Strona 2_redak - Instytut Agrofizyki im. Bohdana DobrzaÅskiego ...

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30 A different approach to the description of interactions between the granules of a granular material has been presented by Góźdź and Pietrow [56] who applied a formalism close to that of quantum mechanics. The formalism permits the creation of a more coherent description of irregularly distributed granules of a medium than is possible in the classical micro-mechanical approach operating with distribution of forces at the contact points of the granules. The basic element of the description is the Hamiltonian operator representing the energy of the system and the interactions between the granules. Another element of the description are vectors of states represented by functions related to the shape and size of granules and to mass distribution. The introduction of strain operators acting on the vectors of states of individual granules ultimately gives the global strain of the whole medium [57]. 3.4. Distinct Element Method Common popularity has been attained by the Distinct Element Method (DEM) developed by Cundall and Strack [39]. The method is used for modelling mechanical processes in granular materials on the basis of elementary interactions between the grains. The method consists in approximated solution of the equation of motion for each grain of the material. The motion takes place as a result of propagation through the material of a disturbance initiated under boundary conditions. The calculation procedure is based on the assumption that during a very short time step ∆t acceleration and speed are constant, and the disturbance of motion of a single grain does not reach further than to the nearest neighbours. This is the key assumption of the method that permits the description of nonlinear interactions occurring among a large number of elements without excessive requirements concerning the calculation memory power. In this approach all the forces acting on a given granule are considered – those resulting from gravity, from interactions with neighbouring granules, and those resulting from the boundary conditions [12]. Then, on the basis of Newton’s second law of dynamics, the acceleration of the granule is determined. Integration in time permits the determination of the new velocity and position. The deformation of individual grain is considered to be infinitely small compared to the deformation of the whole medium. Therefore, it is usually assumed that the grains are rigid and their deformation at the contact points is modelled through their overlapping. The displacements in the normal direction ∆L C n, tangential direction ∆L C s, and those resulting from grain rotation ∆L C ω (fig. 3.10) are considered separately. Modelling of interactions between grains usually involves viscoelastic contact in the normal direction (η n ,k n ) and visco-elastic-frictional contact (η s , k s , µ s ) in the tangential (shear) direction (fig. 3.11). Elasticity models the accumulation of energy in the contact points of the granules, and viscosity and dry friction model the dissipation of energy. The forces of cohesion are neglected.
31 x 3 l C x 1 x 2 a) Initial state b) Normal displacement c) Tangential displacement d) Tangential displacement due to rotation Fig. 3.10. Normal, shear and rotational shear displacements at contact point of grains a) b) k s k n Fig. 3.11. A typical model of contact force between two grains: a) a linear, damped spring element in the normal direction, and b) a linear, damped spring element with a sliding friction in the tangential direction Differential equations of displacement (x) and rotation (ω) of an individual granule of the material, including the visco-elastic contact between granules [39, 103, 146]: mx && + ηx& + kx = 0, 2 2 Iω&& + ηR ω& + kR ω = 0, are approximated with the incremental equations: m[x] && = −η[x] & 2 I[ω] && = −ηR [ω] & t t t − ∆t t − ∆t − k[x] t − ∆t , 2 − kR [ ω ] t − ∆t , (3.7) (3.8)
Page 2 and 3: EC Centre of Excellence AGROPHYSICS
Page 4 and 5: 4 9.3. Factors influencing the coef
Page 6 and 7: 6 BASIC NOTATION c - cohesion [kPa]
Page 8 and 9: 8 Numerous granular materials when
Page 10 and 11: 10 the tensor of plastic strain inc
Page 12 and 13: 12 Fig. 2.1. Yield curves with comp
Page 14 and 15: 14 Stable or unstable behaviour of
Page 16 and 17: 16 In turn, the model of Lade, also
Page 18 and 19: 18 same porosity need not have iden
Page 20 and 21: 20 where: A = ∫ E( β) sin βdβ
Page 22 and 23: 22 (β = 0 o ), positioning the out
Page 24 and 25: 24 grain was poured into the mould
Page 26 and 27: 26 axes of the ellipse characterizi
Page 28 and 29: 28 [176] for the determination of p
Page 32 and 33: 32 where: I - moment of inertia, k
Page 34 and 35: 34 a) b) f s f s m M f n f n Contac
Page 36 and 37: 36 The micropolar model yields resu
Page 38 and 39: 38 Fig. 3.17. Model of shear band f
Page 40 and 41: 40 In the case of agricultural mate
Page 42 and 43: 42 5.2. Density of consolidated mat
Page 44 and 45: 44 (cereal grain, sand, soil), the
Page 46 and 47: 46 Multiple wetting and drying of g
Page 48 and 49: 48 vertical pressure within the ran
Page 50 and 51: 50 of the curve defined by A, where
Page 52 and 53: 52 6. Vertical consolidation refere
Page 54 and 55: 54 poured into the box without vibr
Page 56 and 57: 56 7.2.2. Bulk density Stress-strai
Page 58 and 59: 58 near the box wall and lifted alo
Page 60 and 61: 60 and the upper boundary was 18.5
Page 62 and 63: 62 A -1 5 -1 0 -5 1 0 5 -5 5 1 0 1
Page 64 and 65: 64 Fig. 8.2. Yield locus and effect
Page 66 and 67: 66 in such a way that friction woul
Page 68 and 69: 68 a) b) µ = tg ϕw ϕ w c) d) e)
Page 70 and 71: 70 9.3. Factors influencing the coe
Page 72 and 73: 72 galvanized steel received from d
Page 74 and 75: 74 9.3.4. Velocity of sliding Becau
Page 76 and 77: 76 1+ sinϕ k = . 1 − sinϕ (10.3
Page 78 and 79: 78 N ϕ VLQϕ 3UHVVXUHUDWLRN
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80 10.4.1. Friction force mobilizat
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82 10.4.3. Moisture content With in
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84 Table 10.1. Measured (k s ) and
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86 conventional engineering analyse
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88 a, b - product-dependent coeffic
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90 also be performed as a function
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92 reliable processing and stable v
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94 unconfined yield strength to pow
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96 The Chute Index (CI) is recommen
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98 obtained the maximum deviation o
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100 efficient flow control using op
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102 In the field of experimental in
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104 22. Britton M.G., Moysey E.B.:
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106 69. Horabik J., Rusinek R.: Pre
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108 114. Morland L.W., Sawicki A.,
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110 158. 6WSQLHZVNL$ Influence of m
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112 16. APPENDIX - Physical Propert
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114 16.2. Density and porosity Tabl
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116 Table 16.7. Mean values (±St.
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126 Table 16.16. Cont. 1 2 3 4 Icin
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136 Table 16.25. Cont. 1 2 3 4 Icin
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140 Table 16.29. Cont. Table sugar
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144 Table 16.35. Mean values (± St
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