2.3 Equations of motion and equilibrium for deformable solids
this section, we generalize
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.
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
Since this must hold for every volume of material within a solid, it follows that
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.
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