Some context: I am reading Strang's paper, A Framework for Equilibrium Equations. In it, he gives a simple example with beams and rods. I'm trying to reproduce his arguments for 2D linear elasticity. In the paper, he considers conservation equations of the form \begin{equation} A^Tw = f \end{equation} where here $w$ is the stress variable, related to strains $e$ by Hooke's Law, $w = \mathbb{E} e$.
In the case of linear elasticity, I know that the equilibrium equation is $\mathrm{div} w = f$. This defines the operator $A^T$ as divergence. Integration by parts tells us that then $A$ is $- \nabla$. However, Strang's framework considers the strain $e = -Au$. The problem is, the strain is not defined that way. As far as I am aware, strain is defined as the symmetric part of the displacement gradient, not the full gradient itself.
This has led me to my question. What is the formal adjoint of the symmetric gradient operator? Does it have a nice interpretation? Here's what I have so far. Suppose we have smooth functions, and since we are looking for the formal adjoint, let us disregard boundary terms stemming from integration by parts. Let $u = (u_1, u_2)$ be a vector valued function, and let $\tau$ be a matrix-valued function. Then
\begin{align} ({Au,}{\tau}) &= \left({-\frac{1}{2}\left( \nabla u + \nabla u^\top \right),}{\tau}\right)\\ &= -\frac{1}{2}\left({\nabla u,}{\tau }\right) - \frac{1}{2} \left( \nabla u^\top , \tau \right)\\ &= \frac{1}{2}\left({u,}{\mathrm{div} \tau }\right) + \mathrm{B.T.} + ??\\ & \vdots \\ &= \left(u, A^\top \tau\right). \end{align}
My question arises in how to integrate by parts the term $(\nabla u^\top, \tau)$. I attempt to do so by writing things out component wise, i.e. $(\nabla u^\top, \tau) = \int \sum_{i,j} \partial u_j/\partial x_i \tau_{ij}$. Integrating by parts moves the derivative onto $\tau_{ij}$, but I don't get anywhere afterwards. Can someone provide some insight, and help me to understand why many mathematicians disregard the important symmetric part of the displacement gradient when talking about strain? Thank you in advance!