3.1 Characteristics of Single Particles
3.1.1 Solid Particle Mechanics
The contact force between two particles is usually split into a normal force and a tan-gential force. The physical origin of these forces at the microscopic level involves many phenomena such as surface roughness, local mechanical properties (elasticity, plastic-ity, viscoelasticity) and physical and chemical properties (e.g. surface functionalization, charge) [Andreotti et al. 2013, p. 15]. In the following, the focus is set on the macroscopic laws of solid contact regardless of these physical and chemical properties.
Elastic Contact: The Hertz Law
Figure 3.1 shows two spherical particlesi andj squeezed together by an external normal force Fn. The normal contact force results from elastic deformation of the bodies in contact which were brought together by a distance of 2δ.
Ri
Rj 2a
2δ δ
δ Fn
Fn
i
j
i
j
Figure 3.1.: Elastic contact between two spheres. Two particlesi andj were brought together by a distance of2δby application of a normal force Fn. The radius of the contact area is given bya (modified from Andreotti et al. [2013, p. 16]).
Assuming perfectly elastic and frictionless particles, Fn is given by the Hertz theory of elastic contact [Hertz 1881; Johnson 1985, pp. 84-106]
Fn= 4 3E∗√
R∗δ3/2, (3.1)
whereE∗ and R∗ are the characteristic ’plane-strain’ modulus and radius of the relative
8
Chapter 3. Physical Background
curvature of the two particles given by [Johnson 1985, p. 89]
1
The Poisson’s ratio, ν, is a measure of the negative ratio of the transverse to axial strain of a body under application of uniaxial load. For a spherical particle, the Poisson’s ratio can be approximately determined by [Heikens et al. 1981]
∆Vp
Vp
= (1−2ν) δ
2R, (3.4)
where∆Vp/Vp is the change of material volume and δ/2R is the uniaxial strain.
During compression of a perfectly elastic particle the deformation extends to regions outside of the contact plane. The radius of the contact area, a, as depicted in Figure 3.1, therefore depends on the deformation and the contact pressure distribution within the particle and is given by [Johnson 1985, p. 93]
a=√
δR∗. (3.5)
According to Equation (3.1), the contact force does not depend linearly on the indenta-tion depth δ, although the bodies are considered as perfectly elastic. This nonlinearity results from the increase of the contact area with increasing compression, which by then increases the effective stiffness of the particle [Andreotti et al. 2013, p. 17].
Elastic-plastic Contact
The contact of elastic particles discussed in the previous Section (3.1.1) is generally re-garded as a reversible process and the resulting forces and deformations are independent of the history of loading [Johnson 1985, p. 179]. Real particles however, are not perfectly elastic, but exhibit some irresistibility during the cycle of loading and unloading. The elastic strain energy of a compressed particle can be calculated from the relationship
Chapter 3. Physical Background
between load and compression according to W =
the elastic strain energy (Equation (3.6)) gives
W = 2
The loss of energy during elastic-plastic contact leads to a lower repulsive force F0 < F during the unloading cycle (in the following indicated by primed quantities). The energy of elastic recovery is then given by
W0 = Z
F0dδ0 (3.10)
so that the dissipated energy during loading and unloading is the difference ∆W = W −W0.
Solid Friction
If the friction between the particles is considered, the contact force normally has a tangential component in addition to the normal component. Coulomb’s law of friction still remains the subject of active research [Baumberger and Caroli 2005; Andreotti et al.
2013, p. 17-19] and states the following:
• There is a minimum tangential force to make two bodies in contact slide against each other which is called the friction force and is defined byRs =µsFn. Here, µs
is the static friction coefficient of the two bodies in contact.
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Chapter 3. Physical Background
• When the bodies are sliding against each other the magnitude of the friction force is Rd = µdFn, where µd is the dynamic friction coefficient. The friction force is then exerted in direction opposite to the direction of sliding.
• The friction coefficients µs and µd are constant and depend on the materials and surfaces of the contacting bodies.
The existence of two different friction coefficients can be brought together by the phe-nomenon of "stick-slip" [Baumberger et al. 1994; Andreotti et al. 2013, pp. 20-22]. Thus, the two bodies initially stick together and the tangential force is below the static friction limit of Rs =µsFn. When the threshold is reached, the bodies slide against each other and the dynamic friction resistance isRd=µdFn, with µd< µs.
Rolling Friction
In addition to sliding friction there is also a resistance to rolling motion, i.e. when a sphere rolls on a solid surface. This resistance is called rolling friction [Coulomb 1785;
Johnson 1985; Andreotti et al. 2013].
In analogy to the Coulomb law of friction, the rolling friction of a particle on a rigid surface is defined as
Msr =µrsRnR, (3.11)
whereMsr is the minimal torque needed to put the particle into rolling motion,µrs is the static rolling friction coefficient,Rnis the normal reaction force andRthe particle radius.
Similarly, during rolling motion, the dynamic rolling friction is given byMdr =µrdRnR.
Fn
Rn
λr
Figure 3.2.: Rolling friction of a sphere at the contact surface. Due to an asymmetry of the contact between the particle and the surface, a normal reaction force Rn in a distance of λr can be identified causing a rolling resistance torqueMr=λrRn (modified from Andreotti et al. [2013, p. 24]).
Physically speaking, the rolling resistance originates from an asymmetry of the contact between the particle and the surface when a torque is applied to the particle as indicated
Chapter 3. Physical Background
in Figure 3.2. Before the rolling motion, the static rolling friction may arise from e.g.
adhesive forces between the two contacting surfaces. During rolling motion, the dynamic rolling resistance is caused by viscoelastic or plastic dissipation in the contact zone which creates a hysteretic behavior of compression and repulsion forces [Johnson 1985;
Andreotti et al. 2013, pp. 24-26]. In summary, all these mechanisms induce a shift forwards of the normal reaction Force Rn by a distance λr causing a rolling resistance torque ofMr =λrRn. Based on Equation (3.11), the coefficient of rolling frictionµr can be related to the shift λr by µr = λr/R. Compared to the sliding friction, the rolling friction has less impact on the motion of the particle since λr ∼a (the contact radius) and aR [Andreotti et al. 2013, pp. 24-26].
Collision of Two Particles
So far, the rate of loading on a particle was considered as sufficiently slow for the stresses to be in static equilibrium with the external loads during the time of the contact. During the impact of two particles, which is the case in granular flow, the rate of loading is high and dynamic effects may be important [Johnson 1985, p. 340].
Assume two particles i and j with masses mi and mj move toward each other with velocitiesviandvj, their centers approach each other by a displacement ofδdue to elastic deformation during the impact. The relative velocity is then given by V = vi −vj = dδ/dt. Applying Newton’s second law of motion, gives
m∗d2δ
where V0 is the initial relative velocity of the particles. The maximum compression δ∗
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Chapter 3. Physical Background
The elastic energy stored in the contact zone is in the order of W ∼ F δ∗. The typical collision time is then given by [Andreotti et al. 2013, p. 26-29; Johnson 1985, p. 351-355]
tc∼δ∗/V0 ∼
E∗/ρp is the characteristic speed of bulk elastic waves and ρp the density of the particles [Andreotti et al. 2013, p. 26-29]. Interestingly, the collision time is rather dependent on the material characteristic than on the impact as it only has a weak dependence on the impact velocity V. It is important to note that the above considerations are only valid if the static solution of the equation of elasticity according to Hertz’s law remains valid during the impact. This is the case if the region of elastic deformation belongs to the near field of elastic waves radiated from the contact zone [Johnson 1985, p. 351-360; Andreotti et al. 2013, p. 15-17]. This restriction can be formulated asλ a, where λ is the typical wave length of the radiated waves anda is the radius of the contact zone. Summing up that λ ∼ ctc and a ∼ √
If the two particles collide inelastically, the initial kinetic energy of the system is partially converted into elastic energy but also dissipated, e.g. due to plastic deformation of the material. The application of the Hertz quasi-static approach which is based on the restriction that the particle impact velocity is small compared to the elastic wave speed (Equation (3.18)) also holds during inelastic impact. The plastic deformation of the material reduces the intensity of the impact pressure pulse and thereby diminishes the energy converted into elastic waves [Johnson 1985, p. 361].
Up to the instant maximum compression (∂δ/∂t = 0) the initial kinetic energy is converted into elastic and plastic local deformation of the particles. Hence, the velocity of the particle after the reboundv0 is always smaller than the initial velocityv according
Chapter 3. Physical Background
to
v0 =−ev, (3.19)
where 0 ≤ e ≤ 1 is the coefficient of restitution. The causes of energy dissipation in this case can be manifold, e.g. plastic deformation, viscoelasticity loss, local heating or cracking of the material. The coefficient of restitution, as a measure of energy loss during collision, depends on the size of the particles as well as on the impact velocity, but in many applicationse can be considered as constant [Andreotti et al. 2013, p. 26-29].
Again, based on the initial kinetic energy of the system, e can be calculated by a simple energy balance. Hence, the transformation of the kinetic energy of the system into local particle deformation is given by Equation (3.9) to
1 elastic repulsion, the calculation is similar, but the repulsive force F0 at the onset of rebound is unknown due to energy dissipation, so that
1
2m∗V02 =W0 = Z δ∗
0
F0dδ0. (3.21)
The coefficient of restitution then is the square root of the ratio of the repulsive and initial energy of the particles
e= rW0
W . (3.22)
During almost elastic and viscoelastic contact, where the difference between compression and repulsion is small, the Hertzian compliance relationF(δ)(Equation (3.1)) holds and e can be calculated using the equations above. For large plastic deformations however this relation is no longer valid. Relations of F(δ) covering those deformation regimes are available in literature [Johnson 1985, p. 361-369].