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[[Файл:Flag_MD.jpg|250px|thumb|right| "Flag of Discrete Mechanics" (plate buckling under the action of shear force)]]  
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[[Кафедра "Теоретическая механика"|Department "Theoretical Mechanics"]] > [[Администрация кафедры "Теоретическая механика"| Adminstration ]] > [[Vitaly Kuzkin|V.A. Kuzkin ]] > '''V-model''' <HR>
  
== Introduction ==
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{{DISPLAYTITLE:<span style="display:none">{{FULLPAGENAME}}</span>}}
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<font size="5"> A model for elastic bonds in solids (EVM) </font>
  
The article is based on the paper '''V.A. Kuzkin and I.E. Asonov "[http://pre.aps.org/abstract/PRE/v86/i5/e051301 Vector-based model of elastic bonds for simulation of granullar solids]"// Phys. Rev. E, 86, 051301, 2012'''. ArXive version can be downloaded [[Медиа: Kuzkin_2012_arXive_V-model.pdf‎ |here]].
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[[Файл: Fig1 compos.png|450px|thumb|right| SEM image of composite consisting
 +
of nanoparticles bonded by a copolymer. (N. Preda, et. al., Mater. Res. Bull.
 +
45, 1008 (2010))]]  
  
 +
== Introduction  ==
  
Discrete Element Method (DEM) is widely used for computer simulation of granular materials both solid and free-flowing. In powders and other free-flowing media interactions between particles usually include contact forces, dry and viscous friction, cohesion, electrostatic forces etc. For simulation of solids particles are additionally connected by so-called bonds <ref name="BPM"/>, <ref name="Wang"/>. In general case bond transmits both forces and moments acting between particles. They are responsible for stability, elasticity, strength and other intrinsic  properties that distinguish solids from free-flowing materials. The bonds may have different physical meaning. On the one hand they can specify the law of interaction between different parts of one material represented by the particles. On the other hand  bonds can be considered as a model of some additional glue-like or cement-like material, connecting particles.  
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The article is based on the papers:
 +
* V.A. Kuzkin and I.E. Asonov '''"[http://pre.aps.org/abstract/PRE/v86/i5/e051301 Vector-based model of elastic bonds for simulation of granullar solids]"'''// Phys. Rev. E, 86, 051301, 2012 ([[Медиа: Kuzkin_2012_PRE_proof2.pdf‎ |'''download author's version, pdf''']]).
 +
* Kuzkin V.A., Krivtsov A.M. '''Enhanced vector-based model for elastic bonds in solids''' // Letters on materials 7 (4), 2017, pp. 455-458  [http://arxiv.org/abs/1507.06957 (download ArXiv version)]
  
In practice models allowing to represent forces and moments, acting between particles, as a function of  characteristics of particle motion (linear and angular velocities, rotational matrixes, quaternions, etc.) are required. According to the review presented in paper <ref name="Wang"/> only several models presented in literature describe all possible kinds of deformation of the bond. Bonded-particle model (BPM) was proposed in paper <ref name="BPM"/> for simulation of rocks. The BPM model is widely used in literature for simulation of deformation and fracture of solids in both two and three dimensions. Several drawbacks of BPM model, in particular, in the case of coexistence of bending and torsion of the bond, are discussed in paper <ref name="Wang"/>. It is noted that the main reason for the drawbacks is incremental algorithm used in the framework of BPM model.  Another approach based on decomposition of relative rotation of particles is proposed in paper <ref name="Wang"/>. Forces and moments are represented as functions of angles describing relative turn of the particles.
 
It was shown that method <ref name="Wang"/> is more accurate than incremental procedure of BPM model. However in the framework of model <ref name="Wang"/> potential energy of the bond and its relation to forces and torques are not considered. Though the expression for  potential energy is not required for DEM simulations it is still very important. It is required for control of energy conservation and construction of nonlinear elastic force laws. The approach proposed in paper <ref name="Wang"/> does not guarantee that the forces and moments caused by the bond are conservative. Note that any model for an elastic bond should be perfectly conservative.
 
  
Also let us note the model  recently development by [http://www.dem-solutions.com/academic/edem-academic-workshop-proceedings.php DEM-sloutions Ltd.]. It was proposed to use Timoshenko beam as a model of the bond connecting particle. Though the model has clear physical meaning it is also not the panacea. In particular, it is not straightforward how to implement this model in the case of finite rotations of the particles.  
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The Discrete (or Distinct) Element Method (DEM) is widely used for the computer simulation of solid and free-flowing granular materials. Similarly to classical molecular dynamics, in the framework of DEM the material is represented by a set of many interacting rigid body particles (granules). The equations of the particles' motion are integrated numerically. In free-flowing materials the  particles interact via contact forces, dry and viscous friction forces,  electrostatic forces, etc. Computer simulation of deformation and fracture of granular solids, such as rocks, concrete, ceramics, particle compounds, agglomerates, nanocomposites, etc. is even more challenging. Particles in granular solids are usually connected together by some additional bonding material such as cement or glue. The example of composite material consisting of PbS nanoparticles bonded together by a copolymer is shown in figure above. The copolymer (bonding material) resists the relative translation and rotation of neighboring PbS particles. In DEM simulations bonding material is usually taken into account implicitly using the concept of so-called bonds. Neighboring particles are connected by the bonds that resist to stretching/compression, shear, bending, and torsion. The bonds cause forces and torques acting on the particles along with contact forces.  Mass of the bonding material is usually neglected. The assumption does not influence static properties of the granular material. The influence on the dynamic properties is not so straightforward and should be considered separately. However let us note that in many practical applications the mass of bonding material is much smaller than the mass of the particles (see, for example, figures below). Therefore the mass of bonding
 +
material can be neglected.
  
There are also some other physical drawbacks of the existing bond models that will be highlighted in the paper.  
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{|align="center"
Let us summarize them:
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|-valign="top"
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|[[Файл: Discrete rod exp.png|200px|thumb|right| '''Discrete Rod''': SEM image of 1D assembly of particles embedded in polymer matrix (M. Wang, et.al., Materials Today, Vol. 16, No. 4, 2013)]]
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|[[Файл: Discrete shell exp.png|470px|thumb|right|'''Discrete Shell''': Colloidsome composed of polysterene spheres (A.D. Dismore, et.al. Science, 2002)]] 
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|[[Файл: Aerogel exp.jpg|330px|thumb|right| '''Discrete Solid''': aerogel ([http://www.grc.nasa.gov/WWW/RT/2006/RX/RX19D-meador.html source])]]
 +
|}
  
* energy conservation is not guaranteed ('''[[BPM]], Wang's model''')
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In the paper V.A. Kuzkin and I.E. Asonov "Vector-based model of elastic bonds for simulation of granullar solids"// Phys. Rev. E, 86, 051301, 2012  vector-based model(the V-model) of elastic bonds in solids is developed. The V-model is based on the combination of approaches proposed in works of [[П.А. Жилин | P.A. Zhilin]], E.A. Ivanova, [[А.М. Кривцов | A.M. Krivtsov]], N.F. Morozov <ref name="IvKrMoFi_2003"/>, <ref name="IvKrMo_2007"/> and works of M.P. Allen <ref name="Allen"/>, S.L Price. Equations describing interactions between two rigid bodies in the general case are summarized. The general expression for the potential energy of the bond is represented  via vectors rigidly connected with bonded particles (see figures below).
* stiffnesses of the bond are dependent. BPM contains two independent stiffnesses instead of four ('''[[BPM]]''')
 
* bond connects centers of the particles. In contrast, in reality particles are usually glued by their surfaces. The difference can be crucial, especially for a short bonds ('''[[BPM]], Wang's model, Timoshenko beam''')
 
* large rotations of the particles as well as rigid body rotations of all specimen can not be considered ('''Timoshenko beam''')
 
* low accuracy in the case of short bonds ('''Timoshenko beam''')  
 
* generalization for the case of large nonlinear elastic deformations of the bond is not straightforward. The generalization can be crucial for polymer (for example, rubber) bonds ('''[[BPM]], Wang's model, Timoshenko beam''')
 
* the dependence of forces and moments on orientation of the particles with respect to the bond is not taken into account. Thus if particles rotate with equal angular velocities and there is no relative translation, then forces and moments are equal to zero('''Wang's model''')
 
 
 
== V-model ==
 
 
 
The V-model is based on the combination of approaches proposed in works of [[П.А. Жилин | P.A. Zhilin]], E.A. Ivanova, [[А.М. Кривцов | A.M. Krivtsov]], N.F. Morozov <ref name="IvKrMoFi_2003"/>, <ref name="IvKrMo_2007"/> and works of M.P. Allen <ref name="Allen"/>, S.L Price. The idea of the model is to introduce potential energy of interactions between two bonded particles as a function of particles orientations. The potential energy describes all possible kinds of deformation of the bond: tension/compression, shear, bending and torsion.
 
 
 
Detailed description of the model and procedures for parameters calibration are given in [http://arxiv.org/abs/1202.0001 the paper]. Here let us show some results obtained using V-model.
 
  
 
{|align="center"
 
{|align="center"
 
  |-valign="top"
 
  |-valign="top"
  |[[Файл:Rod instability.jpg|300px|thumb|right| Rod under the action of following shear force, acting on the rod's end]]  
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  |[[Файл:Fig2_single_bond.png|350px|thumb|center| Vectors connected with the particles]]  
  |[[Файл:Plate instability.jpg|200px|thumb|right| Plate under the action of following compressive forces, acting along the edge of the plate]]
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  |[[Файл: Fig3_bond_def.png|600px|thumb|center|Different deformations of the bond.]]
|[[Файл:3D rod torsion.jpg|330px|thumb|right| Torsion of 3D rod-like body]]
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|}
|}
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 +
The vectors are used for description of different types of bond's deformation. The expression for potential energy corresponding to tension/compression, shear, bending, and torsion of the bond is proposed. Forces and torques acting between particles are derived from the potential energy.
 +
Two approaches for calibration of V-model parameters for bonds with different length/thickness ratios are presented. Simple analytical formulas connecting geometrical and elastic characteristics of the bond with parameters of the V-model are derived. Main aspects of numerical implementation of the model are discussed.
  
== Advantages of V-model ==
+
== Advantages of the V-model ==
  
 
Let us summarize advantages of V-model:
 
Let us summarize advantages of V-model:
  
 +
* longitudinal, shear, bending, and torsional stiffnesses of the bond are independent
 
* applicable in the case of large turns of the particles
 
* applicable in the case of large turns of the particles
 
* conservation of energy (bonds are perfectly elastic)
 
* conservation of energy (bonds are perfectly elastic)
* the bond has longitudinal, shear, bending and torsional stiffnesses that can be set independently
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* any non close packed structure, rods and shells can be simulated(see figures above)   
* any non close packed structure, rods and shells can be modeled (see figures above)   
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* forces and torques are calculated as a functions of particles' positions and orientations (more accurate integration of motion equations than in the case of incremental algorithm used for BPM)  
* forces and moments are calculated as a functions of particles' positions and orientations  
 
(more accurate integration of motion equations than in the case of incremental algorithm used for BPM)  
 
 
* description of bonds of any length/thickness ratio. It is shown that in the case of small deformations the behavior of the bond can fit the behavior of Bernulli-Euler rod or Timosheko rod or short cylinder connecting particles (depending on length/thickness ratio)   
 
* description of bonds of any length/thickness ratio. It is shown that in the case of small deformations the behavior of the bond can fit the behavior of Bernulli-Euler rod or Timosheko rod or short cylinder connecting particles (depending on length/thickness ratio)   
 
* simple analytical expressions connecting parameters of V-model with geometrical and mechanical characteristics of the bond  
 
* simple analytical expressions connecting parameters of V-model with geometrical and mechanical characteristics of the bond  
 
* bonds can connect points inside the particles or lying on particles surfaces (not only particle centers)
 
* bonds can connect points inside the particles or lying on particles surfaces (not only particle centers)
* the model can be easily generalized in order to describe nonlinear elastic behavior of the bond  
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* fracture criterion for the bond can be used if required
* fracture criterion used in BPM can be used in couple with V-model
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 +
== Java Script example ==
 +
 
 +
Two dimmensional example: deformation of a discrete rod.
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Author: [[Лапин Руслан|Ruslan Lapin]]
 +
 
 +
Pull the last particle using mouse.
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 +
{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Lapin/v_model.html |width=1200 |height=550 |border=0 }}
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 +
== Some examples ==
 +
'''Press the links under the picture to see the animation.'''
 +
<gallery>
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Файл:Rod instability.jpg | [[Media: Buckling_rod_following_force.gif | Buckling of the discrete rod under the action of following force]]
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Файл: instability under torsion.bmp | [[Media: Instability under torsion.gif | Instability of a discrete rod under torsion ]]
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Файл: Nikolai.jpg  | [[Media: Nikolai paradox.gif | Rod under the action of following torque (Nikolai paradox)]]
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Файл: Torsion .bmp | [[Media: wire_torsion.gif | Torsion of the wire]]
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Файл: Plate instability.bmp | [[Media:  MD flag.gif | Discrete plate under the action of following shear force]]
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Файл: Resid stress.bmp | [[Media: Residualstresses_in_plate.gif | Residual stresses in a discrete plate]]
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Файл: Semisphere_buckling.png | [[Media:Buckling_discrete_shell_light.gif | Buckling of the discrete hemisphere]]
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Файл: Ball_V.bmp | [[Media: Shell_between_walls.gif | Discrete spherical shell bouncing between rigid walls]]
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Файл: Ball vs wall.bmp | [[Media: Shell impact.gif | Discrete spherical shell impacting rigid walls]]
 +
</gallery>
 +
 
 +
== Debugging ==
 +
 
 +
The simplest way of debugging is a solution of the following four test problems for the system of 2 bonded particles. In all four cases forces and torques acting on the particles can be calculated both analytically and numerically. Obviously, the results should coincide.
 +
 
 +
* '''Pure stretching'''. Porition of one particle is fixed. Another particle is moving along the line connecting particles with constant velocity. The force acting on the particles is calculated.
 +
* '''Pure shear'''. Orientations of the particles are fixed. Position of one particle if fixed. Another particle is moving (with constant velocity) along the line, orthogonal to initial direction of the bond. The force and torques acting on the particles are calculated.
 +
* '''Pure bending'''. Positions of both particles are fixed and the particles are rotated along the same axis, orthogonal to the bond, but in opposite directions. The torque acting on the particles is calculated.
 +
* '''Pure torsion'''. Positions of both particles are fixed. Orientation of one particle is fixed. Another particle is rotating around the bond. The torque acting on the particles is calculated.
 +
 
 +
 
 +
== DEM packages using the V-model ==
 +
 
 +
* [http://www.liggghts.com/ '''LIGGGHTS'''], details of implementation and some examples are given [[Implementation_of_V_model_in_LIGGGHTS | here]].
 +
* [http://walberla.net WaLBerla] (implemented by [http://faculty.skoltech.ru/people/igorostanin Igor Ostanin] and Vitaliy Petrov from Skoltech)
  
 
== History and acknowledgements ==
 
== History and acknowledgements ==
  
The idea underlining V-model was first formulated by [[В.А. Кузькин|Vitaly Kuzkin]] during communication with Michael Wolff in Technical University of Hamburg (March, 2011). The first formulation was very simple and coarse, but it works! The results of some test simulations were presented by [[В.А. Кузькин|Vitaly Kuzkin]] on [http://www.apm-conf.spb.ru APM 2011] conference (July, 2011). At the present moment V-model is much more flexible and physically meaningful than its first version. Now it is developed jointly by [[В.А. Кузькин | Vitaly Kuzkin]] and [[И.Е. Асонов | Igor Asonov]].
+
The idea underlining V-model was first formulated by [[В.А. Кузькин|Vitaly Kuzkin]] during communication with Michael Wolff in Technical University of Hamburg (March, 2011). The first formulation was very simple and coarse, but it works! The results of some test simulations were presented by [[В.А. Кузькин|Vitaly Kuzkin]] on [http://www.apm-conf.spb.ru APM 2011] conference (July, 2011). At the present moment V-model is much more flexible and physically meaningful than its first version. Now it is developed jointly by [[В.А. Кузькин | Vitaly Kuzkin]] and [[И.Е. Асонов | Igor Asonov]]. The V-model is implemented in DEM package LIGGGHTS by [[Patrick Fodor]].  
  
 
The authors are deeply grateful to Michael Wolff, [[А.М. Кривцов | Prof. Anton Krivtsov]], Prof. Stephan Heinrich, Dr. Sergiy Antonyuk and Prof. [http://www.williamhoover.info/ William Hoover].
 
The authors are deeply grateful to Michael Wolff, [[А.М. Кривцов | Prof. Anton Krivtsov]], Prof. Stephan Heinrich, Dr. Sergiy Antonyuk and Prof. [http://www.williamhoover.info/ William Hoover].
 +
  
 
== References ==
 
== References ==
  
 
<references>
 
<references>
<ref name="BPM"> Potyondy D.O.,  Cundall P.A. A bonded-particle model for rock // Int. J. of Rock Mech. & Min. Sc., 41, (2004) pp. 1329–1364
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</ref>
 
<ref name="Wang"> Wang Y. A new algorithm to model the dynamics of 3-D bonded rigid
 
bodies with rotations // Acta Geotechnica,  4, (2009), pp. 117–127
 
</ref>
 
 
<ref name="IvKrMoFi_2003"> E.A. Ivanova, A. M. Krivtsov, N. F. Morozov, A. D. Firsova.  Decsription of crystal particle packing considering moment interactions  // Mechanics of Solids. 2003. Vol. 38. No 4, pp. 101-117.
 
<ref name="IvKrMoFi_2003"> E.A. Ivanova, A. M. Krivtsov, N. F. Morozov, A. D. Firsova.  Decsription of crystal particle packing considering moment interactions  // Mechanics of Solids. 2003. Vol. 38. No 4, pp. 101-117.
 
</ref>
 
</ref>
Строка 78: Строка 110:
 
</ref>
 
</ref>
 
</references>
 
</references>
 +
  
 
== Links ==
 
== Links ==
 +
* [[Implementation_of_V_model_in_LIGGGHTS]]
 
* [[Vitaly Kuzkin]]
 
* [[Vitaly Kuzkin]]
 
* [[Игорь Асонов | Igor Asonov]]
 
* [[Игорь Асонов | Igor Asonov]]
* [[Виталий Кузькин: Избранные публикации | Selected publications of Vitaly Kuzkin]]
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* [[Kuzkin_pub | Selected publications of Vitaly Kuzkin]]
  
  
 
[[Category: Потенциальные взаимодействия]]
 
[[Category: Потенциальные взаимодействия]]
 
[[Category: Механика дискретных сред]]
 
[[Category: Механика дискретных сред]]

Текущая версия на 15:30, 29 марта 2018

Department "Theoretical Mechanics" > Adminstration > V.A. Kuzkin > V-model


A model for elastic bonds in solids (EVM)

SEM image of composite consisting of nanoparticles bonded by a copolymer. (N. Preda, et. al., Mater. Res. Bull. 45, 1008 (2010))

Introduction[править]

The article is based on the papers:


The Discrete (or Distinct) Element Method (DEM) is widely used for the computer simulation of solid and free-flowing granular materials. Similarly to classical molecular dynamics, in the framework of DEM the material is represented by a set of many interacting rigid body particles (granules). The equations of the particles' motion are integrated numerically. In free-flowing materials the particles interact via contact forces, dry and viscous friction forces, electrostatic forces, etc. Computer simulation of deformation and fracture of granular solids, such as rocks, concrete, ceramics, particle compounds, agglomerates, nanocomposites, etc. is even more challenging. Particles in granular solids are usually connected together by some additional bonding material such as cement or glue. The example of composite material consisting of PbS nanoparticles bonded together by a copolymer is shown in figure above. The copolymer (bonding material) resists the relative translation and rotation of neighboring PbS particles. In DEM simulations bonding material is usually taken into account implicitly using the concept of so-called bonds. Neighboring particles are connected by the bonds that resist to stretching/compression, shear, bending, and torsion. The bonds cause forces and torques acting on the particles along with contact forces. Mass of the bonding material is usually neglected. The assumption does not influence static properties of the granular material. The influence on the dynamic properties is not so straightforward and should be considered separately. However let us note that in many practical applications the mass of bonding material is much smaller than the mass of the particles (see, for example, figures below). Therefore the mass of bonding material can be neglected.

Discrete Rod: SEM image of 1D assembly of particles embedded in polymer matrix (M. Wang, et.al., Materials Today, Vol. 16, No. 4, 2013)
Discrete Shell: Colloidsome composed of polysterene spheres (A.D. Dismore, et.al. Science, 2002)
Discrete Solid: aerogel (source)

In the paper V.A. Kuzkin and I.E. Asonov "Vector-based model of elastic bonds for simulation of granullar solids"// Phys. Rev. E, 86, 051301, 2012 vector-based model(the V-model) of elastic bonds in solids is developed. The V-model is based on the combination of approaches proposed in works of P.A. Zhilin, E.A. Ivanova, A.M. Krivtsov, N.F. Morozov [1], [2] and works of M.P. Allen [3], S.L Price. Equations describing interactions between two rigid bodies in the general case are summarized. The general expression for the potential energy of the bond is represented via vectors rigidly connected with bonded particles (see figures below).

Vectors connected with the particles
Different deformations of the bond.

The vectors are used for description of different types of bond's deformation. The expression for potential energy corresponding to tension/compression, shear, bending, and torsion of the bond is proposed. Forces and torques acting between particles are derived from the potential energy. Two approaches for calibration of V-model parameters for bonds with different length/thickness ratios are presented. Simple analytical formulas connecting geometrical and elastic characteristics of the bond with parameters of the V-model are derived. Main aspects of numerical implementation of the model are discussed.

Advantages of the V-model[править]

Let us summarize advantages of V-model:

  • longitudinal, shear, bending, and torsional stiffnesses of the bond are independent
  • applicable in the case of large turns of the particles
  • conservation of energy (bonds are perfectly elastic)
  • any non close packed structure, rods and shells can be simulated(see figures above)
  • forces and torques are calculated as a functions of particles' positions and orientations (more accurate integration of motion equations than in the case of incremental algorithm used for BPM)
  • description of bonds of any length/thickness ratio. It is shown that in the case of small deformations the behavior of the bond can fit the behavior of Bernulli-Euler rod or Timosheko rod or short cylinder connecting particles (depending on length/thickness ratio)
  • simple analytical expressions connecting parameters of V-model with geometrical and mechanical characteristics of the bond
  • bonds can connect points inside the particles or lying on particles surfaces (not only particle centers)
  • fracture criterion for the bond can be used if required

Java Script example[править]

Two dimmensional example: deformation of a discrete rod. Author: Ruslan Lapin

Pull the last particle using mouse.

Some examples[править]

Press the links under the picture to see the animation.

Debugging[править]

The simplest way of debugging is a solution of the following four test problems for the system of 2 bonded particles. In all four cases forces and torques acting on the particles can be calculated both analytically and numerically. Obviously, the results should coincide.

  • Pure stretching. Porition of one particle is fixed. Another particle is moving along the line connecting particles with constant velocity. The force acting on the particles is calculated.
  • Pure shear. Orientations of the particles are fixed. Position of one particle if fixed. Another particle is moving (with constant velocity) along the line, orthogonal to initial direction of the bond. The force and torques acting on the particles are calculated.
  • Pure bending. Positions of both particles are fixed and the particles are rotated along the same axis, orthogonal to the bond, but in opposite directions. The torque acting on the particles is calculated.
  • Pure torsion. Positions of both particles are fixed. Orientation of one particle is fixed. Another particle is rotating around the bond. The torque acting on the particles is calculated.


DEM packages using the V-model[править]

History and acknowledgements[править]

The idea underlining V-model was first formulated by Vitaly Kuzkin during communication with Michael Wolff in Technical University of Hamburg (March, 2011). The first formulation was very simple and coarse, but it works! The results of some test simulations were presented by Vitaly Kuzkin on APM 2011 conference (July, 2011). At the present moment V-model is much more flexible and physically meaningful than its first version. Now it is developed jointly by Vitaly Kuzkin and Igor Asonov. The V-model is implemented in DEM package LIGGGHTS by Patrick Fodor.

The authors are deeply grateful to Michael Wolff, Prof. Anton Krivtsov, Prof. Stephan Heinrich, Dr. Sergiy Antonyuk and Prof. William Hoover.


References[править]

  1. E.A. Ivanova, A. M. Krivtsov, N. F. Morozov, A. D. Firsova. Decsription of crystal particle packing considering moment interactions // Mechanics of Solids. 2003. Vol. 38. No 4, pp. 101-117.
  2. E. A. Ivanova, A. M. Krivtsov, N. F. Morozov, Derivation of macroscopic relations of the elasticity of complex crystal lattices taking into account the moment interactions at the microlevel // J. App. Math. and Mech.,Vol. 71, Is. 4, 2007, pp. 543-561.
  3. M.P. Allen, D.J. Tildesley, Computer simulation of liquids, Clarendon Press, Oxford, 1987, p. 385.


Links[править]