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16 In turn, the model of Lade, also applied by Zhang et al. [175] for the description of wheat grain during loading, assumes that the plastic strain increment dε p ij is the sum of two independent components: the plastic strain increment related with the compaction of the material dε c ij and the plastic strain increment related with the dilation of the material dε d ij: dε = dε + dε p ij c ij (2.19) The division of the plastic strain increment into two independent components entails the necessity of adopting also two independent yield functions and two flow rules. For the description of the behaviour of wheat Zhang et al. [175] adopted the following yield functions F c and F d , and plastic potentials G c and G d : d ij . F c = I 2 1 + 2⋅ I 2 − P ⋅( W a c / C ⋅ P ) a 1 q , (2.20) F d = ( I 3 1 / I 3 − 27) ⋅( I / P ) 1 a m − ae d −bW ⋅( W d / P ) a 1 q , (2.21) G c 2 = I 1 + 2 ⋅ I 2 , (2.22) 3 m G (2.23) d = I1 − ( 27 + η ⋅( Pa / I1) ) ⋅ I3, where: a, b, c, m, q, η – material constants, I 1 , I 2 , I 3 – first, second and third invariant of stress tensor, P a – atmospheric pressure, W c , W d – collapse and expansive plastic work. The yield functions F c , responsible for irreversible compaction of material, represents in the space of principal stresses a concave surface with axis of symmetry lined with the axis of isotropic stresses. This condition confirms the known rule that a granular material compacts the easiest under the isotropic stress. In the plastic potential G d , responsible for material expansion, the material constant η represents the slope of the plastic potential surface, and the exponent m represents the curvature of the meridian of the surface. Figure 2.3 presents examples of the application of the models of Ghaboussi and Momen and of Lade for the approximation of the experimental stress-strain relations obtained during tests of monotonic loading of wheat grain samples in triaxial compression apparatus. The presented comparison shows that the model of Lade more accurately approximates the course of the stress-strain relation during monotonic
17 loading. However, it is inferior to the model of Ghaboussi and Momen in the case of description of the behaviour of materials with hysteresis under the conditions of cycling loading. Nevertheless, its simplicity and mathematical coherence make it a highly useful tool for modeling the plastic flow of granular materials. 100 90 80 70 60 50 40 30 20 10 - Eksperimental data Ghaboussi - Momen Lade 0 0 20 40 60 80 100 120 140 x10 -3 Fig. 2.3. Comparisons of calculated and measured stress-strain relations in triaxial compression tests of wheat grain samples [175] 3. MICROSTRUCTURAL APPROACH. DEM MODELLING 3.1. Geometric structure of granular medium The results of geotechnical studies have shown that natural sand deposits formed under the effect of gravity are usually anisotropic. Allen [2] proved that in the course of sand deposit formation grains of sand tend to orient themselves so that their long axes are parallel to the horizontal plane. The result of this is the formation of deposits with a high degree of geometric arrangement of particles. Oda [126] became interested in the effect of the anisotropy of a deposit on its mechanical properties. He conducted a series of laboratory tests on sand samples. The author showed that knowledge of the structure of particle packing is necessary for the determination of the stability of noncohesive soils subjected to external loads. Mechanical phenomena, such as the anisotropy of response to loading, stress-strain relations, strain hardening, strength and porosity, turned out to be dependent on the structure of packing [127-129]. Ever since those experiments it has been known that two samples of the same sand with the
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 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 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
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66 in such a way that friction woul
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68 a) b) µ = tg ϕw ϕ w c) d) e)
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70 9.3. Factors influencing the coe
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72 galvanized steel received from d
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74 9.3.4. Velocity of sliding Becau
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76 1+ sinϕ k = . 1 − sinϕ (10.3
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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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