Merkel Matthias, Baumgarten Karsten, Tighe Brian P, Manning M Lisa
Department of Physics, Syracuse University, Syracuse, NY 13244;
Centre de Physique Théorique (CPT), Turing Center for Living Systems, Aix Marseille Univ, Université de Toulon, CNRS, 13009 Marseille, France.
Proc Natl Acad Sci U S A. 2019 Apr 2;116(14):6560-6568. doi: 10.1073/pnas.1815436116. Epub 2019 Mar 20.
We present an approach to understand geometric-incompatibility-induced rigidity in underconstrained materials, including subisostatic 2D spring networks and 2D and 3D vertex models for dense biological tissues. We show that in all these models a geometric criterion, represented by a minimal length [Formula: see text], determines the onset of prestresses and rigidity. This allows us to predict not only the correct scalings for the elastic material properties, but also the precise magnitudes for bulk modulus and shear modulus discontinuities at the rigidity transition as well as the magnitude of the Poynting effect. We also predict from first principles that the ratio of the excess shear modulus to the shear stress should be inversely proportional to the critical strain with a prefactor of 3. We propose that this factor of 3 is a general hallmark of geometrically induced rigidity in underconstrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Finally, our results may lay important foundations for ways to estimate [Formula: see text] from measurements of local geometric structure and thus help develop methods to characterize large-scale mechanical properties from imaging data.
我们提出了一种方法来理解欠约束材料中几何不相容性导致的刚性，这些材料包括亚等静二维弹簧网络以及用于致密生物组织的二维和三维顶点模型。我们表明，在所有这些模型中，一个由最小长度[公式：见正文]表示的几何准则决定了预应力和刚性的起始。这使我们不仅能够预测弹性材料属性的正确标度，还能预测刚性转变时体积模量和剪切模量不连续性的精确大小以及坡印廷效应的大小。我们还从第一性原理预测，过量剪切模量与剪应力的比值应与临界应变成反比，系数为3。我们提出这个3的系数是欠约束材料中几何诱导刚性的一个普遍标志，可用于在实验中将这种效应与单一组分的非线性力学区分开来。最后，我们的结果可能为从局部几何结构测量中估计[公式：见正文]的方法奠定重要基础，从而有助于开发从成像数据表征大规模力学性能的方法。
