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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'
!
• 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'
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
ensional!area!of!!IZ!
A2
S = τ s − τ 0( ) τ 0 − τ d( )strength!parameter!
for!S!<!0.75!
AU
cri;cal'parameters!|es;mates!of'the!cri;cal!area!(Galis!et!al.!2015)!!
Uenishi!2009!
nonOdim
ensional!area!of!!IZ!
A2
S = τ s − τ 0( ) τ 0 − τ d( )strength!parameter!
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
• introduc;on!!
• cri;cal!parameters!of!the!ini;a;on!zone!(IZ)!!
and!their!es;mates!
!
• effects'of'shape/aspect'ra$o'on'the'IZ'
• verifica;on!!
• influence!of!material!parameters!on!the!IZ!
!!
• op;mal!parameters!of!the!IZ!
!
• summary!
outline'
a b
τ 0
effects!of!shape/aspect!ra;o!on!the!IZ'
a b
τ 0
effects!of!shape/aspect!ra;o!on!the!IZ'
transi;on!!
from'3D''(!init.!controlled!by!area!)!
to'2D''(!init.!controlled!by!length!)!
!
!
a!O!halfOlength!in!inOplane!direc;on!
b!O!halfOlength!in!an;Oplane!direc;on!
a b
τ 0
effects!of!shape/aspect!ra;o!on!the!IZ'
transi;on!!
from'3D''(!init.!controlled!by!area!)!
to'2D''(!init.!controlled!by!length!)!
!
!
effects!of!shape/aspect!ra;o!on!the!IZ'
an$=plane'
in=plane'
Linit
S
Linit
S
Uenishi!&!Rice,!2003!
L2
Uenishi!&!Rice,!2003!
L2
• 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$on'!
• influence!of!material!parameters!of!the!IZ!
!!
• op;mal!parameters!of!the!IZ!
!
• summary!
outline'
verifica;on!|'introduc;on!to!demonstrate!that!!
our'results'and'conclusions'are'not'biased''by!the!choice!of!the!numerical!method,!
we!verify!our!results!using!
● FESD'!2nd!order!finiteOelement!method!
Galis!et!al.!(2008,!2010);!Moczo!et!al.!(2014)!
● SeisSol'ADERODG:!arbitrary!highOorder!deriva;ve!–!discon;nuous!Galerkin!method!
Käser!and!Dumbser!(2006),!De!la!Puente!et!al.!(2009),!Pel;es!et!al.!(2012)!
● SORD''2ndOorder!support!operator!method!
Ely!et!al.!(2008,!2009)!
● WaveQLab3D''6thOorder!summa;onObyOparts!finiteOdifference!method!
Duru!and!Dunham!(2015)!
1.25!
1!
0.75!
100! 200!0! 50!
FEM!
element!!
size!h [m]!
verifica;on!|!high!background!stress,!S'='0.1'nonOdim
ensional!halfOlength!of!IZ!
1.25!
1!
0.75!
100! 200!0! 50! 100!0! 50! 100!0! 50!
FEM! ADERODG!SORD!
50! 100!0! 200!
element!!
size!h [m]!
element!!
size!h![m]! ν = h
deg. of freedom3
verifica;on!|!high!background!stress,!S'='0.1'WQLab3D!
element!!
size!h![m]!
nonOdim
ensional!halfOlength!of!IZ!
1.25!
1!
0.75!
100! 200!0! 50! 100!0! 50! 100!0! 50!
FEM! ADERODG!SORD!
50! 100!0! 200!
element!!
size!h [m]!
element!!
size!h![m]! ν = h
deg. of freedom3
verifica;on!|!high!background!stress,!S'='0.1'
the'cri$cal'size'of'IZ''converge'systema$cally'to'consistent'values'in'all'4'considered'numerical'methods''
(within!resolu;on!of!the!discrete!models)!
element!!
size!h![m]!
nonOdim
ensional!halfOlength!of!IZ!
WQLab3D!
2.0!
1.75!
1.5!
ν = h
deg. of freedom3
the'cri$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
ensional!halfOlength!of!IZ!
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!
• 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'
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
ensional!area!of!!IZ!
density =2700 m3 kgvs = 2000 − 5000 m/sµ = 10.8 − 67.5 GPa
S = τ s − τ 0( ) τ 0 − τ d( )strength!parameter!
AU
cri;cal'parameters!|!influence!of!material!parameters!
A2
nonOdim
ensional!area!of!!IZ!
density =2700 m3 kgvs = 2000 − 5000 m/sµ = 10.8 − 67.5 GPa
S = τ s − τ 0( ) τ 0 − τ d( )strength!parameter!
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
cri;cal'parameters!|!influence!of!material!parameters!
Lfric =µDc
τ s −τ d
nonOdim
ensional!area!of!!IZ!
S = τ s − τ 0( ) τ 0 − τ d( )strength!parameter!
ν = 0 − 0.5ν = 0.2 − 0.35AU
A2
cri;cal'parameters!|!influence!of!material!parameters!
nonOdim
ensional!area!of!!IZ!
S = τ s − τ 0( ) τ 0 − τ d( )strength!parameter!
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
cri;cal'parameters!|!influence!of!material!parameters!
• 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'
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!!
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!!
op;mal!parameters!|!introduc;on!
an!illustra;ve!example!!
rupture!ini;a;ated!by!!'''
slightly'over=cri$cal'parameters!!
● ini;a;on!with!!slightly!overOcri;cal!parameters!
may!take!long!;me!
● shorter!dura;on!of!ini;a;on!!can!be!achieved!by!!
higher'overstress'and/or!!larger'ini$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!
op;mal!parameters!|!high!background!stress,'S'='0.1'long''
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
op;mal!parameters!|!high!background!stress,'S'='0.1'long''
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'
op;mal!parameters!|!low!background!stress,'S'='2.0'
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'
op;mal!parameters!|!low!background!stress,'S'='2.0'
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'
• 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!on!the!IZ!
!!
• op;mal!parameters!of!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]!