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A simulation based on the cosserat continuum model of the vortex structure in granular materials
Acta Mechanica Solida Sinica, Vol. 28, No. 2, April, 2015 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
A SIMULATION BASED ON THE COSSERAT CONTINUUM MODEL OF THE VORTEX STRUCTURE IN GRANULAR MATERIALS Cun Yu
Xihua Chu
Yuanjie Xu
(Department of Engineering Mechanics, Wuhan University, Wuhan 430072, China)
Received 24 June 2013, revision received 6 February 2015
ABSTRACT Displacement fluctuation is the difference between the real displacement and the affine displacement in deforming granular materials. The discrete element method (DEM) is widely used along with experimental approaches to investigate whether the displacement fluctuation represents the vortex structure. Current research suggests that the vortex structure is caused by the cooperative motion of particle groups on meso-scales, which results in strain localization in granular materials. In this brief article, we investigate the vortex structure using the finite element method (FEM) based on the Cosserat continuum model. The numerical example focuses on the relationship between the vortex structure and the shear bands under two conditions: (a) uniform granular materials; (b) granular materials with inclusions. When compared with distributions of the effective strain and the vortex structure, we find that the vortex structure coexists with the strain localization and originates from the stiffness cooperation of different locations in granular materials at the macro level.
KEY WORDS granular materials, vortex structure, strain localization, Cosserat continuum model
I. INTRODUCTION Recently, some researchers using the discrete element method (DEM) have found that granular materials exhibit deformation fluctuations during shearing that arises largely from meso-scale particle interactions of various forms. Williams and Rege[1, 2] suggested that the concept of circulation cells were found to exist at all stages of deformation and thought to be related to shear bands. Kuhn[3] also observed circulation cells, and while a relationship between microbands and circulation cells was hypothesized, the precise nature of that relationship was considered too complex to identify. Thornton and Zhang, and Tordesillas[4, 5] also observed vortices and microbands to form in association with the onset of the peak stress. When the particles were allowed to rotate, they observed three coexisting deformational phases: vorticity cells, rotational bearings, and slip bands. The vortex structures, however, were thought to arise only spontaneously. Tordesillas et al.[4] added the effect of rolling resistance between circular particles and found well formed, co-rotating vortices with slip microbands between them. Buckling force chains were found through the middle and around such vortices. Alonso-Marroquin et al.[6] simulated circular particles at two extremes and compared their behaviors between the two extremes: fully allowing for particle rolling; and precluding particle rolling. The presence of vortices has
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11172216 and 11472196) and the Natural Key Basic Research and Development Program of China (973 Program) (Nos. 2010CB731502 and 2010CB732005).
Vol. 28, No. 2
Cun Yu et al.: A Simulation of the Vortex Structure in Granular Materials
· 183 ·
also been observed experimentally. Utter and Behringer[7] found vortices to appear only occasionally, while quickly dissipating, and the vortices were not considered to have a significant effect on long-term behavior. Using the plane strain test Abedi and Rechenmacher et al.[8] found vortex formation and dissolution in sheared sand. However, all these observations about the vortex structure were based on the DEM on the meso-scale and the experiments on the macro-scale. Kozicki and Niedostakiewicz et al.[9] compared the results of discrete element and finite element simulations of a simple shear test for medium density cohesionless sand, providing useful information about the limitations and possible advantages of micro-polar continuum models for granular media. In this brief article, we observe the deformation fluctuations on the macro scale using the FEM based on the Cosserat continuum model. On the macro scale vortex structures are obvious and closely related to the strain localization.
II. COSSERAT MODEL AND THE DISPLACEMENT FLUCTUATION In the two-dimensional Cosserat model, the stress and strain vector can be defined as σ = σxx
σyy
σzz
σxy
σyx
ε=
εxx
εyy
εzz
εxy
εyx
mzx lc κzx lc
T mzy lc T
κzy lc
(1) (2)
where κzx and κzy are the micro-curvatures in the Cosserat model, mzx and mzy are the coupled stresses conjugated to κzx and κzy , respectively, and lc is the internal length scale. We apply the FEM based on the Cosserat continuum model to observing the displacement fluctuations at all nodes. During the plate compression simulation, we can directly obtain the average strain in the vertical direction (εay ) as follows: Δuy (3) εay = h where Δuy is the loading displacement and h is the height of the plate. And the average volume strain (εaV ) is the arithmetic mean of the values in all integral points εaV =
N 1 p ε N n=1 Vn
(4)
where N is the total number of integral points and εpV is the volume strain at the integral point. As we know, the volume strain is εV = εii (5) So in our simulation, the average volume strain should satisfy the following equation: εaV = εax + εay
(6)
Then we can deduce the average strain in the horizontal direction (εax ) according to Eqs.(3) and (5): εax = εaV − εay =
N 1 p Δuy εVn − N n=1 h
(7)
In the plate we take the geometric center as the reference point, and ln is the branch vector between node n and the reference point. Then we calculate the displacement fluctuation (unr ) at every node by unri = uni − lin εai
(8)
where i stands for x, y and uni , lin εai mean the real displacement and the affine displacement at n node, respectively.
· 184 ·
ACTA MECHANICA SOLIDA SINICA
2015
III. NUMERICAL EXAMPLE
As an example we take a square plate 10 m on a side which is meshed into 24 × 24 elements. The plate is subjected to uniaxial compression between two rigid plates controlled by the vertical displacement as shown in Fig.1. The horizontal displacements of the finite element nodes on the top and bottom boundary are fixed at zero. Material parameters are as follows[10] : E = 5.0 × 107 N/m2 , υ = 0.3, GC = 1.0 × 107 N/m2 , lc = 0.15 m, c0 = 1.5 × 105 N/m2 , hp = −1.5 × 105 N/m2 . In this work, we consider two different conditions: (a) no inclusion in the plate; (b) two inclusions A and B in the plate (the elements’ elastic modulus in the inclusions zone is smaller than other zones, 5.0 × 105 N/m2 ). From Figs. 2 and 3, we can see that there are obvious vortex structures under the two conditions. And it can be seen that the vortex structure is similar to the effective plastic strain distribution. Figures 4 and 5 confirm the relations between them, and we find that the vortex structure still Fig. 1 The notion of the square exists under the condition that there is no plastic strain, yet the elastic plate subjected to uniaxial compresstrain is much smaller. We mainly compare the vortex structure with the sion. effective plastic strain distribution. Since no vortex structure forms in the plate when strain localization does not occur, we can then conclude that the vortex structure is related to the strain localization. The elements’ stiffness cooperation has effects on the real displacement at all nodes and makes a contribution to the displacement fluctuation. As the loading displacement goes from 0.002 m to 0.2 m, the vortex center’s shape and position change with the strain localization’s development, and it can be observed that the magnitude of the vortex center is smaller than that in other zones. In two cases, strain localizations initially form at each corner and then gradually link through ((a) homogeneous and two shear bands both link through; (b) with inclusions, so just one shear band links through) with an increase in the loading displacement . However, the vortex structure does not link through.
Fig. 2. The plate subjected to the loading displacement 0.2 m for the condition with no inclusion.
Fig. 3. The plate subjected to the loading displacement 0.2 m for the condition with two inclusions in the plate.
Vol. 28, No. 2
Cun Yu et al.: A Simulation of the Vortex Structure in Granular Materials
· 185 ·
Fig. 4. The plate subjected to the loading displacement 0.002 m for the condition with no inclusion.
Fig. 5. The plate subjected to the loading displacement 0.002 m for the condition with two inclusions in the plate.
Then, shown in Figs.6 and 7 is the displacement fluctuation vector at different times for each of the two conditions when the plate is subjected to different loading displacements: (a) 0.002 m, (b) 0.036 m, (c) 0.2 m. For the condition with no inclusion, as shown in Fig.6, the vortex center’s zone gradually increases from 0.002 m to 0.2 m and remains steady after 0.2 m. For the other condition, the evolution of the vortex structure is more complex. Under the small loading displacement (0.036 m), there is an obvious vortex structure in the middle of the diagonal line including two inclusions, and this vortex gradually disappears as the loading displacement increases. The vortex structure at each corner also remains steady after 0.2 m.
Fig. 6. The displacement fluctuation vector when the plate is subjected to the different loading displacement for the condition with no inclusion.
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ACTA MECHANICA SOLIDA SINICA
2015
Fig. 7. The displacement fluctuation vector when the plate is subjected to the different loading displacement for the condition with two inclusions in the plate.
IV. CONCLUSION In summary, the FEM based on the Cosserat continuum model is used to investigate the displacement fluctuation in the plain plate. It has been found that there are obvious vortex structures formed during the simulation and the result show that the vortex structure is related to the strain localization. According to these results, it is noted that the vortex structures mainly occur in shear bands, and gradually change as the strain localization evolves.
References [1] Williams,J.R. and Rege,N., Coherent vortex structures in deforming granular materials. Mechanics of Cohesive-frictional Materials, 1997, 2(3): 223-236. [2] Williams,J.R. and Rege,N., The development of circulation cell structures in granular materials undergoing compression. Powder Technology, 1997, 90(3): 187-194. [3] Kuhn,M.R., Structured deformation in granular materials. Mechanics of Materials, 1999, 31(6): 407-429. [4] Tordesillas,A., Force chain buckling, unjamming transitions and shear banding in dense granular assemblies. Philosophical. Magazine, 2007, 7(32): 4987-5016. [5] Tordesillas,A., Muthuswamy,M. and Walsh,S.D.C., Mesoscale measuers of nonaffine deformation in dense granular assemblies. Journal of Engineering Mechanics, 2008, 134(12): 1095-1113. [6] Alonso-Marroquin,F., Vardoulakis,I. and Herrmann,H.J., Effect of rolling on dissipation in fault gouges. Physical Review E, 2006, 74(3): 031306. [7] Utter,B. and Behringer,R.P., Self-diffusion in dense granular shear Flows. Physical Review E, 2004, 69(3): 031308-1-031308-12. [8] Abedi,S., Rechenmacher,A.L. and Orlando,A.D., Vortex formation and dissolution in sheared sands. Granular Matter, 2012, 14: 695-705. [9] Kozicki,J., Niedostatkiewicz,M., Tejchman,J. and Muhlhaus,H., Discrete modelling results of a direct shear test for granular materials versus FE results. Granular Matter, 2013, 15: 607-627. [10] Li,X.K. and Tang,H.X., Pressure-dependent elasto-plastic Cosserat continuum model and finite element simulation of strain. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(9): 1497-1505.
ISSN 0894-9166
A SIMULATION BASED ON THE COSSERAT CONTINUUM MODEL OF THE VORTEX STRUCTURE IN GRANULAR MATERIALS Cun Yu
Xihua Chu
Yuanjie Xu
(Department of Engineering Mechanics, Wuhan University, Wuhan 430072, China)
Received 24 June 2013, revision received 6 February 2015
ABSTRACT Displacement fluctuation is the difference between the real displacement and the affine displacement in deforming granular materials. The discrete element method (DEM) is widely used along with experimental approaches to investigate whether the displacement fluctuation represents the vortex structure. Current research suggests that the vortex structure is caused by the cooperative motion of particle groups on meso-scales, which results in strain localization in granular materials. In this brief article, we investigate the vortex structure using the finite element method (FEM) based on the Cosserat continuum model. The numerical example focuses on the relationship between the vortex structure and the shear bands under two conditions: (a) uniform granular materials; (b) granular materials with inclusions. When compared with distributions of the effective strain and the vortex structure, we find that the vortex structure coexists with the strain localization and originates from the stiffness cooperation of different locations in granular materials at the macro level.
KEY WORDS granular materials, vortex structure, strain localization, Cosserat continuum model
I. INTRODUCTION Recently, some researchers using the discrete element method (DEM) have found that granular materials exhibit deformation fluctuations during shearing that arises largely from meso-scale particle interactions of various forms. Williams and Rege[1, 2] suggested that the concept of circulation cells were found to exist at all stages of deformation and thought to be related to shear bands. Kuhn[3] also observed circulation cells, and while a relationship between microbands and circulation cells was hypothesized, the precise nature of that relationship was considered too complex to identify. Thornton and Zhang, and Tordesillas[4, 5] also observed vortices and microbands to form in association with the onset of the peak stress. When the particles were allowed to rotate, they observed three coexisting deformational phases: vorticity cells, rotational bearings, and slip bands. The vortex structures, however, were thought to arise only spontaneously. Tordesillas et al.[4] added the effect of rolling resistance between circular particles and found well formed, co-rotating vortices with slip microbands between them. Buckling force chains were found through the middle and around such vortices. Alonso-Marroquin et al.[6] simulated circular particles at two extremes and compared their behaviors between the two extremes: fully allowing for particle rolling; and precluding particle rolling. The presence of vortices has
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11172216 and 11472196) and the Natural Key Basic Research and Development Program of China (973 Program) (Nos. 2010CB731502 and 2010CB732005).
Vol. 28, No. 2
Cun Yu et al.: A Simulation of the Vortex Structure in Granular Materials
· 183 ·
also been observed experimentally. Utter and Behringer[7] found vortices to appear only occasionally, while quickly dissipating, and the vortices were not considered to have a significant effect on long-term behavior. Using the plane strain test Abedi and Rechenmacher et al.[8] found vortex formation and dissolution in sheared sand. However, all these observations about the vortex structure were based on the DEM on the meso-scale and the experiments on the macro-scale. Kozicki and Niedostakiewicz et al.[9] compared the results of discrete element and finite element simulations of a simple shear test for medium density cohesionless sand, providing useful information about the limitations and possible advantages of micro-polar continuum models for granular media. In this brief article, we observe the deformation fluctuations on the macro scale using the FEM based on the Cosserat continuum model. On the macro scale vortex structures are obvious and closely related to the strain localization.
II. COSSERAT MODEL AND THE DISPLACEMENT FLUCTUATION In the two-dimensional Cosserat model, the stress and strain vector can be defined as σ = σxx
σyy
σzz
σxy
σyx
ε=
εxx
εyy
εzz
εxy
εyx
mzx lc κzx lc
T mzy lc T
κzy lc
(1) (2)
where κzx and κzy are the micro-curvatures in the Cosserat model, mzx and mzy are the coupled stresses conjugated to κzx and κzy , respectively, and lc is the internal length scale. We apply the FEM based on the Cosserat continuum model to observing the displacement fluctuations at all nodes. During the plate compression simulation, we can directly obtain the average strain in the vertical direction (εay ) as follows: Δuy (3) εay = h where Δuy is the loading displacement and h is the height of the plate. And the average volume strain (εaV ) is the arithmetic mean of the values in all integral points εaV =
N 1 p ε N n=1 Vn
(4)
where N is the total number of integral points and εpV is the volume strain at the integral point. As we know, the volume strain is εV = εii (5) So in our simulation, the average volume strain should satisfy the following equation: εaV = εax + εay
(6)
Then we can deduce the average strain in the horizontal direction (εax ) according to Eqs.(3) and (5): εax = εaV − εay =
N 1 p Δuy εVn − N n=1 h
(7)
In the plate we take the geometric center as the reference point, and ln is the branch vector between node n and the reference point. Then we calculate the displacement fluctuation (unr ) at every node by unri = uni − lin εai
(8)
where i stands for x, y and uni , lin εai mean the real displacement and the affine displacement at n node, respectively.
· 184 ·
ACTA MECHANICA SOLIDA SINICA
2015
III. NUMERICAL EXAMPLE
As an example we take a square plate 10 m on a side which is meshed into 24 × 24 elements. The plate is subjected to uniaxial compression between two rigid plates controlled by the vertical displacement as shown in Fig.1. The horizontal displacements of the finite element nodes on the top and bottom boundary are fixed at zero. Material parameters are as follows[10] : E = 5.0 × 107 N/m2 , υ = 0.3, GC = 1.0 × 107 N/m2 , lc = 0.15 m, c0 = 1.5 × 105 N/m2 , hp = −1.5 × 105 N/m2 . In this work, we consider two different conditions: (a) no inclusion in the plate; (b) two inclusions A and B in the plate (the elements’ elastic modulus in the inclusions zone is smaller than other zones, 5.0 × 105 N/m2 ). From Figs. 2 and 3, we can see that there are obvious vortex structures under the two conditions. And it can be seen that the vortex structure is similar to the effective plastic strain distribution. Figures 4 and 5 confirm the relations between them, and we find that the vortex structure still Fig. 1 The notion of the square exists under the condition that there is no plastic strain, yet the elastic plate subjected to uniaxial compresstrain is much smaller. We mainly compare the vortex structure with the sion. effective plastic strain distribution. Since no vortex structure forms in the plate when strain localization does not occur, we can then conclude that the vortex structure is related to the strain localization. The elements’ stiffness cooperation has effects on the real displacement at all nodes and makes a contribution to the displacement fluctuation. As the loading displacement goes from 0.002 m to 0.2 m, the vortex center’s shape and position change with the strain localization’s development, and it can be observed that the magnitude of the vortex center is smaller than that in other zones. In two cases, strain localizations initially form at each corner and then gradually link through ((a) homogeneous and two shear bands both link through; (b) with inclusions, so just one shear band links through) with an increase in the loading displacement . However, the vortex structure does not link through.
Fig. 2. The plate subjected to the loading displacement 0.2 m for the condition with no inclusion.
Fig. 3. The plate subjected to the loading displacement 0.2 m for the condition with two inclusions in the plate.
Vol. 28, No. 2
Cun Yu et al.: A Simulation of the Vortex Structure in Granular Materials
· 185 ·
Fig. 4. The plate subjected to the loading displacement 0.002 m for the condition with no inclusion.
Fig. 5. The plate subjected to the loading displacement 0.002 m for the condition with two inclusions in the plate.
Then, shown in Figs.6 and 7 is the displacement fluctuation vector at different times for each of the two conditions when the plate is subjected to different loading displacements: (a) 0.002 m, (b) 0.036 m, (c) 0.2 m. For the condition with no inclusion, as shown in Fig.6, the vortex center’s zone gradually increases from 0.002 m to 0.2 m and remains steady after 0.2 m. For the other condition, the evolution of the vortex structure is more complex. Under the small loading displacement (0.036 m), there is an obvious vortex structure in the middle of the diagonal line including two inclusions, and this vortex gradually disappears as the loading displacement increases. The vortex structure at each corner also remains steady after 0.2 m.
Fig. 6. The displacement fluctuation vector when the plate is subjected to the different loading displacement for the condition with no inclusion.
· 186 ·
ACTA MECHANICA SOLIDA SINICA
2015
Fig. 7. The displacement fluctuation vector when the plate is subjected to the different loading displacement for the condition with two inclusions in the plate.
IV. CONCLUSION In summary, the FEM based on the Cosserat continuum model is used to investigate the displacement fluctuation in the plain plate. It has been found that there are obvious vortex structures formed during the simulation and the result show that the vortex structure is related to the strain localization. According to these results, it is noted that the vortex structures mainly occur in shear bands, and gradually change as the strain localization evolves.
References [1] Williams,J.R. and Rege,N., Coherent vortex structures in deforming granular materials. Mechanics of Cohesive-frictional Materials, 1997, 2(3): 223-236. [2] Williams,J.R. and Rege,N., The development of circulation cell structures in granular materials undergoing compression. Powder Technology, 1997, 90(3): 187-194. [3] Kuhn,M.R., Structured deformation in granular materials. Mechanics of Materials, 1999, 31(6): 407-429. [4] Tordesillas,A., Force chain buckling, unjamming transitions and shear banding in dense granular assemblies. Philosophical. Magazine, 2007, 7(32): 4987-5016. [5] Tordesillas,A., Muthuswamy,M. and Walsh,S.D.C., Mesoscale measuers of nonaffine deformation in dense granular assemblies. Journal of Engineering Mechanics, 2008, 134(12): 1095-1113. [6] Alonso-Marroquin,F., Vardoulakis,I. and Herrmann,H.J., Effect of rolling on dissipation in fault gouges. Physical Review E, 2006, 74(3): 031306. [7] Utter,B. and Behringer,R.P., Self-diffusion in dense granular shear Flows. Physical Review E, 2004, 69(3): 031308-1-031308-12. [8] Abedi,S., Rechenmacher,A.L. and Orlando,A.D., Vortex formation and dissolution in sheared sands. Granular Matter, 2012, 14: 695-705. [9] Kozicki,J., Niedostatkiewicz,M., Tejchman,J. and Muhlhaus,H., Discrete modelling results of a direct shear test for granular materials versus FE results. Granular Matter, 2013, 15: 607-627. [10] Li,X.K. and Tang,H.X., Pressure-dependent elasto-plastic Cosserat continuum model and finite element simulation of strain. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(9): 1497-1505.