Lect5 Geodyn Stress

8/18/2019 Lect5 Geodyn Stress

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Stress, Strain, and Viscosity

San Andreas FaultPalmdale


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Solids and Liquids

Solid Behavior:

- elastic- rebound- retain original shape- small deformations are temporary- (e.g. Steel, concrete, wood,

rock, lithosphere)

Liquid Behavior:

- fluid- no rebound- shape changes- permanent deformation- (e.g. Water, oil,

melted chocolate, lava


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Solids andLiquids

Linear Viscous Fluid:

- Rate of deformation is proportional to the applied stress

- !linear" viscous fluid (w.r.t. stress and strain rate)

- lso known as a !#ewtonian $luid"


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Solids > ? > Liquids

%s there anything that behaves in a way between solid and li&uid

Plastic Material- solid- but deforms permanently- malleable

- ductile

uctile or malleable materials are !non-linear"


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Solids and Liquids

re all materials either a solid or a fluid, all the time '

pplied heat can cause solid materials to behave like a fluid

Some material may be elastic when small forces are applied

but deform permanently with larger applied forces

Elastic: eformed materialreturns to original shape

Ductile: Stress eceeds theelastic limit and deformsmaterial permanently


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This rock responded to stress byfolding and flowing by ductile deformation.

Occurs under high heat and high pressure

This rock fractured understress by brittle deformation.

ccurs under low heat and at shallow depths in the Earth's crust.


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*nstressed

cube of rock

Figure .!

!##$% &ohn iley and Sons% (nc.

Three types of stress

+#S%#

S+RSS

/01RSS%#

S+RSS

S2R

S+RSS

Stress) is the force actingn a surface% per unitrea * may be greater inertain directions.


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Strain

/onsider a layer of fluid between 3 plates

+he top plate moves with velocity, 4

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+he shape change can be written as 4 ∆t 7 2

+he rate of shape change is d7dt(shape change)

rate of deformation 8 ε 8 472 (units of 1/time).


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Stress

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1ressure is applied to move the fluid

Stress is described as force per unit area (units of pressure, Pa

σ

= force/area = F/LW

L


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Viscous Fluid

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viscous fluid is defined by the relationship of stress to strain ra

L

σ = 2µε

4iscosity (µ) is the constant of proportionality (units of Pas) +his !constitutive" e&uation describes the mechanical properties

of material


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Viscous Fluid

%f material has a high viscosity (µ),

it will strain less for a given applied stress (σ)

σ = 2µε

µ

!σ

" 2ε


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Physical La#s o$ Motion

%e#ton&s 'st La#: b9ect in motion stays in motion:ction and reaction,

4elocity motion imparted by the top plate induces a reaction of the fluid below


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Physical La#s o$ Motion

%e#ton&s 2nd La#: cceleration of fluid is proportional to the net force

What does this mean?

- %f there is no acceleration, then forces balance, that isthings move but don5t accelerate. %n this case, forces balanceand there is !no net $orce"

$or all viscous fluids, the net force ! ( 4elocities in the arth5s mantle are small and accelerations

are negligible

Momentum is also negligible in slow viscous fluids


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)onservation o$ Mass

0ass is conserved in fluid flow(density changes are negligible)

$luid is !incompressible” .

Rate of fluid flow into bo 8 flow out of bo

#o net accumulation of material


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Stress ;

Strain ;

4iscosity ;

See )lass notes < $ormal treatment of stress and strain


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*ydrostatic Stress

ydrostatic stress is defined as confining pressure

#ormal stresses acting on a particle are e&ual on all sides

=nown as !isotropic"

With no tangential components

σ

'' ! σ

22 ! σ

++ ! P

σ

'2 ! σ

13

! σ2+

! (


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Deviatoric Stress

0antle flow is not driven by hydrostatic pressure

6ut is driven by deviations from it, known as de!iatoric stress

et5s consider an average normal stress

σ

ave = σ

''- σ

22- σ

++ ! σ

ii" +

+

+hen de!iatoric stress is given by

σ

dev = σ

i. − σ

ave

the applied stress - average normal stress 8 difference (deviatoric stres


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Deviatoric Stress

σ

dev =

σ

i. − σ

ave

%f de!iatoric stress is non->ero,than fluid flow proceeds

σ??

< σave

σ?3

σ?@

σ3?

σ33

- σave

σ3@

σ@?

σ@3

σ@@

- σave

%f the diagonals are all e&ual, then there is no deviatoric stress nd there is no fluid flow

+inimumstress

+a,imstres

-eiatoricstress


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Lithostatic Stress

special case of hydrostatic stress

2ydrostatic stress increases with depth in the arth

6ut at each depth, normal stresses are balanced

s a slab sinks into the arth5s interior, it eperiences progressivncreasing hydrostatic stress with depth known as lithostatic stress


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Measurin/ Lithostatic Stress

#ear the surface, stress measurements are stronglyeffected by faults, 9oints, etc.

t deeper depths, pressure closes faults and fractures

and stresses are transmitted across faultsctivityA calculate deviatoric stress in a continental block

#ormal faults in l Salvador Bointing in a rock


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Measurin/ Lithostatic Stress

( e p t h

in et al., 3CCD

$racture density measured in the field in south frica 6oreholes show reduced hydraulic conductivity at deeper depth %ndicating that fractures close with increasing lithostatic stress


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Strain

"train is a measure of deformation

When deformation occurs, different parts of a bodyare displaced by different amounts

p?

p3

0'

02

p@

pE

u' ! b 0

2

isplacements (u) in 0' direction increase with increasing 0

2

u2 ! ( #o displacement in 0

2 directionA


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Strain

p?

p3 0'

02

p@

pE

isplacement of p? and p3 are not e&ual +his implies a gradient of displacement

d u' ! b = # tan αd 0

2

α

Where α is the angle through which the body deforms

Shearing deformation changes along the 02 direction.


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Strain: Solid Body 1otation

p?

p3 0'

02

%f α' ! α

2, then we have rotation only and no deformation

%n this case, the gradient in deformation can be written asA

tan α1 =

α2

α1

d 0'

d u2

tan α2

=

d 02

d u'

"olid body rotation suggests the gradient of deformations are e&u


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Strain:*o# do #e distin/uish deformation and rotation

α2

α1

+hink about how α' and α

2

relate to each other

α1

− α

2

= 0 Rotation only

α1 − α2 = non$ero eformation

α1

+ α

2

= % 1ure shear

α1

+ α

2

C

Simple shear


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Strain:*o# do #e distin/uish deformation and rotation

α2

α1

We generally write this in terms of

the tangent of the rotation angle

d 0'

d u2

'2 ! 3 tan α

1

+ tan α2

= 3 d 0

2

d u'

1otation:

d 02

d u'

De$or5ation:

'2 ! 3 tan α

1

− tan α2

= 3 - d 0

'

d u2


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6he Strain 6ensor

α2

α1

+he strain tensor can them be written

d 0 .

d ui

ε

i& ! 3 - 4

d 0i

d u .

nd the rotation tensor can them be written

d 0i

d u .

ω

i& ! 3 4

d 0 .

d ui


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6he Strain 6ensor : 6est

+he strain tensor can them be written

d 0 .

d ui

ε

i& ! 3 - 4

d 0i

d u .

et5s try a testA %n the case of a bo that is !stretched"

What is i and & in εij

'

+hen find the solution


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6he Strain 6ensor : )ase o$ Volu5e )han/e

%f a cube epands (or shrinks)

V 7 Vo ! ε''

- ε22

- ε++

Vo

Ε

i& ! =

d 0i

d ui u

u %ndicates volume change, dilitation, divergence


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6he Strain 6ensor : )ase o$ Volu5e )han/e

u ! ( 0eans there is no volume changeand fluid is incompressible


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Strain 1ate

Strain rate describes deformation change o!er time

2ow fast can a material deform '

2ow fast can fluid flow '


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Strain 1ate

Strain rate describes deformation change o!er time

2ow fast can a material deform '

2ow fast can fluid flow '

Rates are concerned with !elocity

We can use velocity gradient to measure the rate of fluid flow or shear

d 0 .

d 4i

ε

i&

! 3 - 4 d 0i

d 4 .


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Viscous Fluids

'iscous fluids resist shearing deformation

2ave a linear relationship between stress and strain rate (known as #ewtonian fluids)

$luid eamplesA- air, water viscosities are low

- honey, thick oil viscosities are high


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Viscous Fluids



See /lass #otesA


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Viscous Fluids



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Lect5 Geodyn Stress