148
D. Markauskas and H. Kruggel-Emden
and the effective mass m e f f = m i m j /
m i + m j
.
2.3.2 Tangential Force Model
In addition to the normal force, a force component in the tangential direction for
modeling of occurring friction effects is taken into account in order to prevent particles from sliding apart in a particle bed. Also for the tangential force, different
models are applicable [33]. Similar to the normal force, the most common tangential
models are linear (see e.g. [34–36]). Here, the tangential forces are calculated by
applying a linear spring limited by the Coulomb condition and calculated as
F
t
i j = −min
F
t
spring ,
F
t
coul
= −min
k
t
ξ i j
, μ C
F
n
i j
t i j ,
(7)
where k
t is the tangential stiffness of a linear spring, μ C is the friction coefficient,
ξ i j is the relative tangential displacement and
t i j is the tangential unit vector [33].
For the contact of two spheres k and l of two multi-sphere particles i and j (comp.
Fig. 4b), only the indices are changed in Eq. (7). The tangential spring stiffness k
t is
obtained as
k
t
= κm e f f
π/t
n
2 ,
(8)
where κ is given through the mechanical properties as
κ =
(1 − ν i )/G i +
1 − ν j
/G j
/
(1 − 0.5ν i )/G i +
1 − 0.5ν j
/G j
, (9)
where ν is the Poisson’s ratio and G = E/(2 + 2ν) is the shear modulus of the two
interacting materials of particles i and j depending on Young’s modulus E and again
Poisson’s ratio ν [37].
2.4 Liquid Bridge Forces
For slightly wet or moist particles liquid bridge forces have to be considered [38].
Several researchers have proposed expressions for the determination of capillary
forces (see e.g. [39–41]), viscous forces (see e.g. [42–45]), as well as for the formation, the shape, the liquid volume and the redistribution of liquid due to the rapture
of a liquid bridge [46, 47]. In addition, some expressions in closed form have been
proposed for the accurate calculation of liquid bridges [48, 49] and the derived models have to some extend been used in DEM simulations (see e.g. [50–54]). However,
all these models are limited to the calculation of forces in pendular states, while only
a few researchers also investigated the condition of the funicular state [55–57].
D. Markauskas and H. Kruggel-Emden
and the effective mass m e f f = m i m j /
m i + m j
.
2.3.2 Tangential Force Model
In addition to the normal force, a force component in the tangential direction for
modeling of occurring friction effects is taken into account in order to prevent particles from sliding apart in a particle bed. Also for the tangential force, different
models are applicable [33]. Similar to the normal force, the most common tangential
models are linear (see e.g. [34–36]). Here, the tangential forces are calculated by
applying a linear spring limited by the Coulomb condition and calculated as
F
t
i j = −min
F
t
spring ,
F
t
coul
= −min
k
t
ξ i j
, μ C
F
n
i j
t i j ,
(7)
where k
t is the tangential stiffness of a linear spring, μ C is the friction coefficient,
ξ i j is the relative tangential displacement and
t i j is the tangential unit vector [33].
For the contact of two spheres k and l of two multi-sphere particles i and j (comp.
Fig. 4b), only the indices are changed in Eq. (7). The tangential spring stiffness k
t is
obtained as
k
t
= κm e f f
π/t
n
2 ,
(8)
where κ is given through the mechanical properties as
κ =
(1 − ν i )/G i +
1 − ν j
/G j
/
(1 − 0.5ν i )/G i +
1 − 0.5ν j
/G j
, (9)
where ν is the Poisson’s ratio and G = E/(2 + 2ν) is the shear modulus of the two
interacting materials of particles i and j depending on Young’s modulus E and again
Poisson’s ratio ν [37].
2.4 Liquid Bridge Forces
For slightly wet or moist particles liquid bridge forces have to be considered [38].
Several researchers have proposed expressions for the determination of capillary
forces (see e.g. [39–41]), viscous forces (see e.g. [42–45]), as well as for the formation, the shape, the liquid volume and the redistribution of liquid due to the rapture
of a liquid bridge [46, 47]. In addition, some expressions in closed form have been
proposed for the accurate calculation of liquid bridges [48, 49] and the derived models have to some extend been used in DEM simulations (see e.g. [50–54]). However,
all these models are limited to the calculation of forces in pendular states, while only
a few researchers also investigated the condition of the funicular state [55–57].