Mechanics of cutting in granular media

Abstract

Cutting is an important deformation process encountered during many instances in the engineering of infrastructure, such as trawling, trenching, and excavation. This thesis presents the results of an experimental program on the orthogonal cutting of granular materials, performed under plane strain conditions. The kinematics of granular materials, when subjected to large deformations, is understood through direct imaging and concomitant image analysis. Three suites of experiments were performed to understand the effects of boundary conditions, material systems, and other constraints on the mechanics of cutting.

In the first suite of experiments, we perform cutting on two sets of granular materials, spherical glass beads and angular Cauvery delta sand. We vary the three main parameters of the orthogonal cutting problem, the rake angle/ angle of attack of the tool, the initial depth of cut with respect to the free surface of the granular ensemble, and the cutting speed. The overall deformation of the material reaches a steady state after an initial transient phase, which can be observed from the angle of the repose of the pile that forms in-front of the tool. Instantaneous velocity fields show a bifurcation of flow occurring at the tool tip which results in a velocity jump across planes whose direction coincides with the Coulomb failure surfaces. These velocity jumps manifest as regions of intense shear emanating from the tool tip and has a ``band" like appearance when the effective strain rate contours are plotted. The velocity profiles across the shear band assume a sigmoidal shape which can be fit using an error function and allows us to extract the length scale associated with this velocity jump. We observe that the normalized width of the shear band decreases as particle size increases. The width of the shear band for both granular materials was observed to be insensitive to changes in the rake angle, cutting speed, and also the initial depth of cut and hence is a material constant for a given system size. In some cases, we also observe multiple shear bands forming in the system which run parallel to one another and have similar thickness. The measured volumetric strain rate within the shear bands is much lower than the shear rate. The dilation angle measured is always less than the friction angle suggesting that plastic flow occurring within the shear bands is in a non-associative state. We further observe a dead zone of granular material at the tool interface.

In the second suite of experiments, we use a suite of granular materials such as cylindrical steel beads, rice grains, clay aggregates, polystyrene beads, and smaller glass beads. At the ensemble level, we find that the volume of material removed during cutting increases linearly in time for all granular materials. At the meso-scale, we once again observe similar sigmoidal velocity profiles across the shear band and is further used to determine the shear band width of each granular material. We find the normalized width of the shear bands is a function of particle size. Upon re-scaling the velocity profiles for different granular materials by their corresponding shear band widths, they all collapse onto a single master curve given by the scaled error function. Thus, we witness a remarkable universality in the kinematics of granular materials at multiple length scales during orthogonal cutting. We further measure the temperature within the shear bands and observe a power-law relationship between the temperature and shear rate, with an exponent ~ 0.55, a significant deviation from the predictions of kinetic theory. In-order to further probe the origin of this inherent length scale at the meso-level, we measure the mean squared displacements of the particles in the shear bands and observe that the particles are not diffusive and hence conclude that the length scale corresponding to the shear bands does not have its origin in a diffusion like process.

In the third suite of experiments, a series of constrained cutting experiments were carried out wherein we force the cut granular material to flow through narrow channels by placing a rigid constraint across the cutting tool. We study the formation and evolution of shear bands under such circumstances. By the addition of a constraint, we pin the shear bands between the tool tip and the constraint corner. The separation distance between the tool and the constraint has a direct consequence on the characteristics of the shear bands generated within the ensemble. As the separation between the tool and plate increases, we recover the behavior of unconstrained cutting. Within the channel, the flow fields are very similar to what is observed in vertical hoppers, though the motion of particles is directed against gravity. Like in hopper flows, we observe dead zones along the length of tool and constraint boundaries. We further measure the thickness of this boundary layer using the same error function fit and find that the thickness of the boundary layer is constant across the channel height and nearly symmetric at the two boundaries for small channel widths. As the channel width increases, the boundary layer thickness also increases and becomes asymmetric at the two boundaries.

We also present the results of cutting carried out on a constrained granular ensemble, i.e. two granular chains with different numbers of beads. This addition of a constraint to the granular ensemble suppresses the formation of localized shear bands within the ensemble and the deformation is diffuse and homogeneous. The deformation also propagates to regions faraway from the tool.

Lastly, we provide a numerical underpinning to the cutting experiments using contact dynamics simulations. As in the case of the experiments, we vary the rake angle, initial depth of cut, and cutting speed in simulation. We compare the simulation results with analytical solutions provided by a 2-D model of cutting. We compare the deformation fields obtained from the simulations with our experimental results and deduce the forces on the tool and along the shear band.