Page 269 - Intro to Tensor Calculus

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Compatibility Equations in Terms of Stress

In the generalized Hooke’s law, equation (2.4.29), we can solve for the strain in terms of stress. This
in turn will give rise to a representation of the compatibility equations in terms of stress. The resulting
equations are known as the Beltrami-Michell equations. Utilizing the strain-stress relation

1+ ν ν
e ij = σ ij − σ kk δ ij
E E
we substitute for the strain in the equations (2.4.60) and rearrange terms to produce the result

σ ij,km + σ mk,ji − σ ik,jm − σ mj,ki =
ν (2.4.61)
[δ ij σ nn,km + δ mk σ nn,ji − δ ik σ nn,jm − δ mj σ nn,ki ] .
1+ ν
Now only 6 of these 81 equations are linearly independent. It can be shown that the 6 linearly independent
equations are equivalent to the equations obtained by setting k = m and summing over the repeated indices.
We then obtain the equations
ν
σ ij,mm + σ mm,ij − (σ im,m ) − (σ mj,m ) = [δ ij σ nn,mm + σ nn,ij ] .
,j ,i
1+ ν
Employing the equilibrium equation σ ij,i + %b j = 0 the above result can be written in the form

1 ν
σ ij,mm + σ kk,ij − δ ij σ nn,mm = −(%b i ) ,j − (%b j ) ,i
1+ ν 1+ ν
or
1 ν
2
∇ σ ij + σ kk,ij − δ ij σ nn,mm = −(%b i ) ,j − (%b j ) ,i .
1+ ν 1+ ν
This result can be further simplified by observing that a contraction on the indices k and i in equation
(2.4.61) followed by a contraction on the indices m and j produces the result

1 − ν
σ ij,ij = σ nn,jj .
1+ ν
Consequently, the Beltrami-Michell equations can be written in the form
1 ν
2
∇ σ ij + σ pp,ij = − δ ij (%b k ) ,k − (%b i ) ,j − (%b j ) ,i . (2.4.62)
1+ ν 1 − ν
Their derivation is left as an exercise. The Beltrami-Michell equations together with the linear momentum
(equilibrium) equations σ ij,i + %b j = 0 represent 9 equations in six unknown stresses. This combinations
of equations is difficult to handle. An easier combination of equations in terms of stress functions will be
developed shortly.
The Navier equations with boundary conditions are difficult to solve in general. Let us take the mo-
mentum equations (2.4.27(a)), the strain relations (2.4.28) and constitutive equations (Hooke’s law) (2.4.29)
and make simplifying assumptions so that a more tractable systems results.