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Stress-Strain Theory. Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory. . u(x+dx). dx. dx’. dx. dx’. u(x). x. x’.

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slide1
Stress-Strain Theory

Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory.

slide2
u(x+dx)

dx

dx’

dx

dx’

u(x)

x

x’

Length squared: dl = dx + dx + dx = dx dx

2

2

2

2

i

i

1

2

3

dl = dx’ dx’ = (du +dx )

2

2

i

i

i

i

= du du + dx dx + 2 dudx

i

i

i

i

i

i

Strain Tensor

After deformation

Displacement vector: u(x) = x’- x

slide3
u(x+dx)

dx

dx’

dx

dx’

u(x)

x

x’

Length squared: dl = dx + dx + dx = dx dx

2

2

2

2

i

i

1

2

3

dl = dx’ dx’ = (du +dx )

2

2

i

i

i

i

Length change:dl - dl = du du + 2du dx

2

2

= du du + dx dx + 2dudx

i

i

i

i

i

i

i

i

i

i

du = du dx

Substitute

i

i

j

dx

j

Strain Tensor

After deformation

(1)

into equation (1)

slide4
u(x+dx)

dx

dx’

dx

dx’

u(x)

x

x’

(1)

into equation (1)

Length change:dl - dl = U U

2

2

Strain Tensor

Length change:dl - dl = du du + 2du dx

i

i

2

2

i

i

i

i

(du + du + du du )dx dx

du = du dx

=

i

j

j

k

k

i

Substitute

dx

dx

dx

dx

i

i

j

dx

j

j

i

i

j

Strain Tensor

After deformation

(2)

slide5
1 light year

Problem

V > C

slide6
1 light year

Problem

V > C

V < C

Elastic Strain Theory

Elastodynamics

slide7
dL L’-L

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics

slide8
dL L’-L

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics

slide9
dL L’-L

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics

No Shear Resistance = No Shear Strength

slide10
Tensional

dw

du

dz

Acoustics

dw, du << dx, dz

dx

slide11
dxdz+dxdw+dzdu-dxdz

AreaChange

(dz+dw)(dx+du)-dxdz

=

=

+ O(dudw)

dx dz

dx dz

Area

dw du

=

+

dz

dx

e

+

dw

=

du

xx

U

=

dz

e

zz

Acoustics

really small

big +small

big +small

Infinitrsimal strain

assumption: e<.00001

Dilitation

dx

slide12
k

e

-

P =

(

)

+

Bulk Modulus

xx

U

=

e

dx

zz

1D Hooke’s Law

pressure

strain

-k

du

Infinitrsimal strain

assumption: e<.00001

F/A =

Pressure is F/A of outside

media acting on face of box

slide13
k

F/A =

(

)

+

xx

Compressional

Source or Sink

k

Larger = Stiffer Rock

=

e

e

zz

zz

Hooke’s Law

Dilation

e

k

U

Infinitrsimal strain

assumption: e<.00001

e

k

-

P =

(

)

+ S

+

xx

Bulk Modulus

slide14
Newton’s Law

..

..

-

-

dP

dP

r

r

;

w =

u =

dx

dz

Net force = [P(x,+dx,z,t)-P(x,z,t)]dz

density

x,

..

k

Larger = Stiffer Rock

r

u

P (x,z,t)

P (x+dx,z,t)

ma = F

-dxdz

slide15
Newton’s Law

1st-Order Acoustic Wave Equation

..

P

-

r

u =

u=(u,v,w)

..

..

-

-

dP

dP

r

r

;

w =

u =

dx

dz

density

k

Larger = Stiffer Rock

P (x,z,t)

P (x+dx,z,t)

slide16
Newton’s Law

1st-Order Acoustic Wave Equation

..

P

-

r

u =

(1)

(3)

(4)

..

..

k

P

= -

U

(2)

..

]

P

-

[

u =

1

r

..

k

]

P

-

[

P =

1

r

(Newton’s Law)

(Hooke’s Law)

Divide (1) by density and take Divergence:

Take double time deriv. of (2) & substitute (2) into (3)

slide17
Newton’s Law

2nd-Order Acoustic Wave Equation

..

k

]

P

-

[

P =

1

r

..

k

P

P =

r

k

c =

2

Substitute velocity

r

..

2

P

c

P =

2

Constant density assumption

slide18
Summary

..

..

P

-

k

r

k

]

u =

P

-

[

1. Hooke’s Law: P

P =

= -

U

1

r

2. Newton’s Law:

3. Acoustic Wave Eqn:

k

c =

2

r

..

2

P

c

;

P =

2

Constant density assumption

Body Force Term

+ F

slide19
Problems

1. Utah and California movingE-W apart at 1cm/year.

Calculate strain rate, where distance is 3000 km. Is it e or e ?

2. LA. coast andSacremento moving N-S apart at 10cm/year.

Calculate strain rate, where distance is 2000 km. Is is e or e ?

xx

xx

xy

xy

3. A plane wave soln to W.E. is u= cos

(kx-wt) i.

Compute divergence. Does the volume change

as a function of time? Draw state of deformation boxes

Along path

slide20
U

U

n

dl

U

= lim

k

e

-

A

P =

(

)

+

A 0

xx

+ U(x,z+dz)cos(90)dx

+ U(x,z+dz)cos(90)dx

- U(x,z)dz

dxdz

dxdz

dxdz

dxdz

(x+dx,z+dz)

n

n

e

zz

Divergence

= U(x+dx,z)dz

>> 0

= 0

No sources/sinks inside box.

What goes in must come out

Sources/sinks inside box.

What goes in might not come out

U(x,z)

U(x+dx,z)

(x,z)

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