Geoscience Reference
In-Depth Information
The constants
λ
and
µ
are known as the Lame elastic constants (named after the
nineteenth-century French mathematician G. Lame). In suffix notation, Eqs. (A2.19) are
written as
σ
ij
=
λδ
ij
+
2
µ
e
ij
for
i
,
j
=
x
,
y
,
z
(A2.20)
where the Kronecker delta
1where
i
=
j
0where
i
=
j
The Lame elastic constant
µ
(where
µ
=
σ
xy
/
(2
e
xy
) from Eq. (A2.19)) is a measure of the
resistance of a body to shearing strain and is often termed the
shear modulus
or the
rigidity
modulus
. The shear modulus of a liquid or gas is zero.
Besides the Lame elastic constants, other elastic constants are also used:
Young's modulus
E, Poisson's ratio
σ
(no subscripts) and the bulk modulus
K
.
δ
ij
=
Young's modulus
E
is the ratio of tensional stress to the resultant longitudinal strain for a small cylinder under
tension at both ends. Let the tensional stress act in the
x
direction on the end face of the small
cylinder, and let all the other stresses be zero. Equations (A2.19) then give
σ
xx
=
λ
+
2
µ
e
xx
0
=
λ
+
2
µ
e
yy
0
=
λ
+
2
µ
e
zz
(A2.21)
and
0
=
e
xy
=
e
xz
=
e
yz
(A2.22)
Adding Eqs. (A2.21)gives
σ
xx
=
3
λ
+
2
µ
(A2.23)
Substituting Eq. (A2.23) into Eq. (A2.21)gives
)
µ
e
xx
=
(
λ
+
µ
(A2.24)
Hence, Young's modulus is
σ
xx
e
xx
=
(3
λ
+
2
µ
)
µ
(
λ
+
µ
)
(3
λ
+
2
µ
)
µ
(
λ
+
µ
)
E
=
=
(A2.25)
Poisson's ratio
σ
(named after the nineteenth-century French mathematician Simeon Denis Poisson) is
defined as the negative of the ratio of the fractional lateral contraction to the fractional
longitudinal extension for the same small cylinder under tension at both ends. Using
Eqs. (A2.23) and (A2.21), Poisson's ratio is given by
e
zz
e
xx
=
λ
2
µ
µ
(
λ
+
µ
)
λ
2(
λ
+
µ
)
(A2.26)
σ
=−
=
Consider a small body subjected to a hydrostatic pressure (i.e., the body is immersed in a
liquid). This pressure causes compression of the body. The ratio of the pressure to the
