A local constitutive model for the discrete element method. Application to geomaterials and concrete
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Abstract
This paper presents a local constitutive model for modelling the linear and non linear behavior of soft and hard cohesive materials with the discrete element method (DEM). We present the results obtained in the analysis with the DEM of cylindrical samples of cement, concrete and shale rock materials under a uniaxial compressive strength test, different triaxial tests, a uniaxial strain compaction test and a Brazilian tensile strength test. DEM results compare well with the experimental values in all cases.
Keywords
Local constitutive model Discrete element method Geomaterials Concrete1 Introduction
Extensive research work on the discrete element method (DEM) has been carried out in the last decades since the first ideas were presented by Cundall and [11]. Much of the research efforts have focused on the development of adequate DEM models for accurately reproducing the correct behaviour of non cohesive and cohesive granular assemblies [1, 6, 8, 11, 20, 24, 29, 42, 45, 46, 47], as well as of solid materials [13, 14, 16, 17, 18, 22, 23, 28, 31, 32, 34, 35, 36, 42, 43]. In recent years the DEM has also been effectively applied to the study of multifracture and failure of geomaterials (soils and rocks), concrete, masonry and ceramic materials, among others.
The analysis of solids with the DEM poses a number of difficulties for adequate reproducing the correct constitutive behaviour of the material under linear (elastic) and non linear conditions.
Within the analysis of solids with the DEM the material is typically represented as a collection of rigid particles [spheres in three dimensions (3D) and discs in two dimensions (2D)] interacting among themselves at the contact interfaces in the normal and tangential directions. Material deformation is assumed to be concentrated at the contact points. Appropriate contact laws are defined in order to obtain the desired macroscopic material properties. The contact law can be seen as the formulation of the material model of the underlying continuum at the microscopic level. For frictional cohesive material the contact law takes into account the cohesive bonds between rigid particles. Cohesive bonds can be broken, thus allowing to simulate fracture of the material and its propagation.
A challenge in the failure analysis of solid materials, such as cement, shale rock and concrete, with the DEM is the definition of the limit strengths in the normal and shear directions at the contact interfaces, and the characterization of the non linear relationship between forces and displacements at these interfaces beyond the onset of fracture, accounting for frictional effects, damage and plasticity.
In this work we present a local constitutive model for failure analysis of solid materials typical in geomechanics and concrete applications with the DEM. The model is validated in the analysis of cement, concrete and shale rock samples for several laboratory strength tests. The tests considered include the uniaxial compression strength (UCS) test, triaxial compressive strength tests, the uniaxial strain compaction (USC) test and the Brazilian tensile strength (BTS) test. DEM results compare well with experimental data provided by Weatherford for the cement and shale rock samples [19, 33] and the Technical University of Catalonia (UPC) for the concrete samples [37].
2 Constitutive models for the dem
Standard constitutive models in the DEM are typically characterized by the following parameters:

Normal and shear stiffness parameters \(K_n\) and \(K_s\).

Normal and shear strength parameters \(\mathcal{F}_n\) and \(\mathcal{F}_s\).

Coulomb friction coefficient \(\mu \).

Local damping coefficient \(C_n\).

Global damping coefficient for the translational motion, \(\alpha ^t\).

Global damping coefficient for the rotational motion, \(\alpha ^n\).
The challenge in DEM models for analysis of solids is finding an objective and accurate relationship between the DEM parameters and the standard constitutive parameters of a continuum mechanics model (hereafter called “continuum parameters”): the Young modulus \(E\), the Poisson ratio \(\nu \) and the tension and shear failure stresses \(\sigma _t^f\) and \(\tau ^f\), respectively.
Two different approaches can be followed for determining the DEM constitutive parameters for a cohesive material, namely the global approach and the local approach. In the global approach uniform global DEM properties are assumed for each contact interface in the whole discrete element assembly. The values of the global DEM parameters can be found via different procedures. Some authors have derived analytical relationships between continuum and global DEM parameters [24, 25]. Others have used numerical experiments for determining the relationships between DEM and continuum parameters expressed in dimensionless form [14, 17, 18]. This method has been used by the authors in previous works [21, 22, 23, 31, 34, 35, 36]. Other procedures are based on relating the global DEM and continuum parameters via laboratory tests using inverse analysis techniques [29].
The local approach used in this work assumes that the DEM parameters depend on the local properties of the interacting particles, namely their radii and the continuum parameters at each interaction point. Different alternatives for defining the DEM parameters via a “local approach” have been reported in recent years [13, 14, 16, 31, 32, 42, 43]. A comparative study of several global and local approaches for estimating the DEM constitutive parameters is presented in [36].
In this work we present a new procedure for defining the DEM parameters for a cohesive material in the framework of the local approach. In the next section we describe how the local elastic parameters can be found. Then we define appropriate local failure criteria at the contact interface using an elastodamage model for the normal tensile stress and the shear stress, and an elastoplastic model for the normal compressive stress. The accuracy of the local DEM constitutive model is verified in the analysis of laboratory strength tests for different cohesive materials.
The DEM model presented here can be considered as an extension of that proposed by Donzé and coworkers [13, 16, 38, 43]. Among the distinct features of our model we note the inclusion of the effect of the size of the interacting spheres in the normal and shear parameters, the introduction of a parameter in the constitutive law accounting for the number of contacts and the packaging of particles, the definition of the failure criteria, the estimation of the limit compressive stress at the contact interface, the effect of damage and plasticity in the evolution of the normal and shear parameters and the definition of the material parameters in terms of the uniaxial stress–strain curve obtained from strength tests.
Clearly, the local DEM constitutive model presented in this work is also applicable to standard noncohesive granular material, as a particular case of the more general expressions for the cohesive case. In this sense, the model is able to simulate the frictional behaviour of the particulate material that forms once the bonds between particles are broken.
3 Basic equations
3.1 Equations of motion
The form of the rotational motion of Eq. (2) holds for spheres and cylinders (in 2D) and is simplified with respect to a general form for an arbitrary rigid body with the rotational inertial properties represented by a second order tensor. In the general case it is more convenient to describe the rotational motion with respect to a corotational frame \(\mathbf{x}\) which is embedded at each element, since in this frame the tensor of inertia is constant.
3.2 Integration of the equations of motion
4 Frictional contact conditions
4.1 Contact interface
Let us assume that an individual particle is connected to the adjacent ones by appropriate relationships at the contact interfaces. These relationships define either a perfectly bond or a frictional sliding situation at the interface.
This definition of the contact interface and the interaction domain is motivated by the fact that the two interacting particles can have very different radius for an arbitrary distribution of the particle sizes. The contact interface is thus limited by the size of the smaller of the two particles in contact.
4.2 Interaction range
4.3 Contact search algorithm
Changing contact pairs of elements during the analysis are automatically detected. The simple approach to identify interaction pairs by checking every particle against every other one would be very inefficient, as the computational time is proportional to \(n^2\), where \(n\) is the number of elements. In our formulation the search is performed using a gridbased algorithm. In this case the computation time of the contact search is proportional to \(n\ln {n}\), which allows us to solve large frictional contact systems involving many particles [23].
4.4 Decomposition of the contact force
The relationship between the contact forces \(F_n\), \(F_{s_1}\) and \(F_{s_2}\) and the particle displacements are obtained using the local constitutive model described in the next section.
4.5 Definition of the contact law
In this work we have assumed a proportionality between the normal force at each contact interfaces and the relative displacement and the relative velocity of the contact point. As for the shear force, this has been assumed to be proportional to the relative sliding motion at the contact point. The proportionality coefficients have been estimated starting from the onedimensional stress–strain relationship for the cylindrical contact domain of Fig. 4. This approach has been preferred versus the contact laws proposed by Hertz [15] and Mindlin [27] for modelling the contact interaction between two spheres in the normal and tangential directions, respectively. These laws have been used to model the contact force in granular material with the DEM [4, 46]. For the solids material considered in this work, the existence of a matrix between grains [(modelled via a gap distance, Eq. (16)] prevents in most cases the direct contact between particles, which justifies the more “diffusive” contact model chosen in this work.
A comparison of different contact has shown that simple models in the DEM contact models, such as that presented here, lead to equivalent and, sometimes even better, results than more sophisticated models [12].
5 Local definition of dem elastic constitutive parameters
5.1 Normal force parameters
We note that the Young modulus is assumed to be an intrinsic property of the material. As such it is typically characterized from axial tests on cylindrical samples. Thus, the Young modulus obtained from the experimental axial stress–strain relationship is used for defining the local normal stress–strain relationship at a macroscopic level (23) at the contact interfaces. The same applies to the definition of the Poisson’s ratio of the material used for defining the local shear constitutive relationship in Sect. 5.2.
5.2 Shear force parameters
A similar approach is followed for obtaining the relationship between the shear forces and the relative tangential displacements at each contact interface.
For convenience the upper indices \(i,j\) are omitted hereonwards in the expression of the normal and shear force vectors \(\mathbf{F}_{n}^{ij}\) and \(\mathbf{F}_{s}^{ij}\) at a contact interface.
6 Global background damping force
7 Elastodamage model for tension and shear forces
7.1 Normal and shear failure
Indeed, a coupled failure model in the tensionshear zone can also be used, as shown in Fig. 7b. For the numerical tests presented in the paper the uncoupled model has been used.
7.2 Damage evolution law
Elastic damage under tensile and shear conditions has been taken into account in this work by assuming a linear softening behaviour defined by the softening moduli \(H_n\) and \(H_t\) introduced into the forcedisplacement relationships in the normal (tensile) and shear directions, respectively (Fig. 8).
Damage effects are assumed to start when the strength failure conditions (41) are satisfied. The evolution of the damage parameters from the value zero to one can be defined in a number of ways using fracture mechanics arguments. A key issue is that the area under the line defining the force (relative) displacement relationship in the damaged region (the shadowed area in Fig. 8) equals the specific fracture energy of the material [26, 30, 31].
8 Elastoplastic model for compression forces
The compressive stress–strain behaviour in the normal direction at the contact interface for frictional cohesive materials, such as cement, rock and concrete, is typically governed by an initial elastic law followed by a nonlinear constitutive equation that varies for each material. The compressive normal stress increases under linear elastic conditions until it reaches the limit normal compressive stress \(\sigma _{n_c}^l\) (also called yield stress). This is defined as the axial stress level where the experimental curve relating the axial stress and the axial strain starts to deviate from the linear elastic behaviour. After this point the material is assumed to yield under elasticplastic conditions.
The elastoplastic relationships in the normal compressive direction are defined as
Plasticity effects in the normal compressive direction also affect the evolution of the tangential forces at the interface, as the interface shear strength is related to the normal compression force by Eq. (42).
Figure 10 shows the diagram relating the compressive axial stress and the compressive axial strain used for modelling the elastoplastic constitutive behaviour at the contact interfaces. The form of each diagram is typically obtained from experimental tests on cylindrical samples with the adjustment explained in the next section.
The diagram in Fig. 10 will be used for modelling with the DEM the tests in cement, concrete and shale rock material in this work.
9 Limit compressive stress at the contact interface: experimental adjustment
The limit compressive stress at the contact interface is obtained by correcting the experimental value of the limit compressive stress obtained in a UCS test. The correction is needed for taking into account the micro–macro relationship that relates the limit normal stress at the contact interfaces with the limit compressive stress obtained from experimental tests.
The conclusion of this study is that the limit normal compressive stress at the contact interface, \(\sigma ^l_{n_c}\), is a proportion of the actual limit compressive stress in a experimental test. In our work we have computed \(\sigma ^l_{n_c}\) as \(\sigma ^l_{n_c}=0.62(\sigma ^l_{n_c})_{UCS}\), where \((\sigma ^l_{n_c})_{UCS}\) is the yield stress obtained in a UCS test.
10 Numerical experiments
CPU times (in seconds) and speedups for parallel analysis of UCS test (70,000 spheres) using MPI and OpenMP strategies
No. of processors  OpenMP (time, s)  OpenMP (speedup)  MPI (time, s)  MPI (speedup) 

1  699,25  1  699,25  1 
2  434,03  1,735  372,29  1,878 
4  255,35  3,083  243,92  2,867 
8  151,23  5,301  111,01  6,299 
12  111,05  7,175  71,09  9,725 
16  88,04  8,563  56,35  12,410 
10.1 DEM analysis of laboratory tests on cement samples
10.1.1 UCS and triaxial tests on cement samples
 (1)
A right cylindrical plug is cut from the sample core and its ends ground parallel each other within 0.001 inch. Physical dimensions and weight of the specimen are recorded. The dimensions of the cylindrical samples are 1 inch diameter and 2 inch height. The specimen is tested under saturated condition with water.
 (2)
The specimen is then placed between two endcaps and a heatshrink jacket is placed over the specimen.
 (3)
Axial strain and radial strain devices are mounted in the endcaps and on the lateral surface of the specimen, respectively.
 (4)
The specimen assembly is placed into the pressure vessel and the pressure vessel is filled with hydraulic oil.
 (5)
Confining pressure is increased to the desired hydrostatic testing pressure.
 (6)
Specimen assembly is brought into the contact with a loading piston that allows application of axial load.
 (7)
Increase axial load at a constant rate until the specimen fails or axial strain reaches a desired amount of strain while confining pressure is held constant.
 (8)
Reduce axial stress to the initial hydrostatic condition after sample fails or reaches a desired axial strain.
 (9)
Reduce confining pressure to zero and disassemble sample.
 (a)
The confining pressure is applied up to the desired hydrostatic testing pressure.
 (b)
A prescribed axial motion is applied at the top of the specimen until this fail, or the axial compressive strain strain reaches a desired amount of strain while confining pressure is held constant.
We note that the goal of this study was to reproduce with the DEM model presented the structural behaviour of the sample during the axial compression phase. For this purpose an average Young modulus (deduced from the uniaxial strain compaction (USC) test) was chosen for modelling the the hydrostatic compaction of the sample during the application of the confining pressure. A more detailed study of the non linear behaviour of the cement sample under an hydrostatic load will be presented in a subsequent work.
The stress in the curve in Figure 13 coincides with the effective stress only if we accept that the water pressure in the pores is the same as the external pressure required to enforce the uniaxial strain conditions during the test.
In our work we have used the stress–strain curve obtained in the USC test as the basis for computing the normal compressive force at the contact interfaces (\(\sigma _a\)) for the cement material examples. Note that for saturated conditions this curve already accounts for the effect of water pressure at the pores.
Tables 2, 3 and 4 show the material and DEM parameters for the cement material studied in this work. Table 2 shows the basic material parameters reported in [19]. The tensile strength \(\sigma _t^f\) has been deduced from the BTS test value of \((\sigma _t^f)_{BTS}=2.92\) MPa [19] using the relationship \(\sigma _t^f =1.60 (\sigma _t^f)_{BTS} \simeq 4.80\) MPa as mentioned in Sect. 7.1. On the other hand, \(\tau ^f\) has been taken as \(\tau ^f = \frac{1}{2} (\sigma _{n_c}^f)_{UCS}=8.50\) MPa, where \((\sigma _{n_c}^f)_{UCS}\) is the maximum compressive stress obtained in the UCS test.
Material parameters for cement
\(\rho \) (g/cc)  \(\mu _1\)  \(\mu _2\)  \(E_0\) (GPa)  \(\nu \)  \(\sigma ^f_{t}\) (MPa)  \(\tau ^f\) (MPa) 

1.70  0.30  0.40  3.80  0.20  4.80  8.50 
DEM constitutive parameters for the UCS, USC and BTS tests on cement samples
LCS1 (MPa)  LCS2 (MPa)  LCS3 (MPa)  YRC1  YRC2  YRC3  \(\delta _n\)  \(\delta _s\)  \(\alpha \) 

8.5  9.0  11  3  9  24  0.20  0.2  1.0 
DEM constitutive parameters for triaxial tests on cement samples
Confining pressure (Psi)  LCS1 (MPa)  LCS2 (MPa)  LCS3 (MPa)  YRC1  YRC2  YRC3  \(\delta _n\)  \(\delta _s\)  \(\alpha \) 

500  9.5  11  13  3  9  24  0.20  0.2  1.0 
1000  10  11  14  3  9  24  0.20  0.2  1.0 
2000  13  14  15  3  9  24  0.20  0.2  1.0 
4000  21  23  26  3  9  24  0.20  0.2  1.0 
The confining pressure is directly applied to the spheres that lay on the surface of the specimen. A normal force is applied to each surface particle in the radial direction. The magnitude of the force has been approximated in this work as \(F_{n_i}= \pi r_i^2 p_c\) where \(r_i\) is the particle radius and \(p_c\) is the confining pressure. A more accurate procedure for transferring the confining pressure to the spheres at the boundary of the specimen using the areas of the Voronoi polygon created by the centroids of the spheres is presented in [7].
A small confining pressure of 70 Psi was applied for the analysis of the UCS test. This pressure reproduces the effect of the heatshrink jacket on the lateral deformation of the sample.
Figure 15 shows DEM results of the applied axial versus the axial strain in the cement specimen for the UCS test. Figure 16 shows DEM results for triaxial tests in the cement samples for confining pressures of 500, 1000, 2000 and 4000 Psi using again 42000 spheres. Good correlation between the DEM results and the experimental values [19] is obtained in all cases.
10.1.2 Uniaxial strain compaction (USC) test on cement sample
For the USC test the radial strain is constrained in the sample while a piston presses the sample from the top. The DEM parameters are those given in Tables 2 and 3.
10.1.3 Brasilian tensile strength (BTS) test on cement sample
The BTS test was carried out for a sample of 1.487 in diameter and 0.863 in thickness. The density of the material was 1.70 g/cm\(^{3}\). The experimental value of the maximum load in the BTS test was 847 lb which corresponds to a value of \((\sigma _{t}^f)_{BTS} = 420\) Psi \(\approx \)2.9 MPa. Hence, \(\sigma _t^f= 1.6 (\sigma _{t}^f)_{BTS} = 4.8\) Mpa (Sect. 7.1).
The DEM parameters used are given in Tables 2 and 3.
10.2 DEM analysis of laboratory tests on concrete samples
The experimental tests were carried out at the laboratories of the Technical University of Catalonia (UPC) in Barcelona, Spain. Details of the test are given in [37]. The concrete used in the experimental study was designed to have a characteristic compressive strength of between 32.8 and 38 MPa at 28 days. Standard cylindrical specimens (of 150 mm diameter and 300 mm height) were cast in metal molds and demolded after 24 h for storage in a fog room.
The triaxial tests were prepared with a 3mmthick butyl sleeve placed around the cylinder and an impermeable neoprene sleeve fitted over it. Before placing the sleeves, two pairs of strain gages were glued on the surface of the specimen at midheight. Steel loading platens were placed at the flat ends of the specimen and the sleeves were tightened over them with metal scraps to avoid the ingress of oil.
The tests were performed using a servohydraulic testing machine with a compressive load capacity of 4.5 MN and a pressure capacity of 140 MPa. The axial load from the testing machine is transmitted to the specimen by a piston that passes through the top of the cell. Several levels of confining pressure ranging from 1.5 to 60 MPa were used in order to study the brittle–ductile transition of the response. First the prescribed hydrostatic pressure was applied in the cell, and then the axial compressive load was increased at a constant displacement rate of 0.0006 mm/s.
Two specimens were tested at each confining pressure, and all tests were performed at ages of more than 50 days to minimize the effect of aging response. In addition to the triaxial tests, uniaxial compression tests were also performed.
On the other hand, the value of \(\sigma _t^f\) was estimated using Eq. (43) for a value of \((\sigma _{n_c}^f)_{UCS}=37\) MPa. This gives \(\sigma _t^f \simeq 5\) MPa. As for \(\tau ^f\) we have taken \(\tau ^f = 0.45(\sigma _{n_f}^f)_{UCS}\simeq 16\) MPa.
DEM constitutive parameters for tests on concrete samples
\(\rho \) (g/cc)  \(\mu _1\)  \(\mu _2\)  \(E_0\) (GPa)  \(\nu \)  \(\sigma ^f_{t}\) (MPa)  \(\tau ^f\) (MPa)  

2.5  0.90  0.25  28  0.2  5.0  16 
LCS1 (MPa)  LCS2 (MPa)  LCS3 (MPa)  YRC1  YRC2  YRC3  \(\delta _n\)  \(\delta _s\)  \(\alpha \) 

20  45  70  3  12  22  0.2  0.2  1.0 
10.3 UCS and BTS tests on shale rock material
We have simulated with the DEM code a UCS test and a BTS test on a shale rock material corresponding to a Middle Brown gaseous shale in Devonian formation from Lincoln County, West Virginia. The essential material parameters for the DEM simulations were taken from [33].
DEM constitutive parameters for tests on shale rock samples
\(\rho \) (g/cc)  \(\mu _1\)  \(\mu _2\)  \(E_0\) (GPa)  \(\nu \)  \(\sigma ^f_{t}\) (MPa)  \(\tau ^f\) (MPa)  

2.55  0.7  0.6  30  0.2  5.0  25 
LCS1 (MPa)  LCS2 (MPa)  LCS3 (MPa)  YRC1  YRC2  YRC3  \(\delta _n\)  \(\delta _s\)  \(\alpha \) 

20  30  40  1  1  1  0.2  0.2  1.0 
11 Concluding remarks
We have presented a local constitutive model for the DEM. The model governs the linear and non linear relationships between the normal and tangential forces and the corresponding relative displacements at the contact interfaces between discrete particles. The good behaviour of the model has been verified in its application to the analysis with the DEM of cement, concrete and shale rock samples under different strength tests. DEM results compare well with experimental data for the same tests.
The results obtained in this work show that the DEM model presented yields an accurate and reliable numerical method for linear and nonlinear analysis of geomaterials and concrete under mechanical loading.
Further validation of the DEM model presented is still needed in order to asses its convergence features for the non linear analysis of cohesive material in terms of the number and size distribution of the spheres.
Notes
Acknowledgments
This work was carried out with financial support from Weatherford, the Advanced Grant Projects SAFECON and COMDESMAT of the European Research Council and the BALAMED project (BIA201239172) of MINECO, Spain. The support of CIMNE for making available the codes DEMPACK (www.cimne.com/dempack), KRATOS (www.cimne.com/kratos) and GiD (www.gidhome.com) is gratefully acknowledged.
Supplementary material
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