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From this link, strain compatibility for 2D problem with strains as $$ \varepsilon_{11} = \cfrac{\partial u_1}{\partial x_1} ~;~~ \varepsilon_{12} = \cfrac{1}{2}\left[\cfrac{\partial u_{1}}{\partial x_2} + \cfrac{\partial u_{2}}{\partial x_1}\right]~;~~ \varepsilon_{22} = \cfrac{\partial u_{2}}{\partial x_2} $$

is given by

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

Why the compatibility equation needs to have partial derivatives of strains?

Can we simply write the compatibility equation as

$$ \varepsilon_{11} - 2\varepsilon_{12} + \varepsilon_{22} $$

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    $\begingroup$ You do realize that, mathematically, they're not the same thing, right? $\endgroup$ – Chet Miller Aug 21 '18 at 2:16
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The compatibility conditions are important to ensure the existence of a displacement field $u_i$ when a strain field $\varepsilon_{ij}$ is given. They are integrability conditions. You get them from the definitions of the strains:

$$ \varepsilon_{11}=u_{1,1}\\ \varepsilon_{12}=(u_{1,2}+u_{2,1})/2\\ \varepsilon_{22}=u_{2,2} $$

With the first an last equation you can write $u_1=\int\varepsilon_{11}\ {\rm d}x_1$ and $u_2=\int\varepsilon_{22}\ {\rm d}x_2$. That you can insert in the definition of $\varepsilon_{12}$,

$$ \varepsilon_{12}=\left((\int\varepsilon_{11}\ {\rm d}x_1)_{,2}+(\int\varepsilon_{22}\ {\rm d}x_2)_{,1}\right)/2. $$

Of this expression you take a derivative w.r.t. $x_1$ and $x_2$ to get

$$ \varepsilon_{12,12}=\left(\varepsilon_{11,22}+\varepsilon_{22,11}\right)/2. $$

which is the equation in your question. If this does not hold then you get different $u_1$ when

  • integrating $\varepsilon_{11}$ or
  • when integrating first $\varepsilon_{22}$, inserting $u_2$ from that into $\varepsilon_{12}$ and then integrating for $u_1$.

Now you asked why we would not just take a linear combination of the strains as compatibility condition. The answer is: It does not say anything about the integrability.

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