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In plate theory, there is this equation: $$\epsilon_x=-z\frac{\partial^2 w}{\partial x^2}$$

For example, you can see this equation here.

And there is a conflict, which I'd like to resolve. Wikipedia says that the strain is unitless, i.e. its SI unit is 1. But, in the right side of the quoted equation, there is $z$, which has the unit of meter, and $\frac{\partial^2 w}{\partial x^2}$ is unitless. So, on the right side, meter is the unit, and on the left side, $\epsilon_x$ is unitless.

Where am I wrong?

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The quantity $\frac{\partial^2 w}{\partial x^2}$ is not unitless; it has units of 1/length. Think about how the second derivative is defined:

$$\frac{d^2y}{dx^2}=\lim_{h\to 0}\frac{y(x+h)-2y(x)+y(x-h)}{h^2}$$

Since $h$ is added to $x$ in the argument of $y$, it clearly has the same units as $x$, so if $y$ and $x$ have units of length, then the above quantity is an inverse length.

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  • $\begingroup$ You're absolutely right, I don't know how I missed this trivial thing. Thanks! $\endgroup$ – geza May 16 '18 at 23:21

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