FlexureonVenus

ValeriaReyesOrtega

November21st,2016

ScrippsIns@tu@onofOceanographyUCSanDiego

FactsaboutVenus

•  “Earth’sEviltwin”•  Similar–  Size,massandgravity

•  Notsimilar– Surfacetemperature(~30@meslargerthanEarth),atmospherecomposi@onandtectonicseQng

Geologicfeatures

•  Coronae•  Ridges•  Chasmata•  Volcanoes•  Craters

Objec@ves•  DiscussFlexureinVenus•  Deriveequa@ons2and10fromJohnsonandSandwell(1994)

•  HighlightimportantfindingsonJohnsonandSandwell(1994)

Flexureproblem:BarLoadvs.RingLoad

APPROXIMATION- 2D CARTERSIAN (bar):Applies if the radius a ismany @mes greater thanflexuralparameterα.-RING LOAD: If not, 2Daxisymmetricproblem.

Deriva@onof2DCartesianapproxima@on

•  Thedifferen@alequa@onforflexureproblemswherethereisnoin-planeforceis

•  Ageneralsolu@onfor4thorderODEwhereriarethe4solu@onsofthecharacteris@cequa@on:

! !!!!"! + ∆!"# = 0

!(!) = !!!!!! + !!!!!! + !!!!!! + !!!!!!

!! + ∆!"! = 0

Therootsr1,r2,r3andr4are:

RememberingthatWecanrewriterasHencew(x)is:

(!!)! − !! ∆!"! = 0

(!!)! − ! ∆!"! (!!)! + ! ∆!"

! = 0

! = ± ! ∆!"!

! ! = !

! (1± !)

!! = ± 1± ! !∝ ;

!(!) = !!!(!!!)

!! + !!!(!!!)

!! + !!!!(!!!)

!! + !!!!(!!!)

!!

∝ = !!!∆!

!

UsingWecanrewritew(x)as:

Sincetheappliedloadiszeroexceptatx=x0,andthesymmetryrespecttox0wecananalyzetheposi@vepart.Boundarycondi@on:w=0,xà∞Theexponen@altermex/αdoesn’tgotozero,hence

!!"!!!!"! = cos !!

!"!!!!"!! = sin !

! ! = !!! !!!cos !

! + !!! sin !! + !!

!! !!!cos !

! + !!! sin !!

!!! = !!! = 0

•  Thereforethefinalsolu@oncenteredaroundtheloca@onoftheloadx0isgivenby:

! ! = !!!!!!! !!!cos !!!!

! + !!! sin !!!!!

2-DAxisymmetricmodel,BarLoad

•  Flexureduetoabarloadcanbedescribedbyconvolvingabarloadgeometrywiththeresponseduetoalineload:

Where:

! ! = ! ! ∗ !(!)

Barloadgeometry

Responseduetoalineload

! ! = Π !!

! ! = !!

∆!"# !!! cos !

!+ sin !

!

Where

•  Theconvolu@onoftwofunc@onscanbesolvedinthefrequencydomain,

•  ThenweneedtoobtaintheFTofbothfunc@onsmul@plythemanddotheIFTtogetbacktothespacedomain.

Π !! = 1 ! < !/2

0 ! > !/2

! ! = ! ! !(!)

•  FourierTransformofboxcarfunc@on:

ℱ Π !! = Π !

!!

!!!!!!!"#!" = !!!!!"#!"

!!

! !!

= − 1

!"!!"#$ − !!!"#$

2! = ! sin!"#!"# = ! !"#$(!")

•  NowtocomputetheFTofs(x),wewritethecos(x)andsin(x)intheirexponen@alformandaiersimplify:

•  Let’sobtaintheFTbyre-arrangingthetermstogettheform

ofthedeltafunc@on

! ! = !!2∆!"# 1 − i !

!!(1+!) + 1 + ! !

!!(1−!)

! ! = !!∆!"# !

!! cos !

!+ sin !

!

! ! = !!

2∆!"# (1 + !)!−! 2!"−1+!! ! + (1 − !)!−! 2!"+

1−!! !

!

!!!"

! ! = !!

2∆!"# 1 + ! δ 2!" −1 + !! + 1 − ! δ 2!" +

1 − !!

•  Nowlet’scomputeB(k)S(k)

•  Thentaketheinversetransformtogetw(x)

! ! = !!!∆!"# !"#$%(!") 1 + ! δ 2!" − 1+!

! + 1 − ! δ 2!" + 1−!

! !

!! !!2!"#!"

! ! = !!∆!" !−!−!! cos !−!

! − !−!+!! cos !+!

! !"

Highlights•  Barloadappliedinmostcasesexamined.

•  2-DCartesianmodelprovidesandaccuraterepresenta@onoftheflexuralparameter

•  However,anaxisymmetricmodelmustbeusedtoobtainareliablees@mateofload/bendingmoment

•  2-Dthinelas@cplatemodelàbestfitthicknessintherange

12–34km

•  Noconvincingevidenceforflexurearoundsmallcoronae,howeverthereare5possiblecandidates.

•  The5coronae,ifflexureconsidered,yieldmeanelas@cthicknessesfrom6–22km.

•  AdoptedyieldstrengthenvelopebasedondryolivinerheologyandstrainratessimilartoEarth.

•  Theore@callylithosphereshouldexhibitbrillefractureandflow.

•  Mechanicalthicknessesfrom21-37km,greaterthanpreviouspredic@onsbasedonheatflowandthermalmodels.

•  Thicklithosphere,lowtemperaturegradients

•  Ifdryolivineisstronger,mechanicalthicknesses@matesinthisapproachwillbereduced.

THANKYOU!!