Page 202 - Introduction to Continuum Mechanics

Equations of Motion - Principle of Linear Momentum 187

and

Example 4.6.2
Do the previous example for the following state of stress: 7\ 2 = T 2i = 1000 MPa, all other
TIJ are zero,
Solution, From Eq. (4.6.10), we have

Corresponding to the maximum normal stress T\ — 1000 MPa, Eq. (4.6.11) gives

and corresponding to the minimum normal stress TI - -1000 MPa, (i.e., maximum compres-
sive stress),

'The maximum shearing stress is given by

which acts on the planes bisecting the planes of maximum and minimum normal stresses, i.e.,
the ei-plane and the e2-plane in this problem.

4.7 Equations of Motion - Principle of Linear Momentum
In this section, we derive the differential equations of motion for any continuum in motion.
The basic postulate is that each particle of the continuum must satisfy Newton's law of motion.
Fig. 4.8 shows the stress vectors that are acting on the six faces of a small rectangular element
that is isolated from the continuum in the neighborhood of the position designated by */.

Let B = Bffi be the body force (such as weight) per unit mass, p be the mass density atjq
and a the acceleration of a particle currently at the position*/; then Newton's law of motion
takes the form, valid in rectangular Cartesian coordinate systems