe cen di e ence diag am he la ge is he
ela i e e o om he FE solu ion. Tha is why his a ian appea s in nex analyses as he e e ence one o compa ison
wi h a ian s o nodal selec ion om jus one ing; mo eo e , om a close dis ance om he c ack ip. In some cases he
a ian qua 90° (45°) looks also p omising and is compa able wi h he con a ian . Ne e heless, his is ue only o he
chosen
= 0.5. Wo k [22] p o ides de ailed iews also a di e en
alues.
Nex igu es compa e esul s o wo basic a ian s o nodal selec ion om he icini y o he c ack ip wi h he
con 180° (0°) a ian o he nodal selec ion, i.e. om he whole body o he es specimen.
Fig. 6 shows con ou plo s o he ela i e de ia ion o p incipal s ess
1 o a ian s o nodal selec ion con 180° (0°),
ing 5 mm and ing 0.5 mm o he WST specimen wi h ela i e c ack leng h
= 0.5. As a as ou e ms o Williams
expansion o app oxima ion is used he same end in he ela i e de ia ion can be obse ed. Wi h inc easing N he
esul s o ing 0.5 mm a ian appea o be e y inaccu a e which can be seen on he ex ension o he ed colou a ea
(deno ing a la ge p opo ion o 40% di e ence om he FE solu ion).
T
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1 gn
= 0.5 1 2 4 7 11
con 180° (0°)
qua 90° (45°)
exp 90° (80°)
Figu e 4: Rela i e e o o p incipal ensile s ess
1 ield app oxima ion o chosen a ian s o nodal selec ion.
c ack ip
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135
yy gn
= 0.5 1 2 4 7 11
con 180° (0°)
qua 90° (45°)
exp 90° (80°)
Figu e 5: Rela i e e o o c ack opening s ess
yy ield app oxima ion o chosen a ian s o nodal selec ion.
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136
1 gn
= 0.50 1 2 4 7 11
con 180° (0°)
ing, 5 mm
ing, 0.5 mm
Figu e 6: Rela i e e o o p incipal ensile s ess
1 ield app oxima ion o chosen a ian s o nodal selec ion.
J. Sobek e alii, F a u a ed In eg i à S u u ale, 39 (2017) 129-142; DOI: 10.3221/IGF-ESIS.39.14
137
yy gn
= 0.50 1 2 4 7 11
con 180° (0°)
ing, 5 mm
ing, 0.5 mm
Figu e 7: Rela i e e o o c ack opening s ess
yy ield app oxima ion o chosen a ian s o nodal selec ion.