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In mechanics, we have the so called compatibility conditions, which quarantee that when a body deforms, the strains are "compatible" in such a way to no discontinuities or gaps for inside the body as it deforms. Intuitively, this means that infinitesimal pieces of material connected in the undeformed body stay connected, and won't end up inside one another.

From Wikipedia:

..In the continuum description of a solid body we imagine the body to be composed of a set of infinitesimal volumes or material points. Each volume is assumed to be connected to its neighbors without any gaps or overlaps. Certain mathematical conditions have to be satisfied to ensure that gaps/overlaps do not develop when a continuum body is deformed. A body that deforms without developing any gaps/overlaps is called a compatible body. Compatibility conditions are mathematical conditions that determine whether a particular deformation will leave a body in a compatible state.

Reading the Wikipedia page, these equations/conditions are developed:

$$ \frac{\partial^2\epsilon_{11}}{\partial x_2 ^2}-2\frac{\partial^2\epsilon_{12}}{\partial x_1 \partial x_2}+\frac{\partial^2\epsilon_{22}}{\partial x_1 ^2}=0$$

They are derived by stating that

..there are six strain-displacement relations that are functions of only three unknown displacements. This suggests that the three displacements may be removed from the system of equations without loss of information.

Firstly, sure, the above statement does make sense, but what does it have to do with compatibility, as I described above? Why is it that when a few variables are removed from the system, we are now guaranteed to have a deformed state where no material overlaps or develops discontinuities?

Most of other material that I've read online just state the same intuitive idea of compatibility, and then just state that the deformation is compatible when the strain is integrable and proceed to find conditions ensuring that. Alternatively, it is proven that the solution to this equation can be found:

$$\epsilon=\frac{1}{2}[\nabla u +(\nabla u)^T]$$

Again, sure, we need to know that this equation can be solved to have a sensible strain field, but how do we know the solution strain field is compatible, in the intuitive sense described at the beginning?

Why simply being able find a solution displacement field, or being able to integrate the strain field imply that the strain is compatible?

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