Complex Stress and Strain - Structural and Stress Analysis - page 400

Since it is relatively simple to determine U t and U v , we obtain U s by transposing
Eq. (14.43). Hence

U s = U t − U v (14.44)

Initially, however, we shall demonstrate that the deformation of an element of material
may be separated into change of volume and change in shape.

The principal stresses σ I , σ II and σ III acting on the element of Fig. 14.18 may be
written as

1
3 ( σ I +

1
3 (2 σ I −

σ I = σ II + σ III ) + σ II − σ III )

1
3 ( σ I +

1
3 (2 σ II −

σ II = σ II +

+

σ I −

σ III ) σ III )

1
3 ( σ I +

1
3 (2 σ III −

σ III = σ II + σ III ) + σ II − σ I )

or

!

σ I
σ II = σ + σ II
σ III = σ + σ III

σ I =¯ σ +

(14.45)

"

Thus the stress system of Fig. 14.18 may be represented as the sum of two separate
stress systems as shown in Fig. 14.19. The σ stress system is clearly equivalent to

a hydrostatic or volumetric stress which will produce a change in volume but not a
change in shape. The effect of the σ 1

stress system may be determined as follows.

Adding together Eqs (14.45) we obtain

σ I σ II + σ III σ I + σ II + σ III =

¯ + +

3 σ

but

1
3 ( σ I + σ II + σ III )

σ =

σ

1
II

σ

σ II

σ

1
I

σ I

σ

σ

1
III

σ III

σ

F IGURE 14.19
Representation of
principal stresses
as volumetric and
distortional
stresses

σ

σ

1
III

σ III

σ

1
I

σ I

σ

σ

1
II

σ II σ