Editing Linear elasticity (section)

===== The biharmonic equation =====
The elastostatic equation may be written:
<math display="block">(\alpha^2-\beta^2) u_{j,ij} + \beta^2 u_{i,mm} = -F_i.</math>

Taking the [[divergence]] of both sides of the elastostatic equation and assuming the body forces has zero divergence (homogeneous in domain) (<math>F_{i,i}=0\,\!</math>) we have
<math display="block">(\alpha^2-\beta^2) u_{j,iij} + \beta^2u_{i,imm} = 0.</math>

Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have: <math display="block">\alpha^2 u_{j,iij} = 0</math> from which we conclude that: <math display="block">u_{j,iij} = 0.</math>

Taking the [[Laplacian]] of both sides of the elastostatic equation, and assuming in addition <math>F_{i,kk}=0\,\!</math>, we have
<math display="block">(\alpha^2-\beta^2) u_{j,kkij} + \beta^2u_{i,kkmm} = 0.</math>

From the divergence equation, the first term on the left is zero (Note: again, the summed indices need not match) and we have:
<math display="block">\beta^2 u_{i,kkmm} = 0</math>
from which we conclude that:
<math display="block">u_{i,kkmm} = 0</math>
or, in coordinate free notation <math>\nabla^4 \mathbf{u} = 0</math> which is just the [[biharmonic equation]] in <math>\mathbf{u}\,\!</math>.