The concept of force coming directly from physical observation of deforma-tion was first quantified in one dimension by Hooke but intuitively recognized by cavemen.

**The Stress
Tensor **

The concept of force
coming directly from physical observation of deforma-tion was first quantified
in one dimension by Hooke but intuitively recog-nized by cavemen. To
generalize to 3D and introduce the concept of *internal* force per unit
area (stress), requires the idea of equilibrium and the limit pro-cess of
calculus.

Consider again the
standard 3D structural blob in equilibrium from external boundary forces and
slice it to expose a plane defined by its normal vector *n* (Figure 2.3a).
For equilibrium of the remaining piece, there must be a result-ant force and/or
couple acting on the cross-section of area *A*. If they can be calculated
from any slice from the six equilibrium equations (in 3D), the structure is
statically determinate for internal forces and shear, bending moment, torque,
and normal force diagrams can be draw__n.__ However, what the cross-section
actually feels is small increments of force Vector ?*F _{i}* at
various ori-entations distributed in some undetermined (and usually
difficult-to-determine) way. It is this distribution of force and moment over
the area (the stress field) that we hope to discover (stress analysis) and, if
we are clever, then change and control (design). As we shall see, this stress
distribution is almost never determinate.**

For now let *n* be in the *x*
direction so that the exposed surface is a *yz* pla__ne__. Over a small
area ?*A _{x}* = ?

(inclined at an
arbitrary angle) with components ?*F _{xx}* ?

The average 'intensity of force' or
average stress is defined as the ratio ?*F*/ ?*A* analogous to
'average strain' *d _{ij}* for

If we now slice the structure parallel
to the *xy* and *xz* planes through 0, then, by analogy, we find that
the complete state of stress is defined by 9 compo-nents (3 vectors) given by:

as shown in Figure 2.3 e. Shortly we
will show that the stress tensor is, in fact, 'naturally' symmetric in that the
three moment equilibrium equations for a differential element reduce to:

Thus, only six vector components are
actually involved in defining a state of stress at a point.

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Civil : Principles of Solid Mechanics : Strain and Stress : The Stress Tensor |

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