4.2 Linear Elasticity—Generalized Hooke’s Law
107
Also,
− p = K (ε x + ε y + ε z )
− p = K
−3 p(1 − 2ν)
E
or E = K [3(1 − 2v)]
Therefore,
K =
E
3(1 − 2ν)
(4.29)
Lames constant λ can be denoted as
λ =
ν E
(1 + ν) (1 − 2ν)
(4.30)
Thus among the four elastic constants E, ν, G and K only two constants can be
replaced to the other two ones. Therefore, any two of these constants can be taken
as the independent constants. Further, it is to be noted from Eq. (4.29) that for the
bulk modulus to be positive, the value of Poisson’s ratio ν cannot be greater than 0.5.
Hence, for ν = 0.5, Eq. (4.29) becomes
K = ∞ and G =
E
3
(4.31)
Hence, materials having Poisson’s ratio equal to 0.5 are called as incompressible
materials, since, for such materials, the volumetric stain is zero. Table 4.1 gives the
relations between various elastic constants and Table 4.2 presents typical values of
elastic constants of some materials.
4.3 Elastic Strain Energy for Uniaxial Stress
In mechanics, energy is defined as the capacity to do work, and work is the product of
force and the distance, in the direction, the force moves. In solid, deformable bodies,
stresses multiplied by their respective areas, forces and deformations are distances.
The product of these two quantities is the internal work done in a body by externally
applied forces. This internal work is stored in a body as the internal elastic energy
of deformation or the elastic strain energy.
Consider an infinitesimal element as shown in Fig. 4.1a, subjected to a normal
stress σ x. The force acting on the right or the left face of this element is σ x dydz.
This force causes an elongation in the element by an amount ε x dx, where ε x is the
107
Also,
− p = K (ε x + ε y + ε z )
− p = K
−3 p(1 − 2ν)
E
or E = K [3(1 − 2v)]
Therefore,
K =
E
3(1 − 2ν)
(4.29)
Lames constant λ can be denoted as
λ =
ν E
(1 + ν) (1 − 2ν)
(4.30)
Thus among the four elastic constants E, ν, G and K only two constants can be
replaced to the other two ones. Therefore, any two of these constants can be taken
as the independent constants. Further, it is to be noted from Eq. (4.29) that for the
bulk modulus to be positive, the value of Poisson’s ratio ν cannot be greater than 0.5.
Hence, for ν = 0.5, Eq. (4.29) becomes
K = ∞ and G =
E
3
(4.31)
Hence, materials having Poisson’s ratio equal to 0.5 are called as incompressible
materials, since, for such materials, the volumetric stain is zero. Table 4.1 gives the
relations between various elastic constants and Table 4.2 presents typical values of
elastic constants of some materials.
4.3 Elastic Strain Energy for Uniaxial Stress
In mechanics, energy is defined as the capacity to do work, and work is the product of
force and the distance, in the direction, the force moves. In solid, deformable bodies,
stresses multiplied by their respective areas, forces and deformations are distances.
The product of these two quantities is the internal work done in a body by externally
applied forces. This internal work is stored in a body as the internal elastic energy
of deformation or the elastic strain energy.
Consider an infinitesimal element as shown in Fig. 4.1a, subjected to a normal
stress σ x. The force acting on the right or the left face of this element is σ x dydz.
This force causes an elongation in the element by an amount ε x dx, where ε x is the