A boundary integral for gradient averaging in two dimensions: application to polygonal regions in granular materials

A boundary integral is derived for averaging the gradients of a function within a two-dimensional (2D) region. The double integral uses the boundary derivative of the function to compute the average gradients, and it accounts for possible discontinuities in the function along the boundary. The integral reduces to a simple matrix expression for a polygonal region. When applied to a 2D granular material, the matrix expression can be used to compute the averaged local strains within polygonal void regions (particle clusters). In this situation, a realistic calculation of strain must account for the discontinuous movements among rigid particles along the polygon sides, as might occur if the particles are rotating as well as translating. The matrix expression provides a simple and efficient means of correcting the average strain to account for the discontinuous movements. For a cluster of circular disks, the correction is a consequence of the rolling and sliding among particles. The significance of the correction is illustrated with an example simulation of a large dense assembly of circular disks. Although the paper applies the boundary integral to the kinematics of granular regions, the integral will likely find other applications in 2D situations that involve discontinuous fields