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Strain in the x and y directions are defined by the following equations:

$ε_x=\frac{du}{dx}+\frac{1}{2}[(\frac{du}{dx})^2 + (\frac{du}{dy})^2]$

$ε_y=\frac{dv}{dx}+\frac{1}{2}[(\frac{dv}{dx})^2 + (\frac{dv}{dy})^2]$

My question is when the strain is negative, the first term will be negative and the terms in the square brackets will always be positive. However, this does not occur if the strain is positive, does this mean that the terms in the square brackets need to be directional?

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I presume you got these equations from a derivation similar to https://en.wikipedia.org/wiki/Deformation_(mechanics)#Normal_strain

For small (infinitesimal) strains, the second order (squared) terms are negligible compared with $\partial u/\partial x$, and are simply ignored. This gives what is commonly called "engineering strain".

For large strains, you need a more careful definition of "strain" that gives zero strain for arbitrarily large rigid body rotations of a small element of material. See https://en.wikipedia.org/wiki/Finite_strain_theory for details. The Cauchy-Green strain tensor is commonly used to model situations where small elastic deformations of the material are superimposed on large rigid body translations and rotations.

As a general comment, there is no single "theoretically correct" definition of strain. If you assume the material properties (Young's modulus, Poisson's ratio, etc) are constant in the stress-strain equations, you will get different results depending on which strain measure you take - but in real life, the material properties are strain-dependent in any case, so you can make an accurate (but nonlinear) model of the material with any reasonable definition of strain.

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