10-05-2012, 03:01 PM
On Invariants of the Stress–Strain State in Mathematical Models for Mechanics of Continua
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It is generally believed in models that a change only
in the second invariant is essential; effects of the first
one for incompressible media and the third one associated
with the type of a stress state are often ignored. As
a result, the information on the principal-direction trihedron
is lost. Thus, choosing the first and second
invariants (for the stress deviator) does not call for
information on principal directions.
A representation of invariants, which is similar
to (1), is also used for the symmetric strain tensor or the
strain rate tensor. In this case, a similarity of the stress
tensor and strain (strain rate) one is adopted to relate
invariants of a stress state to those of a strain (strain
rate) state. Note that deviations from the similarity are
both possible and natural in actual situations [3–5]
(Fig. 1).
Taking into account these facts and proceeding from
physical interpretation of the energy dissipation processes
in models of plastic and viscous media, we propose
to introduce another set of invariants related to
areas of the maximum tangential stresses T and principal
shears
G (shear-strain rate). Data from solid-state
physics and recent results obtained in mesomechanics
[6] indicate these directions of investigation [7, 8].
In this case, in performing identity transformations,
it is easy to pass from (2) to substantiating this new set
of invariants:
The first comment is associated with the fact that the
condition of isotropy for set (1), which is understood
that it does not depend on directions of the principal
axes, was never discussed while introducing invariants
(except for rare cases when the third invariant should be
taken into account). Actually, allowing for the first and
second invariants did not require an account for the orientation
of the principal-direction trihedron of the
stress tensor.
In contrast to this, introducing set (4) definitely indicates
such a dependence, because the invariant T is
related to area elements for maximum tangential
stresses, which bisect angles between the first and the
third principal directions. The invariant
well as T, is closely related to the orientation of the
principal axis and indicates the influence of other (two)
extreme tangential stresses