Modelling of elasticity properties of solids by the discrete element method ; Deformuojamo erdvinio kūno tampriųjų savybių modeliavimas diskrečiųjų elementų metodu

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Algirdas MAKNICKAS MODELLING OF ELASTICITY PROPERTIES OF SOLIDS BY THE DISCRETE ELEMENT METHOD Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) 1637-M Vilnius 2009 VILNIUS GEDIMINAS TECHNICAL UNIVERSITY Algirdas MAKNICKAS MODELLING OF ELASTICITY PROPERTIES OF SOLIDS BY THE DISCRETE ELEMENT METHOD Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) Vilnius 2009 Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2004–2009. The dissertation is being defended as an external work. Scientific Consultant Prof Dr Habil Rimantas KAČIANAUSKAS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T). The dissertation is defended at the Council of Scientific Field of Mecha-nical Engineering at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Rimantas BELEVIČIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T).

Sujets

• Discrete element method
• Elastic modulus
• Mechanical engineering

Informations

• Publié par: vilnius_gediminas_technical_university
• Publié le: 01 janvier 2009
• Nombre de lectures: 35
• Poids de l'ouvrage: 1 Mo

Extrait

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    Algirdas MAKNICKAS     MODELLING OF ELASTICITY PROPERTIES OF SOLIDS BY THE DISCRETE ELEMENT METHOD    Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)       

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Vilnius 2009

1637-M

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VILNIUS GEDIMINAS TECHNICAL UNIVERSITY          Algirdas MAKNICKAS     MODELLING OF ELASTICITY PROPERTIES OF SOLIDS BY THE DISCRETE ELEMENT METHOD    Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)       

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Vilnius2009

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Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2004–2009. The dissertation is being defended as an external work.  Scientific Consultant Prof Dr Habil Rimantas KAIANAUSKAS Gediminas (Vilnius Technical University, Technological Sciences, Mechanical Engineering – 09T). The dissertation is defended at the Council of Scientific Field of Mecha- nical Engineering at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Rimantas BELEVIIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T).  Members: Dr Valentin ANTONOVI Gediminas Technical University (Vilnius Institute of Thermal Insulation, Technological Sciences, Materials Engineering – 08T), Prof Dr Habil Rimantas BARAUSKAS (Kaunas University of Technology, Technological Sciences, Mechanical Engenering – 09T), Prof Dr Habil Algimantas BUBULIS(Kaunas University of Technology, Technological Sciences, Mechanical Engenering – 09T), Prof Dr Habil Genadijus KULVIETIS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T).  Opponents: Prof Dr Habil Juozas ATKOINAS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T), Dr Habil Evaldas TORNAU(Semiconductor Physics Institute, Physical Sciences, Physic – 02P). The dissertation will be defended at the public meeting of the Council of Scientific Field of Mechanical Engineering in the Senate Hall of Vilnius Gediminas Technical University at 10 a. m. on 15 June 2009. Address: Saultekio al. 11, LT-10223 Vilnius, Lithuania. Tel.: [phone]; fax [phone]; e-mail: [e-mail] The summary of the doctoral dissertation was  distributed on 14 May 2009. A copy of the doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saultekio al. 14, LT-10223 Vilnius, Lithuania). © Algirdas Maknickas, 2009  

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VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS          Algirdas MAKNICKAS    DEFORMUOJAMO ERDVINIO KNO TAMPRIJSAVYBIMDOLEIAVIMAS DISKREIJELEMENTMETODU    Daktaro disertacijos santrauka Technologijos mokslai, mechanikos inžinerija (09T)        

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Vilnius2009

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Disertacija  rengta  2004–2009  metais  Vilniaus  Gedimino  technikos universitete. Disertacija ginama eksternu. Mokslinis konsultantas prof. habil. dr. Rimantas KAIANAUSKAS Gedimino (Vilniaus technikos universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Disertacija ginama Vilniaus Gedimino technikos universiteto Mechanikos inžinerijos mokslo krypties taryboje: Pirmininkas prof. habil. dr. Rimantas BELEVIIUS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Nariai: dr. Valentin ANTONOVI (Vilniaus Gedimino technikos universiteto Termoizoliacijos institutas, technologijos mokslai, medžiag – inžinerija 08T), prof. habil. dr. Rimantas BARAUSKAS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija – 09T ), prof. habil. dr. Algimantas BUBULIS(Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija – 09T). prof. habil. dr. Genadijus KULVIETIS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Oponentai: prof. habil. dr. Juozas ATKOINAS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos inžinerija – 09T), habil. dr. Evaldas TORNAU(Puslaidininki fizikos institutas, fiziniai mokslai, fizika – 02P). Disertacija bus ginama viešame Mechanikos inžinerijos mokslo krypties tarybos posdyje 2009 m. birželio 15 d. 10 val. Vilniaus Gedimino technikos universiteto senato posdžisalje. Adresas: Saultekio al. 11, LT-10223 Vilnius, Lietuva. Tel.: [phone]; faksas (8 5) 270 0112; el. paštas [e-mail] Disertacijos santrauka išsiuntinta 2009 m. gegužs 14 d. Disertacij galima peržirti Vilniaus Gedimino technikos universiteto bibliotekoje (Saultekio al. 14, LT-10223 Vilnius, Lietuva). VGTU leidyklos „Technika“ 1637-M mokslo literatros knyga. © Algirdas Maknickas, 2009

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Introduction  Topicality of the problem.Development of numerical methods and computation environments opened the possibility of new, more sophisticated mechanical objects modelling. In this context it is natural desire of the researchers to describe macroscopic mechanical characteristics of the materials by their microstructure, which can be adapted for simulation of the existing and future materials. For this purpose researchers are using intensively experimental and numerical methods for the development of which the highest priority is given. Numerical experiments are used because they are cheaper and allow the interpretation of already known results of experiments and provide information to new investigations. One of the methods used for modelling of macroscopic properties modelling is based on microscopic properties of material is discrete element method (DEM). The DEM traditionally was applied for the granular materials. The basic idea of DEM is that any physical structure could be described as a system of moving particles. This idea could be also applied to the description of solid deformable body. Particles forming solid body and existing interaction between them are of different nature than the granular materials because their models are often the result of physical and mathematical abstraction. The modelling of solid deformable body with the discrete elements is just at the initial stage and the unified approach to discrete elements models doesn’t exist. There are several versions of models, based on various approaches. The first step in applications of DEM to a solid body could be modelling of elastic properties. It is an actual problem which has not been solved yet.  The object of investigations.The object of investigations of current work is the three-dimensional deformable solid and their elasticity indicators.  Aims and tasks of the work.The aims of the present work are as follow: to determine the relationship between macroscopic elastic properties of a solid deformable body as a particle system and microscopic properties of the structure; to develop DEM algorithms and software for modelling of the solid deformable body as a particle system. For this purpose the following tasks have to be performed: 1.  Development and investigation of methodology and physically observed three-axial compacting scenario of deformable solid body generated by using sets of mono and poly dispersed spherical discrete elements.

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2.Determination of the relationships between elasticity modulus and Poisson’s ratio of the 3D body and properties of particles system by conducting of uniaxial deformation experiment. 3.  Development of modelling algorithms and software of the 3D elastic body as a particle system and creating of two different versions working in both single processor and multiprocessor environment.  Methodology of researchinvolves analysis of theoretical and experimental investigations by applying analytical and numerical methods and numerical experiments.  Scientific novelty  1.  The introduction of the physical observed scenario for simulation of elastic, isotropic solid body by three-axial compacting of particles. 2.  The evaluation of indicators of transitions states of a particle system in case of three-axial compressions. 3.  The theoretical derived relationships between macroscopic properties of the tree-dimensional elastic solid body and microscopic properties of discrete elements was investigated numerical by using sets of mono and poly-dispersed discrete elements. 4.  The evaluation of analytical relationships of the pressure variation in the solid body as a discrete elements system to the walls was obtained. 5.  New algorithms for modelling of 3D body by discrete elements were created and adapted for sequential and parallel implementations.  Practical value.The obtained result of investigations were summarised and could be used for the development of new DEM models of solid body and investigation of their elastic properties and fracture. New software for modelling of 3D body by discrete elements were created and adapted for sequential and parallel calculations.  Defended propositions  1.  The proposed physically observed methods of generation of a deformable solid body. 2.  The proposed analytical relationship between micro and macro elasticity properties. 3.  The developed software for the simulation of a solid body.   

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The structure of the scientific work.The scientific work consists of the general characteristics of the dissertation, 5 chapters, conclusions, list of literature, list of publications and appendix. The total volume of the dissertation – 122 pages, 148 pictures, 5 tables and 1 addenda.  1. Models and application areas of the discrete element method  Among various numerical techniques, the discrete (distinct) element method (DEM) became widely recognised after the pioneering work published by Cundall and later the work of Cundall and Strack (1979). The representation of granular media as an assembly of contacting particles termed hereafter as discrete elements was seen as a more realistic approach compared to continuum models. In the DEM, the particles of granular media are treated as individual objects and all dynamical state variables of each particle are tracked during the simulation. The DEM allows the simulation of motion and interaction between the particles, taking into account the microscopic geometry and various constitutive models. The main advantage of DEM is a possibility to model highly complex poly-dispersed systems using the basic data on individual particles without making oversimplifying assumptions. This makes DEM different from the conventional discretization methods used in the continuum mechanics, such as the finite difference, finite element and boundary element methods, helping to avoid difficulties encountered in describing the microscopic nature of the media at the continuum level. Various aspects and problems are faced in simulation of solid bodies by applying DEM technique. It is widely recognised that the macroscopic properties of the particulate assemblies depend on their single particle properties and the interaction between contiguous particles, while modelling of the mechanical behaviour of particulates can be reflected by inter-particle stiffness models. The fundamental issues such as selection of realistic inter – particle modulus, evaluation of relationships between microscopic characteristics of particles and macroscopic characteristics of the solid body are related to the generation of particle composition. Various models have been investigated in the works of Hentz et. al. (2004), Antonyuk et al. (2005), Mishra and Thornton (2001), D‘Addetta (2004) and Ramm (2002) Simulation of a solid body by particles requires generation of the initial state of particles, which may be regarded as generation of the initial conditions. In any case, artificial simulation stage will be required for these purposes. This may be done by slightly different

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approaches. It is observed that this initial simulation stage may affect the final results, however, a unified approach is still under development.  2. Models and dependencies of the 3D body  Assumptions and concept of discretization Assumptions employed for discretization the body are: discrete elements are spherical particles; normal contact between particles is determined by interaction model; there is tangential interaction between particles; the space between particles is considered void; the theoretical influence factor of material (structure) is dimensionless bonding coefficient.  Interaction of particles and equations of discrete element method The particle deformation due to collision is assumed to be approximated by the overlap area of the spheres. The contact pointC ij  is defined to be in the centre of the overlap area with the position vectorX cij . The depth of overlaph ij  is defined as the difference between the sum of particle radiuses and distance between the centre of particles, where sum of radiiR i andR j could be multiplied by dimensional less overlap factor 1 . DEM models of granular materials use coefficient1 . When overlap parameters are known, contact forces acting between two particles may be evaluated explicitly as summation of normal elastic components of attraction or propulsion, viscous component and tangential component. The dynamical behaviour of an individual particleiis considered by applying Newton’s second law. Three independent translationsx i  and three independent rotations  i  in terms of the forces and torques at the expressed centre of the particle are as follows  m i d d 2 t 2 x i F i , (1)

d 2   I i i  i . (2) dt 2 T Herem i , andI i are mass and inertia moments. Vectors  F i andT i present the sum of external,  contact force and gravity force  as well as the corresponding torques.   

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Characteristics of discrete elasticity In general, stress tensor can be expressed as linear function of deformation tensor kl as follow  ij C ijkl kl , (3) whereC ijkl  is the material elasticity constants tensor. Stiffness constants tensor C ijkl , the general case, has 81 components. Linear approximation allows using of the principle of independent force additivity for the linear and shear stresses which determines the spatial deformation of the body in the desired direction. Stiffness constants tensorC ijkl  just two none zero component in has isotropic media, E elasticity modulus and coefficient, where their Poisson relation is expressed with the dependence EC 1111 C 1122 1   ,  C 1122  (4)   1122  1111 . These relationships are used in evaluation of elasticity modulus and Poisson's ratios values of the specimen. The basic idea of the spring network models is grounded on formulation of equivalence principle of potential energy of particles in the network and potential energy accumulated in a solid body U net U solid . Using this principle, material elasticity constants tensor can be expressed through potential energy of particles concentrated in the network. Differentiating potential energy derivative twice under elongations variable gives next stiffness constants tensor expression: ij 1  2 diskr  C kl  V  U   . (5) ij kl  Micro – macro relationships As it is known particle interactions contact force may vary according to Hooke's or Hertz laws. If the particle interactions contact force varies according to Hooke's law, the potential energy of specimen now belongs to a quadratic power of deformation. Then the elasticity constants tensor (7) could be found as 9

i  ij j  Z kl  C kl E Mikro 4  vid 1 2 , (6) which shows that the stiffness tensor component is directly proportional to micro modulus and coordination number, where ijkl is the proportionality factor or unit Fabric tensor, which is independent of the number of particles in the system and particle size, but depends on the present network topology of interacting particles. For isotropic solid there is such equality   E Z  1111   1122   1    E 4  M 1 ik  ro  2 m a i v c g ro , (7)

1122    1122   1111 . (8) On the basis of eq. (9–10) derived expression general conclusion could be made that macroscopic elastic modulus of solids formed from the discrete elements depends directly on the elasticity modulus and Poison’s ratio of discrete elements and the average of coordination number, where ijkl  unit Fabric tensor. 3. Numerical investigation of solid body  Modelling concept Solid state discretisation requires initial configurations of particles, which are used for entering the mutual interaction forces of the coupling particles to obtain result by the original rigid body condition of further calculation. This assumption is based on a natural phenomenon of formation of a rigid body, when the solid body is formed from a liquid phase material by establishing of atomic relations between the molecules. The following scenario of a solid deformable body generation is based on this assumption. At first, discrete elements are compacted, and than attraction forces are switched on inside of interpatricle connections.  Scenario of generation of solid deformable body The initial configuration particles have been generated in some different ways. In case of uniform radiuses of discrete elements all the entire limiting area of moving borders has been split in cells, in which centers each particle was placed. In case of irregular radiuses of discrete elements algorithm was

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