CONTINUUM MECHANICS for ENGINEERS

from which if we define
and
( )
µ 3λ + 2µ
E = (6.2-6a)
λ + µ
λ
v = (6.2-6b)
2 λ + µ
( )
we obtain the following form of Hooke’s law for isotropic behavior in terms
of the engineering constants E and v,
[ ( ) − ]
1
1
E
εij = + v σ v
(6.2-7)
ij δijσkk Here E is called Young’s modulus, or simply the modulus of elasticity, and v is
known as Poisson’s ratio. By suitable combinations of these two constants,
we may define two additional constants of importance in engineering elasticity.
First, the shear modulus, or modulus of rigidity, is defined as
E
G = = µ
(6.2-8a)
21+ v
( )
which, as noted, is identical to the Lamé constant µ. Second, the bulk modulus
is defined as
E
K = (6.2-8b)
31−2v ( )
For isotropic elastic materials, any two elastic constants completely define
the material’s response. In addition to that, any elastic constant can be
determined in terms of any two other constants. A listing of all elastic
constants in terms of other pairs of constants is given in Table 6.1-1.
The physical interpretations of the constants E, v, G, and K introduced
above can be determined from a consideration of the special states of stress
displayed in Figure 6-2. In the case of a uniaxial state of stress (tension or
compression), say in the x 1 direction with σ 11 = ±σ 0, and all other stress
components zero (Figure 6.2a), Eq 6.2-7 yields (since σ ii = ±σ 0),