Tensors for Physics

16.2 Elasticity 309

16.2.5 Microscopic Expressions for Elasticity Coefficients.

Consider a system ofNparticles, located at the positionsri,i= 1 , 2 ,...Nwithin
a volumeV. The interaction potential is denoted byΦ=Φ(r^1 ,r^2 ,...,rN). When
the potential is pairwise additive, one hasΦ =

∑

i<jφ

ij=∑

i<jφ(r

ij), where

rij=ri−rj, andφ=φ(r)is the binary interaction potential. In thermal equilibrium,
at the temperatureT, the configurational part of the free energyFpotis given by

βFpot=−lnZpot, Zpot=

∫

exp[−βΦ]dr{N},β=

1

kbT

,

whereZpotis the configurational partition integral,dr{N}is the 3N-dimensional
volume element. The difference between the free energy, where the position vectors
are displaced according torνi→rνi+rμiuμνand the original free energy,δFpot=

Vppotμνuμνyields the expression

Vppotμν=

〈

Φμν

〉

,Φμν=

∑

i

rμi∂νiΦ=−

∑

i

rμIFνi,∂νi=

∂

∂rνi

, (16.29)

for the potential contribution to the pressure tensor. HereFνiis the force acting on
particlei. In the absence of external forces, the total force vanishes:

∑

iF

i
ν=0. The
bracket〈···〉indicates the configurational canonical average

〈...〉=Zpot−^1

∫

...exp[−βΦ]dr{N}.

The change of the pressure tensor (16.29) under a deformationrjλ→rλj+rκjuκλ
yields a relation between the potential contributionσμνof the stress tensor and
the deformation tensor, and consequently a microscopic expression for the elastic
moduli. Starting from

σμν=−

(

p

pot,def

μν −p

pot, 0

μν

)

=−δp

pot

μν=δ

(

V−^1 〈Φμν〉

)

,

whereppotμν,defandppotμν,^0 are the pressure tensors in the strained and in the unstrained
states, and usingσμν=Gμν,λκuλκ, one obtains

VGμν,λκ=〈Φμν,λκ〉 0 +Vppotδμν,δλκ−β

(

〈ΦμνΦλκ〉 0 −〈Φμν〉 0 〈Φλκ〉 0

)

.

(16.30)

The subscript 0 in〈...〉 0 indicates the average in the undeformed state. The first term
viz.
Φμν,λκ=

∑

j

∑

i

rμi∂νirλj∂κjΦ (16.31)