|     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               |