Page 204 - Mechanical Behavior of Materials

Section 5.3 Elastic Deformation 205

z
σ
z

τ yz
τ
zx
σ
y
τ
xy y
σ
x
x

Figure 5.9 The six components needed to completely describe the state of stress at a point.

The situation can be summarized by the following table:

Resulting Strain
Each Direction
Stress x y z
σ x νσ x νσ x
σ x − −
E E E
νσ y σ y νσ y
σ y − −
E E E
νσ z νσ z σ z
σ z − −
E E E
Adding the columns in this table to obtain the total strain in each direction gives the following
equations:

1
ε x = σ x − ν σ y + σ z (a)
E
1
ε y = σ y − ν σ x + σ z (b) (5.26)
E
1
ε z = σ z − ν σ x + σ y (c)
E
The shear strains that occur on the orthogonal planes are each related to the corresponding shear
stress by a constant called the shear modulus, G:

τ xy τ yz τ zx
γ xy = , γ yz = , γ zx = (5.27)
G G G
Note that the shear strain on a given plane is unaffected by the shear stresses on other planes. Hence,
for shear strains, there is no effect analogous to Poisson contraction. Equations 5.26 and 5.27, taken
together, are often called the generalized Hooke’s law.