Submitted Paper
Inserted: 15 jul 2026
Last Updated: 19 jul 2026
Year: 2026
Abstract:
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 $\lvert {{\rm Div}\,{\bf A}}\rvert(S_{\bf u}\setminus J_{\bf u})=0$ which is necessary for the distributional definition. This allows the treatment of stress fields interacting with diffuse micro-cracking.
Keywords: Slicing, Gauss-Green formula, pairing, divergence-measure tensor fields
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