Month: October 2016

The Bradley Model for Cohesion

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One of the initial cohesive models was proposed by Bradley. This model neglects the deformation of contact and considers the van der Waals forces. Bradley showed that the adhesive force for rigid spheres follows a force derived from the Lennard-Jones potential—a mathematical model that approximates the interactions between a pair of neutral atoms or molecules. The force of adhesion between two rigid spheres is described by:

f_n = \frac{8\pi\gamma_{sur} R^*}{3}[\frac{1}{4}(\frac{\delta_n}{\delta_0})^{-8}-(\frac{\delta_n}{\delta_0})^{-2}]

where \delta_0 is the equilibrium separation distance between the particles [1]. Two particles separate when the forces pulling the particles is reached at \delta_n = \delta_0. This model uses the van der Waals forces as the interaction creating the cohesive force. At the macroscopic scale, these forces are negligible when compared to the gravitational force experience by the particles. The figure below provides an example of the Bradley cohesion model and illustrates the attraction force between particles at a distance.

bradley

The Bradley Model [2]

[1] Bradley R. S. “The cohesive force between solid surfaces and the surface energy of solids”, Philosophical Magazine, vol. 13, no. 86, pp. 853-862, 1932.

[2] Farhang R., Dubois R., Discrete-Element Modeling of Granular Materials, John Wiley & Sons Inc. 2011

Macroscopic Cohesion

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The yield strength of the macro-scale cohesion of particulate materials can be described by Coulomb cohesion. The Mohr-Coulomb failure criterion, described by:

\sigma_t = \pm (\tan \varphi \ \sigma_n + c)

divides the mechanical strength of the material into the angle of internal friction  and the macroscopic cohesion of the material c [1]. Here \sigma_n and \sigma_t  are the normal and tangential stresses. These parameters characterize the material at particular stress states and can extend the effects of cohesion into the elastic domain in the stress plane. The angle of internal friction and the macroscopic cohesion can be determined with physical tests of shear, compression or tension. Particulate materials are typically tested under compressive loading. Under a uniaxial compression test, the yield strength \sigma_y of the material can be derived by:

\sigma_y = \frac {\cos \varphi}{1- \sin \varphi}c

The figure provides a graphical representation of Mohr-Coulomb’s criterion where the parameters \varphi and c characterize the strength of the material. The straight line represents the linear failure envelope that is obtained from the shear strength of a material at a particular state of stress in the material.

morhcircle

Morh-Coulomb Criterion [2].

  1. Nedderman R. M., Statics and Kinematics of Granular Materials, Cambridge Univeristy Press, Cambridge, 1992.
  2. Farhang R., Dubois F., Discrete-Element Modeling of Granular Materials, John Wiley & Sons Inc. 2011

Introduction to Cohesive Forces

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Cohesive forces stem from cohesive interactions between particles and can typically be classified into three levels of cohesion: adhesion, capillary and cementation [1]. Here, interactions between particles are limited to surface interactions such as in physico-chemical interactions with very short range or through solid or liquid bridges at the particle contact. Electrical forces such as van der Waals forces are considered negligible in macroscale mechanical behavior. This is due to the gravitational forces dominating the bulk forces in material flow. The cohesive forces between contacting particles form an association with the contacting normal force and the contact overlap or separation . Upon the application of a tensile force between the particles, the adhesive force resists the direction of separation and for a distance the bond is still held. The figure below shows two particles in contact through loading and unloading. The cohesive force resists the tensile force.  The distance at which the cohesive bond is broken differs from the distance for which the cohesive contact bond is formed. The distance at which the cohesive bond breaks leads to a hysteresis phenomenon.

hysteresis.jpg

(a) Formation of a cohesive contact. (b) Tensile strength due to the presence of cohesion. (c) Failure of the cohesive bond with a value of \delta_n > 0. (d) The evolution of the normal force as a function \delta_n of \gamma where represents the energy per unit area to break the cohesive contact [1].

[1] Farhang R., Dubois F., Discrete-Element Modeling of Granular Materials, John Wiley & Sons Inc. 2011