Art4. ECT Simplified : Equilibrium Equations
Sandhu G
ELASTIC CONTINUUM THEORY OF NATURE
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Ordinary material bodies, under stress, will generally be
in a state of static equilibrium. However, in the Elastic
Continuum, the equilibrium in a strained state is essentially
dynamic. In a steady state or static equilibrium, not only the
resultants of all forces acting on an infinitesimal volume
element dV should vanish but the resultant moment of all forces
should also vanish to ensure that pure stresses and strains do
not give rise to rigid body motions and rotations. In the static
equilibrium of a material body under stress, vanishing of
resultant moments can be ensured by the symmetry of stress and
strain components. However, this condition is not applicable for
the Elastic Continuum where there is neither static equilibrium
nor rigid body motions & rotations. Let us consider an
infinitesimal volume element dV = dx.dy.dz in the shape of a
rectangular parallelepiped, with point P(x,y,z) as its center
and faces parallel to coordinate planes.
At any instant of time t, let us examine total forces
acting on all faces due to the combined effect of shear and
normal stresses acting on these faces. For equilibrium of the
infinitesimal volume element dV, under the action of all
resultant surface forces due to spatial stress components and
the inertial body forces, we have to ensure the vanishing of
resultant forces along each of the coordinate directions. The
final equilibrium relations obtained this way involve a sum of
covariant derivatives of stress tensor components and second
order partial derivative of corresponding displacement component
with respect to time. Combining the stress - strain relations for
the Elastic Continuum, we finally obtain a set of equilibrium
equations, which are second order partial differential equations
in terms of displacement vector components. These equilibrium
equations in the Elastic Continuum turn out to be identical to
the vector wave equation of the electromagnetic field theory. In
conventional Cartesian coordinate system (x,y,z), with physical
components of the displacement vector U given by Ux, Uy and Uz
these equilibrium equations reduce to a set of three independent
second order partial differential equations, involving dependent
variables Ux, Uy and Uz respectively. However, when transformed
to spherical or cylindrical coordinate systems, these equilibrium
equations reduce to a set of three simultaneous second order
partial differential equations, involving various displacement
vector components which may no longer be independent.
Generally, the study of Elastic Continuum involves the
solution of such second order simultaneous partial differential
equations subject to appropriate boundary and stability
conditions.
Next we shall discuss the Solution of Equilibrium Equations in
the Elastic Continuum.
The detailed theory, including five mathematical papers are
available at URL:
<http://www.geocities.com/ResearchTriangle/Forum/9850/index.html>
G S Sandhu
<san...@ch1.dot.net.in>