2026/07/19 by Virginia De Cicco, Giovanni Scilla
#math.FA #math.AP
The classical Kohn-Temam stress-strain pairing (\bf A:E\bf u) for symmetric tensors \bf A and \bf u∈ 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∈ BD, we introduce a slicing pairing ((\bf A:E\bf u))Ξ for tensor fields satisfying a directional BV-type condition with respect to a finite frame Ξ. 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 Ξ, 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∖ J\bf u)=0 which is necessary for the distributional definition. This allows the treatment of stress fields interacting with diffuse micro-cracking.