To understand this cartoon, you have to be a physicist, but you must also understand a little about baseball!

Elastic Properties of SolidsTopics Discussed in Kittel, Ch. 3, pages 73-85

Analysis of Elastic Strains

Continuum approximation: good for λ > 30A.Description of deformation (Cartesian coordinates):

Material point i i i ix x x u r r r u r

Ref: L.D.Landau, E.M.Lifshitz, “Theory of Elasticity”, Pergamon Press (59/86)

Displacement vector field u(r).

i id x dx d d d r r r r r r u r rNearby point d d r r u r r u r

ii i j

j

udx dx dxx

222dl d d d r r r u r

d d d r r r u r

2 2d d d d d r r r u r r u r r u r

2 2 i k ki j i j

j i j

u u udl dx dx dx dxx x x

2 2 ji k ki j

j i i j

uu u udl dl dx dx

x x x x

2 i j i ju dx dx

12

ji k ki j

j i i j

uu u uu

x x x x

= strain tensor

12

jii j

j i

uuux x

= linear strain tensor

Dilation

uik is symmetric → diagonalizable → principal axes such that

2 2i j i j i jdl u dx dx 1 2 32 2 21 2 31 2 1 2 1 2u dy u dy u dy

1 2 ii idy u dy → (no summation over i ) 1 i

iu dy

ii

dV dy 1 ii

i

u dy 1 i

i

u dV 1 iiu dV

iidV dV u

dV

Trace of uikFractional volume change

StressTotal force acting on a volume element inside solid

V

dVf f force density

Newton’s 3rd law → internal forces cancel each other → only forces on surface contribute

i ki

k

fx

This is guaranteed if so that i i k kV S

f dV dS σ stress tensor

σik ith component of force acting on the surface element normal to the xk axis.

Moment on volume element

ik i k k iM f x f x dV i j k jk i

j j

x x dVx x

k ii j k k j i i j k j

j j j

x xx x dV dVx x x

i j k k j i j i k k ix x dS dV

Only forces on surface contribute → i k k i (σ is symmetric)

Elastic Compliance & Stiffness Constants

σ and u are symmetric → they have at most 6 independent components

Compact index notations (i , j) → α :(1,1) → 1, (2,2) → 2, (3,3) → 3, (1,2) = (2,1) → 4, (2,3) = (3,2) → 5, (3,1) = (1,3) → 6

1, 2,3for

4,5,6ii

i j j i

uu

u u

u S

S α β elastic compliance constants

12 i j k l i j k lU C u uElastic energy density: 1

2C u u

where

i j k lC C elastic stiffness constants

i , j , k, l = 1,2,3α , β = 1,2,…,6

elastic modulus tensor

Stress: m nm n

Uu

12 i j k l mi n j k l i j m k nlC u u

12 mnk l k l i j mn i jC u C u m n k l k lC u

i j k l j i k l i j l k j il k

k l i j l k i j k l j i l k j i

C C C C

C C C C

i jU C uu

uik & uki treated as independent

21

Elastic Stiffness Constants for Cubic Crystals

2 2 2 2 2 21111 11 22 33 1122 11 22 22 33 33 11 1212 12 23 31

1 22

U C u u u C u u u u u u C u u u

Invariance under reflections xi → –xi C with odd numbers of like indices vanishes

Invariance under C3 , i.e.,

1111iiiiC C

x y z x x z y x

x z y x x y z x

All C i j k l = 0 except for (summation notation suspended):

1122ii k kC C 1212i k ikC C

2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6

1 12 2

C u u u C u u u u u u C u u u

1 111 12 12

2 212 11 12

3 312 12 11

4 444

5 544

6 644

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

uC C CuC C CuC C CuCuCuC

111 12 12 11 12 12

12 11 12 12 11 12

12 12 11 12 12 11

44 44

44 44

44 44

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

C C C S S SC C C S S SC C C S S S

C SC S

C S

where 11 12

1111 12 11 122

C CSC C C C

12

1211 12 11 122

CSC C C C

44

44

1SC

111 12 11 12S S C C 1

11 12 11 122 2S S C C

Bulk Modulus & Compressibility

2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6

1 12 2

U C u u u C u u u u u u C u u u

Uniform dilation:

211 12

1 26

U C C

1 2 3 3u u u

4 5 6 0u u u

δ = Tr uik = fractional volume change

212

B B = Bulk modulus

1 VV p

11 121 23

B C C = 1/κ κ = compressibility

See table 3 for values of B & κ .

Elastic Waves in Cubic Crystals

Newton’s 2nd law:2

2i ki

k

ut x

don’t confuse ui with uα

i k ik j l j lC u →2

2jli

i k j lk

uu Ct x

2212

jli k j l

k j k l

uuC

x x x x

2l

ik j lk j

uC

x x

2 22 2 2 2 2 23 31 1 2 2 1 1

1111 1122 1133 1212 1221 1313 13312 2 2 21 1 2 1 3 2 1 2 3 1 3

u uu u u u u uC C C C C C Ct x x x x x x x x x x x

2 22 2 2 2 23 31 2 2 1 1

1111 1122 12122 2 21 1 2 1 3 2 1 2 3 1 3

u uu u u u uC C Cx x x x x x x x x x x

22 2 2 2 2

31 1 2 1 111 12 44 442 2 2 2

1 1 2 1 3 2 3

uu u u u uC C C Ct x x x x x x x

Similarly 22 2 2 2 2

32 2 1 2 211 12 44 442 2 2 2

2 2 3 2 1 1 3

uu u u u uC C C Ct x x x x x x x

2 2 2 22 2

3 3 3 32 111 12 44 442 2 2 2

3 3 2 3 1 2 1

u u u uu uC C C Ct x x x x x x x

Dispersion Equation2 2

2i l

i k j lk j

u uC

t x x

0

i ti iu u e k r

→2

0 0i i k j l k j lu C k k u

20 0il i k j l k j lC k k u

2 0i l i k j l k jC k k dispersion equation

2 0I kC i j imn j m nC k kkC

Waves in the [100] direction

2 0I kC i j i mn j m nC k kkC

1,0,0kk → 211i j i jC kkC

1111

22112

3113

0 00 00 0

Ck C

C

kC11

244

44

0 00 00 0

Ck C

C

11L

C k

0 1,0,0u Longitudinal

44T

C k

0 0,1,0u

Transverse, degenerate 0 0,0,1u

Waves in the [110] direction

2 0I kC i j i mn j m nC k kkC

1,1,02

kk →

2

11 12 21 22 2i j i j i j i j i jkC C C C kC

1111 1221 1122 12122

2121 2211 2112 2222

3113 3223

00

20 0

C C C Ck C C C C

C C

kC11 44 12 442

12 44 11 44

44

00

20 0 2

C C C Ck C C C C

C

11 12 441 2

2L C C C k

0 1,1,0u Lonitudinal

442T

C k

0 0,0,1u

Transverse 1 11 121

2T C C k

0 1, 1,0 u

Transverse