A slicing approach to stress-strain duality
The classical Kohn-Temam stress-strain pairing $({\bf A}:E{\bf u})$ for symmetric tensors ${\bf A}$ and ${\bf u}\in BD$ is typically formulated under summability assumptions on the divergence of ${\bf A}$. This excludes stress fields whose divergence has singular surface contributions, as occurs at cracks and material interfaces in continuum mechanics. We define and study stress-strain pairings for bounded symmetric divergence-measure tensor fields. For general ${\bf u}\in BD$, we introduce a slicing pairing $(({\bf A}:E{\bf u}))_\Xi$ for tensor fields satisfying a directional $BV$-type condition with respect to a finite frame $\Xi$. The definition is based on a one-dimensional disintegration strategy, and despite this construction, the new pairing enjoys analogous properties of the usual pairing $({\bf A}:E{\bf u})$, such as the absolutely continuity with respect to $|E{\bf u}|$ and the Gauss-Green formulas. We also identify several situations in which the pairing is independent of the choice of frame $\Xi$, including the relevant case in which the stress field ${\bf A}$ belongs to $BV$. While a distributional stress-strain pairing can be defined naturally for bounded $BD$ functions, it cannot be extended to the unbounded setting, since the truncation techniques available in $BV$ fail in $BD$. The slicing pairing is consistent with the distributional one whenever the latter is defined, while being more general even for bounded ${\bf u}$. Indeed, its existence does not require the compatibility condition $|{\rm Div}\,{\bf A}|(S_{{\bf u}}\setminus J_{\bf u})=0$ which is necessary for the distributional definition. This allows the treatment of stress fields interacting with diffuse micro-cracking.