Stress-Strain Stress-Strain TheoryTheory

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 TensorStrain Tensor

xx x’x’

dxdx 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

22

3

2 2

ii

After deformation

dl = dx’ dx’ = (du +dx ) i i i i

22

= du du + dx dx + 2 du dxi i i i i i

dx’dx’dxdx

Strain TensorStrain Tensor

xx x’x’

dxdx dx’dx’u(x)

u(x+dx)

Length squared:Length squared: dl = dx + dx + dx = dx dx 21

22

3

2 2

ii

After deformation

dl = dx’ dx’ = (du +dx ) i i i i

22

= du du + dx dx + 2 du dxi i i i i i

Length change:Length change: dl - dl = du du + 2du dxi2 2

i i i

du = du dx

dx

i j

j

i

SubstituteSubstitute

(1)

into equation (1)

dx’dx’dxdx

Strain TensorStrain Tensor

xx x’x’

dxdx dx’dx’u(x)

u(x+dx)After deformation

Length change:Length change: dl - dl = du du + 2du dxi2 2

i i i

du = du dx

dx

i j

j

i

SubstituteSubstitute

Length change:Length change: dl - dl = U Ui2 2

i

(1)

into equation (1)

(du + du + du du )dx dxi j

dx dx dx dxj

j i i

i

j

kk=

Strain Tensor(2)

dx’dx’dxdx

ProblemProblem

1 light year

V > CV > C

ProblemProblemV > CV > C

1 light year

V < CV < C

Elastic Strain TheoryElastic Strain Theory

ElastodynamicsElastodynamics

AcousticsAcoustics

LLL

L’L’L’

== xxxx ==dL L’-LdL L’-L

L LL L==

Length ChangeChange

Length Length

AcousticsAcoustics

LLL

L’L’L’

== xxxx ==dL L’-LdL L’-L

L LL L==

Length ChangeChange

Length Length

AcousticsAcoustics

LLL

L’L’L’

== xxxx ==dL L’-LdL L’-L

L LL L==

Length ChangeChange

Length Length

No Shear Resistance = No Shear StrengthNo Shear Resistance = No Shear Strength

AcousticsAcoustics

dx

dz

du

dw

dw, du << dx, dz

TensionalTensional

AcousticsAcoustics

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

dx dzdx dz

Area ChangeChange

Area Area

dx

dz

du

dw

==dxdz+dxdw+dzdu-dxdzdxdz+dxdw+dzdu-dxdz

dx dzdx dz+ O(dudw)+ O(dudw)

dw dudw du

dzdz==

dx dx ++

zzzz xxxx== ++

== U

Infinitrsimal strain

assumption: e<.00001

Dilitation

big +smallbig +small really smallreally smallbig +smallbig +small

1D Hooke’s Law1D Hooke’s Law== U

Bulk Modulus

Infinitrsimal strain

assumption: e<.00001

zzzz xxxx++( )P = -

Pressure is F/A of outside

media acting on face of box

F/A = dudx

strainpressure

Hooke’s LawHooke’s Law== U

Infinitrsimal strain

assumption: e<.00001

zzzz-

zzzz xxxx++( )F/A = xxxx++( )Bulk Modulus

Larger = Stiffer RockLarger = Stiffer Rock

P =

DilationDilation

+ S+ S

Source or SinkSource or Sink

CompressionalCompressional

Newton’s LawNewton’s Law

Larger = Stiffer RockLarger = Stiffer Rock

ma = F

-dPdPdxu = u = .. -

dPdPdzw = w = ..;

density

P (x+dx,z,P (x+dx,z,tt))P (x,z,P (x,z,tt))

Net force = [P(x,+dx,z,t)-P(x,z,t)]dzNet force = [P(x,+dx,z,t)-P(x,z,t)]dzxx,,u u ..-dxdz

-dPdPdxu = u = .. -

dPdPdzw = w = ..;

density

Larger = Stiffer RockLarger = Stiffer Rock

P (x+dx,z,P (x+dx,z,tt))P (x,z,P (x,z,tt))

.. - P P u = u =

Newton’s LawNewton’s Law11stst-Order Acoustic Wave Equation-Order Acoustic Wave Equation

u=(u,v,w)u=(u,v,w)

.. - P P u = u =

Newton’s LawNewton’s Law11stst-Order Acoustic Wave Equation-Order Acoustic Wave Equation

= - = - UPP (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 =

1[ ](4)

.. - P P u = u = 1[ ]

.. ..

Newton’s LawNewton’s Law2nd-Order Acoustic Wave Equation2nd-Order Acoustic Wave Equation

..- P P P = P =

1[ ]

P P P = P = ..

Constant density assumptionConstant density assumption

c = c =

22Substitute velocitySubstitute velocity

P P P = P = ..

cc2 2 2 2

SummarySummary

Constant density assumptionConstant density assumption

= -= - U1. Hooke’s Law: P1. Hooke’s Law: P

2. Newton’s Law: 2. Newton’s Law: .. - P P u = u =

..- P P P = P =

1[ ]

3. Acoustic Wave Eqn:3. Acoustic Wave Eqn:

P P P = P = ..

cc2 2 2 2

;c = c =

22 + F+ F

Body Force TermBody Force Term

ProblemsProblems

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

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

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

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

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 changeCompute divergence. Does the volume change

as a function of time? Draw state of deformation boxesas 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 outWhat goes in must come out

Sources/sinks inside box. Sources/sinks inside box.

What goes in might not come outWhat goes in might not come out zzzz xxxx++( )P = -

(x+dx,z+dz)(x+dx,z+dz)