discrete element method (dem) for modeling solid and particulate media

8/10/2019 Discrete Element Method (DEM) for Modeling Solid and Particulate Media

1/6

D i s c r et e E l e m e n t M e t h o d D E M ) fo r M o d e l in g S o l id a n d P a rt ic u l at e M e d i a

Feder ico A. Tavarez , 1 M ichael E. P lesha , 1 and Law rence C. Ba nk2

I n t r o d u c t i o n

The f racture behav ior of so l ids such as concre te and rock under im pact loading has

becom e an area of increased research in recent years . These mater ia ls are complex

and extreme ly heterogeneous , especia l ly af ter they degrade f rom sol id to par t icu la te .

Con t inuum based m ethods such as the f in i te e lement method are cha l lenging to ap ply

to th is c lass of problems , and are p lagued by the need for cont inuum cons t i tu t ive

mod els , severe e lem ent d is tor t ion and f requent remeshing. Fur thermore , e ros ion of

e lements impl ies loss of mass , which may a l ter the so lu t ion for problems wi th

extens ive damage. This paper presents a d iscre te e lement method (DEM) for the

analys is of such problems , in w hich so l ids are a l lowed to progress ively dam age in to

par t icu la tes . The s tudy is conducted us ing a modif ied vers ion of the program

TR UB AL, or ig inal ly developed by Cund al l and S track (1979), for mod el ing the

mechanical behavior of granular materials . Jensen et al . (1999) introduced the

concep t o f clusters to obta in a bet ter representa t ion of par t ic le shape for m odel ing

granular mater ia ls . Clus ters are formed by combining a number of c i rcular -shaped

particles in a sem i-r igid configuration. In add it ion to o ur work in c luster ing, there has

also been s ignificant new act iv i ty by o thers a long these l ines , such as Qiu and Cruse

(1997), Saw am oto et al. (1998), an d Thom as and Bra y (1999).

M e g a c l u s t e r i n g t o M o d e l S o l i d s

To m odel a so l id such as concre te , th is concept can be extended to

meg clusters

to

represent the entire geometry. For modeling damage, Jensen et al . (2001) also

introduced a s l iding energy cri ter ion for cluster fai lure. However, by using only

norm al an d tangen t ia l springs to model in tere lemen t in teract ions, there are num erous

scenar ios in w hich c lus ters can brea k-off and eac h o f these mus t b e ant ic ipated and

accou nted for in the analysis . W hile the procedure reported in Jensen et al. (2001) is

1 Dept . o f Engineer ing Phys ics , Univ . of W iscons in-Madison, Madison, W I 53706,

U.S.A. Co ntact e-mail : plesha@ engr.wisc.edu

2 Dept . o f Civ i l and Envr . Engr. , U niv . of W iscons in-Madison, M adison, W I 53706,

U.S.A.

155

Discrete Element Methods

Downloadedfromascelibrary.orgby

UNIVERSIDADEFEDERALDEPAR

ANAon05/15/14.CopyrightASCE.F

orpersonaluseonly;allrightsreserved

.


2/6

156 D I S C R E T E E L E M E N T M E T H O D S

effect ive for c lus ters wi th a smal l number of e lements , a more fu l ly au tomat ic

procedure for megaelus ters i s des i rab le .

The approach descr ibed here in uses beam f in i te e lements for model ing

intracluster interact ions; therefore, no special cri terion is needed to al low for

declus ter ing . Th e in i t ia l so lid medium i s assembled by consol idat ing par t i c le c lus ters

in to the des i red geomet ry . Af ter the medium i s consol idated , the cent ro id of each

par t ic le i s con nected to the cent ro id o f i ts contacting neighbors us ing ma ss less beam

fin i te e lements , as show n in F igure l (a) . Phys ica l parameters gov ern ing DE M par t ic le

in teract ions may be determined by val idat ing numerica l s imula t ions of s tandard

labora tory tests with actual expe rimen tal resul ts .

B y us ing the proposed D EM approach, nonl inear pheno me na such as cracking

and the progress ive damage of so l ids in to par t i cu la tes are s imula ted au tomat ica l ly .

Severa l cr i t er ia are poss ib le to a l low for f ragmenta tion , and w e p resent ly use two.

The f i rs t a l lows for damag e due to f r i c t ional s lid ing and the second a l lows for s t ress

dep ende nt fai lures such as fracture. T hese cri teria are described in grea ter detai l soon.

Once the beam element fa i l s, it i s removed f rom the ca lcu la t ion , as show n in F igure

1 b) , and i f needed , e i ther imm ediate ly or subsequent ly , i s rep laced b y a c onvent ional

f r ic t ional contact. As c lus ters damage, the mass o f the medium i s conserved becau se

i t i s concent ra ted in the d i scre te par t ic les and not in the beam elements . Tw o d i f ferent

colors are used in F igure 2 to help v i sual ize the decluster ing . Once d eclus ter ing

occurs , the par t i c les on the n ew ly created surface area can fur ther in teract f r i c t ional ly

w i th o ther D EM elements , as show n in F igure 1 c) .

F i gu re 1 . (a) Cons t ruct ion of megaclus ters us ing beam elements ; (b) Fracture of a

cluster into two sep arate clusters; (c) Subsequ ent fr ict ional contac t betw een th e two

clusters

a 0 0 -a 0 0

0 12b 6Lb 0 - 1 2 b 6Lb

0 6Lb 4L2b 0 -6Lb 2L2b

a 0 0 a 0 0

0 - 1 2 b

-6Lb 0

12b

-6Lb

0 6Lb 2L2b 0 -6Lb 4L2b

U

1

V

1

0

U

2

V

2

0

9 2

F

x l

F

y l

M

1

F

x 2

F

y 2

M

2

(1)

Fi gu re 2 . Tw o-dimens ional beam elem ent and d i sp lacement - in ternal force re la t ion






.


3/6

D I S C R E T E E L E M E N T M E T H O D S 157

In the local coordinates of each beam element , th ree degrees of f reedom are

ass igned a t each end , as shown in F igure 2 . Thus , a t each t ime s tep , the e lement

in ternal forces are computed us ing the usual beam element s t i f fness express ions

(Co ok, et al ., 2001).

C l u s te r D a m a g e D u e to F r a c t u r e

F a i lu re o f beam e l emen ts m ay be caused by combined s t r e tch ing and bend ing , and

the fo l low ing form of the von M ises p las t i c i ty cr i ter ion m ay be used to de termine

fa i lure (Kun an d Herm ann, 1996).

6/=+ max0d'[e21)~


4/6

158 D I S C R E T E E L E M E N T M E T H O D S

F i g u r e 3. (a) Spring-d am per mode l for frictional surface contacts; (b) Load-

displacem ent relations for particle surface contact

wh ere fi is the tangential force at the contact,

d~ s

is the increment of relat ive

tangential s l iding displacement between the element and a contacting neighbor. The

total work is obtained by

w,= J dW, (5)

Thus, w hen Equa tion (5) fails to hold, the element separates from the cluster.

The calculations performed in DEM alternate between the application of

Newton's second law to each particle, and internal force calculations at each contact.

The equations o f motion are integrated in t ime using the central difference method as

follows:

M i . + f~ .t = f~ , (6)

1

A t

(7)

2 -1 ext int

- f ~ ) + 2 x .

~+ I = At M f ,~ - xn_ 1

8)

where f int incorporates contact, friction, and viscous (damping) forces. Since explicit

integration is conditionally stable, an ad hoc scheme for time step selection is used

where the critical time step is computed using

~t cri t = a~Jmmi kma~

(9)

wh ere k,,ax is the largest inter-particle spring stiffness, mmin is the m ass of the sma llest

particle, and hence

~kmr~/mm~

is a crude est imate of the highest natural frequen cy o f

vibration for the mo del, and a is a used selected param eter. Com putationa l

experience shows that values of a near 0.1 are typically satisfactory to provide a

stable computation (Jensen, 1999).






.


5/6

D I S C R E T E E L E M E N T M E T H O D S 159

Our appl ica t ion of the proposed method is for model ing impact and

penetra t ion events, w i th specia l in teres t on concre te members . Such s imulat ions are

need ed for assessing vu lnerab il i ty o f s tructures , transportat ion facil i t ies , nuc lear

reactor con tainm ent vessels , and so on, to accidents or terroris t at tacks. Th e various

impact response character is t ics and fa i lure mechanisms , such as mater ia l

den sif ication and spall ing at the front face, scabbing at the rear face, projecti le m otio n

through the target , energy transfer processes , and projecti le shape change due to

eros ion are num er ica l ly evaluated . F igure 4 shows a typical DE M m odel where the

m edium was assembled by consol idat ing three-par tic le c lus ters in to a rec tangular

beam with f ixed-end condi t ions . As a result , the so l id medium conta ins ir regular i ties

and void space, w hich are s im i lar ly present in rea l concre te . As seen in F igure 4 , the

project i le was a lso modeled by a coarse DE M cluster . D amag e to the concre te beam

as w el l as the project i le is qu ant i ta tive ly and qual i ta t ive ly evaluated in the num er ica l

s imulation.

Figure 4. Concrete beam w ith f ixed end condi t ions subjected to project i le

penetra t ion

Acknowledgments

Financia l suppor t for Feder ico A. Tavarez is provided through a Dwight D.

Eisenhow er Transpor ta t ion Fel lowship f rom the Nat ional High way Ins t itu te . Dr .

Richard P . Jen sen is acknowledged for h is valuable ass is tance and col laboration .

References

Cook, R.D., Malkus, D.S. , Plesha, M.E. , and Witt , R.J . (2001) Concepts and

Applications o f Finite Element Analysis, 4 h ed., Jo hn W iley an d Sons.

Cundall, P.A., and Strack, O.D.L. (1979) The D&tinct Element Metho d as a Too l fo r

Resea rch in G ran ular Media, Pa r t 11,

NSF Rep ort Grant ENG76-20711.






.


6/6

1 60 D I S C R E T E E L E M E N T M E T H O D S

Jensen, R.P., Bosscher, P.J., Plesha, M .E., and Edil, T.B. (1999) "DE M Simulation o f

Granu lar M edia - Structure Interface: Effects o f Surface Roughness and Particle

Shape,"

lnt. J. N um. An al Methods in Geom echanics,

No. 23, pp. 531-547.

Jensen, R .P., Plesha, M.E., Edil, T.B., and Bosscher, P.J. (2001) "DE M Simulation of

Particle D am age in Granular Med ia - Structure Interfaces," lnt. Journal of

Geomechanics.,

Vol. 1, No . 1, pp. 21 -39 .

Kun, F., and Hermann, H.J. (1996) A Study of Fragmentation Processes using a

Discrete Element Method,"

Co mp. Methods in A ppl. M ech. and Engineering,

No.

138, pp. 3 -18 .

Qiu, x. , and Kruse, D. (1997) "Analysis of Flow of Ore Materials in a Co nveyor

Transfer Chute Using Discrete Element Method,"

Joint ASME, ASCE, and SES

Sum mer Meeting, Northwestern University, Chicago, IL, June 29-Oct. 1.

Sawamoto, Y., Tsubota, H., Kasai, Y., Koshika, N, and Morikawa, H. (1998)

"Analytical Studies on Local Damage to Reinforced Concrete Structures Under

Impact Loading by Discrete Element Method,"

Nuclear Eng r. an d Design,

Vol. 179,

pp. 157-177.

Thom as, P.A ., and Bray, J.D. (1999) "Capturing Nonspherical Shape o f Gran ular

M aterial w ith D isk C lusters,",/ .

Geo tech. Eng.,

Vol. 125 , 169-178.





.

discrete element method (dem) for modeling solid and particulate media

Documents