AI 中文总结
研究有界对称散度测度张量场的应力 - 应变对偶性，引入基于一维分解策略的切片配对$(({\bf A}:E{\bf u}))_\Xi$，该配对具有类似经典配对的性质，与分布配对兼容且更具一般性，能处理与弥散微裂纹相互作用的应力场。
AI 中文摘要
经典的科恩 - 特马姆应力 - 应变配对$({\bf A}:E{\bf u})$通常是在对${\bf A}$散度的可和性假设下针对对称张量${\bf A}$和${\bf u}\in BD$制定的，这排除了散度具有奇异表面贡献的应力场。我们定义并研究有界对称散度测度张量场的应力 - 应变配对。对于一般的${\bf u}\in BD$，我们引入切片配对$(({\bf A}:E{\bf u}))_\Xi$，它基于一维分解策略，新配对具有与通常配对$({\bf A}:E{\bf u})$类似的性质，如关于$|E{\bf u}|$的绝对连续性和高斯 - 格林公式。我们还确定了配对与框架$\Xi$选择无关的几种情况。分布应力 - 应变配对虽能为有界$BD$函数自然定义，但不能扩展到无界情形，而切片配对与之兼容且更具一般性，它的存在不需要分布定义所需的兼容性条件，可用于处理与弥散微裂纹相互作用的应力场。
英文摘要
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 $(({\bf A}:E{\bf u}))_Ξ$ 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 $|{\rm Div}\,{\bf A}|(S_{\bf u}\setminus J_{\bf u})=0$ which is necessary for the distributional definition. This allows the treatment of stress fields interacting with diffuse micro-cracking.
Comments49 pages, 1 figure
