jurnal mohsen pemodelan material

Journal of Solid Mechanicsand MaterialsEngineering

Vol.2, No.12, 2008

Discrete Element Model for ContinuumDynamic Problems∗

Mohsen GAEINI∗∗, Sandro MIHRADI∗∗∗ and Hiroomi HOMMA∗∗∗∗ Toyohashi University of Technology

ICCEED, Toyohashi University of Technology, 1–1 Tempaku-cho Toyohashi 441–8580 Japan

E-mail: [email protected], [email protected]∗∗∗ Toyohashi University of Technology

(now, Institute of Technology Bandung, Jl. Ganesa 10, Bandung, Indonesia)

E-mail: [email protected]

AbstractIn this study the discrete element method (DEM) is developed in the framework of theconstitutive law of elastic isotropic materials for two-dimensional plane stress analysisof solids. Contact stiffnesses (normal and tangential spring constants) are theoreticallyderived for hexagonal elements in an arbitrary arrangement as a function of thicknessand material parameters including Young’s modulus and Poisson’s ratio. Moreover, anew method is presented to calculate stress components within an element. To validatethe accuracy and efficiency of our discrete model several test problems are given. Atfirst, uniaxial tension test using a bar is performed and the convergence of the solu-tion to a definite value in the limit of mesh refinement is validated. In addition, twoexamples of stress wave propagation problems are given. Compressive wave speedis calculated and through comparing the numerical results with the other discrete ele-ment models and also analytic solution, the accuracy of the present DEM model is thendiscussed.

Key words : Discrete Element Method, Constitutive Law, Plane Stress, Tensor, StressWave

1. Introduction

Numerical simulation is an effective measure for studying various mechanical phenom-ena in materials and structures. In this process, the continuum assumption is usually usedas a basis for the idealization of many engineering materials. Finite element (FEM), finitedifference (FDM), and boundary element methods (BEM) are among the numerical solutiontechniques, which are founded on this assumption. However, there are many situations espe-cially in large deformation problems with void nucleation and fracture in engineering materi-als where continuity can not be assumed. In these cases, for the above numerical techniques,special treatments therefore have to be added in the calculation procedures, which make theprocess more complicated.

Discrete element method (DEM) is a member of the numerical analysis family, likeFEM, FDM and BEM, that solves numerically the mathematical formulation for the wholemedium constructed from a constitutive formula for each element discretizing a continuousmedium. However, the DEM is especially designed to deal with even engineering problemsthat are dominated by discontinuous mechanical behaviors. The fundamental assumption usedin DEM distinguishes this method with other numerical methods. It assumes that the mate-rial consists of discrete rigid particles, which may have different shapes and properties. Therigid particles interact with neighboring particles according to interaction laws that are ap-plied at the point of contact. This numerical technique was first proposed by Cundall(1) tosolve the problems in rock mechanics where the continuity between the separate elementsdoes not exist. DEM is capable for modeling multiple interacting continuities, discontinuities

∗Received 22 May, 2008 (No. 08-0382)[DOI: 10.1299/jmmp.2.1478]

1478

Journal of Solid Mechanicsand Materials Engineering

Vol.2, No.12, 2008

or fracturing bodies, and also it has been widely used in geotechnical engineering and powdertechnology(2). Recently, this method has also been used to simulate the failure and damageprocesses of brittle materials(3), (4) and also for some impact problems(5). However, a mainissue in DEM as a tool for modeling continuum problems is to determine the interaction lawsgoverning at points of contact between elements.

This paper aims to present a new approach to establish the contact stiffnesses theoreticallyas a function of thickness and material parameters including Young’s modulus and Poisson’sratio for an elastic isotropic material under the plane stress condition. However, most of thepast researches to develop a mathematical expression for the DEM inter-element parametersrely on special arrangements of discrete elements as a unit cell(6) or seven-disk elements(7).The distinct point of the present approach is to develop a mathematical expression of springconstants by only considering two elements in an arbitrary orientation, which also promisesa positive value of contact stiffnesses for all engineering materials. Moreover, an alternativemethod to the average stress method(9) is proposed to calculate the stresses within element.In addition, several test problems are given and through comparing the results with analyticsolutions, the accuracy and validity of the present DEM model and stress calculation is thendiscussed.

NomenclatureE : Young’s modulus, GPaν : Poisson’s ratioρ : density, kg/m3

kn : normal stiffness, N/mks : shear stiffness, N/mun : normal displacement, mus : shear displacement, mfn : normal spring force, Nfs : tangential spring force, Nδ : thickness, mmr : radius, mm

V : volume , mm3

t : time, μsdt : time increment, μsλ : coefficient of mass damping, s−1

p : applied stress, kPagi : local coordinate basisei : global coordinate basisσ : local stress tensorσi j : local stress components, Paε : local strain tensorεi j : local strain components, Paσ : local stress vectorε : local strain vector

SubscriptsNoE : number of elements el : element

2. Elastic Discrete Model for Solid Materials

2.1. GeneralDEM as a tool for modeling solid materials, discretizes the entire volume of a solid to

rigid elements of simple shape with bonding characterized by normal and tangential springconstants, as shown in Fig. 1. The analysis procedure in DEM calculates the relative dis-placements between elements at each time step for internal force evaluation. Afterwards,the equations of motion for individual elements are constructed and integrated over the timeusing a time integration scheme such as the central difference method to compute elementdisplacements and rotational angle. In the present DEM model, the central difference methodis used for the integration of motion equations and the mass damping given by Liu et al(7) isintroduced to minimize vibration excited in the DEM model.

Fig. 1 two dimensional solid materials and arrangement of discrete elements

1479


Vol.2, No.12, 2008

2.2. Determination of spring constantsThe mechanical relationship between the elastic coefficients for the elastic isotropic ma-

terial and the inter-element parameters (normal and tangential spring constants) governingdiscrete element model has been investigated by different researchers(6), (7). In the presentstudy a new approach is used to establish the inter-element normal and tangential spring con-stants as a function of element size and material parameters including Young’s modulus andPoisson’s ratio. For the reference, first, a seven-disk model(7) as shown in Fig. 2 (a) is consid-ered.

Fig. 2 (a) Arrangement of Seven-disk element and (b) Two disk-element

Taking the assumption that elastic internal energy is totally stored in springs, the relationεn1i j =

uni j

2r and εsi j =usi j

2r ( j = i1, ..., i6), between normal, shear strains and correspondingdisplacements the total internal energy stored in an arbitrary element eli may be given as

Weliint = r2

i6∑j=i1{kni j(εni j)

2 + ksi j(εsi j)2} (1)

Further more, the strain energy density of one disk element eli, is given as follow

Weliint(d) =

Weliint

V(2)

where V is the volume of one single element given by

V = 2√

3r2δ (3)

According to Green’s formula, the stress components can be obtained by

σmn =∂Weli

int(d)

∂εmn(4)

By using Eqs. (1)∼(4), assuming homogenous material properties and the transformation

εn1i j = cos2(αi j){ε11} + sin2(αi j){ε22} + sin(2αi j){ε12} (5)

εsi j = sin(αi j) cos(αi j){ε22 − ε11} + cos(2αi j){ε12} (6)

between strain components in local and global coordinate system, the following relation be-tween stress and strain components will be obtained.

σ11 =

√3

12(8√

3kn1 + kn2 + 3ks2)ε11 +

√3

4(kn2 − ks2)ε22 (7)

σ22 =

√3

4(kn2 − ks2)ε11 +

√3

4(3kn2 + ks2)ε22 (8)

σ12 =

√3

6(3kn2 + 2ks1 + ks2)ε12 (9)

1480


Vol.2, No.12, 2008

Considering Eqs. (7)∼(9), the spring constants for an elastic isotropic material under planestress condition will be obtained as

kn1 = kn2 = kn =

√3δ3

(c12 + c22) =

√3

3· Eδ

1 − ν (10)

ks1 = ks2 = ks =

√3δ3

(c22 − 3c12) =

√3

3· (1 − 3ν)Eδ

1 − ν2 (11)

However, the above spring constants are valid only when each element is surroundedby six disk elements. It is clear that the elements on the boundary, which have less than sixneighbors can not be modeled by seven-disk model. To overcome the above disadvantage, atwo-disk model with hexagon shape boundaries is developed as depicted in Fig. 2 (b). At theinterface of two elements, a local coordinate system is defined along and perpendicular to theedge with unit normal basis g1, g2 and g3 given by

g1 = cosαi j e1 + sinαi j e2 (12)

g2 = − sinαi j e1 + cosαi j e2 (13)

g3 = e3 (14)

According to the cauchy’s theorem the traction force tni j on the each edge of the elementeli is denoted by the unit normal vector ni j in the local coordinate system by

tni j = σni j · ni j (15)

where ni j = g1 and stress tensor σni j is expressed by

σni j= (σni j )mn gm ⊗ gn, for (m, n = 1, 2, 3) (16)

Taking the assumption that internal energy is stored only in springs, the traction force is givenby

tni j = (fni j

δl)g1 + (

fsi j

δl)g2 (17)

where fni j = kni j uni j , fsi j = ksi j usi j and l = 2√

3r/3. By considering Eqs. (15)∼(17) andsymmetry of the stress tensor, the following relations for a plane stress problem are given

σm3 = σ3m = 0 for m = 1, 2, 3 (18)

t1i j = σ11i j = σn1i j =fni j

δl=

kni j uni j

δl=

√3kni j uni j

2δr(19)

t2i j = σ12i j = σ21i j = σsi j =fsi j

δl=

ksi j usi j

δl=

√3ksi j usi j

2δr(20)

Furthermore, the normal and shear strains on the edge may be given by

εn1i j =uni j

2r(21)

2εsi j =usi j

2l=

√3usi j

4r(22)

where they are expressed in the local coordinate system. To achieve the total strain energy ata point on the edge, stress and strain vectors may be expressed in the local coordinate systemas

σ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝σ11

σ22

σ12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝σn1i j

σn2i j

σsi j

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (23)

1481


Vol.2, No.12, 2008

and

ε =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝εn1i j

εn2i j

2εsi j

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . (24)

σ and ε can be related by the constitutive matrix c′ as follow

σ = c′ε (25)

or ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝σn1i j

σn2i j

σsi j

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

c′11 c′12 c′13

c′21 c′22 c′23

c′31 c′32 c′33

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝εn1i j

εn2i j

2εsi j

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (26)

However, as the stress components are given by Eqs. (19)∼(22), σn1i j and σsi j are only afunction of εn1i j and εsi j respectively. Considering the above fact, symmetry of the constitutivelaw, and Eq. (26), it can be deduced

c′12 = c′21 = c′13 = c′31 = c′23 = c′32 = 0 (27)

On the other hand, c′ has a diagonal form and is given by

c′ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝c′11 0 00 c′22 00 0 c′33

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (28)

For an elastic isotropic material under the plane stress condition, the constitutive matrix in theglobal coordinate system is given by

c =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝c11 c12 c13

c21 c22 c23

c31 c32 c33

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (29)

where cmn = cnm for m, n = 1, 2, 3 and are given by

c11 = c22 =E

1 − ν2 (30)

c12 = c21 =Eν

1 − ν2 (31)

c33 =E

2(1 + ν)(32)

c13 = c31 = c23 = c32 = 0 (33)

However, the constitutive matrix should be independent of the coordinate system. On theother hand, to keep the equivalence between c and c′ and considering a diagonal form for c′,the components of c′ may be identified by the eigenvalues of c and c′11, c′22 and c′33 can bedetermined eventually as

c′11 = c11 + c12 (34)

c′22 = c11 − c12 (35)

c′33 = c33 =12

(c11 − c12) (36)

From Eq. (26), σn1i j and σsi j are given by c′11εn1i j and 2c′33εsi j respectively. Considering Eqs.(19)∼(22), (34) and (36), spring constants can be finally determined as

kn =

√3δ3

(c12 + c22) =

√3

3· Eδ

1 − ν (37)

1482


Vol.2, No.12, 2008

ks =δ

4(c11 − c12) =

Eδ4(1 + ν)

(38)

which are independent of the DEM element radius. In addition, assuming non-negative valuesfor contact stiffnesses, equations given here for normal and tangential spring constants can beapplied to all engineering materials whereas shear stiffness given by Liu(7) limits the maximumvalue of Poisson’s ratio to ν = 1

3 .In addition, it should be noted that Eqs. (19) to (22) are dependent on element special

geometry, but by introducing the special coordinate system, geometry parameters vanish fromthe both sides of Eq.( 26) and consequently the derived spring constants are independent ofelement special geometry. However, because the strain components are derived by use of thehexagonal geometry property, the application of the proposed DEM model must be limited tothe hexagonal elements whereas there is no limitation of the type of loading as no special typeof loading is assumed to derive Eqs. (37) and (38).

2.3. Stress Calculation (cross sectional method)In the present method to calculate stresses the hexagonal discrete element is approxi-

mated by a circle which is virtually cut into two halves. Stress components are then calculatedfrom the vertical and horizontal static equilibrium principle. For clarity, the method to cal-culate the normal stress in e1 direction is presented here. At first a particle is cut verticallyinto two halves, as depicted in Fig. 3. σR

11 and σL11 are then calculated by force equilibrium

Fig. 3 Two halves of disk element and the procedure to find σ11

for right and left half circles respectively. To decide normal stress, σ11 in e1 direction, thefollowing conditions are given, where the unbalance force will go to displace the element:If {σR

11 > 0 & σL11 > 0 & (σR

11 > σL11)} → σ11 = σ

L11

If {σR11 > 0 & σL

11 > 0 & (σL11 > σ

R11)} → σ11 = σ

R11

If {σR11 < 0 & σL

11 < 0 & (σR11 > σ

L11)} → σ11 = σ

R11

If {σR11 < 0 & σL

11 < 0 & (σL11 > σ

R11)} → σ11 = σ

L11

If {(σR11 × σL

11) <= 0} → σ11 = 0If {σR

11 = σL11} → σ11 = σ

R11

Similar technique is also applied to calculate σ22 and σ12.

3. Numerical results and Discussions

3.1. Tension in a barConsider a bar with length L of 40 mm, width w of 4 mm and thickness δ of 1 mm, as

shown in Fig. 4 (a). The width is chosen as 4 times bigger than thickness to develop a stateof plane stress in the bar. The right and left hand ends are subjected to force P(t) per unitvolume. The bar is made of steel with Young’s modulus E, 210 GPa, poisson’s ratio ν, 0.27and density ρ, 7800 kg/m3 where the values for normal and tangential spring constants aregiven by Eqs. (37) and (38). Due to symmetry of the geometry and loading conditions,calculations are done for half of the bar with the same load at one end, and displacements atthe other end is bounded in direction of the applied load. DEM simulations are conducted

1483


Vol.2, No.12, 2008

Fig. 4 (a) The uniaxial tension test on a bar under a force P per unit volume, (b) stressacross the cross section for the finest mesh using the presented method andaverage stress method(9), (c) stress distribution across the cross section of thebar for different DEM element size and (d) average strain energy versus timemeasured at the center of the DEM numerical specimen.

for element radii r of 0.05, 0.1 and 0.2 mm with total number of elements NoE of 9125, 2243and 542 respectively. In order to minimize unrealistic element oscillation in the model, thecoefficient of mass damping, λ of 100 s−1 is used in the calculation, which is obtained throughtrial and error process. The time increment dt of 0.01 μs is chosen to ensure the stability ofthe calculation.

Theoretically, engineering stresses for a bar under uniaxial tension across an arbitrarycross section of the bar perpendicular to the longitudinal axis should be equal to the appliedstress. To validate the accuracy of the present DEM method, stresses across a cross section10 mm distant from the end are measured by the present method and then compared to theaverage stress method(9) as shown in Fig. 4 (b). Figure 4 (c) shows the convergence of thesolution to the applied stress as expected by the mesh refinement. Furthermore, it should benoted that stress calculation in the present DEM model is only a post-processing calculationand in fact displacements are primary variables to be used to calculate other parameters likestresses or strains. To show the equivalence between the present stress calculation and cor-responding displacements, the strain energy per unit volume is measured for the element atcenter of the bar. The average strain energy(eint) is calculated by two different approaches aseint =

14V∑(knu2

n j+ksu2s j) for j = 1 to 6 and alternatively as eint =

12 (σ11ε11+σ22ε22+2σ12ε12)

where stresses are calculated by cross sectional method presented in the previous section.Consequently strains are calculated by stresses and the constitutive law. Results for differentelement size are plotted in Fig. 4 (d). A good agreement between stresses and correspondingdisplacements can be easily seen, which further confirms the accuracy of the present methodto calculate stresses.

1484


Vol.2, No.12, 2008

3.2. Stress wave propagation in a finite and semi-infinite plateIn this section two example problems of stress wave propagation are analyzed. Both,

a long bar and a plate with thickness of 1 mm are impacted by a pressure pulse, p(t) =pmax sin(πt/t1), as shown in Figs. 5 (a) and (b). The bar is made of plaster and the plateis made of steel. The material and DEM parameters are given in the Table 1.

Table 1 Material Properties and DEM Parameters

material E ν ρ r NoE pmax(MPa) t1(μs) dt(μs)Plaster 2.98 0.3 1440 0.5 1112 1.65 20 0.1Steel 210 0.27 7800 0.2 3204 250 0.5 0.005

Fig. 5 (a) Example1: Bar specimen geometry (b) Example2: Plate specimen geometry(c) σ11 history at the center of the bar (d) σ22 history at the center of the plate

Spring constants are calculated by Eqs. (37) and (38) for the present model and Eqs.(10) and (11) for seven-disk model. The normal stress history in the direction of applied loadat the center of plates can be successfully analyzed. The stress wave propagation results areshown in Figs. 5 (c) and (d). From the results, the wave propagation inside the plates can beclearly indicated. At first the incident wave travels as the compressive stress wave and afterthe reflection from the back boundary, it is inverted as a tensile stress wave.

To verify the accuracy of the calculation, stress wave velocity in the model is comparedwith the theoretical one. The stress wave velocity in the model can be calculated by dividingthe traveling distance by the traveling time (the interval between the negative and positivepeaks). The result for both examples are given in Table 2.

Table 2 Evaluated Wave Velocity

Specimen Bar PlateTheoretical wave speed (m/s) 1508 5800Presented DEM model (m/s) 1505 5568

Seven-disk model 1437 5446

Deviations of 0.2 % and 4 % from the theoretical longitudinal wave velocity are obtainedby the presented model, for the bar and the plate respectively. In contrast, the seven-disk

1485


Vol.2, No.12, 2008

element model gives a deviation of 4.7 % and 6.1 %. In other words, these results show theeffect of spring constants on wave velocity. As it is given by Eqs. (10), (11), (37) and (38)both models calculate the same value for normal stiffness whereas the shear stiffness is differ-ent.The presented model gives the shear stiffness values of 0.573 KN/mm and 41 KN/mm forplaster and steel respectively, whereas the other model gives 0.189 KN/mm and 24 KN/mmfor both the materials.

4. Conclusion

Summarizing the obtained results can deduce conclusion that the discrete element modelproposed here can be validly used for the numerical analysis of stress wave propagation prob-lems. By comparing the numerical results from the present model with the results from theother model such as seven-disk model(7), the validity and the accuracy of the presented algo-rithm were clearly confirmed. In particular, the spring constants of the present approach in-duce more acceptable wave velocity in the problems of the stress wave propagation. Moreover,the present expressions of contact stiffnesses can be applied to a broader range of materialswhereas corresponding equations given by other researchers(6) – (8) limit the value of Poisson’sratio to ν = 1

3 to ensure a positive value of shear stiffness. However, application of the presentapproach is limited to the problems with small deformation. Further development is neededfor large deformation problem.

References

( 1 ) Cundall, P.A., A Computer Model for Simulating Progressive Large Scale Movement inBlock Rock System, Symp ISRM Proc 2, (1971), pp. 129-136.

( 2 ) Thomas, P.A. and Bray, J.D., Capturing Nonspherical Shape of Granular Materials withDisk Clusters. Journal of Geotechnical and Geoenvironmental Engineering, Vol. 125,No. 3 (1999), pp. 169-178.

( 3 ) Camborde, F., Mariotty, C. and Donze, F. V., Numerical Study of Rock and ConcreteBehaviour by Discrete Element Modeling, Computers and Geotechnics, Vol. 27 (2000),pp. 225-247.

( 4 ) Mihradi, S., and Homma, H., Fracture Simulation of Brittle Material under Impact Load-ing by Discrete Element Method, Proceeding of The 13th International Symposium onPlasticity and Its Current Application, Alaska (2007), pp. 307-309.

( 5 ) Hentz, S., Donze, F. V., and Daudeville, L., Discrete Element Modeling of ConcreteSubmitted to Dynamic Loading at High Strain Rates, Computers and Structures, Vol. 82(2004), pp. 2509-2524.

( 6 ) Tavarez, A.F, and Plesha, M.E., Discrete Element Method for Modelling Solid and Par-ticulate Materials, Int. J. Numer. Meth. Eng., Vol. 70 (2007), pp. 379-404.

( 7 ) Liu, K., Gao, L. and Tanimura, S., Application of Discrete Element Method in ImpactProblems, JSME International Journal, Series A, Vol. 47, No. 2 (2004), pp. 138-145.

( 8 ) Sawamoto, Y., Tsubota, H., Kasai, Y., Koshika, N. and Morikawa, H., Analytical studieson Local Damage to Reinforced Concrete Structures under Impact Loading by DiscreteElement method, Nuclear Engineering and Design, Vol.179 (1998), pp.157-177.

( 9 ) Drescher, A. and Jong, G., Photoelastic Verification of a Mechanical Model for the Flowof a Granular Material, J. Mech. Phys. Solids, Vol. 20 (1972), pp. 337-351.

1486

jurnal mohsen pemodelan material

Documents