Suppose I start with a rectangular elastic (to keep things simple, zero Poisson's ratio) sheet of length $2\pi R$, thickness $h$, and (immaterial) width $W$. I roll it up into a cylinder of radius $R$, and allow it to relax. By symmetry, if it deforms at all, it will do so by uniformly scaling of the cross section to a cylinder of radius $R+\Delta R$.

I'm trying to study the behavior of the cylinder using Kirchoff-Love plate theory, but have gotten myself confused. If I use Cauchy strain, I get that the strain is proportional to $$\frac{z+\Delta R}{R}$$ where $z\in (-h/2, h/2)$ is the normal coordinate. I then get that the energy density of the cylinder depends on $(\Delta R)^2$ but not $\Delta R$, and therefore setting $\Delta R=0$ minimizes energy -- the cylinder neither grows nor shrinks.

However, if I instead use Green strain, the deformation energy has a nonzero linear term in $\Delta R$, and I get that the cylinder shrinks ever so slightly. I find this very counter-intuitive.

Why is using Cauchy strain inappropriate for approaching this problem? If I perform this experiment with a real sheet, will I really observe (very slight) shrinking? Is there intuition for why this occurs?


I suppose that when you say "Cauchy strain" you mean the infinitesimal strain tensor. In general, I would not use this strain measure since it is valid for

$$\frac{\partial u_i}{\partial X_j} \ll 1\, .$$

Now, let's consider the problem that you are proposing. The configuration is given by

\begin{align} &x = (R - Y) \sin\left(X/R\right)\, ,\\ &y = R - (R - Y) \cos\left(X/R\right)\, ,\\ &z = Z\, , \end{align} where $R$ is the radius of curvature of the deformation, $X, Y, Z$ the material coordinates, and $x, y, z$ the spatial ones.

For this problem the deformation gradient is $$F = \left[\begin{matrix}\frac{\left(R - Y\right) \cos{\left(\frac{X}{R} \right)}}{R} & - \sin{\left(\frac{X}{R} \right)} & 0\\- \frac{\left(- R + Y\right) \sin{\left(\frac{X}{R} \right)}}{R} & \cos{\left(\frac{X}{R} \right)} & 0\\0 & 0 & 1\end{matrix}\right]\, ,$$

and the jacobian is

$$J = 1 - \frac{Y}{R}\, .$$

Notice that this is the change of volume of the body. Thus, it does shrink, at least for points above $Y=0$.

Now, let's move on to different strain measures. If we use the Lagrange strain we have the following

$$E = \frac{1}{2}\left(F^T F - I\right) = \left[\begin{matrix} - \frac{Y}{R} + \frac{Y^{2}}{2 R^{2}} & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{matrix}\right]\, ,$$

if we want to compute the extension, we could use

$$E_1 = \sqrt{1 + 2E_{11}} - 1\approx - \frac{Y}{R}\, ,$$

that would be positive/negative depending on the sign of $Y$.

Finally, let's consider the infinitesimal strain tensor.

$$\epsilon_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i}\right)\, ,$$


$$\begin{align} u = x - X\, ,\\ v = y - Y\, ,\\ z = z - Z\, . \end{align}$$


$$\epsilon = \left[\begin{matrix}\cos{\left(\frac{X}{R} \right)} - 1 - \frac{Y \cos{\left(\frac{X}{R} \right)}}{R} & - \frac{Y \sin{\left(\frac{X}{R} \right)}}{2 R} & 0\\- \frac{Y \sin{\left(\frac{X}{R} \right)}}{2 R} & \cos{\left(\frac{X}{R} \right)} - 1 & 0\\0 & 0 & 0\end{matrix}\right]\, ,$$

and considering the first term in the Taylor series

$$\epsilon \approx \left[\begin{matrix}- \frac{Y}{R} & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right]\, .$$

So, we could say that it can shrink or extend depending on $Y$. The extension depends on $Y/R$ linearly.


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