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Figure 10.2. Forces acting around an infinitesimal volumetric mass element.
10.2.2 Stress
The notion of stress was introduced by Cauchy around 1822. Intuitively, the stress
combines strains and physical material properties to determine the internal forces
of the object. In fact, when external forces are applied to the object, these are
transmitted from point to point within the material body, leading to the generation
of internal forces; any small element of matter of the continuum object receives
forces from all around. One way to describe the surrounding internal forces act-
ing locally is to evaluate the force, F , acting on a given surface element of an
infinitesimal element (see Figure 10.2 ) . The component of F along the normal n
is analogous to a pressure applied to a small element of matter of the object. Note
that the stress has the same units as the pressure (pascal (Pa) in SI). The orthog-
onal component of F with respect to n is the force that makes parallel internal
surfaces slide past one another, and it is normally known as the shearing force.
Cauchy extrapolated this idea to the three-dimensional case to define the stress
per volume unit acting on the continuum object as the second-order tensor σ ,
known as the Cauchy stress tensor, which is defined as
,
σ 11
σ 12
σ 13
σ =
σ 21
σ 22
σ 23
σ 31
σ 32
σ 33
where σ 11 , σ 22 ,and σ 33 are normal stresses, and σ 12 , σ 13 , σ 21 , σ 23 , σ 31 ,and σ 32
are shear stresses.
The first index i indicates that the stress acts on a plane normal to the x i axis,
and the second index j denotes the direction in which the stress acts. A stress