Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 59

n

T 11

T 12

t ( n )

T 31

T 22

e 3

T 12

T 32

T 13

e 2

T 23

e 1

T 33

Fig. 3.4 The surface tractions on a tetrahedron

on the tetrahedron. The tetrahedron is selected in such a way that the stress vectors
acting on the mutually orthogonal faces are the stress vectors acting on the coordi-
nate planes. Recall that we have represented the components of the stress vectors
acting on the planes whose normals are e 1 , e 2 , and e 3 by the components of the stress
tensor. The three scalar equations of Newton's second law will suffice for the
determination of the unknown components of t ( n ) acting on the fourth face. For
simplicity we will only derive ( 3.15 ) in the e 1 direction. We let A be the area of the
inclined face with normal n , and h the perpendicular distance from P to the inclined
face. The mean value of t ( n ) will be denoted and defined by

ð

1
A

t ðnÞ 1 ¼ t ðnÞ 1 d A ¼ t ðnÞ 1 ð Q Þ; (3.17)

A

where, as a consequence of the mean value theorem, the point Q lies inside A .
Analogously defined me an values of the components of T over their respective
areas will be denoted by T . The reason for requiring that the stress components be
continuous functions of position in a neighborhood of P is to ensure that the mean
values of the stress components actually occur at certain points always within the
corresponding areas.
Since the area of the inclined face of the tetrahedron may be represented by the
vector An , where A is the magnitude of the area and n is the normal to the plane
containing the area, the areas of the orthogonal faces are each given by An 1 , An 2 ,
An 3 . The fact that the areas of four faces of a tetrahedron, where three of the faces of
the tetrahedron are orthogonal, are A , An 1 , An 2 , and An 3 is a result from solid
geometry. Summing forces in the e 1 direction and setting the result equal to the
mass times the acceleration of the tetrahedron we find that

d 1 Ah x 1 Ah

3 ¼ r €

t ðnÞ 1 A T 11 An 1 T 12 An 2 T 13 An 3 þ r

3 ;

(3.18)

where Ah /3 is the volume of the tetrahedron, d 1 is the action-at-a-distance force
(e.g., gravity) in the e 1 direction and x 1 the acceleration of the tetrahedron in that €