Chiara ZaniniPolitecnico di Torino chiara.zanini@polito.itRATE-INDEPENDENT MODELING IN FRACTURE MECHANICS: A VANISHING VISCOSITY APPROACH

1

Plan

Mathematically how to study the propagationof a crack

2 vanishing viscosity approachO

0in setting 2 D

IEIIInaten.ae

pre existing crack oflength se

time dependent loading Ict

how to model the crack propagation

during some time Interval cost

A A GRIFFITH theory of brittle fracture

calce a macroscopic manifestation ofthe rupture of molecular links at the

atomisti level which is accuratelydescribed through an

energy density ateach point of the crack

Ìn result of the competitionbetween the bulk energy elastic energyof tre unbroken part and surface energyon the crack

3D ad

arealengthens

the system is described

by means of energy functional

9 uasi stahseth.mg no dynamic fractureat each time t the system is in equilibriumninth the loads appliedi e at any time t the configurationdescribing the system is a stationary point

of the energy functional IMINIMIZE R

3the system con pass from unbroken to

broken if this produces a decrease mi the

energyOUR SETTING 62 Cip

µ pt'prescribed crack path of length L

Tco admissible crackHOF connected subset of t oflength o containing theinitial crack Fracture

energy functional TOUGHNESS

Flt no zitta fax slitta Lodiro

un

surface8144,0 back part of part

the energy

reduced functional Itt o miss EH v o

11920 veh triplocitron

1 è t e 0f tra Holt

2 D I t 41 E Ke

RENÉE rate FRACTURE TOUGHNESS

4

3 si ki do.IM O

70 7

à It o the crack can grow only ifthe energy release rate is critical

1998 FRANFORT MARGO i

C t o È Il t.si Ko

3 from Griffith rewrites as

è It D EH 0

Note that CHAINROLE

EH o q.EH.dtD DEH.dtD.lt

3 rewrites as

EH ott GE t alti O

integrating intuire

Ell alti ECO t 9 EG ok dr

5

2 Dot t.sk sia

sqelt.skzOqifoi

sE t o is strictly convex then

2 can be written as

E 6 alti e Ect è toe Colt thP

T h 0

EH è EA ttake the

Final limit mh 0

70

Def quasi static irreversible crack propagationEranofont Man'go 98 D Maso Franofort toadcio

ti ulti ok 410 solo CuoioGwen

anda imiti t.ultl.sk E FA ù E

µ è è sold Ie H ritto

b iRREVERSIB Y.im t non decreasing

c ENERGYB nc.ci f t uCtl oCtI eL1 0 t

l

and EH.ua olttt fcauo.ooI ftqflr uH olrDdiio

A Mielke Theory of sete independentprocesses441,0411

siepe if ti get is a

solution for the lead Atthen 7 a O q at represents the correspondentevolution for the lead list

Example from Mechanics i

mii FIA k lett

getti q let Ascisseg'lett iii Ig letletti list ECA Fiat ai

mieia 0

ci 0 Èci 0 Rid ipORIA È reti Rho Rider In 4

tv

if 4 0 ORIO ricopioRivi RIO per 0 tv

Intero Art

re C 1,1 ORIO 1,1

a so meta ORG Kg ali0 set

oRlopakgollt II.inIepIdentevolutionary sistemi

reparasmeterize by at 2 O

arcaica 1kg dat

H R is pas 1 homogeneous

drlq.cat OR is a homogeneous

ORIA 1kg alt

energy functional EH g 12kg Http

ORIA dgEH.HN 0

This equation can be reformulated by the

Ìn w lati di5 global stability condition

ELENA E lt.at tRlg gltDtqE E It alti ftp ds Elo qo tf Elsglsf

POWER

Pray a quasi static irreversible crack propagationtrafitti alti

is an energetic solution s E

Indeed

Fit alti ott Italo slltl.us KoT

ritt.ultl.HN RHO

R vkv if v o

a if riso

1 Fit alti alti e Flt è E è ottteatrino

9

felt uh ok Ride a 84 un.si tRlo D

E t.ultholtilsEH.it E RE 0 RiottaEH ù E Rio ott

J s

energybalance2 3 E 8

EH alti alti Equo a uh oh d

E It uh ott Rio o E 0,4 1 4 84,44

ftp.lolrddr E

3 result

General schema

time incremental minimum problems i

µ sequence of subdivisions of Edito ti alzi a at t

10lim max 1 È È 0b as a T

define via ok as follows

Cia o luna1 etgminfflti.io

èi 1

tenenteMòpiecewise constant functions

un Lt È mia o.lt 0

uhm i iskra largestinteger sit t at

pass to time continuous setting le co

Notai quasistatic irreversible crack propagation

is a global minimizes for the energy Flt

local

the energy FA is not convex so

no uniqueness of minimize

MoeORIA GEA qui

LE

Fumenti the global stability condition

provides a quasi static propagation whichis not realistic

the instant afta o

te 9il p e altio

0

intuitionFlo

qi910

tiftp.gqltl lultholty

Im define a different motion of crack evolutionbased on a lad stability criterion

infinitesimi E It alt gittataE OT xp R

quasi static evolution D 5 t alti 0

i

i 9e.HR tl0J agdtregularizationviscous termti 9,4 regular

has no jumpsE O g t q t rith

a a solution to dgfct.gl 0

vanishing viscosity limit

Zanini 07

Efendi ev Mielke 06

Define a function tagli as follows

dgflt.gl 0 y dgt 0,9 O

a DI ECO go pas defg NÉ i

gtt via the

implicit function themt dgflt.ci f O

king It i

ti difettionly trascinareall the others stay positiva

iv Daft vb

ftp.jlsi xIFI.co.in c sol

vuol g dgfcts.ge O

dfflts.ge pas def

i dgflt.pt Oi

Y q t piecewiset regular

E 21

14Consider fare E o fixedSÌ Dgt t getti 0

Assume E la t xD Fa È

and t 999 aT.no c 0

Pag 7 g api 3 sol g Ct to

sai dgflt.gl O

Noi 9defined on Cost and uniformly boundedN r.to E

comment J sol 9 t on sometimetira interval 6 from ODE's theory

multiply the agm by 9E 9etdgkkql.ge eIe9e col9eFa

E a c Injgao.com

15

Prap2_iE9ezjs L4o.tI

IndeediEfIefsDgflt.qejje

È 9 t.ge

integrate in time qEco feti EH AHE lista di

Èarm bound iv r.to E

E da 0 s Eco t

Maine let ga be the solution of

sci Da FA gatti 0

add seAssume Ie o

1

16let alt be the function ma defined via

the Implicit function them with 9

Then i

a

9 9 uniformly on compact sets

of Cat i ti 4

I t i 1 k 3 ti ti such that

9,1 ti 1 Es vi Is uniformly on

µ compact setsFIRvi dgfcti.ro sli al alti

7 the graph of 9

em_e

ti

approaches the completion of the graphofqobtained by using the heteroclinic trajectories vi

17

Efendier Mielke Vanishing viscosity finite dimension

energy Itt 9 dissipationRfi TCR

1 0 ORIA Data g È dgflt.gtO9lt BVko.ttRetail Rià Ehila

oedpelcil idgtlt.ge o

oedRGJ isgxdgtdt.ge

Paisleysett ti II G Idv

Isis si is Gals 9,1 6

note città b

OEOR.li dgtlt.ge e o

Ec ore daII'stà

1 It IIII normalization condition

o E E È µo e ORCI Da ICE ftal

can distinguish three regimesi E O it corresponds to a jump of alt

viscous slipsi È e 10,1 If e 10,1 dry friction

o I 1 II 1 0 stickingo

Mielke Rossi salari 08

About vanishing viscosity applied tocrack propagation

MISSING SOMEa Tonda Z 09 E E 4,0 CHARACTERIZATION

µ AT JUMPS

Knees Mielke Z 08

I It o minSfigata slitta veltriRino

Risi Ìl ksi if si 20

if is 0

E o 0 e ORLI sò D I t.se

ardireto prove existence of use time incremental

minimum problems o fixed time stepdefLe

E

givenfor 2 1

TI eAgamin Ike E e Relè

a priori poundsmaine thm iJ o.lt salta OEORs.si tDIltp

ao si o

b K E d I It zo

e K 1 E Da Ict 9 0

20Moreover uhm K Do I E so is

aconstantint foto

E 0

Main them o m BV KatOHI At

anda trio t non decreasing jumpset

of 0

b Ka D I 4,041130 7 tela t o

c ifk D.TK 170 then tedioand f o differentiabilityset

cfrd t testo io e t alta

me have k D I t E 0

equivalent to the BV solution introduced

by Mielke Rossi Savoia

compare the energetic solution with the

vanishing viscosity ore

example I e o E Ico 21

then do t.se E o

sign of ht D Ict o a signof Kat To

sign of fa È.co Eco Itaa sign of È

ÈÈ

NO MOTION OF THEjump CRACK

sL

According to Griffith

if È 70 no crack GROWTH

if _È SO JUMP OF THE CRACK

if Gto 0 slow crack propagation

22

vanishing viscosity solution i

I I ae

É È

L o

o o