Stress in three-dimensional continuous media - Biomechanics: Concepts and Computation - page 149

stress tensor , when the principal stresses and stress directions are known, will

not be outlined here.

σ

8.6 Mohr's circles for the stress state

When the stress tensor σ is given, the stress vector s on an arbitrary oriented plane,
defined by the unit outward normal n can be determined by

s = σ · n . (8.67)

Then, the normal stress s n and shear stress s t can be determined. The normal
stress s n is the inner product of the stress vector s with the unit outward normal n :
s n = s · n and can be either positive or negative (extension or compression). The
shear stress s t is the magnitude of the component of s tangent to that surface, see
Section 8.4 . In this way, it is possible to add to each n a combination ( s n , s t ) that
can be regarded as a mapping in a graph with s n along the horizontal axis and s t
along the vertical axis, see Fig. 8.15 . It can be proven that all possible combina-
tions ( s n , s t ) are located in the shaded area between the drawn circles (the three
Mohr's circles). If the principal stresses

σ 3 are known, Mohr's circles
(the centroid is located on the s n -axis) can be drawn at once! For this it is not nec-
essary to determine the principal stress directions. In the s n s t -coordinate system
the combinations (

σ 1 , σ 2 and

σ 3 , 0) constitute the image points associ-
ated with the faces of the principal stress cube in Fig. 8.14 . Based on Fig. 8.15 ,it
can immediately be concluded that

σ 1 , 0), ( σ 2 , 0) and (

( σ n ) max = σ 3 , ( σ n ) min = σ 1
( σ s ) max = ( σ 3 − σ 1 ) / 2. (8.68)

s t

s n

σ 1 σ 2 σ 3

Figure 8.15
Mohr's circles for the stress.