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Wave propagation
PARTICLES TO CONTINUUM
Wave propagation in elastic solid media
April 19, 2010
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Wave propagation
SUMMARY
I Tools: Vector, Tensors...I Actors: Stress, Strain, Elasticity Relation...I Goal: Wave propagation in elastic media
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Wave propagation
Vectors and Tensors
VECTOR aaa
In the 3-dimensional Euclidean space (O,eee1,eee2,eee3) a VECTOR aaacan be expressed as a vector sum of its 3 components:
aaa = a1eee1 + a2eee2 + a3eee3 =3∑
i=1
aieeei = aieeei
and in matrix form:
[aaa] = [ai ] =
a1a2a3
We need 3 "quantiies" for its complete specification(for a scalar, we need 1).
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Wave propagation
Vectors and Tensors
TENSOR AAA
TENSOR AAA is a linear transformation of the Euclidean vectorspace E into itself, a law that transforms an arbitrary vector aaa ina vector uuu:
AAA · aaa = uuu
in such a way that:
AAA · (αaaa + βbbb) = αAAA · aaa + βAAA · bbb ∀aaa,bbb ∈ E, α, β ∈ R
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Wave propagation
Vectors and Tensors
TENSOR AAAAgain, if (eee1,eee2,eee3) are the base vectors of the coordinate system,then the components are:
Ai,j = (AAA · eeei ) · eeej (i , j = 1,2,3)
and matrix associated to AAA in the coordinate system is:
[AAA] = [Ai,j ] =
A11 A12 A13A21 A22 A23A31 A32 A33
we need 9 "quantities" for its complete specification.The linear transformation
ui =3∑
j=1
Aijaj = Aijaj
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Wave propagation
Vectors and Tensors
VECTOR AND TENSOR CALCULUS
φ(PPP) = φ(x1, x2, x3) = φ(xm) Scalar Field (Tensor order 0)
ψψψ(PPP) =
ψ1 = ψ1(x1, x2, x3)ψ2 = ψ2(x1, x2, x3)ψ3 = ψ3(x1, x2, x3)
= ψi (xm) Vector Field (Tensor o.1)
AAA(PPP) =
A11(xm) A12(xm) A13(xm)A21(xm) A22(xm) A23(xm)A31(xm) A32(xm) A33(xm)
= Aij (xm) Tensor Field (o. 2)
...CCC(PPP) = · · · = Cijkh(xm) Tensor Field (Tensor order 4)...
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Wave propagation
Vectors and Tensors
VECTOR AND TENSOR CALCULUS
gradφ =∇∇∇φ(xm)
[(gradφ)i =
∂φ
∂xi(xm)
]VECTOR
gradψψψ =∇∇∇ψψψ(xm)
[(gradψψψ)ij =
∂ψi
∂xj(xm)
]2nd order TENSOR
divψψψ =∇∇∇ ·ψψψ(xm)
[divψψψ =
∂ψi
∂xi(xm)
]SCALAR
divAAA =∇∇∇ ·AAA(xm)
[(divAAA)i =
∂Aij
∂xj(xm)
]VECTOR
curlψψψ =∇∇∇×ψψψ(xm)
[(curlψψψ)i = εijk
∂ψk
∂xj(xm)
]VECTOR
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Wave propagation
Vectors and Tensors
VECTOR AND TENSOR CALCULUS
It is easy to verify that:
curl(gradφ) = 000 ∇∇∇× (∇∇∇φ) = 000div(curlψψψ) = 0 ∇∇∇ · (∇∇∇×ψψψ) = 0
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Wave propagation
Elasticity
STRESSThe Stress describes the forces acting in the interior of a continuousbody C, in each infinitesimaly deformed configuration. Le be:
I PPP point in the interior of C in the referring configuration;
I π a surface with normal nnn, passing through PPP, cutting C+ and C−;
I ∆A a small portion of the surface π in the vicinity if PPP;
I ∆rrr the force field acting from C+ on C− through ∆A
Eulero-Cauchy STRESS VECTOR
ttt (n) = lim∆A→0
∆r∆A
lim∆A→0
∆mmm∆A
= 0
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Wave propagation
Elasticity
STRESS
t (n)t (n)t (n) = t (n)t (n)t (n)(P,nP,nP,n)
Given a system of rectangular cartesian coordinates (0,eee1,eee2,eee3):
ttt1 = σ11eee1 + σ12eee2 + σ13eee3
ttt2 = σ21eee1 + σ22eee2 + σ23eee3
ttt3 = σ31eee1 + σ32eee2 + σ33eee3
[σij ] =
σ11 σ12 σ13σ21 σ22 σ23σ31 σ32 σ33
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Wave propagation
Elasticity
STRESS
Cauchy-Poisson Theorem
1. σσσ is the Stress Tensor such that tntntn = σσσ · nnnt(n)1
t(n)3
t(n)3
=
σ11 σ12 σ13σ21 σ22 σ23σ31 σ32 σ33
n1n2n3
2. The solid to be in equilibrium:
∇∇∇ · σσσ + ρbbb = 000[∂σij
∂xj+ ρbi = 0
]3. The solid to be in equilibrium σσσ have to be symmetric:
σσσ = σσσT [σij = σji
]
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Wave propagation
Elasticity
STRAIN
Deformation of a body C:
QQQ = f (PPP) [ηi = fi (xj )] ∀PPP ∈ C η1 = η1(x1, x2, x3)η2 = η2(x1, x2, x3)η3 = η3(x1, x2, x3)
Desplacement of the point PPP abody C:
vvv(PPP) = (QQQ)− (PPP) = f (PPP)−PPP
[vi (xj ) = ηi − xi = fi (xj )− xi ]
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Wave propagation
Elasticity
STRAINWe want to analize the deformation of the the vector dPPP in dQQQ:
dQQQ =∇∇∇f (PPP)dPPP =∇∇∇[vvv(PPP) + PPP] · dPPP = (H + IH + IH + I)︸ ︷︷ ︸FFF
·dPPP
with detFFF > 0 FFF = III translation
We focus on Infinetisimal theory: |vvv | ∼ 0 and |HHH| ∼ 0
HHH =12(HHH + HHHT )+
12(HHH −HHHT )
εεε = 12
(HHH + HHHT
)INFINITESIMAL STRAIN TENSOR
RRR = 12
(HHH −HHHT
)INFINITESIMAL ROTATION TENSOR
vvv(P ′P ′P ′) = vvv(PPP)︸ ︷︷ ︸Translaion
+RRR · dPPP︸ ︷︷ ︸Rotation
+ εεε · dPPP︸ ︷︷ ︸PureStrain
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Wave propagation
Elasticity
STRAIN
Similarly to stress, given the direction nnn such that:
dPdPdP = dPnnn in the reference configuration
dQQQ = dP(εεε · nnn) ∀nnn in PPPdQ1dQ3dQ3
= dP
ε11 ε12 ε13ε21 ε22 ε23ε31 ε32 ε33
n1n2n3
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Wave propagation
Elasticity
ELASTICITY
CONSTITUTIVE RELATIONS: to describe the dependence of theSTRESS in a body on kinematic variables, in particular STRAIN.
The simplest relation between them is Linear Elasticity (Hooke’s law):
σ(P)σ(P)σ(P) = CCC · ε(P)ε(P)ε(P) [σij = Cijkhεkh]
!!!!! σσσ depend only on σσσ, the symmetric part of∇ · v∇ · v∇ · v
In a particle sample, having k and k t , as normal and tangentialcontact stiffness (lll = lnnn is the branch vector):
Cijkh =1V
∑p
[k
Nc∑c=1
(2l2)nci nc
j nck nc
h + k tNc∑
c=1
(2l2)nci tc
j nck tc
h
]
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Wave propagation
Elasticity
ELASTICITY
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Wave propagation
The propagation of waves in elastic solid media
EQUATION OF MOTION
∇∇∇ · σσσ + ρbbb︸︷︷︸=0
= ρ∂2uuu∂t2 Newton 2 law applied to ρdV
σσσ = CCC · εεε Elasticity relation
εεε =12
(∇uuu +∇uuu(T )
)Kinematic Compatibility relation
∂σij
∂xj= ρ
∂2ui
∂t2
σij = Cijkhεkh
εij = 12
(∂ui∂xj
+∂uj∂xi
)
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Wave propagation
The propagation of waves in elastic solid media
For ISOTROPIC materials exist symmetries in C, only 3 elasticconstants are different from zero:
C1111 = C2222 = C3333 6= 0C1122 = C1133 = C2233 6= 0C1212 = C1313 = C2323 6= 0
It is possible to reduce to only 2 parameters independent betweenthem, using the Lamè moduli. We call:
G = C1212 = C1313 = C2323
λ = C1122 = C1133 = C2233
and the additional relation is: C1111 = C2222 = C3333 = λ+ 2G
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Wave propagation
The propagation of waves in elastic solid media
The elasticity relation becomes:
σσσ = CCC · εεε = λ(∇ · uuu)III + 2Gεεε = λ(∇ · uuu)III + G(∇uuu +∇uuu(T )) (1)
and inserting (1), the equation of motion:
ρ∂2uuu∂t2 = ∇ · (λ(∇ · uuu)III + G(∇uuu +∇uuu(T )))
= λ∇(∇ · uuu)III + G∇ · (∇uuu +∇uuu(T )))
= (λ+ G)∇(∇ · uuu) + G∇2uuu
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Wave propagation
The propagation of waves in elastic solid media
For every vector uuu we can decompose it:
uuu = uuu(P)+uuu(S) =∇∇∇φ+∇∇∇×ψψψ[ui = u(P)
i + u(S)i =
∂φ
∂xi+ εijk
∂ψj
∂xk
]for the vector calculus properties:
∇∇∇× (∇∇∇φ) =∇∇∇× uuu(P) = 000∇∇∇ · (∇∇∇× ψ) =∇∇∇ · uuu(S) = 0
uuu(P) PRESSURE (IRROTATIONAL) WAVES involving no rotationsuuu(S) SHEAR(EQUIVOLUMINIAL) WAVES involving rotations withoutdilatation
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Wave propagation
The propagation of waves in elastic solid media
So the equations of motion become:
ρ∂2∇∇∇φ∂t2 + ρ
∂2(∇∇∇×ψψψ)
∂t2 = (λ+ G)
∇∇∇2(∇∇∇φ) +∇∇∇ · (∇∇∇×ψψψ)︸ ︷︷ ︸=0
+ G∇∇∇2(∇∇∇φ) + G∇∇∇2(∇∇∇×ψψψ)
[ρ∂2∇∇∇φ∂t2 − (λ+ 2G)∇∇∇2(∇∇∇φ)
]+
[ρ∂2(∇∇∇×ψψψ)
∂t2 −G∇∇∇2(∇∇∇×ψψψ)
]= 0
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Wave propagation
The propagation of waves in elastic solid media
[ρ∂2uuu(P)
∂t2 − (λ+ 2G)∇∇∇2(uuu(P))
]︸ ︷︷ ︸
uuu(P)
+
[ρ∂2uuu(S)
∂t2 −G∇∇∇2uuu(S)
]︸ ︷︷ ︸
uuu(S)
= 0
For a pressure wave and a shear wave, respectively we have:
∂2uuu(P)
∂t2 − (λ+ 2G)
ρ∇∇∇2(uuu(P)) = 0 v (P) =
√λ+ 2G
ρ
∂2uuu(S)
∂t2 − Gρ∇∇∇2uuu(S) = 0 v (S) =
√Gρ
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Wave propagation
The propagation of waves in elastic solid media
waves
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Wave propagation
The propagation of waves in elastic solid media
Some references
The classic!!! A.E.H. Love A treatise on the mathematical theory ofelasticity, Dover, New York (1944).
Contains details on the different anisotropy classes of materials S.G.Lekhnitskii Theory of elasticity of an isotropic elasticbody, Holed day, Inc., San Francisco (1963).
An easy and practical tool! A.J.M. Spencer Continuum Mechanics,Dover Publications, Inc. Mineola, New York (2004).
Full of example on tensor calculus P. Chadwick Continuum Mechanics:Concise Theory and Problems, Dover Publications, Inc.Mineola, New York (1999).
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