The numerical approximation for the indentation of granular materials by a smooth rigid wedge punch
A numerical approximation of the stress equation for the indentation of granular materials by a smooth rigid wedge is studied. Granular materials are prevalent in our daily life, comprising sugar, coffee, grains, sand, gravel, soils, industrial raw materials, and pharmaceuticals. They can be either...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2022
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| Subjects: | |
| Online Access: | http://umpir.ump.edu.my/id/eprint/35908/1/The%20numerical%20approximation%20for%20the%20indentation%20of%20granular%20materials%20by%20a%20smooth%20rigid%20wedge%20punch.ir.pdf |
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| Summary: | A numerical approximation of the stress equation for the indentation of granular materials by a smooth rigid wedge is studied. Granular materials are prevalent in our daily life, comprising sugar, coffee, grains, sand, gravel, soils, industrial raw materials, and pharmaceuticals. They can be either dry or wet. In a solid-state, they can withstand deformations and form heaps; as in a liquids state, they will flow, and in gases state, they exhibit compressibility. These complex behaviour give rise to another state of matter that is poorly understood. A key observation in this study is to achieve accuracy in predicting flow field in general geometries for the double slip and double spin model. Plane strain conditions are assumed, and the materials obey the Mohr-Coulomb yield condition. This numerical approximates the solution for the deformation of granular materials under a wedge smooth rigid punch by a finite difference method. The granular material is assumed to be in a dense, solid-like state. The solution only refers to the initial motion after the punch. The construction of the deformation region is the combination of boundary value problems formed by the network of the α− and β− characteristic lines. These characteristic lines are determined from the solution of the stress equilibrium equation. The construction of the stress and velocity field in the deforming region is presented using the MATLAB program. The results obtained show that the stress, p, in triangle ABC is the major principal stress while it is the minor principal stress in the triangle BED. This result is consistent with the theory stated by the previous work. The approximated solution stress variables (p, ψ) and the corresponding velocity at each point (x, y) along the characteristic lines were then tested using the work rate equation. The positive value of the work rates proved that this solution method is admissible. Meanwhile, the deformation region constructed was consistent with the geometrical result obtained from the previous work. This model can be used if it satisfies the condition θ3 = θ1 + θ2, the angle between α- and β-characteristic lines with the contact surface is π/4 + ϕ/2, and the angle between both α- and β-characteristic lines with the raised surface is π/4−ϕ/2, as determined by the results. This method gives reliable and straightforward algorithms for solving the deformation problems involving stress variables and velocity. The method will consequently help to improve the existing tools and experimental facilities in the related industries and eventually increase efficiency. |
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