Peter MOCZO, Peter PAZAK

Department of Astronomy, Physics of the Earth, and MeteorologyFaculty of Mathematics, Physics and Informatics

Comenius University

LECTURE NOTES FOR STUDENTS OF THE COURSE

INTRODUCTION TO THE PHYSICS OF THE EARTH - PART SEISMOLOGY

Version 08December 2006

BASIC CONCEPTS OF CONTINUUM MECHANICS

Continuum

An application of a force to real object causes some deformation of the object, i.e., change of itsshape. If the deformation is negligibly small, we can work with a concept of a rigid body. Therigid body retains a fixed shape under all conditions of applied forces.

If the deformations are not negligible, we have to consider the ability of an object to undergothe deformation, i.e., its elasticity, viscosity or plasticity. Here, we will restrict ourselves to theelastic behavior.

For the purpose of the macroscopic description of elastic behavior of a material it is usefulto introduce a concept of continuum. In the concept of continuum we do not explicitly consideratomic or molecular structure of the matter. Instead, we consider macroscopic material particlesor elementary particles of continuum: an elastic body can be viewed as a system of materialparticles. At the same time, we want to make use of the mathematical theory of continuousfunctions. The latter means that we want to assign a value of a material parameter to anygeometrical point (position) in a given coordinate system.

How to unify the three aspects: actual atomic/molecular discrete structure of the matter,macroscopic material particles, and theory of continuous functions?

Think of a geometric point/position. If we want to assign a value of a material parameter tothis point/position, we dene such a small macroscopic volume (centered at the point) for whichwe can define average macroscopic value of the material parameter. The size and shape of sucha volume depend on a problem we want to solve. It may be 1mm3 in one case or 1km3 in someother case.

Body forces

Non-contact forces proportional to mass contained in a considered volume of a continuum.

- forces between particles that are not adjacent; e.g., mutual gravitational forces

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- forces due to the application of physical processes external to the considered volume; e.g.,forces acting on buried particles of iron when a magnet is moving outside the consideredvolume

Stress, traction

If forces are applied at a surface S surrounding some volume of continuum, that volume of con-tinuum is in a condition of stress. This is due to internal contact forces acting mutually betweenadjacent particles within a continuum. Consider an internal surface S dividing a continuum intopart A and part B.

d

s

B

A

n

A

B

s

Fd

n

~n – unit normal vector to S

δ ~F – an infinitesimal force acting across an infinitesimal area δS– force due to material A acting upon material B

Cauchy’s hypothesis:

~T (~n) = limδS→0

δ ~F

δS

[~T ]U = N m−2

~T (~n) – traction vector (stress vector)– force per unit area exerted by the material in the direction of ~n across the surface

The part of ~T – that is normal to the surface – normal stress– that is parallel to the surface – shear stress

Properties of the traction vector:~T (−~n) = −~T (~n)

Traction vector depends on the orientation of the surface element S across which contract forceacts.

Therefore, the state of stress at a point has to be described by a tensor.

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Stress tensor

x1

x2

x3

T(n )

nT(-x1)

T(-x2)

T(-x3)

-x1

-x2

-x3

Ti(~n) = Ti(xj)nj

σji = Ti(xj)[σji]U = Nm−2

Cauchy’s stress formulaTi(~n) = σjinj (1)

σji is the i-th component of the traction exerted by a material with greater xj across the planenormal to the j-th axis on material with lesser xj .

Example:

Stress tensor fully describes a state of stress at a given point.Stress tensor is symmetric:

σij = σji

Displacement vector, strain tensor

Lagrangian description follows a particular particle that is specified by its original position atsome reference time.Eulerian description follows a particular spatial position and thus whatever particle that happensto occupy that position.Since a real seismogram is a record of Lagrangian motion, we will use the Lagrangian description.

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Position of the particleat time t

origin

0

origin

u

xx

Position of the particleat time t

Displacement ~u = ~u(~x, t) is the vector distance of a particle at time t from the position ~x of theparticle at some reference time t0 . ~X = ~x + ~u is the new position.

[~u]U = m∂~u∂t – particle velocity, ∂2~u

∂t2– particle acceleration

~u can generally include both the deformation and rigid–body translation and rotation.Deformation itself can be described by a strain tensor:

εij = 12(ui,j −uj ,i ) (2)

Constitutive law

The mechanical behavior of the elastic continuum is defined by the relation between the stressand strain. If forces are applied to the continuum, the stress and strain change together accordingto the stress–strain relation. Such relation is called the constitutive relation.

A linear elastic continuum is described by Hooke’s law, which in Cauchy’s generalized formulationis

σij = cijklεkl

Each component of the stress tensor is a linear combination of all components of the straintensor.cijkl is the 4th – order tensor of elastic coefficients and has 34 = 81 components.

In the case of an isotropic medium, Hooke’s law includes only two elastic coefficients (moduli):

σij = λεkkδij + 2µεij (3)

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Equation of Motion for a Continuous Medium

Consider a material volume V of continuum with surface S in which material parameters arecontinuous. Inside of V consider an arbitrary volume Ω with surface SΩ. Let ~nΩ be a normalvector to surface SΩ pointing from interior of volume Ω outward. The configuration is shown inFig. 1.

Fig. 1. Material volume V of a smooth continuum bounded by surface S. External traction ~p(~n) acts at surface

S, body force ~f acts in volume V . Volume Ω with surface SΩ is a testing volume considered in derivation of theequation of motion.

Consider body force ~f(xk, t) acting in volume Ω and traction ~pΩ(xk, t) acting at surface SΩ.An application of Newton’s second law to volume Ω gives

ddt

∫

Ωρ∂ui

∂tdV =

∫

SΩpΩ

i dS +∫

ΩfidV . (4)

Throughout the text dV and dS will be used for volume and surface elements, respectively.Because Ω and SΩ move with particles, the particle mas ρdΩ does not change with time. Theequation can be written as

∫

ΩρuidV =

∫

SΩpΩ

i dS +∫

ΩfidV . (5)

At surface SΩ, traction ~p is related to the stress tensor σij :

pΩi = σijn

Ωj . (6)

Assuming continuity of the stress tensor throughout volume Ω, Gauss’s divergence theorem canbe applied to the surface integral:

∫

SΩpΩ

i dS =∫

SΩσijn

Ωj dS =

∫

Ωσij ,j dV . (7)

Equation (5) can be then written as∫

Ω(ρui − σij ,j −fi)dV = 0 . (8)

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Equation (8) is valid for any volume Ω inside V . We want to show that then the integranditself is equal to zero: ρui − σij ,j −fi = 0. Assume that ρui − σij ,j −fi > 0 at some point insideV . Because the integrand is continuous throughout V , it is possible to find such volume Ω(containing that point) for which ρui − σij ,j −fi > 0 and thus also

∫Ω(ρui − σij ,j −fi)dV > 0.

This, however, would be in contradiction with eq. (8). Thus,

ρui − σij ,j −fi = 0 (9)

everywhere in V . Equation (9) together with boundary condition at surface S,

pi = σijnj , (10)

make a strong formulation for the considered problem.

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ELASTIC WAVES IN UNBOUNDED HOMOGENEOUSISOTROPIC MEDIUM

Wave potentials and separation of the equation of motion.Wave equations for P and S waves

Consider an unbounded homogeneous isotropic medium. Substituting Hooke’s law (3) into theequation of motion (eq. 9) gives

ρui − (λεkkδij + 2µεij),j −fi = 0 . (11)

Considering Lame coefficients λ and µ as well as density ρ to be spatial constants we obtain

ρ ui = (λ + µ)uk,ki + µui,jj + fi . (12)

where we also used the definition of the strain tensor (2).

Another form of equation of motion which we will consider later is obtained by using thevector identity rot rot ~u = grad div ~u−∇2~u, which in index notation reads

εijk εklm um,lj = uk,ki−ui,jj .

Substituting the identity into (12) yields

ρ ui = (λ + 2µ)uk,ki − µεijk εklm um,lj + fi (13)

Omit the body-force term in (12)

ρui = (λ + µ)uj,ji + µui,jj (14)

Apply now the Helmholtz decomposition to the displacement vector ~u:

ui = Φ,i + εilkΨk,l (15a)

i.e., ~u = ∇Φ +∇× ~Ψ (15b)

Φ and ~Ψ are scalar and vector Helmholtz potentials. Find the divergence of ~u. Apply the diver-gence to eq. (15a):

ui,i = Φ,ii + εilkΨk,li

Since εilk Ψk,li = 0 (∇.∇× ~Ψ = 0) (16)ui,i = Φ,ii (∇.~u = ∇.∇Φ) (17)

Find now the rotation of ~u. Apply the rotation to eq. (15a):

εmni ui,n = εmni Φ,in + εmni εilk Ψk,ln

εmni Φ,in =12(εmni Φ,in + εmin Φ,ni) =

12(εmni Φ,in − εmni Φ,in) = 0 (18a)

(∇×∇ Φ = 0) (18b)

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εmni ui,n = εmniεilk Ψk,ln (∇× ~u = ∇×∇× ~Ψ) (19)

Since the divergence of ~u equals ∇.∇Φ 6= 0 and the rotation of ~u equals ∇ × ∇ × ~Ψ , theHelmholtz decomposition of ~u means the decomposition into the part (∇Φ) which causes onlyvolume changes and the part (∇ × ~Ψ) which causes only shear changes without any volumechange (i.e., changes in form).

Since eq. (15a assigns 4 functions Φ, Ψ1 , Ψ2 and Ψ3 to 3 components of the displacement vector,one additional condition for the four functions is necessary. We can use

Ψi,i = 0 (∇.~Ψ = 0) (20)

since the use of the rotation in eq. (15a) means, in fact, that we give up any part of ~Ψ whichwould have a nonzero divergence.Insert now decomposition (15a) into eq. (14):

ρ Φ,i + ρ εilk Ψk,l = (λ + µ)(Φ,jji + εjlk Ψk,lji) + µ(Φ,ijj + εilk Ψk,ljj)

εjlk Ψk,lji = (εjlk Ψk,lj),i = (0),i = 0εilk Ψk,ljj = εilk(Ψk,jj),l

ρ Φ,i + ρ εilk Ψk,l = (λ + 2µ)Φ,jji + µ εilk(Ψk,jj),l

(ρ Φ− (λ + 2µ)Φ,jj),i + εilk(ρ Ψk − µ Ψk,jj),l = 0

This equation (together with appropriate boundary conditions) implies that

ρ Φ− (λ + 2µ)Φ,jj = 0 (21a)and ρ Ψk − µ Ψk,jj = 0 (22a)

i.e.,

ρ Φ− (λ + 2µ)∇2Φ = 0 (21b)

ρ ~Ψ − µ∇2~Ψ = 0 (22b)

Define

α =

√λ + 2µ

ρ(23)

andβ =

õ

ρ(24)

Then eq. (21a, 22a) and (21b, 22b) become

Φ = α2 Φ,jj (25a)Ψk = β2 Ψk,jj (26a)

and

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Φ = α2∇2Φ (25b)

~Ψ = β2∇2~Ψ (26b)

Equations (25a) and (26a) are the wave equations. Eq. (25a, 25b) describes propagation of awave with speed α while eq. (26a, 26b) describes propagation of a wave with speed β. Sinceα > β, the wave propagation with speed α is faster and arrives in a given place as the first ofthe two waves. Therefore, it is called the P wave according to the Latin word primae. The othertype of wave is called the S wave according to the Latin word secundae.We have found an interesting and important result:

– The equation of motion for an unbounded homogeneous isotropic medium can be separatedinto two wave equations.

– Two independent types of waves can propagate in the unbounded homogeneous isotropicmedium - one with speed α - the P wave, and the other with speed β - the S wave. Propagationof the P wave is accompanied with changes in volume, propagation of the S wave accompaniedwith changes in form.

The separation of the equation of motion into two wave equations is possible also in the case ofpresence of the body-force term in the equation,

ρ ui = (λ + µ)uj,ji + µ ui,jj + fi (27)

The Helmholtz decomposition can be applied also to ~f :

fi = g,i + εilk qk,l (28a)

i.e., ~f = ∇g +∇× ~q (28b)

Then we can get (analogously with the previous case)

Φ = α2 Φ,jj +1ρ

g (29a)

Ψk = β2 Ψk,jj +1ρ

qk (30a)

and

Φ = α2∇2Φ +1ρ

g (29b)

~Ψ = β2∇2~Ψ +1ρ

~q (30b)

Recalling decomposition (15a),

ui = Φ,i + εilk Ψk,l

define ~uP and ~uS :

uPi = Φ,i , uS

i = εilk Ψk,l (31)ui = uP

i + uSi (32)

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Then it follows from the wave equations (25a) and (26a) that ~uP and ~uS also satisfy waveequations

uPi = α2 uP

i,jj (33)

uSi = β2 uS

i,jj (34)

Relations (17) and (19) mean that

uSi,i = 0 (∇.~uS = 0) (35)

and εmni uPi,n = 0 (∇× ~uP = 0) (36)

Recall now equation of motion in the form (13) with ~f = 0

ρ ui = (λ + 2µ)uk,ki − µεijk rk,j (37)

where rk = εklm um,l. Apply the divergence to the above equation (37). We get

ρ ui,i = (λ + 2µ)uk,kii

since εijk rk,ji = 0

ρ(ui,i),tt = (λ + 2µ)(ui,i),kk

or

(ui,i),tt = α2(ui,i),kk (38a)i.e., (∇.~u),tt = α2∇2(∇.~u) (38b)

Recall now equation of motion in the form (14 or 12 with ~f ≡ 0).

ρ ui = (λ + µ)uj,ji + µ ui,jj

Apply the rotation to the equation:

ρ εmni ui,n = (λ + µ)εmni uj,jin + µ εmni ui,jjn

since εmni uj,jin = 0

ρ(εmni ui,n),tt = µ(εmni ui,n),jj

or

(εmni ui,n),tt = β2(εmni ui,n),jj (39a)i.e., (∇× ~u),tt = β2∇2(∇× ~u) (39b)

We see that eqs. (38a, 38b) and (39a, 39b) are the wave equations for∇.~u and∇×~u, respectively.Using definitions (23) and (24) we can rewrite e.g. eq. (13) in the form

ui = α2 uk,ki − β2 εijkεklm um,lj +fi

ρ(40)

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The ratio between the P-wave and S-wave speeds is

α

β=

√λ + 2µ

µ(41)

More important than relation (41) is the relation between α/β and Poisson’s ratio σ .Poisson’s ratio is the ratio of radial to axial strain when an uniaxial stress is applied.For example

τ11 6= 0 , τ22 = τ33 = 0e22 = e33

σ = −e22e11

= λ2(λ+µ) (42)

It follows from (41) and (42) thatα

β=

√2(1− σ)1− 2σ

(43)

If µ = 0 (fluid, no shear resistance), σ = 0.5.If the solid has an infinite shear resistance, σ = 0.Thus

0 < σ < 0.5 (44)

Relations (43) and (44) imply a very important relation

α

β>√

2 (45)

The case of µ = λ defines Poisson’s body. Then

α

β=√

3 and σ = 0.25 (46)

Plane waves

Consider the wave equation for the P wave

Φ = α2 Φ,jj (47)and uP

i = Φ,i (48)

Assume a solution in a form

Φ(xi, t) = A f(ϑ) (49a)where ϑ(xi, t) = t− τ(xi) (49b)and τ(xi) = plxl (49c)

and A and pl (l = 1, 2, 3) are real constants. ϑ is the phase function or phase. Let ϑ = ϑ0 fortime t = t0. Then

ϑ0 = t0 − plxl (50)

or

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plxl + ϑ0 − t0 = 0 (51)

Since eq. (51) is the equation of a plane, the surface of constant phase ϑ0 at time t0 is a plane.Therefore, solution (49a) of the equation (47) is called the plane P wave. Let f(ϑ) 6= 0 forϑ ∈< ϑ1, ϑ2 > and f(ϑ) = 0 outside the interval. Then the plane ϑ(xi, t) = ϑ1 is called thewavefront since it separates the region in motion from the region which is in rest.Insert (49a) into eq. (47):

A f ′′ = α2A f ′′(p21 + p2

2 + p23)

where

f ′′ =d2f

dϑ2

p21 + p2

2 + p23 =

1α2

(52)

Defineνi = αpi (53)

Thenν21 + ν2

2 + ν23 = 1 (54)

It follows from (54) that νi is a directional cosine of some vector ~N .Relation (49c) implies

τ(xi) =1α

νj xj (55)

Consider now a scalar product of ∇τ and ~N :

τ,i νi =1α

νj xj,i νi =1α

νj νi δij

=1α

νi νi =1α

Consider now a vector product of ∇τ and ~N :

εikl τ,k νl = εikl1α

νj xj,k νl

=1α

εikl δjk νj νl

=1α

εijl νj νl = 0

We have found that vector ~N is parallel to ∇τ , i.e., it is perpendicular to the surface τ(xi) =const. Then it follows from (49b) that ~N is perpendicular to the surface (plane) of the constantphase.Vector ~p = (p1, p2, p3) ; |~p| = 1

α is frequently used in the theory of elastic waves and is called theslowness vector.Relations (48) and (49a) imply

uPi = (A f(t− 1

ανj xj)),i

uPi = −A

1α

f ′(t− 1α

νj xj)νi (56)

Relation (56) means that the displacement vector of the plane P wave has the direction of vector~N and is perpendicular to the plane of a constant phase (i.e., a wavefront).

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Consider now a special case:~N = ~N(1, 0, 0)

Then Φ = A f(t− x1

α

)

and ϑ = t− x1α

The plane of the constant phase ϑ0 = t− x1α propagates in the direction x1 with speed α since

dx1

dt= α

It is easy to verify that

Φ(xi, t) = A f(t− 1α

νi xi) + B g(t +1α

νi xi) (57)

also satisfies eq. (47). Φ(xi, t) represents two plane waves. The first one, described by

A f(t− 1α

νi xi)

propagates in the direction of ~N with speed α, the second one, described by

B g(t +1α

νi xi)

propagates in the direction of − ~N with speed α.Consider now the wave equation for the S wave

Ψi = β2 Ψi,jj (58)

anduS

i = εilk Ψk,l (59)

Assume solution in the form of the plane wave

Ψi(xj , t) = Ci f(ϑ) (60a)ϑ(xj , t) = t− τ(xj) (60b)

τ(xj) = plxl (60c)

Insert (60a) into eq. (58):

Ci f ′′ = β2 Ci f ′′(p21 + p2

2 + p23)

p21 + p2

2 + p23 = 1

β2 (61)

Defineνi = βpi (62)

Thenν21 + ν2

2 + ν23 = 1 (63)

It follows from eq. (59) that

uSi = εilk(Ck f(t− 1

βνj xj)),l

uSi = − 1

βεilk Ck f ′(t− 1

βνj xj)νl (64)

Relation (64) means that the displacement of the plane S wave is perpendicular to the directionof propagation (i.e., particles oscillate perpendicularly to the direction of propagation).

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