# Compatibility equations in elastcity

## Main Question or Discussion Point

Strain field in deformable solid (continuum) has to follow compatibility equations so as to ensure single valued continuous displacement field.
There are in all 81 such equations and most of them are repeated, finally we are left with 6 equations.
It is quoted in text books that only 3 out of these 6 equations are independant.
If it is so why we need to use six equations to ensure possibility of strain field?
Only three equations should be enough.

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Mech_Engineer
Gold Member
If I'm understanding your question correctly I think the difference lies in compression/tension vs. shear. There are three tension directions and three shear planes (for a total of 6 values), but only three of the six values are needed to find all other values.

Just to clarify there are 3 normal and 6 shear values,

σxx, σyy, σzz,

τxy, τyx
τzy, τyz
τxz, τzx

and the corresponding strain ε values to go with them

but symmetries reduce these.

AlephZero