ini$a$on'of'dynamic'ruptures''...

Ini$a$on'of'Dynamic'Ruptures''in'Numerical'Simula$ons'''

!Mar$n'GALIS'1,'Jean=Paul'AMPUERO!2,'P.'Mar$n'MAI!1,'Jozef'KRISTEK!3,4,'Peter'MOCZO!3,4'

'''''

1!!KAUST,!Thuwal,!Saudi!Arabia!

2!!California!Ins;tute!of!Technology,!Pasadena,!USA!!

3!!Comenius!University!Bra;slava,!Slovakia!

4!!Slovak!Academy!of!Sciences,!Bra;slava,!Slovakia!!

!NMEM!2015!

Smolenice!castle!|!Slovakia!|!5!–!9!July!2015!

eOmail:!mar;[email protected]!

•  introduc$on''•  cri;cal!parameters!of!the!ini;a;on!zone!(IZ)!!

and!their!es;mates!

!

•  effects!of!shape/aspect!ra;o!on!the!IZ!

•  verifica;on!!

•  influence!of!material!parameters!of!the!IZ!

!!

•  op;mal!parameters!of!the!IZ!

!

•  summary!

outline'

introduction

ar;ficial!procedures!are!used!!

to!ini;ate!dynamic!ruptures!!

under!linear!slipOweakening!fric;on!law!

!

the!ar;ficial!ini;a;on!may!have!significant!impact!!

on!the!resul;ng!dynamic!rupture!propaga;on!!

'therefore,!!

it'is'desirable'to'understand'and'then'minimize'side'effects'of'the'ini$a$on''

''here!we!discuss!ini;a;on!

using'an'overstressed'asperity,!i.e.,!region!with!ini;al!trac;on!higher!than!sta;c!trac;on!

•  introduc;on!!

•  cri$cal'parameters'of'the'ini$a$on'zone'(IZ)''and'their'es$mates'

!




!!


!

•  summary!

outline'

nonOdim

ensional!area!of!!IZ!

S = τ s − τ 0( ) τ 0 − τ d( )strength!parameter!

''runaway!ruptures!successful!ini;a;on!

''''''''stopping'ruptures!!'unsuccessful!ini;a;on!

the'ini$a$on'is'controlled'by'the'area'of'the'ini$a$on'zone'(not!by!the!halfOlength!or!shape)!

cri;cal'parameters!|'the!cri;cal!area!of!the!IZ!!(Galis!et!al.!2015)!!

nonOdim


A2


for!S!<!0.75!

AU

cri;cal'parameters!|es;mates!of'the!cri;cal!area!(Galis!et!al.!2015)!!

Uenishi!2009!

nonOdim


A2


for!S!>!0.75!

A2 = π 3

161

f min4

(τ s − τ d )2

(τ 0 − τ d )4 µ 2Dc2

f (x) = x 1+

τ 0i − τ 0

τ 0 − τ d1− 1− 1 / x 2⎛

⎝⎞⎠

⎛

⎝⎜

⎞

⎠⎟

AU

Acrit = max(AU, A2)

cri;cal'parameters!|es;mates!of'the!cri;cal!area!(Galis!et!al.!2015)!!

Uenishi!2009!

Galis et al., 2015


•  cri;cal!parameters!of!the!ini;a;on!zone!(IZ)!!

and!their!es;mates!

!

•  effects'of'shape/aspect'ra$o'on'the'IZ'


•  influence!of!material!parameters!on!the!IZ!

!!


!

•  summary!

outline'

a b

τ 0

effects!of!shape/aspect!ra;o!on!the!IZ'

a b

τ 0


transi;on!!

from'3D''(!init.!controlled!by!area!)!

to'2D''(!init.!controlled!by!length!)!

!

!

a!O!halfOlength!in!inOplane!direc;on!

b!O!halfOlength!in!an;Oplane!direc;on!

a b

τ 0


transi;on!!

from'3D''(!init.!controlled!by!area!)!

to'2D''(!init.!controlled!by!length!)!

!

!


an$=plane'

in=plane'

Linit

S

Linit

S

Uenishi!&!Rice,!2003!

L2

Uenishi!&!Rice,!2003!

L2



and!their!es;mates!

!


•  verifica$on'!


!!


!

•  summary!

outline'

verifica;on!|'introduc;on!to!demonstrate!that!!

our'results'and'conclusions'are'not'biased''by!the!choice!of!the!numerical!method,!

we!verify!our!results!using!

●  FESD'!2nd!order!finiteOelement!method!

Galis!et!al.!(2008,!2010);!Moczo!et!al.!(2014)!

●  SeisSol'ADERODG:!arbitrary!highOorder!deriva;ve!–!discon;nuous!Galerkin!method!

Käser!and!Dumbser!(2006),!De!la!Puente!et!al.!(2009),!Pel;es!et!al.!(2012)!

●  SORD''2ndOorder!support!operator!method!

Ely!et!al.!(2008,!2009)!

●  WaveQLab3D''6thOorder!summa;onObyOparts!finiteOdifference!method!

Duru!and!Dunham!(2015)!

1.25!

1!

0.75!

100! 200!0! 50!

FEM!

element!!

size!h [m]!

verifica;on!|!high!background!stress,!S'='0.1'nonOdim

ensional!halfOlength!of!IZ!

1.25!

1!

0.75!

100! 200!0! 50! 100!0! 50! 100!0! 50!

FEM! ADERODG!SORD!

50! 100!0! 200!

element!!

size!h [m]!

element!!

size!h![m]! ν = h

deg. of freedom3

verifica;on!|!high!background!stress,!S'='0.1'WQLab3D!

element!!

size!h![m]!

nonOdim


1.25!

1!

0.75!

100! 200!0! 50! 100!0! 50! 100!0! 50!

FEM! ADERODG!SORD!

50! 100!0! 200!

element!!

size!h [m]!

element!!

size!h![m]! ν = h

deg. of freedom3

verifica;on!|!high!background!stress,!S'='0.1'

the'cri$cal'size'of'IZ''converge'systema$cally'to'consistent'values'in'all'4'considered'numerical'methods''

(within!resolu;on!of!the!discrete!models)!

element!!

size!h![m]!

nonOdim


WQLab3D!

2.0!

1.75!

1.5!

ν = h

deg. of freedom3

the'cri$cal'size'of'IZ''converge'systema$cally'to'consistent'values'in'all'4'considered'numerical'methods''

(within!resolu;on!of!the!discrete!models)!

verifica;on!|!low!background!stress,!S'='2.0'nonOdim


100! 200!0! 50! 100!0! 50! 100!0! 50!

FEM! ADERODG!SORD!

50! 100!0! 200!

element!!

size!h [m]!

element!!

size!h![m]!

element!!

size!h![m]!

WQLab3D!



and!their!es;mates!

!



•  influence'of'material'parameters'of'the'IZ'!!


!

•  summary!

outline'

LU 3

a ≅ 0.624 C(ν ) 11− ν

µ.Dcτ s − τ d

LU 3

b ≅ 0.624 C(ν )µ.Dc

τ s − τ d

C(ν ) = E(k) + (1− ν )K (k)

2 − ν

k = ν (2 − ν )

Uenishi!2009!

E(k) and K(k) are!complete!ellip;c!integrals!!

of!the!first!and!second!kind!

A2 = π 3

161

f min4

(τ s − τ d )2

(τ 0 − τ d )4 µ 2Dc2

f (x) = x 1+

τ 0i − τ 0

τ 0 − τ d1− 1− 1 / x 2⎛

⎝⎞⎠

⎛

⎝⎜

⎞

⎠⎟

Galis et al., 2015

cri;cal'parameters!|!influence!of!material!parameters!

nonOdim


density =2700 m3 kgvs = 2000 − 5000 m/sµ = 10.8 − 67.5 GPa


AU


A2

nonOdim


density =2700 m3 kgvs = 2000 − 5000 m/sµ = 10.8 − 67.5 GPa


µ!does!not!affect!!the!nonOdimensional!cri;cal!area!!

if!Lfric!is!used!for!normaliza;on!

!

!

!

!

no!maner!which!quan;ty!!

is!used!for!normaliza;on,!

the!intersec;on!s;ll!remains!at!S!~0.75.!

AU

A2


Lfric =µDc

τ s −τ d

nonOdim



ν = 0 − 0.5ν = 0.2 − 0.35AU

A2


nonOdim



ν = 0 − 0.5ν = 0.2 − 0.35

the!es;mate!A2 does!not!depend!on!v!!!

the!es;mate!AU !depends!on!v but!varies!only!marginally!!

for!v-range!typical!for!crust!

AU

A2




and!their!es;mates!

!




!!

•  op$mal'parameters'of'the'IZ'!

•  summary!

outline'

op;mal!parameters!|!introduc;on!

an!illustra;ve!example!!

rupture!ini;a;ated!by!!'''

slightly'over=cri$cal'parameters!!

●  ini;a;on!with!!slightly!overOcri;cal!parameters!

may!take!long!;me!

op$mal!parameters!!






may!take!long!;me!

op$mal!parameters!!






may!take!long!;me!

●  shorter!dura;on!of!ini;a;on!!can!be!achieved!by!!

higher'overstress'and/or!!larger'ini$a$on'area''

●  if!they!are!too'large'they!can!affect'resul$ng'selfOsustained!dynamic'rupture'

●  we!examine!rela;ons!between!

the!ini;a;on!area,!overstress!

and!dura;on!of!the!ini;a;on!

to!find!op;mal!parameters!

op;mal!parameters!|!high!background!stress,'S'='0.1'long''

short'

rupture!!

propaga;on!


short'

rupture!!

propaga;on!

1.0' 2.2'1.1' 3.9'1.4' 6.8'

Ainit Ainit0

0.005' 0.9'0.05' 1.6'0.3' 2.5'

Δτ 0 Δτ E


short'

rupture!!

propaga;on!

overstress'<'strength'excess'

1.0' 2.2'1.1' 3.9'1.4' 6.8'

Ainit Ainit0

0.005' 0.9'0.05' 1.6'0.3' 2.5'

Δτ 0 Δτ E

long''

short'

op;mal!parameters!|!low!background!stress,'S'='2.0'

rupture!!

propaga;on!

long''

short'


1.0' 1.4'1.1' 1.7'1.2' 2.5'

Ainit Ainit0

0.005' 0.9'0.05' 1.6'0.3' 2.5'

Δτ 0 Δτ E

rupture!!

propaga;on!

long''

short'


1.0' 1.4'1.1' 1.7'1.2' 2.5'

Ainit Ainit0

0.005' 0.9'0.05' 1.6'0.3' 2.5'

Δτ 0 Δτ E

rupture!!

propaga;on!

area'of'IZ'<'1.2'x'cri$cal'area'

effects!of!ini;a;on!on!ground!mo;on!|!introduc;on'

Galis!et!al.,!2015!!

analyzed!effects!of!ini;a;on!on!rupture!propaga;on!

!

we!now!extend!the!analysis!to!effects!on!ground!mo;on!

15km!

10km!

7.5km!

ini;a;on!zone!

fault!plane!

receiver!

effects!of!ini;a;on!on!ground!mo;on!|!high!background!stress,'S'='0.1'

1.0' 0.005'1.0' 0.3'1.0' 1.0'

AinitAinit0

Δτ 0Δτ E

1.0' 0.005'1.0' 0.1'1.0' 0.45'1.0' 1.45'

AinitAinit0

Δτ 0Δτ E

1.0' 0.005'0.66' 0.1'0.39' 0.45'0.19' 1.45'

AinitAinit0

Δτ 0Δτ E

superOcri;cal!ini;a;on! slightly!overcri;cal!ini;a;on!

op;mal!ini;a;on!

;me![s]!

;me![s]! ;me![s]!

no!observable!!

hypoOcentral!PO!or!SO!waves!

!

op$mal'ini$a$on:'ovestress!<!strength!excess!

comparison'of'magnitude'of'par$cle'velocity'

1.0' 0.005'1.1' 0.005'1.2' 0.005'

AinitAinit0

Δτ 0Δτ E

1.0' 0.005'1.0' 0.1'1.0' 0.45'1.0' 1.45'

AinitAinit0

Δτ 0Δτ E

1.0' 0.005'0.66' 0.1'0.39' 0.45'0.19' 1.45'

AinitAinit0

Δτ 0Δτ E

superOcri;cal!ini;a;on! slightly!overcri;cal!ini;a;on!

op;mal!ini;a;on!

;me![s]!

;me![s]! ;me![s]!

effects!of!ini;a;on!on!ground!mo;on!|!low!background!stress,'S'='2.0'

super=cri$cal'ini$a$on:''strong!hypoOcentral!PO!or!SO!waves!

!

slightly'overcri$cal'and'op$mal'ini$a$on:''marginal!varia;ons!in!!

hypoOcentral!PO!or!SO!waves!

!

op$mal'ini$a$on:'area!<!1.2!x!cri;cal!area!

comparison'of'magnitude'of'par$cle'velocity'



and!their!es;mates!

!



•  influence!of!material!parameters!on!the!IZ!

!!


!

•  summary'

outline'

● !!for!a!fixed!overstress!and!aspect!ra;o!close!to!1!the'ini$a$on'is'controlled'by'the'area'of'the'ini$a$on'zone'

however,!'if!one!side!of!IZ!should!be!shorter!!than!the!corresponding!cri;cal!halfOlength!

ini$a$on'is'controlled'by'half=length'

summary'

●  the'cri$cal'area'can!be!es;mated!by!!

!!!!!!AU!:!!es;mate!by!Uenishi,!2009!

A2!:!!es;mate!by!Galis!et!al.,!2015!

●  'efficient'ini$a$on'with'minimized'side'effects!on!rupture!propaga;on!and!ground!mo;on!

can!be!achieved!

high'background'stress'(low'S)' low'background'stress'(high'S)'overstress!<!strength!excess! area!of!IZ!<!1.2!x!cri;cal!area!

Acrit = max(AU, A2)

Thank'you''

eOmail:!mar;[email protected]!

ini$a$on'of'dynamic'ruptures''...

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