I calculated the stress $\sigma$ and strain $\varepsilon$ for a solid plate with dirichlet boundary conditions $u = g$, where $u$ is the displacement. With these I want to calculate $\nabla_n u = t$ on the boundary where $\nabla_n$ describes the normal derivative. Is there any easy way to do this?

I have the displacement $u$ but since it is a vector with real numbers I cannot calculate the normal derivative.

Edit: With some research I figured out that $\sigma = C\varepsilon = C \nabla u$ and with that $\nabla_n u = C^{-1}\sigma \cdot n$. My question is, how can I calculate a map between $u = g$ and $\nabla_n u$ which gives me $t$.

  • $\begingroup$ Why do you need the normal derivative of the displacements? $\endgroup$
    – nicoguaro
    May 29 '19 at 14:10
  • $\begingroup$ Also, I don't think that your expression is correct. Due to the major symmetry in the $C$ tensor you are only getting the symmetric part of the gradient and projecting it. $\endgroup$
    – nicoguaro
    May 29 '19 at 16:19
  • $\begingroup$ @nicoguaro thank you for your answer. I am trying to calculate the so called Dirichlet-to-Neumann map. If my expression is incorrect, how could I define a correct way for a map which gives me the normal derivative of the displacement on the dirichlet boundary? $\endgroup$
    – user29088
    May 30 '19 at 9:56
  • $\begingroup$ For elasticity the Neumann boundary conditions are given by $C\nabla_n u$, i.e., the normal projection of the stress tensor. $\endgroup$
    – nicoguaro
    May 30 '19 at 13:04
  • $\begingroup$ Thank you @nicoguaro. Where can I find the derivation of your expression? And is there a way to express $\nabla_n u$ with $\sigma$ or $\varepsilon$? $\endgroup$
    – user29088
    May 30 '19 at 17:10

Short answer

You can calculate the normal derivative of a vector, it is given by the projection of the gradient of the vector (a second-order tensor) in the normal direction. In cartesian coordinates, this is

$$ \nabla_n u = \nabla u \cdot n = \left[\begin{matrix}\frac{\partial}{\partial x} u_x & \frac{\partial}{\partial y} u_x & \frac{\partial}{\partial z} u_x\\ \frac{\partial}{\partial x} u_y & \frac{\partial}{\partial y} u_y & \frac{\partial}{\partial z} u_y\\ \frac{\partial}{\partial x} u_z & \frac{\partial}{\partial y} u_z & \frac{\partial}{\partial z} u_z\end{matrix}\right] \left[\begin{matrix}n_{x}\\n_{y}\\n_{z}\end{matrix}\right] = \left[\begin{matrix}n_{x} \frac{\partial}{\partial x} u_x + n_{y} \frac{\partial}{\partial y} u_x + n_{z} \frac{\partial}{\partial z} u_x\\n_{x} \frac{\partial}{\partial x} u_y + n_{y} \frac{\partial}{\partial y} u_y + n_{z} \frac{\partial}{\partial z} u_y\\n_{x} \frac{\partial}{\partial x} u_z + n_{y} \frac{\partial}{\partial y} u_z + n_{z} \frac{\partial}{\partial z} u_z\end{matrix}\right] $$

Long answer

In linear elasticity, the constitutive relation (Hooke's law) relates the strain (a second-order tensor) to the stress (a second-order tensor) by the stiffness tensor (a fourth-order tensor), i.e.

$$\sigma = C \varepsilon\, ,$$

or, in index notation,

$$\sigma_{ij} = C_{ijkl} \varepsilon_{kl}\, ,$$

where summation over repeated indices is implied. The stiffness tensor has the following symmetries $C_{ijkl} = C_{ijlk} = C_{jikl} = C_{klij}$.

Furthermore, the strain tensor is given by the symmetric part of the gradient of the displacement vector:

$$\varepsilon = \frac{1}{2}\left[\nabla u + (\nabla u)^T\right]\, ,$$

or, in index notation,

$$\varepsilon = \frac{1}{2}(u_{i,j} + u_{j, i})\, ,$$

where the comma represents a spatial derivative.

Thus, due to symmetries in the stiffness tensor, we can say that

$$\sigma = C \varepsilon = C\nabla u\, .$$

But, that does not imply that we can invert the relation to directly obtain the gradient of $u$ from $\sigma$.

This is why:

$$\sigma = C\nabla u = C \varepsilon + \underbrace{C\omega}_{=0} \, ,$$


$$S\sigma = SC \varepsilon = \varepsilon \, ,$$

where $S$ is the compliance tensor, and $S_{ijkl}C_{klrs}=\frac{1}{2}(\delta_{ir}\delta_{js} + \delta_{is}\delta_{jr})$.

  • $\begingroup$ Thank you for the detailed answer. So what is the problem now, since it is an equality, to take the inverse of $C$ and multiply the whole equation with the normal direction $n$? $\endgroup$
    – user29088
    May 31 '19 at 14:39
  • $\begingroup$ If you multiply by the inverse of $C$ you would recover the symmetric part of the gradient. $\endgroup$
    – nicoguaro
    May 31 '19 at 15:14
  • $\begingroup$ Thank you, especially for your patience. But I have a big problem (as a mathematican) to understand "would recover the symmetric part of the gradient". I hope I understand correctly that $C \omega = 0 \, \Leftrightarrow \, \sigma = C \nabla u = C \varepsilon$. Is that correct? $\endgroup$
    – user29088
    May 31 '19 at 18:54
  • 1
    $\begingroup$ @Kerem, well, I have never thought it that way. But, I would say that $C \omega = 0 \, \Leftrightarrow \, C \nabla u = C \varepsilon \equiv \sigma$. $\endgroup$
    – nicoguaro
    May 31 '19 at 18:56

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