Stresses in bulk solids in wedge hoppers: A flexible formulation of the co-ordinate specific, Lame–Maxwell equations for circular arc, principal stress systems

A 2-D model of stress distribution within bulk solids, with circular arc principal stress orientation, in a wedge hopper was developed in a previous paper [Matchett, O'Neill, & Shaw, Stress distributions in 2-dimensional, wedge hoppers with circular arc stress orientation — a co-ordinate-specific Lamé–Maxwell model, Powder Technology, 187(2008) 298–306]. This model worked in an orthogonal, curvilinear co-ordinate system co-incident with the principal stress trajectories: (x − ψo) space.This paper presents an equivalent model in (x − ε) space. This allows backward numerical integration of the force balance equations, enabling surface and wall boundary conditions to be modelled. This was not possible in the original model.The equations are first-order, and boundary conditions can only be specified at single surfaces. Thus, if a stable, cohesive arch is proposed, the surface overpressure is determined by the model. Calculated overpressures have reasonable physical values.The present model was integrated backwards from the surface downwards and it was found that the integration was very sensitive to the surface overpressure stresses.Likewise, wall boundary conditions were specified with backwards integration in ε.The minimum outlet for flow was calculated from the model and compared with the experimental data of Berry et al. Wall normal stresses in a wedge hopper from Schulze and Schwedes were also compared to model predictions. In both cases there was reasonable agreement between measurements and model predictions.

Stresses were modelled in wedge hoppers assuming circular arc principal stress orientation — see figure. Equations were derived which are flexible forms of the Lame–Maxwell equations, allowing backward numerical integration with the use of surface and wall boundary conditions. The viability of stress conditions was determined by a Mohr–Coulomb relationship.The model was used to calculate the minimum outlet for flow. Data were compared with the experimental results of Berry et al and Schulze & Schwedes with reasonable agreement.Figure optionsDownload as PowerPoint slide