  This paper extends our earlier work [arXiv:2309.13595] on the $L^p$
approximation of the shape tensor by Laurain and Sturm. In particular, it is
shown that the weighted $L^p$ distance to an affine space of admissible
symmetric shape tensors satisfying a divergence constraint provides the shape
gradient with respect to the $L^{p^\ast}$-norm (where $1/p + 1/p^\ast = 1$) of
the elastic strain associated with the shape deformation. This approach allows
the combination of two ingredients which have already been used successfully in
numerical shape optimization: (i) departing from the Hilbert space framework
towards the Lipschitz topology approximated by $W^{1,p^\ast}$ with $p^\ast > 2$
and (ii) using the symmetric rather than the full gradient to define the norm.
Similarly to [arXiv:2309.13595], the $L^p$ distance measures the shape
stationarity by means of the dual norm of the shape derivative with respect to
the above-mentioned $L^{p^\ast}$-norm of the elastic strain. Moreover, the
Lagrange multiplier for the momentum balance constraint constitute the steepest
descent deformation with respect to this norm. The finite element realization
of this approach is done using the weakly symmetric PEERS element and its
three-dimensional counterpart, respectively. The resulting piecewise constant
approximation for the Lagrange multiplier is reconstructed to a shape gradient
in $W^{1,p^\ast}$ and used in an iterative procedure towards the optimal shape.
