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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’. - PowerPoint PPT Presentation
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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 2ii
After deformation
dl = dx’ dx’ = (du +dx ) i i i i22
= 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 2ii
After deformation
dl = dx’ dx’ = (du +dx ) i i i i22
= 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 dxdx
i j
j
iSubstituteSubstitute
(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 dxdx
i j
j
iSubstituteSubstitute
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’-LL LL L ==
Length ChangeChange
Length Length
AcousticsAcoustics
LLL
L’L’L’
== xxxx ==dL L’-LdL L’-LL LL L ==
Length ChangeChange
Length Length
AcousticsAcoustics
LLL
L’L’L’
== xxxx ==dL L’-LdL L’-LL 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)-dxdzdx 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 dudzdz
==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 dlA
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)
