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
130
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
132
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

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
134

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
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
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.
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
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.