Stress-Strain TheoryTheor

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

Strain Tensor
Strain Tensor
x
x
x’
x’
dx
dx
dx’
dx’
u(x)
u(x+dx)
Displacement vector:

Displacement vector: u(x) = x’- x
Length squared:
Length squared: dl = dx + dx + dx = dx dx
21
2
2
3
2
2
i
i
After deformation
dl = dx’ dx’ = (du +dx )
i i i i
2
2
= du du + dx dx + 2 du dx
i
i
i
i
i
i
dx’
dx’
dx
dx

Strain Tensor
Strain Tensor

x
x
x’
x’
dx
dx
dx’
dx’
u(x)
u(x+dx)
Length squared:
Length squared: dl = dx + dx + dx = dx dx
21
2
2
3
2
2
i
i
After deformation
dl = dx’ dx’ = (du +dx )
i i i i
2
2
= du du + dx dx + 2 du dx
i
i
i
i

i
i
Length change:
Length change: dl - dl = du du + 2du dx
i
2 2
i i i
du = du dx
dx
i
j
j
i
Substitute
Substitute
(1)
into equation (1)
dx’
dx’
dx
dx

Strain Tensor
Strain Tensor
x
x
x’
x’
dx
dx

dx’
dx’
u(x)
u(x+dx)
After deformation
Length change:
Length change: dl - dl = du du + 2du dx
i
2 2
i i i
du = du dx
dx
i
j
j
i
Substitute
Substitute
Length change:
Length change: dl - dl = U U
i
2 2
i
(1)
into equation (1)
(du + du + du du )dx dx
i
j
dx dx dx dx
j

j
i i
i
j
k
k
=
Strain Tensor
(2)
dx’
dx’
dx
dx

Problem
Problem
1 light year
V > C
V > C

Problem
Problem
V > C
V > C
1 light year
V < C
V < C
Elastic Strain Theory
Elastic Strain Theory
Elastodynamics

Elastodynamics

Acoustics
Acoustics
L
L
L
L’
L’
L’
=
=
ε
ε
xx
xx
=
=
dL L’-L
dL L’-L
L L
L L
=
=
Length
Change
Change
Length
Length

Acoustics
Acoustics
L
L
L
L’
L’
L’
=
=
ε
ε
xx
xx
=
=
dL L’-L
dL L’-L
L L
L L
=
=
Length
Change
Change
Length
Length

Acoustics
Acoustics

L
L
L
L’
L’
L’
=
=
ε
ε
xx
xx
=
=
dL L’-L
dL L’-L
L L
L L
=
=
Length
Change
Change
Length
Length
No Shear Resistance = No Shear Strength
No Shear Resistance = No Shear Strength

Acoustics
Acoustics

dx
dz
du
dw
dw, du << dx, dz
Tensional
Tensional

Acoustics
Acoustics
=
=
(dz+dw)(dx+du)-dxdz
(dz+dw)(dx+du)-dxdz
dx dz
dx dz
Area

Change
Change
Area
Area
dx
dz
du
dw
=
=
dxdz+dxdw+dzdu-dxdz
dxdz+dxdw+dzdu-dxdz

dx dz
dx dz
+ O(dudw)
+ O(dudw)
dw du
dw du
dz
dz
=
=
dx
dx
+
+
ε
ε
ε
ε
zz
zz
xx
xx
=
=
+
+
=
=
U
Infinitrsimal strain

assumption: e<.00001
Dilitation
big +small
big +small
really small
really small
big +small
big +small

1D Hooke’s Law
1D Hooke’s Law
=
=
U
κ
Bulk Modulus
Infinitrsimal strain
assumption: e<.00001
−κ
ε
ε
ε
ε
zz
zz
xx
xx
+
+
( )

P =
-
Pressure is F/A of outside
media acting on face of box
F/A =
du
dx
strain
pressure

Hooke’s Law
Hooke’s Law
=
=
U
κ
Infinitrsimal strain
assumption: e<.00001
ε
ε
ε
ε
zz
zz
κ
-
ε
ε
ε
ε

zz
zz
xx
xx
+
+
( )
F/A =
κ
xx
xx
+
+
( )
Bulk Modulus
κ
Larger = Stiffer Rock
Larger = Stiffer Rock
P =
Dilation
Dilation
+ S
+ S
Source or Sink
Source or Sink
Compressional
Compressional

Newton’s Law
Newton’s Law

κ
Larger = Stiffer Rock
Larger = Stiffer Rock
ma = F
-
dP
dP
dx
u =
u =
ρ
ρ

-
dP
dP
dz
w =
w =
ρ
ρ

;
density
P (x+dx,z,
P (x+dx,z,
t
t
)
)

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

-dxdz

-
dP
dP
dx
u =
u =
ρ
ρ

-
dP

dP
dz
w =
w =
ρ
ρ

;
density
κ
Larger = Stiffer Rock
Larger = Stiffer Rock
P (x+dx,z,
P (x+dx,z,
t
t
)
)
P (x,z,
P (x,z,
t
t
)
)

-
P
P
u =
u =

ρ
ρ
Newton’s Law
Newton’s Law
1
1
st
st
-Order Acoustic Wave Equation
-Order Acoustic Wave Equation
u=(u,v,w)
u=(u,v,w)


-
P
P
u =
u =
ρ
ρ
Newton’s Law
Newton’s Law
1
1
st
st
-Order Acoustic Wave Equation
-Order Acoustic Wave Equation
= -

= - U
κ
P
P
(Hooke’s Law)
(Hooke’s Law)
(Newton’s Law)
(Newton’s Law)
(1)
(2)
Divide (1) by density and take Divergence:
Divide (1) by density and take Divergence:
(3)
Take double time deriv. of (2) & substitute (2) into (3)
Take double time deriv. of (2) & substitute (2) into (3)

-
P
P
P =
P =
ρ
ρ
1
[
]
κ
(4)

-

P
P
u =
u =
ρ
ρ
1
[
]



Newton’s Law
Newton’s Law
2nd-Order Acoustic Wave Equation
2nd-Order Acoustic Wave Equation

-
P
P
P =
P =
ρ
ρ
1
[
]
κ
P
P

P =
P =
κ

ρ
ρ
Constant density assumption
Constant density assumption
κ
ρ
ρ
c =
c =
2
2
Substitute velocity
Substitute velocity
P
P
P =
P =

c
c
2
2
2
2

Summary

Summary
Constant density assumption
Constant density assumption
= -
= -
U
κ
1. Hooke’s Law: P
1. Hooke’s Law: P
2. Newton’s Law:
2. Newton’s Law:

-
P
P
u =
u =
ρ
ρ

-
P
P
P =
P =
ρ
ρ
1
[
]

κ
3. Acoustic Wave Eqn:
3. Acoustic Wave Eqn:
P
P
P =
P =

c
c
2
2
2
2
;
κ
ρ
ρ
c =
c =
2
2
+ F
+ F
Body Force Term
Body Force Term

Problems
Problems
1.

1.
Utah and California movingE-W apart at 1
Utah and California movingE-W apart at 1


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


xx
xx
xy
xy
2.
2.
LA. coast andSacremento moving N-S apart at 10
LA. coast andSacremento moving N-S apart at 10


cm/year.
cm/year.
Calculate strain rate, where distance is 2000 km. Is is e or e
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
3. A plane wave soln to W.E. is u= cos
(kx-wt) i.
(kx-wt) i.
Compute divergence. Does the volume change
Compute divergence. Does the volume change


as a function of time? Draw state of deformation boxes
as a function of time? Draw state of deformation boxes
Along path
Along path

Divergence
Divergence
U
U
= lim
U
n
dl
A
A 0
n
U(x+dx,z)

U(x+dx,z)
U(x,z)
U(x,z)
= U(x+dx,z)dz
= U(x+dx,z)dz
dxdz
dxdz
+ U(x,z+dz)cos(90)dx
+ U(x,z+dz)cos(90)dx
dxdz
dxdz
-
-
U(x,z)dz
U(x,z)dz
dxdz
dxdz
+ U(x,z+dz)cos(90)dx
+ U(x,z+dz)cos(90)dx
dxdz
dxdz
n
= 0
= 0
>> 0
>> 0
(x,z)
(x,z)
No sources/sinks inside box.
No sources/sinks inside box.

What goes in must come out
What goes in must come out
Sources/sinks inside box.
Sources/sinks inside box.
What goes in might not come out
What goes in might not come out
ε
ε
ε
ε
zz
zz
xx
xx
+
+
( )
P =
κ
-
(x+dx,z+dz)
(x+dx,z+dz)