Lecture 9 Elasticity - Indian Institute of...

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Lecture 9 Elasticity

Transcript of Lecture 9 Elasticity - Indian Institute of...

Page 1: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

Lecture 9

Elasticity

Page 2: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

•  Einstein summation convention: repetition of index in a term implies summation over the repeated index

aibi = a1b1 + a2b2 + a3b3

σkk = σ11 + σ22 + σ33

•  Kronecker delta

1 if i = j δij = 0 if i ≠ j

Page 3: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

•  Stress tensor is symmetrical

•  Units :

Stress = Force / Area

Newton / m2 = Pascal (Pa) 1 MPa = 106 Pa = 10 bars

Page 4: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

Deviatoric Stress:

ti j = si j �13

skkdi j

r∂2Q∂t2 = (l+2µ)—2Q

r ∂2

∂t2

✓∂u∂y

� ∂v∂x

◆= µ—2

✓∂u∂y

� ∂v∂x

r ∂2

∂t2

✓∂v∂z

� ∂w∂y

◆= µ—2

✓∂v∂z

� ∂w∂y

r ∂2

∂t2

✓∂w∂x

� ∂u∂z

◆= µ—2

✓∂w∂x

� ∂u∂z

u = —f+—⇥y

—2f =1

V 2p

∂2f∂t2

—2y =1

V 2s

∂2y∂t2

Vp =

sl+2µ

r=

sK + 4

3µr

1

Principal Stress (σ1, σ2, σ3) SHmax or most compressive horizontal principal axis Differential stress (σ1 - σ3)

Page 5: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

World Stress Map

Page 6: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

World Stress Map

Page 7: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

Strain: Response of a body to stress

x δx

L M

δx+δu x+u

L’ M’

exx = ∂ux / ∂x eyy = ∂uy / ∂y ezz = ∂uz / ∂z exy = eyx = ½ (∂ux / ∂y + ∂uy / ∂x) exz = ezx =½ (∂ux / ∂z + ∂uz / ∂x) eyz = ezy = ½ (∂uy / ∂z + ∂uz / ∂y)

Page 8: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

Hooke’s law

Stress is proportional to Strain σij = λΘδij + 2µeij

Θ = exx + eyy + ezz δij è Kronecker delta λ and µ è Lame’s parameters

µ è shear modulus

Page 9: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

•  Young’s modulus, E = σxx / exx

•  Bulk modulus, K = p / Θ = λ + ⅔ µ

•  Hydrostatic pressure, p = ⅓ (σxx + σyy + σzz)

•  Poisson’s ratio, σ = - ezz / exx

Page 10: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

Stress and strain• Stress-strain relation:

– Elastic domain:

• Stress-strain relation is linear

• Hooke’s law applies

– Beyond elastic domain:

• Initial shape not recovered when stress is

removed

• Plastic deformation

• Eventually stress > strength of material =>

failure

– Failure can occur within the elastic domain

= brittle behavior

• Strain as a function of time under stress:

– Elastic = no permanent strain

– Plastic = permanent strain

• Our goal: find the relation between stress

and strain

•  In elastic domain - stress and strain linear -  Hooke’s law applies

•  Beyond elastic domain -  initial shape not recovered when stress removed -  plastic deformation

-  eventually failure

No permanent strain

Permanent strain

Page 11: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

Flexure - subduction

• In addition to load of overriding plate:

– Sediments

– Non-elastic response

Fowler: The Solid Earth

Page 12: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

Elasticity• Let’s assume:

– A rectangular prism with 3 sides defining (O,x,y,z)

– A uniform tension Nz exerted on 2 sides perpendicular to

(O,z)

• When the prism is stretched along (O,z):

– Change in length is proportional to tension: εz = Δh/h ∝ Nz

– One can show experimentally that:

– E = Young’s modulus

– Units of stress = km/m2

– Small E => more elastic

• If the prism is stretched along (O,z), it must shrink along

(O,x,y) to conserve mass:

– One can show experimentally that contraction:

– ν = Poisson’s ratio (dimensionless)

!

"z

=#h

h=1

EN

z

!

"x = "y = #$

ENz x

y

z

O

-Nz

Nz

Elasticity• Let’s assume:

– A rectangular prism with 3 sides defining (O,x,y,z)

– A uniform tension Nz exerted on 2 sides perpendicular to

(O,z)

• When the prism is stretched along (O,z):

– Change in length is proportional to tension: εz = Δh/h ∝ Nz

– One can show experimentally that:

– E = Young’s modulus

– Units of stress = km/m2

– Small E => more elastic

• If the prism is stretched along (O,z), it must shrink along

(O,x,y) to conserve mass:

– One can show experimentally that contraction:

– ν = Poisson’s ratio (dimensionless)

!

"z

=#h

h=1

EN

z

!

"x = "y = #$

ENz x

y

z

O

-Nz

Nz

Elasticity• Let’s assume:

– A rectangular prism with 3 sides defining (O,x,y,z)

– A uniform tension Nz exerted on 2 sides perpendicular to

(O,z)

• When the prism is stretched along (O,z):

– Change in length is proportional to tension: εz = Δh/h ∝ Nz

– One can show experimentally that:

– E = Young’s modulus

– Units of stress = km/m2

– Small E => more elastic

• If the prism is stretched along (O,z), it must shrink along

(O,x,y) to conserve mass:

– One can show experimentally that contraction:

– ν = Poisson’s ratio (dimensionless)

!

"z

=#h

h=1

EN

z

!

"x = "y = #$

ENz x

y

z

O

-Nz

Nz

Young’s modulus, E = σz / εz

Elasticity• Let’s assume:

– A rectangular prism with 3 sides defining (O,x,y,z)

– A uniform tension Nz exerted on 2 sides perpendicular to

(O,z)

• When the prism is stretched along (O,z):

– Change in length is proportional to tension: εz = Δh/h ∝ Nz

– One can show experimentally that:

– E = Young’s modulus

– Units of stress = km/m2

– Small E => more elastic

• If the prism is stretched along (O,z), it must shrink along

(O,x,y) to conserve mass:

– One can show experimentally that contraction:

– ν = Poisson’s ratio (dimensionless)

!

"z

=#h

h=1

EN

z

!

"x = "y = #$

ENz x

y

z

O

-Nz

Nz

Poisson’s ratio, Perfectly incompressible material,

Poisson’s ratio

• Poisson's ratio = ratio oftransverse to longitudinalnormal strain underuniaxial stress, in thedirection of stretchingforce, with:

– Tensile deformation positive

– Compressive deformationnegative

• All common materialsbecome narrower in crosssection when they arestretched => Poisson’s ratiopositive.

!

"z

=#h

h=1

EN

z

!

"x = "y = #$

ENz

!

"# = $%x

%z Poisson’s ratio

• If Vs = 0, ν = 0.5:

– Either a fluid (shear waves

do not propagate through

fluids)

– Or material that maintains

constant volume regardless

of stress = incompressible.

• Vs ~ 0 is characteristic of

a gas reservoir.

!

" =(Vp

2# 2Vs

2)

2(Vp

2#Vs

2)

A negative Poisson's ratio change is associated with the top of a gas zone,

Poisson’s ratio

Examples:

– Perfectly incompressible material => ν = 0. 5 (no volume change)

– If ν > 0.5 => volume increase under compression = dilatant.

– If ν < 0 => become thicker when stretched = auxetic materials (somepolymer foams)

– Rubber: ν = 0.5, Cork: ν = 0 (why is cork used to close glassbottles?)

– Earth’s interior ν = 0.24-0.27

– Granite: ν = 0.2-0.3

– Carbonate rocks ν ~ 0.3

– Sandstones ν ~ 0.2

– Shale ν > 0.3

– Coal ν ~ 0.4.

E. Calais notes

Page 13: Lecture 9 Elasticity - Indian Institute of Scienceceas.iisc.ac.in/~aghosh/Teaching/Lecture9_elasticity.pdf · Lecture 9 Elasticity • Einstein summation convention: repetition of

•  Young’s modulus, E = σxx/εxx

•  Shear modulus, μ = ½ σxy / εxy

•  Poisson’s ratio, ν = εxx /εzz

•  Lame’s parameter, λ

•  Bulk modulus, K = p / Θ where hydrostatic pressure, p = ⅓ (σxx + σyy + σzz)

There are 5 elastic constants:

K = E / 3(1-2ν)

μ = E / 2(1+ν)

λ = Eν / (1+ν)(1-2ν)