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implement tangential force history in mindlin/force and mindlin_rescale/force

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Jibril B. Coulibaly
2020-08-10 10:53:30 -05:00
parent a6cd4a935e
commit bf724332d4
2 changed files with 158 additions and 59 deletions
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doc/src/pair_granular.rst
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@ -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`
5. *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
@ -272,7 +274,7 @@ 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:
@ -294,7 +296,7 @@ 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\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}_{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}}\|` .
@ -314,22 +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,
or accumulated tangential force (i.e. contact history). This is discussed
in details below for accumulated tangential displacement in the context
of the *linear_history* option. The same treatment of the accumulated
displacement, or accumulated force, applies to the other options as well.
The remaining tangential options all use accumulated tangential
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:
@ -390,27 +394,18 @@ overlap region) to induce a torque on each particle according to:
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 stiffness depends
on the radius of the contact region and the force must therefore be computed
incrementally. The accumulated tangential force characterizes the contact history.
The increment of the elastic component of the tangential force :math:`\mathbf{F}_{te}`
is given by:
of :math:`a`, the radius of the contact region. The tangential force is given by:
.. math::
\mathrm{d}\mathbf{F}_{te} = -k_t a \mathbf{v}_{t,rel} \mathrm{d}\tau
The tangential force is given by:
\mathbf{F}_t = -\min(\mu_t F_{n0}, \|-k_t a \mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
.. math::
\mathbf{F}_t = -min(\mu_t F_{n0}, \|\mathbf{F}_{te} + \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 = 8G_{eff}`, where :math:`G` is the effective shear modulus given by:
:math:`k_t = 8G_{eff}`, where :math:`G_{eff}` is the effective shear modulus given by:
.. math::
@ -424,23 +419,80 @@ normal contact model that specifies material parameters :math:`E` and
case, mixing of the shear modulus for different particle types *i* and
*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::
\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 force is re-scaled as the contact
magnitude of the tangential displacement is re-scaled as the contact
unloads, i.e. if :math:`a < a_{t_{n-1}}`:
.. math::
\mathbf{\xi} = \mathbf{\xi_{t_{n-1}}} \frac{a}{a_{t_{n-1}}}
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>` ).
.. 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}}}
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.
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.
----------
@ -477,7 +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
@ -529,7 +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
@ -794,3 +846,15 @@ 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.