Isostaticity theory (IT) provides stress field equations for marginally rigid packs of non-cohesive particles, interacting through hard-core forces. Its main advantage over strain-based theories is by closing the stress equations with stress–structure, rather than stress–strain, relations, which enables modelling the stress chains, often observed in experiments and simulations. The usefulness of IT has been argued to extend beyond its applicability at marginal rigidity. It has been shown to be derivable from first principles in two-dimensions, with the structure quantified by a fabric tensor that couples to the stress field. However, upscaling IT to the continuum is done currently empirically by postulating convenient closure equations. The problem is that a volume average of the fabric tensor vanishes in the continuum limit, trivialising the closure equation. This poses an unusual upscaling problem, necessitating a new approach. Such an approach is developed here, resolving the problem for planar granular assemblies. The new method is developed initially for idealised ‘unfrustrated’ packs by coarse-graining first to the two-grain scale, after which a conventional coarse-graining can be used. It is then extended to general realistic systems, by introducing an intermediate ‘de-frustration’ procedure. The applicability of the method is illustrated with a tractable example.

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中文翻译：

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粒状物质等静性理论的超常问题

等静压理论（IT）提供了通过硬核力相互作用的非粘性颗粒的边缘刚性包装的应力场方程。与基于应变的理论相比，它的主要优势是通过用应力-结构关系而不是应力-应变关系来封闭应力方程，从而可以对应力链进行建模，而应力链通常在实验和模拟中都可以观察到。有人认为，IT的实用性超出了边际刚性的适用性。它被证明是从二维的第一原理派生而来的，其结构是通过与应力场耦合的织物张量来量化的。但是，目前通过经验证明，通过假设方便的闭合方程式，可以将IT升级到连续体。问题是织物张量的体积平均值在连续极限内消失，简化闭合方程。这带来了一个异常的扩展问题，因此需要一种新的方法。这里开发了这种方法，解决了平面颗粒组件的问题。这种新方法最初是针对理想化的“未挫败”包装而开发的，方法是先进行粗粒化至两粒级，然后再使用常规的粗粒化。然后通过引入中间的“挫折”程序将其扩展到一般的现实系统。该方法的适用性以易于处理的例子说明。之后，可以使用常规的粗粒度。然后通过引入中间的“挫折”程序将其扩展到一般的现实系统。该方法的适用性以易于处理的例子说明。之后，可以使用常规的粗粒度。然后通过引入中间的“挫折”程序将其扩展到一般的现实系统。该方法的适用性以易于处理的例子说明。

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