Fundamentals - Preventive Biomechanics - page 51

n 1Hj þ

r 11 r Hj r 12 n 2Hj þ r 13 n 3Hj ¼ 0

n 2Hj þ

r 12 n 1Hj þ r 22 r Hj r 23 n 3Hj ¼ 0 ð j ¼ 1 ; 2 ; 3 Þ

n 3Hj ¼ 0

r 13 n 1Hj þ r 23 n 2Hj þ r 33 r Hj

ð 3 : 104 Þ

According to linear algebra, the homogenous system of equations ( 3.103 ) has a
nontrivial solution n Hj 6¼ 0 ; if, using

r 11 r Hj r 12 r 13

¼ ! r 22 r Hj

det ð S r Hj I Þ¼ r 12 r 23 0 ð 3 : 105 Þ

r 33 r Hj

r 13 r 23

the coefficient determinant of the system of equations vanishes (solvability con-
dition). Execution of operation ( 3.105 ) leads to the following characteristic
polynomial in the form of a cubic equation for the eigen-values r Hj

P ð r Hj Þ¼ det ð S r Hj I Þ¼ r Hj S I r Hj þ S II r Hj S III ¼ 0 : ð 3 : 106 Þ

In ( 3.106 ), the S i (i = I, II, III) are the three basic invariants of the stress tensor
S which read, using S ¼ S T

according to ( 3.128 ):

S I ¼ trS ¼ r 11 þ r 22 þ r 33
S II ¼ 1

2 ð S I trS 2 Þ¼ r 11 r 22 þ r 11 r 33 þ r 22 r 33 ð r 12 þ r 13 þ r 23 Þ
S III ¼ det S ¼ r 11 r 22 r 33 þ 2r 12 r 23 r 13 ð r 11 r 23 þ r 33 r 12 þ r 22 r 13 Þ:

ð 3 : 107 Þ

At known coordinates r ij of the stress tensor S, the three invariants ( 3.107 ) and
the eigen-values (principal stresses) r Hj can be derived by using ( 3.106 ), by
solving the cubic equation. Finally, the eigen-vectors n Hj can be derived using
( 3.104 ) by solving the algebraic system of equations.

3.2.5 Balance Equations

In the previous sections strain and stress measures have been introduced inde-
pendently, and no relation between the two (kinematic and dynamic) measures has
been shown. In generating material equations, where these two measures must be
related, the first law of thermodynamics, as a balance equation, gains a predom-
inant role. Balance equations are relations where quantities of equal 'quality' are
balanced. The best known balance equation in mechanics (as a special case of the
principle of linear momentum) is the description of the motion of the mass centre
of a system of particles, where the sum of all external forces is equated (balanced)
with the inertia force of a body (''force equal mass times acceleration''). Balance