Matrix tensor notation part I. Rectilinear orthogonal coordinates

Hassenpflug, W.C.

doi:10.1016/0898-1221(93)90110-h

Cited by 5 publications

(3 citation statements)

References 5 publications

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“…The subscripts i, j and k are used, assuming values of 1, 2 and 3, unless stated otherwise. In the latter part of the appendix, Hassenpflug [31] notation is used. A column vector is indicated by a overbar, x, a row vector by a underbar, x and a matrix by both, x.…”

Section: Appendix: the Material-point Methodsmentioning

confidence: 99%

Discrete and continuum modelling of soil cutting

Coetzee

2014

Comp. Part. Mech.

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Both continuum and discrete methods are used to investigate the soil cutting process. The Discrete Element Method (dem) is used for the discrete modelling and the Material-Point Method (mpm) is used for continuum modelling. Mpmis a so-called particle method or meshless finite element method. Standard finite element methods have difficulty in modelling the entire cutting process due to large displacements and deformation of the mesh. The use of meshless methods overcomes this problem. Mpm can model large deformations, frictional contact at the soil-tool interface, and dynamic effects (inertia forces). In granular materials the discreteness of the system is often important and rotational degrees of freedom are active, which might require enhanced theoretical approaches like polar continua. In polar continuum theories, the material points are considered to possess orientations. A material point has three degrees-of-freedom for rigid rotations, in addition to the three classic translational degrees-of-freedom. The Cosserat continuum is the most transparent and straightforward extension of the nonpolar (classic) continuum. Two-dimensional dem and mpm (polar and nonpolar) simulations of the cutting problem are compared to experiments. The drag force and flow patterns are compared using cohesionless corn grains as material. The corn macro (continuum) and micro (dem) properties were obtained from shear and oedometer tests. Results show that the dilatancy angle plays a significant role in the flow of material but has less of an influence on the draft force. Nonpolar mpm is the most accurate in predicting blade forces, blade-soil interface stresses and the position and orientation of shear bands. Polar mpm fails in predicting the orientation of the shear band, but is less sensitive to mesh size and mesh ori-C. J. Coetzee (B) Department of Mechanical and Mechatronic Engineering, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa e-mail: ccoetzee@sun.ac.za entation compared to nonpolar mpm. dem simulations show less material dilation than observed during experiments.

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Section: Appendix: the Material-point Methodsmentioning

confidence: 99%

Discrete and continuum modelling of soil cutting

Coetzee

2014

Comp. Part. Mech.

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“…The subscripts i, j and k are used, assuming values of 1, 2 and 3, unless stated otherwise. In the latter part of the appendix, Hassenpflug [24] notation is used. A column vector is indicated by a overbar, x, a row vector by a underbar, x and a matrix by both, x.…”

Section: Appendix a The Material-point Methodsmentioning

confidence: 99%

Discrete and continuum modelling of excavator bucket filling

Coetzee

Basson

Vermeer

2007

Journal of Terramechanics

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“…The subscripts i; j and k are used, assuming values of 1, 2 and 3, unless stated otherwise. In the latter part of the section, Hassenpflug [11] notation is used. A column vector is indicated by a overbar, x; a row vector by a underbar, x and a matrix by both, x: Plane strain conditions are assumed.…”

Section: The Materials Point Methodsmentioning

confidence: 99%

The modelling of anchors using the material point method

Coetzee

Vermeer

Basson

2005

Num Anal Meth Geomechanics

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SUMMARYThe ultimate capacity of anchors is determined using the material point method (MPM). MPM is a socalled meshless method capable of modelling large displacements, deformations and contact between different bodies. A short introduction to MPM is given and the derivation of the discrete governing equations. The analysis of a vertically loaded anchor and one loaded at 458 is presented. The loaddisplacement curves are compared to that obtained from experiments and the effect of soil stiffness and anchor roughness is investigated. The results of the vertically loaded anchor are also compared to an analytical solution. The displacement of the soil surface above the anchor was measured and compared to the numerical predictions.Convergence with mesh refinement is demonstrated and the effect of mesh size and dilatancy angle on the shear band width and orientation is indicated.The results show that MPM can model anchor pull out successfully. No special interface elements are needed to model the anchor-soil interface and the predicted ultimate capacities were within 10% of the measured values.

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Matrix tensor notation part I. Rectilinear orthogonal coordinates

Cited by 5 publications

References 5 publications

Discrete and continuum modelling of soil cutting

Discrete and continuum modelling of soil cutting

Discrete and continuum modelling of excavator bucket filling

The modelling of anchors using the material point method

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