Introduction to Acoustics

34 Part A Propagation of Sound
Part A 3.2
initially a small box (∆x by ∆y by ∆z)is
δ[Work]= �
� �
∂
σii δξi ∆x∆y∆z
∂xi
i
+ �
� �
∂
σij δξi ∆x∆y∆z , (3.58)
∂x j
i�= j
where here σij is interpreted as the i-th component of the
force per unit area on a surface whose outward normal
is in the direction of increasing x j. With use of the
symmetry of the stress tensor and of the definition of the
strain tensor components, one obtains the incremental
strain energy per unit volume as
δÍ = �
σijδɛij . (3.59)
ij
If the reference state, where Í = 0, is taken as that in
which the strain is zero, and given that each stress component
is a linear combination of the strain components,
then the above integrates to
Í = �
ij
1
2 σijɛij . (3.60)
Here the stresses are understood to be given in terms of
the strains by an appropriate stress–strain relation of the
general form of (3.57).
Because of the symmetry of both the stress tensor
and the strain tensor, the internal energy Í per unit
can be regarded as a function of only the six distinct
strain components: ɛxx, ɛyy, ɛzz, ɛxy, ɛyz, andɛxz. Consequently,
with such an understanding, it follows from
the differential relation (3.59)that
∂Í
= σxx ;
∂ɛxx
∂Í
∂ɛxy
= 2σxy , (3.61)
with analogous relations for derivatives with respect to
the other distinct strain components.
Isotropic Solids
If the properties of the solid are such that they are
independent of the orientation of the axes of the Cartesian
coordinate system, then the solid is said to be
isotropic. This idealization is usually regarded as good
if the solid is made up of a random assemblage of
many small grains. The tiny grains are possibly crystalline
with directional properties, but the orientation
of the grains is random, so that an element composed
of a large number of grains has no directional orientation.
For such an isotropic solid the number of
y
x
�(x)
x+ �(x)
Fig. 3.5 Displacement field vector ξ in a solid. A material
point normally at x is displaced to x + ξ(x)
different coefficients in the stress–strain relation (3.57)
is only two, and the relation can be taken in the general
form
σij = 2µLɛij + λLδij
3�
ɛkk . (3.62)
k=1
This involves two material constants, termed the
Lamè constants, and here denoted by λL and µL
(the latter being the same as the shear modulus
G).
Equivalently, one can express the strain components
in terms of the stress components by the inverse of the
above set of relations. The generic form is sometimes
written
ɛij =
1 + ν
E σij − ν
E
�
σkkδij , (3.63)
k
where E is the elastic modulus and ν is Poisson’s ratio.
The relation of these two constants to the Lamè constants
is such that
νE
,
(1 + ν)(1 − 2ν)
(3.64)
E
µL = G = .
2(1 + ν)
(3.65)
λL =
In undergraduate texts on the mechanics of materials,
the relations (3.63) are often written out separately
for the diagonal and off-diagonal strain elements, so
x