Page 261 - Mechanical Behavior of Materials

Section 6.6 Complex States of Strain 263

6.6.1 Principal Strains

Principal normal strains and principal shear strains occur in a similar manner as for stresses. For
plane strain, where ε z = γ yz = γ zx = 0, modifying Eqs. 6.6 and 6.7 according to Eq. 6.37 gives the
axis rotations and values for the principal normal strains:

γ xy
tan 2θ n =
ε x − ε y

2
ε x + ε y ε x − ε y γ xy 2
ε 1 ,ε 2 = ± + (6.38)
2 2 2
Equations 6.8 through 6.10 are similarly modified to obtain the axis rotation and value for the
principal shear strain in the x-y plane, and also the accompanying normal strain:

ε x − ε y
tan 2θ s =−
γ xy

ε x + ε y
2
2
γ 3 = (ε x − ε y ) + (γ xy ) , ε γ 3 = (6.39)
2
As for the stress equations, θ is positive counterclockwise. Positive normal strains correspond
to extension, negative ones to contraction. Positive shear strain causes a distortion corresponding
to a positive shear stress, in that the long diagonal of the resulting parallelogram has a positive
slope. (Look ahead to Fig. E6.8(a) for an example of a positive shear strain.) Direct use of these
equations can be replaced by Mohr’s circle in a manner similar to its use for stress. In accordance
with Eq. 6.37, the σ-axis becomes an ε-axis, and the τ-axis becomes a γ/2-axis.
For three-dimensional states of strain, the principal strains can be obtained by modifying
Eqs. 6.26 and 6.18 with the use of Eq. 6.37:
γ xy γ zx
(ε x − ε)
2 2

γ xy γ yz

(ε y − ε) = 0 (6.40)

2 2

γ zx γ yz

(ε z − ε)
2 2
γ 1 = |ε 2 − ε 3 | , γ 2 = |ε 1 − ε 3 | , γ 3 = |ε 1 − ε 2 | (6.41)
6.6.2 Special Considerations for Plane Stress
For cases of plane stress, σ z = τ yz = τ zx = 0, the Poisson effect results in normal strains ε z
occurring in the out-of-plane direction, so that the state of strain is three-dimensional. If the material
is isotropic, or if the material is orthotropic and a material symmetry plane is parallel to the x-y