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

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34 a) b) f s f s m M f n f n Contact point Contact surface Fig. 3.13. Idealization of contact behaviour in MDEM by Iwashita and Oda [72] a) two mechanisms of particle rotation, b) contact points (DEM) and contact surfaces (MDEM) behaviour The accuracy of representation of the simulated processes increases as the models are equipped with more and more precise values of the moduli of elasticity, rigidity, micro-hardness, roughness, of contact friction coefficient, and damping coefficients. 3.5. Model of micropolar continuum The model of micropolar continuum includes rotation into the kinematics of material. The starting point is the derivation of the field of displacement u(x, t) and rotations ω(x, t) vectors (fig. 3.14). Deformation of infinitesimal element of the material occurs as a result of superposition of displacement and rotations. Interaction between elements of the structure of material on elementary surface dS takes place not only through the force vector, but also through the couple force vector. An elementary area of the material is affected not only by force stresses, but also by couple stresses. The theory of non-symmetrical elasticity was formulated by the brothers E. and F. Cosserat. At present the theory draws considerable attention from researchers, and one of the areas of its application is the mechanics of granular materials [79]. x 2 A R Fig. 3.14. Displacement and rotation in a micropolar continuum x 1
35 The micropolar elasto-plastic constitutive model of a granular material with isotropic hardening and softening differs from the classical theory of plasticity by the presence of rotations, couple stresses, and a characteristic length corresponding to the mean grain diameter. Due to the introduction of rotations into the kinematics, each material point in the 3D case has three translational and three rotational degrees of freedom, while in 2D and in axis-symmetrical cases two translational and one rotational degree of freedom. The gradient components of the rotation cause curvatures that are associated with the couple stresses. This makes the stress and strain tensors non-symmetric, and the constitutive equation contains the characteristic length. The micropolar elasto-plastic model in Mühlhaus’s approach [119] was formed by the extension of the non-associated elasto-plastic flow rule of Drucker- Prager with isotropic hardening and softening by the Cosserats’ rotations, curvatures, couple stresses, and mean grain diameter. As a result, the micropolar model includes the characteristic length and at the same time retains the essence of the continuous medium. The constitutive model of granular materials formulated by Mühlhaus contains a number of constants and of material functions that have to be determined experimentally. These include the modulus of elasticity, Poisson constant, cohesion, dependence of internal friction angle on plastic strain, dependence of dilatation angle on plastic strain, mean grain diameter, and micropolar constants. a) b) c) d) Fig. 3.15. Comparison of the particle displacement (a, b) and rotation (c, d) fields obtained from the discrete method (a, c) and the continuum method (b, d) for the case of loading by (a, b) normal and shear stress (c, d) couple stresses [32]
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 30 and 31: 30 A different approach to the desc
Page 32 and 33: 32 where: I - moment of inertia, k
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
Page 80 and 81: 80 10.4.1. Friction force mobilizat
Page 82 and 83: 82 10.4.3. Moisture content With in
Page 84 and 85:
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
Page 108 and 109:
108 114. Morland L.W., Sawicki A.,
Page 110 and 111:
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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