Tighe Brian P, Socolar Joshua E S
Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, North Carolina 27708, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Mar;77(3 Pt 1):031303. doi: 10.1103/PhysRevE.77.031303. Epub 2008 Mar 13.
We study the nonlinear elastic response of a two-dimensional material to a localized boundary force, with the particular goal of understanding the differences observed between isotropic granular materials and those with hexagonal anisotropy. Corrections to the classical Boussinesq result for the stresses in an infinite half space of a linear, isotropic material are developed in a power series in inverse distance from the point of application of the force. The breakdown of continuum theory on scales of order of the grain size is modeled with phenomenological parameters characterizing the strengths of induced multipoles near the point of application of the external force. We find that the data of Geng [Phys. Rev. Lett. 87, 035506 (2001)] on isotropic and hexagonal packings of photoelastic grains can be fitted within this framework. Fitting the hexagonal packings requires a choice of elastic coefficients with hexagonal anisotropy stronger than that of a simple ball-and-spring model. For both the isotropic and hexagonal cases, induced dipole and quadrupole terms produce propagation of stresses away from the vertical direction over short distances. The scale over which such propagation occurs is significantly enhanced by the nonlinearities that generate hexagonal anisotropy.
我们研究二维材料对局部边界力的非线性弹性响应，特别旨在理解各向同性颗粒材料与具有六边形各向异性的材料之间所观察到的差异。针对线性各向同性材料无限半空间中应力的经典布辛涅斯克结果的修正，是在与力的作用点的逆距离的幂级数中展开的。连续介质理论在晶粒尺寸量级尺度上的失效，是用表征外力作用点附近感应多极子强度的唯象参数来建模的。我们发现，耿[《物理评论快报》87, 035506 (2001)]关于光弹性颗粒的各向同性和六边形堆积的数据可以在此框架内得到拟合。拟合六边形堆积需要选择具有比简单球 - 弹簧模型更强的六边形各向异性的弹性系数。对于各向同性和六边形情况，感应偶极子和四极子项都会使应力在短距离内从垂直方向传播开来。这种传播发生的尺度会因产生六边形各向异性的非线性而显著增大。
