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Merge pull request #2196 from jibril-b-coulibaly/mindlin_rescale

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Implement force history in Mindlin granular pair styles
This commit is contained in:
Axel Kohlmeyer
2020-08-26 11:42:06 -04:00
committed by GitHub
parent e2fd95d8ed 921b6d8135
commit 4ad68c98a0
3 changed files with 234 additions and 73 deletions
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143
doc/src/pair_granular.rst
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@ -93,7 +93,7 @@ on particle *i* due to contact with particle *j* is given by:
.. math::
\mathbf{F}_{ne, Hooke} = k_N \delta_{ij} \mathbf{n}
\mathbf{F}_{ne, Hooke} = k_n \delta_{ij} \mathbf{n}
Where :math:`\delta_{ij} = R_i + R_j - \|\mathbf{r}_{ij}\|` is the particle
overlap, :math:`R_i, R_j` are the particle radii, :math:`\mathbf{r}_{ij} = \mathbf{r}_i - \mathbf{r}_j` is the vector separating the two
@ -106,7 +106,7 @@ For the *hertz* model, the normal component of force is given by:
.. math::
\mathbf{F}_{ne, Hertz} = k_N R_{eff}^{1/2}\delta_{ij}^{3/2} \mathbf{n}
\mathbf{F}_{ne, Hertz} = k_n R_{eff}^{1/2}\delta_{ij}^{3/2} \mathbf{n}
Here, :math:`R_{eff} = \frac{R_i R_j}{R_i + R_j}` is the effective
radius, denoted for simplicity as *R* from here on. For *hertz*\ , the
@ -123,7 +123,7 @@ Here, :math:`E_{eff} = E = \left(\frac{1-\nu_i^2}{E_i} + \frac{1-\nu_j^2}{E_j}\r
modulus, with :math:`\nu_i, \nu_j` the Poisson ratios of the particles of
types *i* and *j*\ . Note that if the elastic modulus and the shear
modulus of the two particles are the same, the *hertz/material* model
is equivalent to the *hertz* model with :math:`k_N = 4/3 E_{eff}`
is equivalent to the *hertz* model with :math:`k_n = 4/3 E_{eff}`
The *dmt* model corresponds to the
:ref:`(Derjaguin-Muller-Toporov) <DMT1975>` cohesive model, where the force
@ -140,7 +140,7 @@ where the force is computed as:
\mathbf{F}_{ne, jkr} = \left(\frac{4Ea^3}{3R} - 2\pi a^2\sqrt{\frac{4\gamma E}{\pi a}}\right)\mathbf{n}
Here, *a* is the radius of the contact zone, related to the overlap
Here, :math:`a` is the radius of the contact zone, related to the overlap
:math:`\delta` according to:
.. math::
@ -167,7 +167,7 @@ following general form:
\mathbf{F}_{n,damp} = -\eta_n \mathbf{v}_{n,rel}
Here, :math:`\mathbf{v}_{n,rel} = (\mathbf{v}_j - \mathbf{v}_i) \cdot \mathbf{n} \mathbf{n}` is the component of relative velocity along
Here, :math:`\mathbf{v}_{n,rel} = (\mathbf{v}_j - \mathbf{v}_i) \cdot \mathbf{n}\ \mathbf{n}` is the component of relative velocity along
:math:`\mathbf{n}`.
The optional *damping* keyword to the *pair_coeff* command followed by
@ -259,7 +259,9 @@ tangential model choices and their expected parameters are as follows:
1. *linear_nohistory* : :math:`x_{\gamma,t}`, :math:`\mu_s`
2. *linear_history* : :math:`k_t`, :math:`x_{\gamma,t}`, :math:`\mu_s`
3. *mindlin* : :math:`k_t` or NULL, :math:`x_{\gamma,t}`, :math:`\mu_s`
4. *mindlin_rescale* : :math:`k_t` or NULL, :math:`x_{\gamma,t}`, :math:`\mu_s`
4. *mindlin/force* : :math:`k_t` or NULL, :math:`x_{\gamma,t}`, :math:`\mu_s`
5. *mindlin_rescale* : :math:`k_t` or NULL, :math:`x_{\gamma,t}`, :math:`\mu_s`
6. *mindlin_rescale/force* : :math:`k_t` or NULL, :math:`x_{\gamma,t}`, :math:`\mu_s`
Here, :math:`x_{\gamma,t}` is a dimensionless multiplier for the normal
damping :math:`\eta_n` that determines the magnitude of the tangential
@ -268,11 +270,11 @@ coefficient, and :math:`k_t` is the tangential stiffness coefficient.
For *tangential linear_nohistory*, a simple velocity-dependent Coulomb
friction criterion is used, which mimics the behavior of the *pair
gran/hooke* style. The tangential force (\mathbf{F}_t\) is given by:
gran/hooke* style. The tangential force :math:`\mathbf{F}_t` is given by:
.. math::
\mathbf{F}_t = -min(\mu_t F_{n0}, \|\mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
\mathbf{F}_t = -\min(\mu_t F_{n0}, \|\mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
The tangential damping force :math:`\mathbf{F}_\mathrm{t,damp}` is given by:
@ -294,8 +296,8 @@ keyword also affects the tangential damping. The parameter
literature use :math:`x_{\gamma,t} = 1` (:ref:`Marshall <Marshall2009>`,
:ref:`Tsuji et al <Tsuji1992>`, :ref:`Silbert et al <Silbert2001>`). The relative
tangential velocity at the point of contact is given by
:math:`\mathbf{v}_{t, rel} = \mathbf{v}_{t} - (R_i\Omega_i + R_j\Omega_j) \times \mathbf{n}`, where :math:`\mathbf{v}_{t} = \mathbf{v}_r - \mathbf{v}_r\cdot\mathbf{n}{n}`,
:math:`\mathbf{v}_r = \mathbf{v}_j - \mathbf{v}_i`.
:math:`\mathbf{v}_{t, rel} = \mathbf{v}_{t} - (R_i\mathbf{\Omega}_i + R_j\mathbf{\Omega}_j) \times \mathbf{n}`, where :math:`\mathbf{v}_{t} = \mathbf{v}_r - \mathbf{v}_r\cdot\mathbf{n}\ \mathbf{n}`,
:math:`\mathbf{v}_r = \mathbf{v}_j - \mathbf{v}_i` .
The direction of the applied force is :math:`\mathbf{t} = \mathbf{v_{t,rel}}/\|\mathbf{v_{t,rel}}\|` .
The normal force value :math:`F_{n0}` used to compute the critical force
@ -314,21 +316,24 @@ form:
.. math::
F_{n0} = \|\mathbf{F}_ne + 2 F_{pulloff}\|
F_{n0} = \|\mathbf{F}_{ne} + 2 F_{pulloff}\|
Where :math:`F_{pulloff} = 3\pi \gamma R` for *jkr*\ , and
:math:`F_{pulloff} = 4\pi \gamma R` for *dmt*\ .
The remaining tangential options all use accumulated tangential
displacement (i.e. contact history). This is discussed below in the
context of the *linear_history* option, but the same treatment of the
accumulated displacement applies to the other options as well.
displacement (i.e. contact history), except for the options
*mindlin/force* and *mindlin_rescale/force*, that use accumulated
tangential force instead, and are discussed further below.
The accumulated tangential displacement is discussed in details below
in the context of the *linear_history* option. The same treatment of
the accumulated displacement applies to the other options as well.
For *tangential linear_history*, the tangential force is given by:
.. math::
\mathbf{F}_t = -min(\mu_t F_{n0}, \|-k_t\mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
\mathbf{F}_t = -\min(\mu_t F_{n0}, \|-k_t\mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
Here, :math:`\mathbf{\xi}` is the tangential displacement accumulated
during the entire duration of the contact:
@ -356,7 +361,7 @@ work:
.. math::
\mathbf{\xi} = \left(\mathbf{\xi'} - (\mathbf{n} \cdot \mathbf{\xi'})\mathbf{n}\right) \frac{\|\mathbf{\xi'}\|}{\|\mathbf{\xi'}\| - \mathbf{n}\cdot\mathbf{\xi'}}
\mathbf{\xi} = \left(\mathbf{\xi'} - (\mathbf{n} \cdot \mathbf{\xi'})\mathbf{n}\right) \frac{\|\mathbf{\xi'}\|}{\|\mathbf{\xi'} - (\mathbf{n}\cdot\mathbf{\xi'})\mathbf{n}\|}
Here, :math:`\mathbf{\xi'}` is the accumulated displacement prior to the
current time step and :math:`\mathbf{\xi}` is the corrected
@ -372,7 +377,7 @@ discussion):
.. math::
\mathbf{\xi} = -\frac{1}{k_t}\left(\mu_t F_{n0}\mathbf{t} + \mathbf{F}_{t,damp}\right)
\mathbf{\xi} = -\frac{1}{k_t}\left(\mu_t F_{n0}\mathbf{t} - \mathbf{F}_{t,damp}\right)
The tangential force is added to the total normal force (elastic plus
damping) to produce the total force on the particle. The tangential
@ -387,27 +392,68 @@ overlap region) to induce a torque on each particle according to:
\mathbf{\tau}_j = -(R_j - 0.5 \delta) \mathbf{n} \times \mathbf{F}_t
For *tangential mindlin*\ , the :ref:`Mindlin <Mindlin1949>` no-slip solution is used, which differs from the *linear_history*
option by an additional factor of *a*\ , the radius of the contact region. The tangential force is given by:
For *tangential mindlin*\ , the :ref:`Mindlin <Mindlin1949>` no-slip solution
is used which differs from the *linear_history* option by an additional factor
of :math:`a`, the radius of the contact region. The tangential force is given by:
.. math::
\mathbf{F}_t = -min(\mu_t F_{n0}, \|-k_t a \mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
\mathbf{F}_t = -\min(\mu_t F_{n0}, \|-k_t a \mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
Here, *a* is the radius of the contact region, given by :math:`a =\sqrt{R\delta}`
Here, :math:`a` is the radius of the contact region, given by :math:`a =\sqrt{R\delta}`
for all normal contact models, except for *jkr*\ , where it is given
implicitly by :math:`\delta = a^2/R - 2\sqrt{\pi \gamma a/E}`, see
discussion above. To match the Mindlin solution, one should set :math:`k_t = 4G/(2-\nu)`, where :math:`G` is the shear modulus, related to Young's modulus
:math:`E` by :math:`G = E/(2(1+\nu))`, where :math:`\nu` is Poisson's ratio. This
can also be achieved by specifying *NULL* for :math:`k_t`, in which case a
discussion above. To match the Mindlin solution, one should set
:math:`k_t = 8G_{eff}`, where :math:`G_{eff}` is the effective shear modulus given by:
.. math::
G_{eff} = \left(\frac{2-\nu_i}{G_i} + \frac{2-\nu_j}{G_j}\right)^{-1}
where :math:`G` is the shear modulus, related to Young's modulus :math:`E`
and Poisson's ratio :math:`\nu` by :math:`G = E/(2(1+\nu))`. This can also be
achieved by specifying *NULL* for :math:`k_t`, in which case a
normal contact model that specifies material parameters :math:`E` and
:math:`\nu` is required (e.g. *hertz/material*\ , *dmt* or *jkr*\ ). In this
case, mixing of the shear modulus for different particle types *i* and
*j* is done according to:
*j* is done according to the formula above.
.. note::
The radius of the contact region :math:`a` depends on the normal overlap.
As a result, the tangential force for *mindlin* can change due to
a variation in normal overlap, even with no change in tangential displacement.
For *tangential mindlin/force*, the accumulated elastic tangential force
characterizes the contact history, instead of the accumulated tangential
displacement. This prevents the dependence of the tangential force on the
normal overlap as noted above. The tangential force is given by:
.. math::
1/G = 2(2-\nu_i)(1+\nu_i)/E_i + 2(2-\nu_j)(1+\nu_j)/E_j
\mathbf{F}_t = -\min(\mu_t F_{n0}, \|\mathbf{F}_{te} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
The increment of the elastic component of the tangential force
:math:`\mathbf{F}_{te}` is given by:
.. math::
\mathrm{d}\mathbf{F}_{te} = -k_t a \mathbf{v}_{t,rel} \mathrm{d}\tau
The changes in frame of reference of the contacting pair of particles during
contact are accounted for by the same formula as above, replacing the
accumulated tangential displacement :math:`\xi`, by the accumulated tangential
elastic force :math:`F_{te}`. When the tangential force exceeds the critical
force, the tangential force is directly re-scaled to match the value for
the critical force:
.. math::
\mathbf{F}_{te} = - \mu_t F_{n0}\mathbf{t} + \mathbf{F}_{t,damp}
The same rules as those described for *mindlin* apply regarding the tangential
stiffness and mixing of the shear modulus for different particle types.
The *mindlin_rescale* option uses the same form as *mindlin*\ , but the
magnitude of the tangential displacement is re-scaled as the contact
@ -421,9 +467,32 @@ Here, :math:`t_{n-1}` indicates the value at the previous time
step. This rescaling accounts for the fact that a decrease in the
contact area upon unloading leads to the contact being unable to
support the previous tangential loading, and spurious energy is
created without the rescaling above (:ref:`Walton <WaltonPC>` ). See also
discussion in :ref:`Thornton et al, 2013 <Thornton2013>` , particularly
equation 18(b) of that work and associated discussion.
created without the rescaling above (:ref:`Walton <WaltonPC>` ).
.. note::
For *mindlin*, a decrease in the tangential force already occurs as the
contact unloads, due to the dependence of the tangential force on the normal
force described above. By re-scaling :math:`\xi`, *mindlin_rescale*
effectively re-scales the tangential force twice, i.e., proportionally to
:math:`a^2`. This peculiar behavior results from use of the accumulated
tangential displacement to characterize the contact history. Although
*mindlin_rescale* remains available for historic reasons and backward
compatibility purposes, it should be avoided in favor of *mindlin_rescale/force*.
The *mindlin_rescale/force* option uses the same form as *mindlin/force*,
but the magnitude of the tangential elastic force is re-scaled as the contact
unloads, i.e. if :math:`a < a_{t_{n-1}}`:
.. math::
\mathbf{F}_{te} = \mathbf{F}_{te, t_{n-1}} \frac{a}{a_{t_{n-1}}}
This approach provides a better approximation of the :ref:`Mindlin-Deresiewicz <Mindlin1953>`
laws and is more consistent than *mindlin_rescale*. See discussions in
:ref:`Thornton et al, 2013 <Thornton2013>`, particularly equation 18(b) of that
work and associated discussion, and :ref:`Agnolin and Roux, 2007 <AgnolinRoux2007>`,
particularly Appendix A.
----------
@ -460,7 +529,7 @@ exceeds a critical value:
.. math::
\mathbf{F}_{roll} = min(\mu_{roll} F_{n,0}, \|\mathbf{F}_{roll,0}\|)\mathbf{k}
\mathbf{F}_{roll} = \min(\mu_{roll} F_{n,0}, \|\mathbf{F}_{roll,0}\|)\mathbf{k}
Here, :math:`\mathbf{k} = \mathbf{v}_{roll}/\|\mathbf{v}_{roll}\|` is the direction of
the pseudo-force. As with tangential displacement, the rolling
@ -512,7 +581,7 @@ is then truncated according to:
.. math::
\tau_{twist} = min(\mu_{twist} F_{n,0}, \tau_{twist,0})
\tau_{twist} = \min(\mu_{twist} F_{n,0}, \tau_{twist,0})
Similar to the sliding and rolling displacement, the angular
displacement is rescaled so that it corresponds to the critical value
@ -763,3 +832,15 @@ Technology, 233, 30-46.
.. _WaltonPC:
**(Otis R. Walton)** Walton, O.R., Personal Communication
.. _Mindlin1953:
**(Mindlin and Deresiewicz, 1953)** Mindlin, R.D., & Deresiewicz, H (1953).
Elastic Spheres in Contact under Varying Oblique Force.
J. Appl. Mech., ASME 20, 327-344.
.. _AgnolinRoux2007:
**(Agnolin and Roux 2007)** Agnolin, I. & Roux, J-N. (2007).
Internal states of model isotropic granular packings.
I. Assembling process, geometry, and contact networks. Phys. Rev. E, 76, 061302.