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This is a follow up question to my question yesterday Structural mechanics traction boundary condition question
Does a traction free boundary condition also mean that the displacement gradient is 0 where the boundary condition is applied? My thinking is that it does not because it seems you can have traction free (zero stress) surface with a non-zero displacement gradient.
The traction is defined as
$$ \mathbf{t} = \mathbf{n} \cdot \boldsymbol{\sigma} $$
In terms of components, the zero-traction condition is
$$ t_j = \sum_i n_i \sigma_{ij} = 0 $$
From the above you can see that the stress components don't necessarily have to be zero for the traction to be zero.
For a linear elastic material,
$$ \boldsymbol{\sigma} = \mathsf{C}:\nabla{\mathbf{u}} $$
In component form,
$$ \sigma_{ij} = \sum_k \sum_l C_{ijkl} \frac{\partial u_k}{\partial x_l} $$
Once again, since there is no requirement that all the $C_{ijkl}$ values have to be positive, you don't need all the displacement gradients to be zero for the stresses to be zero (which is not strictly necessary for zero-tractions anyway).
However, zero displacement gradients will lead to zero tractions.
