Application of Discrete Element Method for Continuum Dynamic Problems

Arch Appl Mech (2006) 76: 229–243DOI 10.1007/s00419-006-0018-8

ORIGINAL

K. Liu · W. Liu

Application of discrete element method for continuumdynamic problems

Received: 13 May 2005 / Accepted: 9 March 2006 / Published online: 25 April 2006© Springer-Verlag 2006

Abstract A new method based on the principle of minimum potential energy is presented, aiming to overcomesome weakness of the present discrete element method (DEM). Our primary research is to put forward theDEM with a tight theory base and a fit technique for treating continuum dynamic problems. By using thismethod, we can not only extend the existing seven-disc model, but also establish a new nine-disc model in ageneral way. Moreover, the equivalences of two kinds of models have been verified. In addition, three numer-ical examples of stress wave propagation problems are given in order to validate accuracy and efficiency ofthe present DEM models and their algorithms. Finally, the dynamic stress concentration problem of a squareplate with a circular hole is analyzed.

Keywords Computation mechanics · Continuum dynamics · Discrete element Method · Numerical model ·Stress wave Propagation

1 Introduction

Various mechanical phenomena can be observed in materials and structures under impact loading, such as stresswave propagation, large deformation, damage and failure [1]. Numerical simulation is an effective measurefor studying those problems. Among the numerical algorithms, finite difference method (FDM), finite volumeelement method (FVM) [11–13], finite element method (FEM), boundary element method (BEM) and themethod of characteristics are suitable for analyzing the dynamic behaviors of continuum, and for forecastingthe failure region of materials accurately. However, it is difficult for the above methods to simulate the entirefailure process. Additional special treatments (such as remeshing and contact judgment) have to be consideredin the procedure of calculation, where damage or fracture appears. The problems demand further research.

The discrete element method (DEM), which was first proposed by Cundall [2] in early 1970’s, is provedto be a successful tool for modeling non-continuum, such as rocks or granular materials. It has been widelyused in geotechnical engineering and powder technology [3–5]. However, accuracy of the traditional DEMis not so good as FEM for continuum, thus the DEM–FEM combination algorithms have been introduced[6,7]. FEM is adopted in the continuum part, while DEM is used where damage appears. Since two proceduresof DEM and FEM are involved in the algorithms, the programs and the logical relation may be too complex.Furthermore, artificial judgements are often needed in the fractured area, which are unfavorable to practicalapplication. Therefore, a special DEM model for both continuum and non-continuum could be set up naturally

This work was supported by Nation Natural Science Foundation of China (nos. 10232040 and 10572002).

K. Liu (B) · W. LiuLTCS and Department of Mechanics and Engineering Science,Peking University, Beijing 100871, People’s Republic of ChinaE-mail: [email protected].: +86-10-62765844Fax: +86-10-62751812

230 K. Liu, W. Liu

whose accuracy matches that of FEM for continuum. In our model, the connective links between destroyedelements simply change to contact links of the traditional DEM and the element arrangement patterns keepsunchangable. In this research field, some successful simulations on the transmforming process from contimu-um to non-continuum is reported in Ref. [1,8,9], such as the transient responses of a steel warhead penetratinga concrete disc harrow and the damage process of concrete block under impact loading. However, all thoseDEM models are developed for specific problems. Lack of rigorous theoretical foundation and flexibility ofarrangement patterns in modeling seriously limits the wide use of DEM, especially for continuum dynamicproblems.

In this paper, a new method to establish the DEM model for continuum dynamic problems is presentedbased on the principle of minimum potential energy. The present method not only extends the exsiting seven-disc model [1,8], but also puts forward a new nine-disc model in a general way. Moreover, the equivalencesof these two kinds of models have been verified. By mean of the models and the corresponding numericalscheme, stress wave propagations due to a longitudinal pulse are calculated in an isotropic, an anisotropic anda layered plate. And the dynamic stress concentration problem on a square plate with a hole is investigated.Comparing the numerical results with the corresponding results obtained by FEM, FVM, and the method ofcharacteristics, accuracy and efficiency of the models and their algorithms are examined.

2 Discrete model for continuum

2.1 Basic formulation

The plane stress problem is considered here. A elastic plate is subdivided into many rigid disc elements, whichare linked by two kinds of springs (a normal spring and a tangential spring) as shown in Fig. 1. There aretwo possible kinds of compact arrangement patterns of discs, type A and type B. Type A is called seven-discmodel, and type B is called nine-disc model. For type A, semi-disc elements are needed on the neat boundary.Type C in Fig. 1, an arbitrary arrangement disc model, is not considered, because the model is not suitable fortreating continuum problems.

Assuming the elastic plate is subdivided into N disc elements, the total potential energy of the disc-springsystem is given as follows:

� =N∑

i

(Ui Vi ) +N∑

i

(uxiρuxi Vi + uyiρu yi Vi ) −N∑

i

(uxi fxi Vi + uyi fyi Vi ) −N∑

i

(uxi Txi Si + uyi Tyi Si ),

(1)

A

ij

C

i

j

Bnk

sk

j

i

nju

niu

yiusju

xiu

yju

xjuiO

siuiO jO

jO n

s

x

y

oo

′

a

. . . . .. .

.....

.....

. .. . .

. . . . .. .

. ..

.....

. .

. .

. ..

...

.

.

.

.

..

..

.

..

. . . . . . .

........ . .

. .

. . .

... .....

. .

. .

. . ...

Fig. 1 Discrete element method models based on rigid disc elements

Application of discrete element method for continuum dynamic problems 231

where Ui is the average strain energy around disc i , Vi the volume of disc i , ρ the mass density, Si the boundaryarea of external force on disc i . uxi , uyi and uxi , u yi are the displacements and the accelerations of disc i inthe horizontal and the vertical directions, respectively. fxi , fyi and Txi , Tyi are the components of body forceand surface force on disc i in the horizontal and the vertical directions, respectively.

The deformation effect of a elastic body is performed through the deformation of springs in the DEMmodels. Supposing that disc i connects with p discs (see Fig. 1), the average strain energy around disc i iswritten as follows:

Ui = 1

Vi

p∑

j

12

[ 12 kni j (unj − uni )

2 + 12 ksi j (us j − usi )

2] (2)

where kni j and ksi j are the spring constants between discs i and j along the normal and the tangential direc-tions, respectively. uni and usi are the normal and the tangential displacements of disc i . unj and us j are thenormal and the tangential displacements of disc j .

As shown in Fig. 1, assuming the rotation angle between x-axes and the normal direction of the spring asα, l = cos(α), and m = sin(α), we obtain that

un = uxl + uym, us = uyl − ux m. (3)

Substituting Eq. (3) into Eq. (2), the following result can be yielded

Ui = 1

4Vi

p∑

j

kni j [li j (ux j − uxi ) + mi j (uyj − uyi )]2 + 1

4Vi

p∑

j

ksi j [−mi j (ux j − uxi ) + li j (uyj − uyi )]2.

(4)

Substituting Eq. (4) into Eq. (1) and according to variational calculus ∂�/∂uxi = 0, ∂�/∂uyi = 0, uxiand u yi are given by

uxi = 1ρVi

{fxi Vi + Txi Si +

p∑j

kni j [l2i j (ux j − uxi ) + li j mi j (uyj − uyi )]

+p∑j

ksi j [m2i j (ux j − uxi ) − li j mi j (uyj − uyi )]

},

u yi = 1ρVi

{fyi Vi + Tyi Si +

p∑j

kni j [li j mi j (ux j − uxi ) + m2i j (uyj − uyi )]

+p∑j

ksi j [−li j mi j (ux j − uxi ) + l2i j (uyj − uyi )]

}.

(5)

Therefore, [uxi ]t and[u yi

]t at moment t are obtained by Eq. (5), where [ ]t denotes the physical quantity

at moment t . At moment t + �t , [uxi ]t+�t ,[u yi

]t+�t , [uxi ]t+�t and

[uyi

]t+�t can be obtaind according to

Euler formula{

[uxi ]t+�t = [uxi ]t + [uxi ]t�t[u yi ]t+�t = [u yi ]t + [u yi ]t�t

{[uxi ]t+�t = [uxi ]t + [uxi ]t�t,[uyi ]t+�t = [uyi ]t + [u yi ]t�t,

(6)

where �t is a time increment.

2.2 The relation between relative displacement and strain

The relative displacement between two points (i and j) can be expressed as dui = (εi j +ωi j )dx j . Consideringa small-rotation problem (rotation tensor ωi j are ignored), the following equations are obtained.

{ux j − uxi = (x j − xi )εx + y j −yi

2 γxy = (ri + r j )lεx + ri +r j2 mγxy,

uyj − uyi = (y j − yi )εy + x j −xi2 γxy = (ri + r j )mεx + ri +r j

2 lγxy,(7)

232 K. Liu, W. Liu

where ri and r j are the radius of disc i and disc j , respectively. εx , εy and γxy are the engineering straincomponents.

Substituting Eq. (7) into Eq. (3), the relations between the relative displacement and strain at two pointsare given by

{unj − uni = (ri + r j )(l2εx + m2εy + lmγxy),

us j − usi =(

ri +r j2

)[2lm(εy − εx ) + (l2 − m2)γxy]. (8)

3 The relation between spring constants and elastic constants

Considering two disc elements that are linked by the normal spring and the tangential spring, the relationbetween spring constants and elastic constants can be certainly established if plastic deformation and fracturedo not take place. As shown in Fig. 1, taking disc i into consideration, the average strain energy is expressedwith the elastic potential energy of all normal and all tangential springs. Substituting Eq. (8) into Eq. (2), theaverage strain energy over disc i is obtain by

Ui = 1

Vi

p∑

j

[kni j r

2i

(l2i jεxi + m2

i jεyi + li j mi jγxyi

)2 + 14 ksi j r

2i (2li j mi j (εyi − εxi ) + (l2

i j − m2i j )γxyi )

2]

(9)

The subscript i of Ui is omitted, as disc i is an arbitrary disc. Making

A =∑

j

kni j r2i l4

i j , B =∑

j

kni j r2i m4

i j , C = C1 =∑

j

kni j r2i l2

i j m2i j , C2 =

∑

j

ksi j r2i l2

i j m2i j ,

D =∑

j

ksi j r2i (l2

i j − m2i j )

2, E =∑

j

kni j r2i l3

i j mi j , F =∑

j

kni j r2i li j m

3i j ,

G =∑

j

ksi j r2i li j mi j (l

2i j − m2

i j ). (10)

Equation (9) can be written as

U = 1

V

[(A + C2)ε

2x + (B + C2)ε

2y + (2C1 − 2C2)εxεy +

(C1 + D

4

)γ 2

xy + (2E − G)εxγxy

+(2F + G)εyγxy

]. (11)

The average strain energy formula in anisotropic materials for the plane stress problem is written as

U = 12 (c11ε

2x + c22ε

2y + c66γ

2xy + 2c12εxεy + 2c16εxγxy + 2c26εy, γxy), (12)

where ci j are the elastic coefficients.The average strain energy in Eq. (11) and Eq. (12) are equivalent. So, the conditions of the DEM model

are given by

V c11 = 2A + 2C2, V c22 = 2B + 2C2, V c66 = 2C1 + D/2,

V c12 = 2C1 − 2C2, V c16 = 2E − G, V c26 = 2F + G. (13)

The relation between spring constants and elastic constants are obtained, if the arrangement patterns of discssatisfy Eq. (13).

Application of discrete element method for continuum dynamic problems 233

3.1 Seven-disc model and its spring constant determination

For type A shown in Fig. 1, each disc is surrounded by other six discs, which forms regular hexagon lattices.As shown in Fig. 2, the seven discs are numbered and all disc radius are r . Suppose kn1 and ks1 are the normaland the tangential spring constants between disc 0 and disc 1, and between disc 0 and disc 4, respectively; kn2and ks2 are the normal and the tangential spring constants between disc 0 and disc 2, and between disc 0 anddisc 5, respectively; kn3 and ks3 are the normal and the tangential spring constants between disc 0 and disc 3,and between disc 0 and disc 6, respectively. Finally, Eq. (10) is changed to

A = r2 (2kn1 + 1

8 kn2 + 18 kn3

), B = r2 ( 9

8 kn2 + 98 kn3

), C1 = r2 ( 3

8 kn2 + 38 kn3

),

C2 = r2 ( 38 ks2 + 3

8 ks3), D = r2 (

2ks1 + 12 ks2 + 1

2 ks3), E = r2

(√3

8 kn2 −√

38 kn3

),

F = r2(

3√

38 kn2 − 3

√3

8 kn3

), G = r2

(−

√3

4 ks2 +√

34 ks3

). (14)

Then substituting Eq. (14) into Eq. (13), the normal and the tangential spring constants for anisotropic materialsare obtained by

kn1 =√

36 (3c11 + 2c12 − c22)δ,

kn2 = kn3 =√

33 (c12 + c22 ± √

3c16 ± √3c26)δ,

ks1 = 2√

33 (3c66 − c22)δ,

ks2 = ks3 =√

33 (c22 − 3c12 ± 3

√3c16 ∓ √

3c26)δ,

kn1 =√

36 (3c11 + 2c12 − c22)δ,

kn2 =√

33 (c12 + c22 + √

3c16 + √3c26)δ,

kn3 =√

33 (c12 + c22 − √

3c16 − √3c26)δ,

ks1 = 2√

33 (3c66 − c22)δ,

ks2 =√

33 (c22 − 3c12 + 3

√3c16 − √

3c26)δ,

ks3 =√

33 (c22 − 3c12 − 3

√3c16 + √

3c26)δ,

(15)

where δ = V/(2√

3r2) is the thickness of the disc.For orthotropic materials, c16 = 0 and c26 = 0, Eq. (15) is changed to

kn1 =√

36 (3c11 + 2c12 − c22)δ,

kn2 = kn3 =√

33 (c12 + c22)δ,

ks1 = 2√

33 (3c66 − c22)δ,

ks2 = ks3 =√

33 (c22 − 3c12)δ.

(16)

0 1

2 3

4

5 6

Fig. 2 Seven-disc model

234 K. Liu, W. Liu

For isotropic materials, c11 = c22 = (E/1 − ν2), c12 = (νE/1 − ν2), and c66 = G = (E/2(1 + ν)), Eq. (16)is changed to

{kn = kn1 = kn2 = kn3 = Eδ√

3(1−ν),

ks = ks1 = ks2 = ks3 = Eδ(1−3ν)√3(1−ν2)

,(17)

where E is elastic modulus, v is Poisson’s ratio. For the cases of orthotropic materials and isotropic materials,the results about the normal and the tangential spring constants are the same as the research results based onGreen’s formula [1].

3.2 Nine-disc model and its spring constant determination

For type B shown in Fig. 1, each disc is surrounded by other eight discs, which forms regular square lattices.As shown in Fig. 3, the nine discs are numbered and all disc radius are r . Suppose kn1 and ks1 are the normaland the tangential spring constants between disc 0 and disc 1, and between disc 0 and disc 5, respectively; kn3and ks3 are the normal and the tangential spring constants between disc 0 and disc 3, and between disc 0 anddisc 7, respectively; kn2 and ks2 are the normal and the tangential spring constants between disc 0 and disc 2,between disc 0 and disc 4, between disc 0 and disc 6, and between disc 0 and disc 8, respectively. Finally, Eq.(10) is changed to

A = 2r2(kn1 + kn2), B = 2r2(kn3 + kn2), C1 = 2r2kn2, C2 = 2r2ks2,

D = 2r2(ks1 + ks3), E = F = G = 0. (18)

This model is not suitable for general anisotropic materials since E = F = G = 0. However, it can solve theproblem of orthotropic materials for c16 = 0 and c26 = 0. When ks = ks1 = ks2 = ks3, substituting Eq. (18)into Eq. (13), the normal and the tangential spring constants for orthotropic materials are obtained by

kn1 = 13 (3c11 + c12 − 4c66)δ,

kn2 = 13 (c12 + 2c66)δ,

kn3 = 13 (c12 + 3c22 − 4c66)δ,

ks = 23 (c66 − c12)δ,

(19)

where δ = V/(4r2).For isotropic materials, c11 = c22 = (E/1 − ν2), c12 = (νE/1 − ν2), and c66 = G = (E/2(1 + ν)),

Eq. (20) is changed to

kn1 = kn3 = 1+3ν3(1−ν2)

Eδ,

kn2 = 13(1−ν2)

Eδ,

ks = 1−3ν3(1−ν2)

Eδ,

(20)

0 1

3

5

7 8

2 4

6

Fig. 3 Nine-disc model

Application of discrete element method for continuum dynamic problems 235

4 Numerical results and discussion

4.1 Stress wave propagation in anisotropic half-space

For validating accuracy and efficiency of the DEM models and the numerical algorithm, an anisotropic dynamicproblem is calculated by our DEM algorithm and ANSYS LS-DYNA, respectively. As shown in Fig. 4, the stresspropagation process in an anisotropic half-space under a pressure pulse is simulated. The elastic coefficientsare c11 = 4.52385 GPa, c12 = 0.88425 GPa, c22 = 4.52385 GPa, c66 = 1.61195 GPa, c16 = 0.39005 GPa,and c26 = 0.39005 GPa. The mass density is ρ = 1, 244 kg/m3. The initial and the boundary conditions aregiven by

{σx = σy = σxy = 0, vx = vy = 0 for t = 0,

σy = −p(x, t) = −p0 H(a − |x |) exp(−α t2) sin(β t) for y = 0,(21)

where H(x) denotes the Heaviside function, σx , σy and σxy are the stress components, vx and vy are thevelocities in the horizontal and the vertical directions, α = 5.689 × 109 s−2, β = 2, 659 s−1, p0 = 3.31 GPa,and the loaded area is a = 20 mm.

For DEM, the seven-disc model is used with the radius r = 1 mm and the time step �t = 0.8 µs, where thespring constants are given by Eq. (15). The dimension of the numerical model is 600 × 300 × 1 mm3(Length ×Width × Thickness), where the number of discs is 51,814, and the total time is t = 200 µs. It takes about210 s to solve the problem with our PC (An Intel Pentium IV 1.6G CPU).

Using ANSYS LS-DYNA, the number of nodes is 45,451, and the time step is �t ≈ 0.8 µs. The wholecomputation consumes 310 s (the computer is the same). Thus, the efficiency of our DEM algorithm is a littlehigher than ANSYS LS-DYNA. Figs. 5 and 6 show the contour lines of the horizontal and the vertical displace-ments at time t = 104 µs calculated respectively by our DEM algorithm and ANSYS LS-DYNA. ComparingFig. 5 with Fig. 6, we can find that distributions of the displacements are almost the same. From Figs. 5 and 6,it is seen the typical characters that stress wave propagation in anisotropic media is not symmetrical, and thespeeds of stress waves are different along each direction. It testifies accuracy and efficiency of the DEM modelsand our program in calculating continuum dynamic problems.

4.2 Comparison of the numerical results by seven-disc model and nine-disc model

In this paper, two kinds of DEM models, seven-disc model and nine-disc model, are proposed for continuumdynamic problems in order to enrich arrangement patterns of discs and enhance flexibility in practical appli-cation. Now we validate consistency of the two models through a numerical example. The geometric modeland the boundary conditions of the numerical example are the same as those in Sect. 4.1. But the material isisotropic, and parameters are E = 4.6375 GPa, ν = 0.3, and ρ = 1, 244 kg/m3.

2a( , )p x t

y

o x

Fig. 4 Semi-infinite plate and coordinate system

236 K. Liu, W. Liu

Fig. 5 contour lines of the horizontal and the vertical displacements at t = 104 µs calculated by present method. (a) The contourlines of the vertical displacement. (b) The contour lines of the horizontal displacement

Fig. 6 The contour lines of the horizontal and the vertical displacements at t = 104 µs calculated by ANSYS LS-DYNA. (a) Thecontour lines of the vertical displacement. (b) The contour lines of the horizontal displacement

Application of discrete element method for continuum dynamic problems 237

The number of discs is 51,814 for the seven-disc model and 45,000 for the nine-disc model with the radiusr = 1 mm. The time step is �t = 0.2 µs, and the total time is t = 200 µs. Figure 7 shows the contour linesof the vertical displacement at time t = 104 µs calculated by the seven-disc model and the nine-disc model,respectively. Because the above problem is symmetrical about y-axis, only the part in the positive direction ofx-axis is presented. Comparing Fig. 7a with Fig. 7b, it can be seen that the numerical results are the same.

Figure 8 shows the distribution of the dimensionless maximum shear stress τmax/pmax at t = 104 µs forthe two DEM models, respectively, where pmax = max[p(x, t)] is shown in Eq. (21), τmax is the maximumshear stress. From the figure, we can clearly identify the quasi-longitudinal wave, the quasi-transverse wave,the von Schmidt wave and the two peaks of Rayleigh wave. And the numerical results are the same by twomodels. Comparing the results shown in Fig. 8 with the corresponding result for the same example obtained bythe method of characteristics [10] (the figure is omitted here), we can find that the distributions of τmax/pmaxare almost the same, except that the two peak values of Rayleigh wave are slightly different.The reason isbecause the boundary value is usually substituted by the value of the inner discs in DEM. It not only showsthat accuracy of two kinds of models is the same, but also verifies accuracy of DEM in continuum again.

4.3 Elastic wave propagation in a layered medium

In order to estimate correctness and capability of the DEM models, stress wave propagation in a layered mediumshould be simulated. Berezovski and Maugin has concerned the numerical example in a layered medium by

Fig. 7 The contour lines of the vertical displacement at t = 104 µs. (a) By the seven-disc model. (b) By the nine-disc model.

238 K. Liu, W. Liu

Fig. 8 Distribution of the dimensionless maximum shear stress τmax/pmax at t = 104 µs. (a) By seven-disc model. (b) Bynine-disc model

using the FVM[11]. In this section, the same example in Ref. [11] is calculated by using our program. Ashort-time excitation by sinusoidal normal stress at a part of the bottom boundary with linear retardation isconsidered. All other boundaries are stress-free in a layered medium. Figure 9 describes the density distribu-tion, where the gray and the black represent the copper and aluminum, respectively. The number of discs is117,896 for the seven-disc model with the radius r = 0.5 mm. The time step is �t = 0.08 µs, and the totaltime is t = 80 µs. Figure 10 shows the contour plot of the total displacement at time t = 65 µs calculated bythe seven-disc model. The picture is asymmetric because of asymmetric loading at the boundary. Comparingthe result shown in Fig. 10 with the result by the FVM in Ref. [11] (the figure is omitted here), we can findthat the snapshot of the elastic stress wave propagation is roughly same. Although the result by FVM is moreaccuracy, the numerical result obtained by DEM is satisfied.

4.4 Stress wave propagation in a square plate with a hole

The accuracy of the present DEM models is ensured through the examples in Sects. 4.1 and 4.2. In this sec-tion, the problem of dynamical stress concentration is calculated. As shown in Fig. 11, the stress propagationprocess in an orthotropic square plate with a circular hole under a pressure pulse is simulated. The elastic

Application of discrete element method for continuum dynamic problems 239

Fig. 9 Density distribution in a layered medium

Fig. 10 Contour plot of elastic wave propagation in a layered medium

coefficients are c11 = 70.72 GPa, c12 = 21.84 GPa, c22 = 101.9 GPa, and c66 = 24.40 GPa. The mass densityis ρ = 2, 488 kg/m3. The dimension of a square plate is 200 × 200 × 1 mm3(Length × width × thickness).A hole with a radius of 10 mm lies in the center of the plate. The initial and the boundary conditions are given by

{σx = σy = σxy = 0, vx = vy = 0 for t = 0,

σy = −p(x, t) = −p0 H(a − |x |)(1 − 0.5(αt − β)2)e−(αt−β)2for y = 0,

(22)

where H(x) denotes the Heaviside function, α = 1 × 106 s−1, β = 2.5, p0 = 500 MPa, and the loaded areais a = 10 mm.

240 K. Liu, W. Liu

2a( , )p x t

x

y

o

A(-0.001,0.115)

B(-0.001,0.085)

CD

Fig. 11 Sketch map of a square plate model with a hole at the center

Fig. 12 The stress wave field of stress σy at t = 20 µs

The problem is numerically calculated by the nine-disc model, where the spring constants are given by Eq.(19). The disc radius is r = 0.25 mm, and the number of elements is 158376. The time step is �t = 0.05 µs.Figures 12 and 13 show the stress wave fields of stress σy at time t = 20 and 28 µs, respectively. The lon-gitudinal wave arrived the center of the square plate at time t = 20 µs. After t = 20 µs, the longitudinalwave is disconnected in the hole, and the reflected wave has occurred on top of the hole. When t = 28 µs, thelongitudinal wave is rejoined again, that shows the phenomenon of diffraction. The shape of the stress waveis close to a half ellipse, because the material is orthotropic and the vertical wave speed is fast. The reflectedwave produced at the boundary of the hole is interfered with the transverse wave. These interactivities betweenstress waves result in the occurrence of stress concentrations.

The comparative curves at points A(−0.001, 0.115) and B(−0.001, 0.085) in Fig. 11 with the hole andwithout hole are obtained for analyzing the phenomenon of dynamic stress concentration. Figures 14 and 15show the time variation of σy at points A and B, respectively. It can be seen from Fig. 14 that the maximumtensile stress occurs at point A when t = 18 µs. Because of the reflection wave aroused at the hole, the peakvalue of σy with the hole is twice as the value without hole, which shows the phenomenon of dynamic stressconcentration clearly. When t = 20 µs, the longitudinal wave has arrived at the middle of the square plate.The values of σy at points A and B are small, and the maximum tensile stress occurs at points C(−0.015, 0.1)

Application of discrete element method for continuum dynamic problems 241

Fig. 13 The stress wave field of stress σy at t = 28 µs

Fig. 14 Time variation comparison of σy at A

and D(0.015, 0.1) from Fig. 12. But the values of σy with the hole and without hole are very close. Whent = 22 µs, the tensile stress does not occur at point B due to the existence of the hole, but have a smaller valuelater on. After this, the stress values are smaller around the hole as the stress wave has passed by.

4.5 Stability of the numerical scheme

Because our DEM is a explicit method, the solution is only stable if the time step �t is approximated as

�t ≤ �tcr = 2

ωmax, (23)

where �tcr is the critical value of the time step, ωmax = √kmax/m is the maximal circular frequency of a

mass-spring system, m is the mass of the smallest particle, kmax is the maximum normal contact spring stiffness.As the effect of boundary condition and other factors have to be taken into account, �t is always less than

�tcr. In our numerical examples, the stability is evaluated by examining the relative energy error Err in the

242 K. Liu, W. Liu

Fig. 15 Time variation comparison of σy at B

Fig. 16 Time curves of the relative energy error

system, which is defined as

Err = Ein − Et

Ein(24)

where Ein is the input energy and Et is the total energy.For the example by seven-disc model in Sect 4.2, Fig. 16 shows time curves of the relative energy error

with different time steps �t = 0.050, 0.822, 0.858, 1.000 µs. As can be seen, for �t = 0.822 µs which isobtained from Eq. (23), not only the relative energy error is steady, but also the curve is almost the same as thecurve calculated with the less time step �t = 0.050 µs.

5 Conclusions

Based on the above results, we assert that the present DEM models is efficient for the numerical analysis ofdynamic problems in continuum. It has extended the range of application of DEM to solving anisotropism

Application of discrete element method for continuum dynamic problems 243

problems. The usage of two kinds of models, seven-disc model and nine-disc model, has strengthened theflexibility of practical application. The same numerical results are obtained by seven-disc model and nine-discmodel for impact problems. Furthermore, comparing the numerical results obtained by DEM, FEM, FVM andthe method of characteristics, validity and accuracy of our DEM models are clearly demonstrated. Moreover,the stress wave propagation on a square plate with a hole is simulated in order to analyze the phenomenon ofdynamical stress concentration and wave diffraction. Therefore, DEM not only is applied in non-continuum(it widely used in geotechnical engineering and powder technology), but also is suitable for continuum (asFEM and FVM). The research on DEM, however, has just started from the principle of continuum, and a lotof work need be launched. The development of the DEM models for plastic materials and elastic-viscoplasticmaterials will be our future work.

References

1. Liu, K., Gao, L., TANIMURA, S.: Application of discrete element method in impact problems. JSME Int J Ser A 47, 138–145(2004)

2. Cundall, P.A.: A computer model for simulating progressive large scale movement in block rock system. Symp ISRM Proc2, 129–136 (1971)

3. Herrmann, H.J., Luding, S., Modeling granular media on the computer. Continuum Mech Thermodyn 10, 189–231 (1998)4. Tsuji, Y.: Activities in discrete particle simulation in Japan. Powder Technol 113, 278–286 (2000)5. Liu, K., Gao, L.: A review on the discrete element method (in Chinese). Adv mech 33, 483–49 (2003)6. Onate, E., Rojek, J.: Combination of discrete element and finite element methods for dynamic analysis of geomechanics

problems. Comput Methods Appl Mech Eng 193, 3087–3128 (2004)7. Mohammadi, S., Owen, D.R.J., Peric, D.: A combined finite/discrete element algorithm for delamination analysis of com-

posites. Fin Elem Anal Des 28, 321–336 (1998)8. Sawamoto, Y., Tsubota, H., Kasai, Y., Koshika, N., Morikawa, H.: Analytical studies on local damage to reinforced concrete

structures under impact loading by discrete element method. Nucl Eng Des 179, 157–177 (1998)9. Liu, K., Gao, L.: The application of discrete element method in solving three dimensional impact dynamics problems. Acta

Mech Sol 16, 256–261 (2003)10. Liu, K., Li, X.: Numerical simulation of stress wave propagation in an orthotropic plate. In: Proceedings of the 2nd interna-

tional symposium on impact engineering, pp. 19–24 (1996)11. Berezovski, A., Maugin, G.A.: Simulation of thermoelastic wave propagation by means of a composite wave-propagation

algorithm. J Comput Phys 168, 249–264 (2001)12. LeVeque, R.J.: Wave Propagation algorithms for multiple hyperbolic systems. J Comput Phys 131, 327–353 (1997)13. Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical simulation of two-dimensional wave propagation in functionally

graded materials. Eur J Mech A Solids 22, 257–265 (2003)