Wave propagation

PARTICLES TO CONTINUUM

Wave propagation in elastic solid media

April 19, 2010

Wave propagation

SUMMARY

I Tools: Vector, Tensors...I Actors: Stress, Strain, Elasticity Relation...I Goal: Wave propagation in elastic media

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).

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

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

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)...

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

Wave propagation

Vectors and Tensors

VECTOR AND TENSOR CALCULUS

It is easy to verify that:

curl(gradφ) = 000 ∇∇∇× (∇∇∇φ) = 000div(curlψψψ) = 0 ∇∇∇ · (∇∇∇×ψψψ) = 0

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

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

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

]

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 ]

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

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

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

]

Wave propagation

Elasticity

ELASTICITY

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

)

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

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

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

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

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

Wave propagation

The propagation of waves in elastic solid media

waves

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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