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Chapter 2
Governing Equations
2.3 Equations of motion
and equilibrium for deformable solids
In
this section, we generalize Newton’s laws of motion (conservation of linear and
angular momentum) to a deformable solid.
2.3.1 Linear momentum balance in terms
of Cauchy stress
Let  denote the Cauchy stress distribution within
a deformed solid. Assume that the
solid is subjected to a body force ,
and let and denote the displacement, velocity and
acceleration of a material particle at position in the deformed solid.
Newton’s third law of motion (F=ma) can be expressed
as
Written out in full
Note that the derivative is taken with respect to position
in the actual, deformed solid. For the special (but rather common) case of a
solid in static equilibrium in the absence of body forces
Derivation: Recall that the resultant force
acting on an arbitrary volume of material V
within a solid is
where
T(n) is the internal traction acting on the surface A with normal n that bounds V.
The linear momentum of
the volume V is
where
v is the velocity vector of a
material particle in the deformed solid. Express T in terms of and set
Apply
the divergence theorem to convert the first integral into a volume integral, and
note that one can show (see Appendix D) that
so
Since this must hold for
every volume of material within a solid, it follows that
as
stated.
2.3.2 Angular
momentum balance in terms of Cauchy stress
Conservation
of angular momentum for a continuum requires that the Cauchy stress satisfy
i.e. the
stress tensor must be symmetric.
Derivation: write down the equation for
balance of angular momentum for the region V within the deformed
solid
Here, the left hand side is the resultant moment (about the
origin) exerted by tractions and body forces acting on a general region
within a solid. The right hand side is
the total angular momentum of the solid about the origin.
We can
write the same expression using index notation
Express T in terms of and re-write the first integral as a volume
integral using the divergence theorem
We may
also show (see Appendix D) that
Substitute
the last two results into the angular momentum balance equation to see that
The
integral on the right hand side of this expression is zero, because the
stresses must satisfy the linear momentum balance equation. Since this holds for any volume V, we conclude that
which is the result we
wanted.
2.3.3 Equations of
motion in terms of other stress measures
In terms
of nominal and material stress the balance of linear momentum is
Note
that the derivatives are taken with respect to position in the undeformed solid.
The
angular momentum balance equation is
To
derive these results, you can start with the integral form of the linear
momentum balance in terms of Cauchy stress
Recall
(or see Appendix D for a reminder) that area elements in the deformed and
undeformed solids are related through
In
addition, volume elements are related by . We can use these results to re-write the
integrals as integrals over a volume in the undeformed solid as
Finally,
recall that and that to see that
Apply
the divergence theorem to the first term and rearrange
Once
again, since this must hold for any material volume, we conclude that
The
linear momentum balance equation in terms of material stress follows
directly, by substituting into this equation for in terms of
The angular momentum
balance equation can be derived simply by substituting into the momentum
balance equation in terms of Cauchy stress
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