12.005 Lecture Notes 3
Tensors
Most physical quantities that are important in continuum mechanics like temperature,
force, and stress can be represented by a tensor.
Temperature can be specified by stating
a single numerical value called a scalar and is called a zerothorder tensor.
A force,
however, must be specified by stating both a magnitude and direction.
It is an example of
a firstorder tensor.
Specifying a stress is even more complicated and requires stating a
magnitude and two directions—the direction of a force vector and the direction of the
normal vector to the plane on which the force acts.
Stresses are represented by second
order tensors.
Stress Tensor
Representing a force in three dimensions requires three numbers, each referenced to a
coordinate axis.
Representing the state of stress in three dimensions requires nine
numbers, each referenced to a coordinate axis and a plane perpendicular to the coordinate
axes.
Returning to determining traction vectors on arbitrary surfaces.
Consider two surfaces S
1
and S
2
at point Q.
Figure 3.1
T
(1)
S
1
Q
S
2
T
(2)
Q
n
1
n
1
ramp
n
2
n
2
page
Example
Figure by MIT OCW.
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 Fall '10
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 Force, stress tensor, MIT OCW

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