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bobinthebox

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I am studying the finite bending of a rubber-like block, assuming Neo-Hookean response. In the following, ##l_0##,##h##, ##\bar{\theta}## are parameters, while the variables are ##r## and ##\theta##.

The Cauchy stress tensor is

##T= - \pi I + \mu(\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r \otimes e_r + \frac{4 \bar{\theta}^2}{l_0^2}r^2 e_{\theta} \otimes e_{\theta} - I)##

Now I need to solve ##div(T)=0##, where the divergence has to be computed in cylindrical coordinates. The author says:

I assume ##T_r## means ##e_r \cdot T e_r##, right? If so, I obtained

##T_r = - \pi + \mu \frac{l_0^2}{4 \bar{\theta}^2 r^2} -1 ##

##T_{\theta} = - \pi + \mu \frac{4 \bar{\theta}^2 r^2}{l_0^2} -1 ##

Unfortunately, the first equilibrium equation I obtain is different from the one of the book, which is attached to this message.

I obtain ##\frac{\partial \pi}{ \partial r} + \mu \frac{4 \bar{\theta}^2 r}{l_0^2}=0##

I'd like to have a check about this, because I think I computed correctly the two principal stresses, so maybe there's a mistake in the book.

Best regards,

Bob

The Cauchy stress tensor is

##T= - \pi I + \mu(\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r \otimes e_r + \frac{4 \bar{\theta}^2}{l_0^2}r^2 e_{\theta} \otimes e_{\theta} - I)##

Now I need to solve ##div(T)=0##, where the divergence has to be computed in cylindrical coordinates. The author says:

Since there are only two non-null principal stresses, ##T_r## and ##T_θ## , equilibrium becomes ##\frac{\partial T_r}{\partial r} + \frac{T_r - T_{\theta}}{r}=0 \quad \frac{\partial T_{\theta}}{\partial \theta} = 0##

**Question:**I assume ##T_r## means ##e_r \cdot T e_r##, right? If so, I obtained

##T_r = - \pi + \mu \frac{l_0^2}{4 \bar{\theta}^2 r^2} -1 ##

##T_{\theta} = - \pi + \mu \frac{4 \bar{\theta}^2 r^2}{l_0^2} -1 ##

Unfortunately, the first equilibrium equation I obtain is different from the one of the book, which is attached to this message.

I obtain ##\frac{\partial \pi}{ \partial r} + \mu \frac{4 \bar{\theta}^2 r}{l_0^2}=0##

I'd like to have a check about this, because I think I computed correctly the two principal stresses, so maybe there's a mistake in the book.

Best regards,

Bob

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