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3D Dynamic analysis of soil–tool interaction using the finite element method
Journal of Terramechanics 40 (2003) 51–62 www.elsevier.com/locate/jterra
3D Dynamic analysis of soil–tool interaction using the finite element method Mootaz Abo-Elnora, R. Hamiltonb,*, J.T. Boyleb a Military Technical College, Cairo, Egypt Department of Mechanical Engineering, University of Strathclyde, UK
b
Abstract Previous experimental and finite element studies have shown the influence of both soil initial conditions and blade operating conditions on cutting forces. However, most of these finite element analyses (FEA) are limited to small blade displacements to reduce element distortion which can cause solution convergence problems. In this study a dynamic threedimensional FEA of soil–tool interaction was carried out based on predefined failure surfaces to investigate the effect of cutting speed and angle on cutting forces over large blade displacements. Sandy soil was considered in this study and modeled using the hypoplastic constitutive model implemented in the commercial FEA package, ABAQUS. Results reveal the validity of the concept of predefined failure surfaces in simulating soil–tool interaction and the significant effect of cutting acceleration on cutting forces. # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Dynamic soil–tool interaction; Hypoplasticity; Sand modeling
1. Introduction The productivity of mine-clearing and earth-moving equipment, such as motor graders, scrappers and bulldozers, is dependant on the cutting process speed along with other initial conditions (blade geometry, soil–metal coefficient of friction and soil type) and operating conditions (cutting angle and depth). The effect of initial and operating conditions have been shown experimentally to have a great effect on machine productivity [1–7]. However, experimental study of soil–tool interaction is expensive and limited to certain cutting speeds and results are highly dependant on the accuracy of the measuring devices. Several dynamic models of soil–tool interaction have been developed analytically using the limit equilibrium method [3,4,8– * Corresponding author. Tel.: +44-141-548-2046; fax: +44-141-552-2105. E-mail address: [email protected] (R. Hamilton). 0022-4898/$20.00 # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jterra.2003.09.002
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10]. Due to the difficulty of conducting adequate tests and the limitations of analytical methods, the soil dynamic response to a tool has not been well predicted [11]. However with increasing computational power and development of more sophisticated material models, the finite element technique shows more promise in analyzing soil–tool interaction. The finite element method (FEM) is a relatively new and effective numerical method. Several researchers have performed finite element simulations of the soil–tool interface process and studied factors affecting cutting forces [6–8,12]. Most of the available models are applicable to low cutting speeds (< 5 mm/s). The objective of this study is to investigate the dynamic effects of the soil–tool interface process on cutting forces using the finite element technique. The dynamic effects such as blade cutting speed and acceleration are also taken into account.
2. Dynamic constitutive models for soils The existing models for predicting cutting forces under dynamic conditions are based on the hypothesis that a large portion of the increase in cutting forces due to increasing tool speed can be attributed to (1) the increase of the soil shear strength and soil–metal friction with increasing shear rate [8], (2) the inertial forces arise when soil is accelerated from a state of rest to a certain velocity. Investigation of the relationship between soil–metal friction and sliding speed has been carried out [13]. A model of soil shear strength as a function of shear rate and normal pressure has been developed by Dechao [14] and verified with a series of laboratory tests under shear rate range of 5–30 s1. Experimental results showed that an increase in soil shear strength is intensified by a rise in shear rate. The experimental results also showed that the relation between soil shear strength and normal pressure is non linear at high shear rate. this suggested that the Mohr–Coulomb criterion could only be applied under low shear rate conditions. To account for the effect of the shear rate on soil shear strength and soil–metal friction; dynamic models for soil–tool interaction have been developed [7,11] based on a hyperbolic stress–strain model developed by Duncan and Chang [15]. However, results obtained from a numerical procedure that have used inappropriate constitutive laws can be of limited or doubtful validity. Hence, selecting the most appropriate material model is more likely to produce reasonable results. The mechanical behaviour of granular soils, typically ranging from silt to gravel, can be modeled by various different theories. Hypoplasticity stems from the framework of rational mechanics starting from principal requirements on the model properties; a single equation describing many important features of the behaviour of granular soils was obtained [16]. Based on the general concept of hypoplasticity various aspects of the mechanical behaviour of granular materials have been investigated in the past few years, e.g. shear banding [17,18], rate-dependence and cohesion [19]. Critical states have been incorporated in order to better describe the influence of pressure level and density on the material behaviour [18,20,21]. It has been demonstrated in these papers that the hypoplastic model can describe the dependence of
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the material behaviour on the pressure level and density with a single set of material constants. The Wolffersdorf version of the hypoplastic model [22] was modified for monotonic loading and implemented into the finite element code ABAQUS [23] as a user defined material and used in this study to simulate a fully drained sandy soil. 2.1. Effect of strain rate (rate dependency) The hypoplastic equation is a rate type tensorial function but it is rate independent from the material behaviour point of view. In fact, the mechanical behaviour of granular materials is commonly assumed to be time independent. However, investigations have been done [24] to study the effect of strain rate using standard triaxial tests. Fig. 1 presents the stress–strain relationship for a loose sand specimen at different strain rates. The figure reveals an insignificant effect of the strain rate on the stress in the case of sandy soil. However, in clay soils, draft force, acting on the blade in the horizontal direction, is generally not sensitive to inertial forces, but shear strength increases substantially with increasing shear rates [7].
3. Material and methods The hypoplastic model, implemented in ABAQUS as a user defined material using the UMAT user subroutine, was used to describe sand behaviour. Table 1 represents the hypoplastic model parameters of the simulated sand soil with initial void ratio of e0=0.75.
Fig. 1. Rate effect on sand behaviour, Prisco and Imposimato [24].
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Table 1 Parameters of the hypoplastic model for Karlsruhe sand [26] ’c (deg)
hs (MPa)
n
ed0
ec0
ei0
30
5800
0.28
0.53
0.84
1.0
0.13
1.0
3.1. Model description The 3D model dimensions are presented in Fig. 2 and listed in Table 2, where ‘W1’ is the lateral bound width, ‘W2’ is the blade cutting width, ‘L1’ is the distance between the blade and the left-hand boundary, ‘L’ is the soil bin length, ‘h’ is the soil bin height, ‘d’ is the cutting depth and ‘’ is the cutting angle. W2 and W1 were varied to study the effect of blade width and lateral bound width of the simulated sample respectively as shown later, the remaining dimensions were held constant. A total of 30 separate finite element models were run [25]. Most of the analyses were carried out through 50 mm of blade displacement in the horizontal plane along the X axis direction shown in Fig. 2. A blade displacement of 50 mm was chosen as a compromise to cut down solution times. For a typical 3D model solution time was
Fig. 2. Soil–tool interface model dimensions.
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Table 2 Dimensions of the three-dimensional soil–tool interaction model L1 (mm)
L (mm)
h (mm)
d (mm)
W1 (mm)
W2 (mm)
( )
200
900
700
200
400
500
75
about 50 h to account for 200 mm of blade displacement using a dual Xenon 1 GHz processor PC with 512 MB of memory. However, it takes 18 h to account for 50 mm of blade displacement. To verify the behaviour of the cut soil in front of the cutting blade, one analysis was run for a blade displacement of 200 mm. 3.2. Finite element mesh and boundary conditions A three dimensional 8-node linear brick continuum element (C3D8) [23] was selected to represent both the sand and the blade in the finite element model. 1082 elements were used to simulate the sand, and 288 elements were used to simulate the blade which was defined as a rigid body with a reference node. Simulating the blade using the *RIGID BODY feature in ABAQUS enables the calculation of the resultant reaction forces acting on the entire blade at a single reference node. The *Dynamic solution feature in ABAQUS was used to switch to dynamic simulation. Due to the symmetric geometry of the model, one half of the model was simulated but all the results consider the complete model. Two failure surfaces were predefined, one along the horizontal plane in front of the blade-cutting tip and the other along a vertical plane at a distance of ‘W2/2’ from the symmetric plane and of height ‘d’, i.e. along the blade vertical boundary, as shown in Fig. 2. The concept of master and slave contact in ABAQUS was used to simulate the interface between the cutting blade and the soil; and soil itself along the predefined failure surfaces. Relative motion was allowed with friction along the soil–tool and soil–soil interface surfaces. The model was meshed in a manner that increased the mesh density near the blade and the predefined failure surfaces as shown in Fig. 3. The boundary conditions of the model are (Fig. 3): 1. Bottom base nodes, at Y=0, are fully constrained. 2. Nodes on vertical boundaries parallel to the Y–Z plane, at X=0 and X=L, are constrained in the horizontal direction along X axis. 3. Nodes on vertical boundaries parallel to the X–Y plane, at Z=0 and Z ¼ ðW2=2Þ þ W1 (symmetry plane), are constrained in the lateral direction along Z axis. 4. The blade is constrained in the vertical direction and from any rotation but it is free to move in the horizontal direction. Gravity effect was taken into account by applying the gravity acceleration as a body load to simulate the sand weight for a single step. This allowed contact to establish between the soil–blade and soil–soil interface.
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Fig. 3. Finite element mesh and boundary conditions.
3.3. Model validation The validation of the present 3D soil–tool interface model was carried out through investigation of various aspects as follows: predefined failure surfaces validation; sand failure and shear band formation.
3.4. Effect of cutting speed To investigate the effect of cutting speed on blade cutting forces, a series of finite element models were carried out at different constant cutting speeds (V =10, 30, 50, 100, 200 mm/s) through a blade displacement of 50 mm in the horizontal direction along the X axis. 3.5. Effect of cutting acceleration To investigate the effect of cutting acceleration on blade cutting forces, a series of finite element models were carried out at different cutting accelerations (a=1, 9, 25, 100, 400 mm/s2) through a blade displacement of 50 mm in the horizontal direction along the X-axis. The blade was assumed to be initially at rest.
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4. Results and discussion As mentioned before, the hypoplastic model described sand behaviour through a rate independent constitutive relation, i.e. the cutting speed should not show any significant effect on the cutting forces. However from the kinetics point of view, the inertia of the cutting blade and the inertia of the sand should have importance for sandy, frictional soils [7]. Hence moving the blade with different constant velocities (zero acceleration) should not affect the cutting forces but accelerating the blade at different rates should affect the cutting forces. 4.1. Model validation 4.1.1. Predefined failure surfaces validity The validity of the assumed predefined failure surfaces was examined first. The von Mises stress distribution at zero blade displacement and the vertical (X–Y) shear stress distribution at 50 mm blade displacement were plotted in Fig. 4a and b, respectively. The continuity of the contours plotted in Fig. 4a reveals the homogeneity of the stress distribution after the soil gravity effect was applied and just before cutting. During cutting the most significant variable that could judge the performance of the contact mechanism is the shear stress distribution. As shown in Fig. 4b, the continuous contours of the shear stress distribution along the predefined failure surfaces reveals the validity of predefined failure surfaces assumption under condition that the gravity effect has to be considered. 4.1.2. Sand failure and shear band formation The formation of a shear band during cutting, characterized by high void ratio, is obvious as a conical zone or a hemisphere around the blade as shown in Fig. 5. Fig. 5 represents a plot of the void ratio distribution of the simulated sand after a blade displacement of 200 mm. Void ratio is a parameter that indicates the soil compaction state (dense or loose). Loose sand is characterized by a high void ratio
Fig. 4. (a) Von Mises stress distribution at zero blade displacement. (b) Vertical shear stress distribution at 50 mm blade displacement.
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and dense sand is characterized by a low void ratio. The blade was set graphically to be virtually transparent in Fig. 5 to enable ease identification of the shear zone. Soil failure around and behind the blade can be easily seen from the figure. 4.2. Effect of cutting speed Fig. 6 represents the progress in draft cutting forces as the blade moves horizontally with different constant velocities the cutting forces increase rapidly at the
Fig. 5. Void ratio distribution after 200 mm of blade displacement.
Fig. 6. Draft force predicted for a narrow blade at several cutting speeds in a sandy soil.
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beginning of blade displacement and are then ‘‘damped out’’ after a while and then increase linearly as the blade moves further. Hence at the beginning of the motion the cutting speed showed a very significant effect on the cutting forces and then this effect vanishes as the blade displaced. The rapid change of the cutting forces with cutting speed at the beginning of the motion may be due to the momentum change between the two contacting bodies, the blade with an initial velocity and the soil at rest. This rapid change in momentum is dependent on the cutting speed and increases as the speed increases. However in general, the cutting speed seems to have no effect on the blade cutting forces after the initial transient phase of momentum transfer. The trend shown in Fig. 6 has been obtained numerically in the literature [27] (Fig. 7). This study was limited to a blade displacement of 5 mm at high cutting speeds ( > 0.1 m/s). This gives a rather unclear view of the effect of cutting speed on cutting forces since the behaviour is dominated by the initial transient response. However in the present model the cutting forces can be predicted through a longer distance of blade displacement, given a clear, less ambiguous view of the effect of cutting speed on cutting forces. This shows both the initial transient response and the subsequent ‘‘steady state’’ die out to the same value of cutting force. 4.3. Effect of cutting acceleration Fig. 8 represents the progress in draft cutting forces as the blade moves horizontally with different accelerations. It is clear that the cutting acceleration forces increase with blade acceleration. This again reveals the significant effect of inertial forces on soil–tool interaction particularly for sandy soil [7].
Fig. 7. Blade draft vs. displacement at different speeds, after [27].
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Fig. 8. Effect of cutting acceleration on draft forces.
5. Conclusion Three-dimensional dynamic finite element analyses have been carried out to simulate soil–tool interaction and study the effect of cutting speed and cutting acceleration on predicted cutting forces. The so called hypoplastic constitutive model was used to describe the behaviour of the simulated sand in monotonic loading during soil–tool interface process. A series of models were analyzed with various cutting speeds and cutting accelerations using three-dimensional models. Results showed the significant effect of cutting acceleration on cutting forces. From the various 3D analyses carried out, some concluding remarks can be made as follows: (1) The concept of predefined failure surfaces has shown to be suitable for modeling the 3D soil–tool interaction problem. (2) The cutting speed has an insignificant effect of the cutting forces. (3) The cutting acceleration has a significant effect on the cutting forces in that as the cutting acceleration increases the cutting forces increase.
Acknowledgements The authors acknowledge the HKS inc. for use of an ABAQUS educational license. The main author also acknowledges the Egyptian government for its support.
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References [1] Yong RN, Hanna AW. Finite element analysis of plane soil cutting. Journal of Terramechanics 1977;14(3):103–25. [2] Spektor M, Katz M. Experimental study of frontal resistance force in soil cutting. Journal of Terramechanics 1985;22(3):127–33. [3] Dechao Z, Yusu Y. A dynamic model for soil cutting by blade and tine. Journal of Terramechanics 1992;29(3):317–27. [4] Qinsen Y, Shuren S. A soil–tool interaction model for bulldozer blades. Journal of Terramechanics 1994;31(2):55–65. [5] AboElnor M, El-Aziz SM, Shokry MK. Experimental study of blade–soil interaction forces. 8th Int. Applied Mechanics and Mechanical Engineering Conference, 1998, Cairo, Egypt. [6] Mouazen AM, Nemenyi M. Finite element analysis of subsoiler cutting in non-homogeneous sandy loam soil. Soil & Tillage Research 1999;51:1–15. [7] Rosa UA, Wulfsohn D. Constitutive model for high speed tillage using narrow tools. Journal of Terramechanics 1999;36:221–34. [8] Swick WC, Perumpral JV. A model for predicting soil–tool interaction. Journal of Terramechanics 1988;25(1):43–56. [9] AboElnor M, El-Aziz SM, Shokry MK. Development of a mathematical model for predicting blade– soil interaction forces. 8th Int. Applied Mechanics and Mechanical Engineering Conference, 1998, Cairo, Egypt. [10] Luengo O, Singh S. Modeling and identification of soil–tool interaction in automated excavation. IEEE/RSJ International Conference on Intelligent Robotic Systems. 1998. Canada, Victoria, BC. [11] Kushwaha RL, Shen J. Finite element analysis of the dynamic interaction between soil and tillage tool. Transactions of the ASAE 1995;37(5):1315–9. [12] Terzaghi K. Theoretical soil–tool mechanics. New York: J. Wiley & Sons Inc.; 1959. [13] Yusu Y, Dechao Z. Investigation of the relationship between soil–metal friction and sliding speed. Journal of Terramechanics 1990;27(4):283–90. [14] Dechao Z, Yusu Y. Investigation on the relationship between soil shear strength and shear rate. Journal of Terramechanics 1991;28(1):1–10. [15] Duncan JM, Chang CY. Nonlinear analysis of stress and strain in soils. J Soil Mech Found Div, ASCE 1970;96(SM5):1629–53. [16] Kolymbas D. Introduction to hypoplasticity. Rotterdam: Balkema; 1999. [17] Bauer E, Huang W. Effect of initial anisotropy on shear banding in granular materials. In: Numerical Models in Geomechanics—NUMOG. Rotterdam: Balkema; 1999. [18] Tejchman J, Herle I, Wehr J. FE-Studies on the influence of initial void ratio, pressure level and mean grain diameter on shear localization. International Journal For Numerical and Analytical Methods in Geomechanics 1999;23:2045–74. [19] Bauer E. Calibration of a comprehensive hypoplastic model for granular materials. Soils and Foundations 1996;36(1):13–26. [20] Wu W, Bauer E, Kolymbas D. Hypoplastic constitutive model with critical state for granular materials. Mechanics of Materials 1996;23:45–69. [21] Wu W. On a simple critical state model for sand. In: Numerical models in geomechanics—NUMOG. Rotterdam: Balkema; 1999. [22] Wolffersdorf P-A. A hypoplastic relation for granular materials with a predefined limit state surface. Mechanics of Cohesive-Frictional Materials 1996;1:251–71. [23] Hibbit, Karlsson, Sorensen. ABAQUS/standard user’s manual, Version 6.1. 2000, Pawtucket, RI, USA. [24] Prisco C-d, Imposimato S. Time dependent mechanical behaviour of loose sands. Mechanics of Cohesive-Frictional Materials 1996;1:45–73. [25] AboElnor M. Advanced numerical techniques to simulate soil–tool interface problems. PhD thesis, Strathclyde University, Glasgow, UK; 2002.
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[26] Herle I, Gudehus G. Determination of parameters of a hypoplastic constitutive model from properties of grain assemblies. Mechanics of Cohesive-Frictional Materials 1999;4:461–86. [27] Shen J, Kushwaha RL. Soil–machine interaction: a finite element perspective. Marcel Dekker, Inc; 1998.
3D Dynamic analysis of soil–tool interaction using the finite element method Mootaz Abo-Elnora, R. Hamiltonb,*, J.T. Boyleb a Military Technical College, Cairo, Egypt Department of Mechanical Engineering, University of Strathclyde, UK
b
Abstract Previous experimental and finite element studies have shown the influence of both soil initial conditions and blade operating conditions on cutting forces. However, most of these finite element analyses (FEA) are limited to small blade displacements to reduce element distortion which can cause solution convergence problems. In this study a dynamic threedimensional FEA of soil–tool interaction was carried out based on predefined failure surfaces to investigate the effect of cutting speed and angle on cutting forces over large blade displacements. Sandy soil was considered in this study and modeled using the hypoplastic constitutive model implemented in the commercial FEA package, ABAQUS. Results reveal the validity of the concept of predefined failure surfaces in simulating soil–tool interaction and the significant effect of cutting acceleration on cutting forces. # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Dynamic soil–tool interaction; Hypoplasticity; Sand modeling
1. Introduction The productivity of mine-clearing and earth-moving equipment, such as motor graders, scrappers and bulldozers, is dependant on the cutting process speed along with other initial conditions (blade geometry, soil–metal coefficient of friction and soil type) and operating conditions (cutting angle and depth). The effect of initial and operating conditions have been shown experimentally to have a great effect on machine productivity [1–7]. However, experimental study of soil–tool interaction is expensive and limited to certain cutting speeds and results are highly dependant on the accuracy of the measuring devices. Several dynamic models of soil–tool interaction have been developed analytically using the limit equilibrium method [3,4,8– * Corresponding author. Tel.: +44-141-548-2046; fax: +44-141-552-2105. E-mail address: [email protected] (R. Hamilton). 0022-4898/$20.00 # 2003 ISTVS. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jterra.2003.09.002
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10]. Due to the difficulty of conducting adequate tests and the limitations of analytical methods, the soil dynamic response to a tool has not been well predicted [11]. However with increasing computational power and development of more sophisticated material models, the finite element technique shows more promise in analyzing soil–tool interaction. The finite element method (FEM) is a relatively new and effective numerical method. Several researchers have performed finite element simulations of the soil–tool interface process and studied factors affecting cutting forces [6–8,12]. Most of the available models are applicable to low cutting speeds (< 5 mm/s). The objective of this study is to investigate the dynamic effects of the soil–tool interface process on cutting forces using the finite element technique. The dynamic effects such as blade cutting speed and acceleration are also taken into account.
2. Dynamic constitutive models for soils The existing models for predicting cutting forces under dynamic conditions are based on the hypothesis that a large portion of the increase in cutting forces due to increasing tool speed can be attributed to (1) the increase of the soil shear strength and soil–metal friction with increasing shear rate [8], (2) the inertial forces arise when soil is accelerated from a state of rest to a certain velocity. Investigation of the relationship between soil–metal friction and sliding speed has been carried out [13]. A model of soil shear strength as a function of shear rate and normal pressure has been developed by Dechao [14] and verified with a series of laboratory tests under shear rate range of 5–30 s1. Experimental results showed that an increase in soil shear strength is intensified by a rise in shear rate. The experimental results also showed that the relation between soil shear strength and normal pressure is non linear at high shear rate. this suggested that the Mohr–Coulomb criterion could only be applied under low shear rate conditions. To account for the effect of the shear rate on soil shear strength and soil–metal friction; dynamic models for soil–tool interaction have been developed [7,11] based on a hyperbolic stress–strain model developed by Duncan and Chang [15]. However, results obtained from a numerical procedure that have used inappropriate constitutive laws can be of limited or doubtful validity. Hence, selecting the most appropriate material model is more likely to produce reasonable results. The mechanical behaviour of granular soils, typically ranging from silt to gravel, can be modeled by various different theories. Hypoplasticity stems from the framework of rational mechanics starting from principal requirements on the model properties; a single equation describing many important features of the behaviour of granular soils was obtained [16]. Based on the general concept of hypoplasticity various aspects of the mechanical behaviour of granular materials have been investigated in the past few years, e.g. shear banding [17,18], rate-dependence and cohesion [19]. Critical states have been incorporated in order to better describe the influence of pressure level and density on the material behaviour [18,20,21]. It has been demonstrated in these papers that the hypoplastic model can describe the dependence of
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the material behaviour on the pressure level and density with a single set of material constants. The Wolffersdorf version of the hypoplastic model [22] was modified for monotonic loading and implemented into the finite element code ABAQUS [23] as a user defined material and used in this study to simulate a fully drained sandy soil. 2.1. Effect of strain rate (rate dependency) The hypoplastic equation is a rate type tensorial function but it is rate independent from the material behaviour point of view. In fact, the mechanical behaviour of granular materials is commonly assumed to be time independent. However, investigations have been done [24] to study the effect of strain rate using standard triaxial tests. Fig. 1 presents the stress–strain relationship for a loose sand specimen at different strain rates. The figure reveals an insignificant effect of the strain rate on the stress in the case of sandy soil. However, in clay soils, draft force, acting on the blade in the horizontal direction, is generally not sensitive to inertial forces, but shear strength increases substantially with increasing shear rates [7].
3. Material and methods The hypoplastic model, implemented in ABAQUS as a user defined material using the UMAT user subroutine, was used to describe sand behaviour. Table 1 represents the hypoplastic model parameters of the simulated sand soil with initial void ratio of e0=0.75.
Fig. 1. Rate effect on sand behaviour, Prisco and Imposimato [24].
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Table 1 Parameters of the hypoplastic model for Karlsruhe sand [26] ’c (deg)
hs (MPa)
n
ed0
ec0
ei0
30
5800
0.28
0.53
0.84
1.0
0.13
1.0
3.1. Model description The 3D model dimensions are presented in Fig. 2 and listed in Table 2, where ‘W1’ is the lateral bound width, ‘W2’ is the blade cutting width, ‘L1’ is the distance between the blade and the left-hand boundary, ‘L’ is the soil bin length, ‘h’ is the soil bin height, ‘d’ is the cutting depth and ‘’ is the cutting angle. W2 and W1 were varied to study the effect of blade width and lateral bound width of the simulated sample respectively as shown later, the remaining dimensions were held constant. A total of 30 separate finite element models were run [25]. Most of the analyses were carried out through 50 mm of blade displacement in the horizontal plane along the X axis direction shown in Fig. 2. A blade displacement of 50 mm was chosen as a compromise to cut down solution times. For a typical 3D model solution time was
Fig. 2. Soil–tool interface model dimensions.
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Table 2 Dimensions of the three-dimensional soil–tool interaction model L1 (mm)
L (mm)
h (mm)
d (mm)
W1 (mm)
W2 (mm)
( )
200
900
700
200
400
500
75
about 50 h to account for 200 mm of blade displacement using a dual Xenon 1 GHz processor PC with 512 MB of memory. However, it takes 18 h to account for 50 mm of blade displacement. To verify the behaviour of the cut soil in front of the cutting blade, one analysis was run for a blade displacement of 200 mm. 3.2. Finite element mesh and boundary conditions A three dimensional 8-node linear brick continuum element (C3D8) [23] was selected to represent both the sand and the blade in the finite element model. 1082 elements were used to simulate the sand, and 288 elements were used to simulate the blade which was defined as a rigid body with a reference node. Simulating the blade using the *RIGID BODY feature in ABAQUS enables the calculation of the resultant reaction forces acting on the entire blade at a single reference node. The *Dynamic solution feature in ABAQUS was used to switch to dynamic simulation. Due to the symmetric geometry of the model, one half of the model was simulated but all the results consider the complete model. Two failure surfaces were predefined, one along the horizontal plane in front of the blade-cutting tip and the other along a vertical plane at a distance of ‘W2/2’ from the symmetric plane and of height ‘d’, i.e. along the blade vertical boundary, as shown in Fig. 2. The concept of master and slave contact in ABAQUS was used to simulate the interface between the cutting blade and the soil; and soil itself along the predefined failure surfaces. Relative motion was allowed with friction along the soil–tool and soil–soil interface surfaces. The model was meshed in a manner that increased the mesh density near the blade and the predefined failure surfaces as shown in Fig. 3. The boundary conditions of the model are (Fig. 3): 1. Bottom base nodes, at Y=0, are fully constrained. 2. Nodes on vertical boundaries parallel to the Y–Z plane, at X=0 and X=L, are constrained in the horizontal direction along X axis. 3. Nodes on vertical boundaries parallel to the X–Y plane, at Z=0 and Z ¼ ðW2=2Þ þ W1 (symmetry plane), are constrained in the lateral direction along Z axis. 4. The blade is constrained in the vertical direction and from any rotation but it is free to move in the horizontal direction. Gravity effect was taken into account by applying the gravity acceleration as a body load to simulate the sand weight for a single step. This allowed contact to establish between the soil–blade and soil–soil interface.
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Fig. 3. Finite element mesh and boundary conditions.
3.3. Model validation The validation of the present 3D soil–tool interface model was carried out through investigation of various aspects as follows: predefined failure surfaces validation; sand failure and shear band formation.
3.4. Effect of cutting speed To investigate the effect of cutting speed on blade cutting forces, a series of finite element models were carried out at different constant cutting speeds (V =10, 30, 50, 100, 200 mm/s) through a blade displacement of 50 mm in the horizontal direction along the X axis. 3.5. Effect of cutting acceleration To investigate the effect of cutting acceleration on blade cutting forces, a series of finite element models were carried out at different cutting accelerations (a=1, 9, 25, 100, 400 mm/s2) through a blade displacement of 50 mm in the horizontal direction along the X-axis. The blade was assumed to be initially at rest.
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4. Results and discussion As mentioned before, the hypoplastic model described sand behaviour through a rate independent constitutive relation, i.e. the cutting speed should not show any significant effect on the cutting forces. However from the kinetics point of view, the inertia of the cutting blade and the inertia of the sand should have importance for sandy, frictional soils [7]. Hence moving the blade with different constant velocities (zero acceleration) should not affect the cutting forces but accelerating the blade at different rates should affect the cutting forces. 4.1. Model validation 4.1.1. Predefined failure surfaces validity The validity of the assumed predefined failure surfaces was examined first. The von Mises stress distribution at zero blade displacement and the vertical (X–Y) shear stress distribution at 50 mm blade displacement were plotted in Fig. 4a and b, respectively. The continuity of the contours plotted in Fig. 4a reveals the homogeneity of the stress distribution after the soil gravity effect was applied and just before cutting. During cutting the most significant variable that could judge the performance of the contact mechanism is the shear stress distribution. As shown in Fig. 4b, the continuous contours of the shear stress distribution along the predefined failure surfaces reveals the validity of predefined failure surfaces assumption under condition that the gravity effect has to be considered. 4.1.2. Sand failure and shear band formation The formation of a shear band during cutting, characterized by high void ratio, is obvious as a conical zone or a hemisphere around the blade as shown in Fig. 5. Fig. 5 represents a plot of the void ratio distribution of the simulated sand after a blade displacement of 200 mm. Void ratio is a parameter that indicates the soil compaction state (dense or loose). Loose sand is characterized by a high void ratio
Fig. 4. (a) Von Mises stress distribution at zero blade displacement. (b) Vertical shear stress distribution at 50 mm blade displacement.
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and dense sand is characterized by a low void ratio. The blade was set graphically to be virtually transparent in Fig. 5 to enable ease identification of the shear zone. Soil failure around and behind the blade can be easily seen from the figure. 4.2. Effect of cutting speed Fig. 6 represents the progress in draft cutting forces as the blade moves horizontally with different constant velocities the cutting forces increase rapidly at the
Fig. 5. Void ratio distribution after 200 mm of blade displacement.
Fig. 6. Draft force predicted for a narrow blade at several cutting speeds in a sandy soil.
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beginning of blade displacement and are then ‘‘damped out’’ after a while and then increase linearly as the blade moves further. Hence at the beginning of the motion the cutting speed showed a very significant effect on the cutting forces and then this effect vanishes as the blade displaced. The rapid change of the cutting forces with cutting speed at the beginning of the motion may be due to the momentum change between the two contacting bodies, the blade with an initial velocity and the soil at rest. This rapid change in momentum is dependent on the cutting speed and increases as the speed increases. However in general, the cutting speed seems to have no effect on the blade cutting forces after the initial transient phase of momentum transfer. The trend shown in Fig. 6 has been obtained numerically in the literature [27] (Fig. 7). This study was limited to a blade displacement of 5 mm at high cutting speeds ( > 0.1 m/s). This gives a rather unclear view of the effect of cutting speed on cutting forces since the behaviour is dominated by the initial transient response. However in the present model the cutting forces can be predicted through a longer distance of blade displacement, given a clear, less ambiguous view of the effect of cutting speed on cutting forces. This shows both the initial transient response and the subsequent ‘‘steady state’’ die out to the same value of cutting force. 4.3. Effect of cutting acceleration Fig. 8 represents the progress in draft cutting forces as the blade moves horizontally with different accelerations. It is clear that the cutting acceleration forces increase with blade acceleration. This again reveals the significant effect of inertial forces on soil–tool interaction particularly for sandy soil [7].
Fig. 7. Blade draft vs. displacement at different speeds, after [27].
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Fig. 8. Effect of cutting acceleration on draft forces.
5. Conclusion Three-dimensional dynamic finite element analyses have been carried out to simulate soil–tool interaction and study the effect of cutting speed and cutting acceleration on predicted cutting forces. The so called hypoplastic constitutive model was used to describe the behaviour of the simulated sand in monotonic loading during soil–tool interface process. A series of models were analyzed with various cutting speeds and cutting accelerations using three-dimensional models. Results showed the significant effect of cutting acceleration on cutting forces. From the various 3D analyses carried out, some concluding remarks can be made as follows: (1) The concept of predefined failure surfaces has shown to be suitable for modeling the 3D soil–tool interaction problem. (2) The cutting speed has an insignificant effect of the cutting forces. (3) The cutting acceleration has a significant effect on the cutting forces in that as the cutting acceleration increases the cutting forces increase.
Acknowledgements The authors acknowledge the HKS inc. for use of an ABAQUS educational license. The main author also acknowledges the Egyptian government for its support.
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References [1] Yong RN, Hanna AW. Finite element analysis of plane soil cutting. Journal of Terramechanics 1977;14(3):103–25. [2] Spektor M, Katz M. Experimental study of frontal resistance force in soil cutting. Journal of Terramechanics 1985;22(3):127–33. [3] Dechao Z, Yusu Y. A dynamic model for soil cutting by blade and tine. Journal of Terramechanics 1992;29(3):317–27. [4] Qinsen Y, Shuren S. A soil–tool interaction model for bulldozer blades. Journal of Terramechanics 1994;31(2):55–65. [5] AboElnor M, El-Aziz SM, Shokry MK. Experimental study of blade–soil interaction forces. 8th Int. Applied Mechanics and Mechanical Engineering Conference, 1998, Cairo, Egypt. [6] Mouazen AM, Nemenyi M. Finite element analysis of subsoiler cutting in non-homogeneous sandy loam soil. Soil & Tillage Research 1999;51:1–15. [7] Rosa UA, Wulfsohn D. Constitutive model for high speed tillage using narrow tools. Journal of Terramechanics 1999;36:221–34. [8] Swick WC, Perumpral JV. A model for predicting soil–tool interaction. Journal of Terramechanics 1988;25(1):43–56. [9] AboElnor M, El-Aziz SM, Shokry MK. Development of a mathematical model for predicting blade– soil interaction forces. 8th Int. Applied Mechanics and Mechanical Engineering Conference, 1998, Cairo, Egypt. [10] Luengo O, Singh S. Modeling and identification of soil–tool interaction in automated excavation. IEEE/RSJ International Conference on Intelligent Robotic Systems. 1998. Canada, Victoria, BC. [11] Kushwaha RL, Shen J. Finite element analysis of the dynamic interaction between soil and tillage tool. Transactions of the ASAE 1995;37(5):1315–9. [12] Terzaghi K. Theoretical soil–tool mechanics. New York: J. Wiley & Sons Inc.; 1959. [13] Yusu Y, Dechao Z. Investigation of the relationship between soil–metal friction and sliding speed. Journal of Terramechanics 1990;27(4):283–90. [14] Dechao Z, Yusu Y. Investigation on the relationship between soil shear strength and shear rate. Journal of Terramechanics 1991;28(1):1–10. [15] Duncan JM, Chang CY. Nonlinear analysis of stress and strain in soils. J Soil Mech Found Div, ASCE 1970;96(SM5):1629–53. [16] Kolymbas D. Introduction to hypoplasticity. Rotterdam: Balkema; 1999. [17] Bauer E, Huang W. Effect of initial anisotropy on shear banding in granular materials. In: Numerical Models in Geomechanics—NUMOG. Rotterdam: Balkema; 1999. [18] Tejchman J, Herle I, Wehr J. FE-Studies on the influence of initial void ratio, pressure level and mean grain diameter on shear localization. International Journal For Numerical and Analytical Methods in Geomechanics 1999;23:2045–74. [19] Bauer E. Calibration of a comprehensive hypoplastic model for granular materials. Soils and Foundations 1996;36(1):13–26. [20] Wu W, Bauer E, Kolymbas D. Hypoplastic constitutive model with critical state for granular materials. Mechanics of Materials 1996;23:45–69. [21] Wu W. On a simple critical state model for sand. In: Numerical models in geomechanics—NUMOG. Rotterdam: Balkema; 1999. [22] Wolffersdorf P-A. A hypoplastic relation for granular materials with a predefined limit state surface. Mechanics of Cohesive-Frictional Materials 1996;1:251–71. [23] Hibbit, Karlsson, Sorensen. ABAQUS/standard user’s manual, Version 6.1. 2000, Pawtucket, RI, USA. [24] Prisco C-d, Imposimato S. Time dependent mechanical behaviour of loose sands. Mechanics of Cohesive-Frictional Materials 1996;1:45–73. [25] AboElnor M. Advanced numerical techniques to simulate soil–tool interface problems. PhD thesis, Strathclyde University, Glasgow, UK; 2002.
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[26] Herle I, Gudehus G. Determination of parameters of a hypoplastic constitutive model from properties of grain assemblies. Mechanics of Cohesive-Frictional Materials 1999;4:461–86. [27] Shen J, Kushwaha RL. Soil–machine interaction: a finite element perspective. Marcel Dekker, Inc; 1998.