Energy-conserving contact interaction models for ...

Comput. Methods Appl. Mech. Engrg. 205–208 (2012) 169–177

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Energy-conserving contact interaction models for arbitrarily shaped discrete elements Y.T. Feng ⇑, K. Han, D.R.J. Owen Civil and Computational Engineering Centre, College of Engineering, Swansea University, SA2 8PP, UK

a r t i c l e

i n f o

Article history: Received 15 July 2010 Received in revised form 2 January 2011 Accepted 9 February 2011 Available online 18 February 2011 Keywords: Discrete element Normal contact model Energy conservation Arbitrarily shaped bodies

a b s t r a c t This work aims to establish a unified theoretical framework for normal contact of arbitrarily shaped discrete elements in discrete element modelling solely based on the energy-conservation principle for elastic contact. It will show that the normal force must be a potential field vector associated with a potential. With the construction of an appropriate potential function, a complete normal contact model, including the magnitude and direction of the normal force and the normal contact line, can be unambiguously derived for a pair of any shaped bodies. The model in its final form is simple and has a clear geometric perspective for 2D bodies. It can also recover some well-known models for bodies with simple geometric shapes. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The discrete element method (DEM), originated in the 1970’s by Cundall and Strack [1], has now been established as a powerful numerical technique for modelling a wide range of physical problems involving discrete/particulate phenomena. The unique feature of the method is its ability to account for the motion of individual discrete elements in a system. In the classical sense, all discrete elements are assumed rigid and their localised contact behaviour in terms of contact forces is determined by the so-called contact interaction laws which are developed within the framework of penalty methods [2], i.e. the traditional impenetrability constraint is only approximately satisfied, thereby allowing a small amount of overlap between the contacting bodies to occur. The numerical procedures and algorithmic details of DEM can be found in [1] and elsewhere [4]. The key to the success of the method primarily relies on two modelling aspects: (1) the realistic representation of real shapes of discrete bodies; and (2) the employment of physically correct interaction laws between the bodies. In many existing DEM developments and applications, only circular/spherical bodies are used. It is well known that circular/spherical bodies lack the angularity that is essential to capture some physical behaviour in many engineering applications. Therefore it is practically important to employ irregular shaped elements in DEM. This, however, imposes

⇑ Corresponding author. E-mail address: [email protected] (Y.T. Feng). 0045-7825/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2011.02.010

considerable challenges on the use of proper contact interaction laws for such elements. Strictly speaking, a contact interaction law, which defines the relationship between the contact forces and the contact overlap characteristics, should be physically based. This can be generally achieved in three different ways. One is to obtain an analytical expression, and the others are to find an empirical expression by experimentation or by numerical modelling. However, it is well known that only a very limited number of analytical formulations are available for use as contact interaction laws. Even the most significant one, the Hertz theory, is valid only for two linear elastic (dry) spheres with their surfaces being continuous up to second derivatives in the contact region. Its extension, even to two 2D circular disks (or cylinders in 3D), is problematic, as explained in [9]. Thus there is a serious problem in purely relying on the analytical approach to establish contact interaction laws for bodies with sharp corners. In this case experimental or computational approaches may have to be adopted. It is, however, impossible to obtain a universal interaction model valid for all possible combinations of shapes and sizes. For the above reasons, ‘artificial’ or numerical contact interaction models are often proposed in the DEM to deal with contact situations where no existing analytical or empirical contact models are available. At present, most of the numerical contact models are of a heuristic nature. Very often, a linear ‘‘spring’’ contact model is used regardless of the actual contact behaviour. The issues discussed above can be illustrated by a corner/corner contact scenario arising from two polygons/polyhadra in contact. Although polygons/polyhadra are more realistic shapes than

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Y.T. Feng et al. / Comput. Methods Appl. Mech. Engrg. 205–208 (2012) 169–177

circles/spheres, especially in rock or geomechanics applications, their application in the DEM has been rather limited. This can be attributed, at least partially, to the lack of simple and robust interaction models for corner-corner contact of two polygonal/polyhadral bodies. A particular difficulty is how to determine the direction of the normal contact force. Because of the directional discontinuity at the corners, the normal direction is often not evolved in a smooth way but presents a discontinuous jump when a small relative movement occurs between the two contacting polygons/polyhedra. This numerical defect often introduces a certain amount of artificial energy into the computation, which, when accumulated and propagated, may cause severe numerical errors or a total simulation failure. Although a scheme could be designed to account for all possible contact scenarios as in the discontinuous deformation analysis (DDA) [6], it is very unlikely that such a scheme can be sufficiently robust to resolve the ambiguity of normal contact directions for all cases. Also the well known ‘common plane’ model proposed in [5] is defined in a rather heuristic manner. More importantly, it cannot guarantee a smooth evolution of the normal contact direction during the continuous relative movement of a contact pair. In addition, the commonly used nodal/facet contact model in the finite element modelling of contact has also been proved inadequate in tackling general polyhedral contact situations [2]. Note that a similar scheme to this work where overlapping areas are used to treat non-smooth contact is proposed in [3]. Consequently, a fundamental issue arises on how to assess these numerical interaction models, or what basic properties the numerical models should possess. This has been partially addressed recently in [11] in the context of upscaling of discrete element models, in which four principles are proposed to assess the suitability of an interaction model. In the current work, another principle, energy-conservation under elastic contact, is introduced. Although energy-conservation is a fundamental physical principle, some existing numerical models may violate this principle, thereby introducing artificial energy into a system and leading to erroneous results. More significantly, based on the energy-conservation principle, this study will provide a unified theoretical framework for developing numerical contact interaction models for arbitrarily shaped contacting bodies. The derived numerical contact model will uniquely and unambiguously determine not only the magnitude but also the direction of the normal contact forces, as well as the surface on which the reference contact point (where the contact forces should be applied) lies. This is a further extension of our previous work on the development of energy based contact models for polygons and polyhedra [7,8], although the main results remain the same. The paper is organised as follows. In the next section, a general theoretical framework will be established for developing an energy-conserving contact model. Then the existing interaction laws for circular disks and spheres will be examined under the energyconservation principle. Sections 4 and 5 introduce the energy-conserving normal contact models for polygons and polyhedra contact, respectively. Further extension of these contact interaction models to arbitrarily shaped 2D/3D bodies will be discussed in Section 6. Note that normal contact between two frictionless and dry bodies are considered only.

Fig. 1. Two arbitrarily shaped bodies in contact.

The angular orientations of the bodies are represented by h and h0 , respectively. In the context of contact mechanics [9], the two bodies will deform under the action of forces to form a common contact line (in 2D) or surface (in 3D). A distributed contact pressure will act on body I from body II on the line/surface in the normal direction. The resultant force Fn of the pressure distribution is the normal contact force that is exerted on body I due to body II. An equal reaction force F0n ¼ Fn acts on body II from body I. It is always possible to find a particular point, referred to as the contact point, on the common line/surface at which both Fn and F0n act so that no resultant moment needs to be added to either body. In the context of the DEM, both bodies are assumed to be rigid. A contact interaction law has to be defined to determine the magnitude and direction of the normal contact forces, Fn and F0n , in terms of the spatial configurations of the two contacting bodies described by x, x0 , h and h0 , together with the shapes of the bodies, i.e. Fn = Fn(x, x0 , h, h0 ), and F0n ¼ Fn ðx; x0 ; h; h0 Þ. The position of the contact point should also be determined. This point and the normal direction defines the (nominal) contact/tangential plane on which the tangential contact force acts. Assume that body I is prescribed to move along an arbitrary closed path L translationally only, while body II is fully fixed. Then the normal force Fn will be a function of x only, Fn = Fn(x). The total work done by the normal force along the entire path can be expressed as

WðLÞ ¼

I

Fn ðxÞ dx:

When the contact is elastic, clearly

WðLÞ ¼ 0 8L ðclosed pathÞ: 2. Energy-conserving contact interaction models: general framework Consider two arbitrarily shaped (2D or 3D) discrete bodies, I and II, in contact as shown in Fig. 1, where a 2D illustration is used for simplicity. In a fixed global coordinate system, two points O and O0 from the two bodies are arbitrarily chosen as the two reference points of the bodies and have coordinates of x and x0 , respectively.

ð1Þ

L

ð2Þ

If Fn(x) is continuous, then following a theorem of calculus [12], to satisfy the above condition Fn has to be a potential vector field, i.e. there exists a scalar potential function /(x) such that

Fn ðxÞ ¼ r/ðxÞ ¼ FðxÞn with

FðxÞ ¼ kFn k;

n ¼ Fn =kFn k;

ð3Þ


where F(x) is the magnitude, and n is the unit vector of the force or the normal direction, which is also normal to the contact plane. The contact point, however, is yet to be defined. For the force given by (3), an equivalent version of Condition (2) states that the work done by the force when the body moves from a point A, xA , to another point B, xB , will be path-independent:

WðLA!B Þ ¼

Z

B

Fn ðxÞ dx ¼ /ðxA Þ /ðxB Þ:

ð4Þ

A

Similarly, when the rotational motion of body I is permitted, the normal force Fn should be a function of angular vector h as well: Fn(x, h). Then for an elastic contact, the total work done by the force along any closed path, now in the (x, h) space, should be zero:

WðLÞ ¼

I

½Fn ðx; hÞ dx þ ðrc Fn ðx; hÞÞ dh ¼ 0;

ð5Þ

L

where rc = xc x and xc are the coordinates of the contact point c. Following a similar argument to the previous non-rotational contact, it can be concluded that condition (5) is equivalent to the existence of a potential /(x, h) that defines the normal force as

@/ðx; hÞ Fn ¼ rx /ðx; hÞ ¼ @x

ð6Þ

and a nominal ‘contact’ moment M as

M ¼ rh /ðx; hÞ ¼

@/ðx; hÞ @h

ð7Þ

such that

M ¼ rc Fn :

ð8Þ

Eq. (8) provides the condition to determine the position of the reference contact point c, xc. However, it can be deduced from (8) that

M Fn ¼ M rc ¼ 0:

ð9Þ

When M and Fn are given by (6) and (7), xc cannot be uniquely determined from (8), but a general solution can be obtained instead:

xc ¼ x þ

nM þ kn ðn ¼ Fn =kFn kÞ; kFn k

ð10Þ

where k is a parameter, which represents a straight line in the normal direction n, indicating that any point along the line can be chosen as the contact point c. Thus this line is termed the normal contact line. Because the contact point determines the position of the contact plane, this means that the actual position of the contact plane cannot be uniquely determined by the energy conservation principle alone. This is not an issue, however, for the normal contact forces because the bodies will be subjected to the same loading effect as long as the normal forces act on the normal contact line. Thus the position of the contact plane will be important only for tangential forces, which is beyond the scope of the current work. Practically the position of the plane can often be determined in an ad hoc manner, or for a geometric convenience. The above discussion applies equally to the second body, resulting in the normal force defined by

F0n ¼ rx0 /ðx0 ; h0 Þ

ð11Þ

0

0

from which a contact point c, metric form:

n 0 M0 0 0 x0c ¼ x0 þ F0 þ k n n

ð14Þ

and both (10) and (13) should represent the same normal contact line. In summary, based on the energy conservation principle, a generic numerical framework is fully established to define a complete normal contact interaction model for two arbitrary bodies, including the magnitude and direction of the normal contact forces and the normal contact line, provided that the potential function / can be defined in the first place. Therefore it is of paramount importance that a proper potential function / can be constructed for a given pair of contacting bodies. There are some basic mathematical properties that /(x, x0 , h, h0 ) should possess. The force equilibrium condition (14) requires

rx / þ rx0 / ¼ 0:

ð15Þ

The condition (8) and a similar one for F0 and M0 require

rx / rh / ¼ 0;

rx0 / rh0 / ¼ 0:

ð16Þ

In the subsequent sections, some particular forms of the potential function will be proposed to recover the existing numerical normal contact models for disk/disk or sphere/sphere, polygon/ polygon and polyhedron/polyhedron. For disk/disk or sphere/ sphere contact, the potential is chosen as a function of the overlap penetration; while for other generally shaped 2D/3D bodies, including polygon/polygon and polyhedron/polyhedron, the potential is taken to be a function of the overlap area or volume. 3. Disk/disk or sphere/sphere contact models Consider two disks or spheres with radii of R and R0 , and the position vectors of their centres, x and x0 , respectively. The overlap or penetration, d(P0), is defined as

d ¼ ðR þ R0 Þ kx x0 k:

ð17Þ

For disk/disk or sphere/sphere contact, all the existing contact models, including the Hertz model for two linear elastic spheres [9], take the line linking the two centres as the normal direction n:

n ¼ ðx x0 Þ=kx x0 k:

ð18Þ

The magnitude of the force, Fn, is expressed as a function of the penetration d as

F n ¼ F n ðdÞ

ð19Þ

which is independent of the rotation of the disks/spheres. In most cases, Fn(d) is chosen to be in power law form

F n ðdÞ ¼ kn db ;

ðkn and b are parametersÞ

ð20Þ

which includes, as special cases, a linear model (b = 1) and the Hertz model (b = 3/2). We will demonstrate that the above contact model can be derived from the framework established in the previous section and thus is energy conserving, provided that Fn(d) is a single-valued continuous function. In this case, it is possible to construct a function /(d)

Z

d

F n ðxÞdx:

ð21Þ

0

0

M ¼ rh /ðx ; h Þ 0

Fn þ F0n ¼ 0

/ðdÞ ¼

and the ‘contact’ moment by

171

where n0 ¼ F0n =F0n and k0 are parameters. It is expected that

ð12Þ x0c ,

is defined by a straight line in para-

ð13Þ

For instance, for the power law force of (19), /(d) = kndb + 1/(b + 1). Then, following the procedure developed in the previous section, the normal force can be obtained as

Fn ¼

d/ðdÞ dd ¼ /0 ðdÞ : dx dx

ð22Þ

172


From (17), 0

dd ðx x Þ ¼ ¼n dx kx x0 k

ð23Þ

and (21) gives /0 (d) = Fn(d). Then it follows that

Fn ¼ FðdÞn:

ð24Þ

Furthermore, as /(d) is independent of the rotation of the disks/ spheres,

M¼

/ðdÞ ¼0 dh

ð25Þ

from (10), the normal contact line is given by

xc ¼ x þ kn ðk is a parameterÞ

ð26Þ

which passes through the two disk/sphere centres, i.e. the normal forces will pass through the centres, as expected. In conclusion, the standard normal models for disks/spheres can be fully recovered from the energy-conservation principle if the potential / is a function of the penetration d. On the other hand, it will be shown in Section 6 that an alternative contact model can be established for disks/spheres, based on the assumption that the potential is a function of the overlap area/volume.

If the penetrating vertices p and q are taken as the two reference points of the two bodies, respectively, then the configurations of the bodies can be described by the coordinates of the reference points, xp and xq, and the two rotation angles, hp and hq, of the bodies. Assume that the two sets of edges connected to the penetrating vertices p and q intersect at points g and h with coordinates of xg = (xg, yg) and xh = (xh, yh), respectively. Then the overlap area, A, formed by the two penetrating polygons as shown in Fig. 2, is determined by the four points p, h, q and g:

A¼

1 ðxp xq Þ ðxg xh Þ: 2

ð27Þ

Since the points g and h are determined not only by xp and xq, but also by hp and hq, the overlap area A is therefore a function of xp, xq, hp and hq, i.e. A = A(xp, xq, hp, hq). 4.1. Normal contact forces Now by virtue of (6), the normal force Fn can be obtained as

Fn ¼ rxp /ðAÞ ¼

@/ðAÞ d/ðAÞ @A ¼ ¼ /0 ðAÞ rxp A @xp dA @xp

ð28Þ

or

4. Normal contact models for polygon/polygon contact

Fn ¼ F n n;

Consider two rigid polygonal bodies I and II that are in a typical corner–corner contact situation (Fig. 2). Other contact situations arising from two polygon contact can be treated as either a special case, such as a corner-edge contact, or its extensions for more complex contact scenarios. As is first proposed in [7], the potential / is chosen to be a monotonically increasing function of the overlap area A, instead of the penetration. This is because, unlike the disk or sphere contact, the penetration is not always a well defined contact feature for non-circular/spherical bodies. Following the framework established in Section 2, an energy-conserving contact interaction model for two polygons can be established. The derivation of the contact model will be outlined below and further details can be found in [7].

where

n¼

ð29Þ

r xp A krxp Ak

with rxp A ¼

@A @xp

ð30Þ

and

F n ¼ kFn k ¼ /0 ðAÞkrxp Ak

ð31Þ

in which n defines the normal direction, and Fn is the magnitude determined by both /0 (A) and krxp Ak. Let t be the unit vector of the direction from h to g:

t¼

xg xh 1 ¼ fDxgh ; Dygh g kxg xh k bw

ð32Þ

in which Dxgh = xg xh,Dygh = yg yh; and bw is the distance between the two intersection points:

bw ¼ kxg xh k ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dx2gh þ Dy2gh :

ð33Þ

It is shown in [7] that

rxp A ¼ fDygh ; Dxgh g

ð34Þ

thus

krxp Ak ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dy2gh þ Dx2gh ¼ bw :

ð35Þ

Consequently

F n ¼ /0 ðAÞkrxp Ak ¼ bw /0 ðAÞ

ð36Þ

and

n¼

r xp A 1 fDygh ; Dxgh g: ¼ krxp Ak bw

ð37Þ

Clearly n t = 0, i.e. the normal contact direction is perpendicular to the line linking the two intersection points. In other words, t defines the tangential direction. To locate the position of the reference contact point C or the normal contact line, the nominal ‘contact’ moment is calculated as

M hp ¼ Fig. 2. Two polygons in corner/corner contact.

@/ðAÞ @A ¼ /0 ðAÞ : @hp @hp

ð38Þ

173


It is shown in [7] that, if Mhp and z-direction, then

@A @hp

are treated as vectors in the

@A ¼ bw rm n; @hp

ð39Þ

where

rm ¼ x m x p

ð40Þ

in which

xm ¼

1 ðxg þ xh Þ 2

ð41Þ

is the coordinates of the middle point of the line gh, m. Thus

M hp ¼ /0 ðAÞ

@A ¼ rm Fn @hp

ð42Þ

which indicates that xm is a particular solution to (10). Thus the normal contact line can be expressed as

xc ¼ xm þ kn;

ð43Þ

where k is a parameter. Obviously xc = xm, i.e. the middle point of the line gh is a preferred choice of the contact point as it has a clear geometric interpretation. Then line gh is the contact (tangential) line. Its length bw is now referred to as the contact width. These features are illustrated in Fig. 3. The normal force to the second body F0n can be determined in a similar fashion. Not surprisingly, F0n ¼ Fn . Also the line of the contact point is proved to be the same as (43). These imply that both conditions (15) and (16) are satisfied. Finally, when body I is continuously moving from the left position to the right (see Fig. 4), the normal direction defined above will smoothly change from n1 to n2. Therefore no directional jump occurs at the corner.

4.2. Normal force magnitude and choice of potential function Different choices of the potential function, /(A), will result in different magnitudes of the normal forces, while the normal direction will remain the same since it is determined by rx/. As suggested in [7], several possible options for /(A), and thus /0 (A), are listed in Table 1, in which kn is the penalty coefficient. It is noted that the commonly used overlap distance/gap is not explicitly present in the model. Instead, its usual role is replaced by the contact width, bw, which is a well defined characteristic contact feature. It is highlighted, however, that the magnitude of the normal force has to be computed strictly following the formula given by (31). Otherwise the energy-conservation property cannot be satisfied. For a numerical illustration, consider the contact of two polygons as shown in Fig. 5, where the second body is fully fixed while the first one is moving horizontally from the position O to the position O0 . The normal direction is determined by (37), which ensures a smooth transition of the normal direction from O to O0 . The magnitude Fn has the following three choices: (1) Fn = bw; (2) Fn = bwA, and 3) Fn = A. The first two correspond to /(A) = A, and A2, respectively, while no function /(A) is associated with the third one. Thus it is expected that the first two will be energy-conserving but not the third one. This can be simply checked by calculating the total work done by the contact force moving from O to O0 ,W(O ? O0 ). As positions O and O0 correspond to two non-contact situations, W(O ? O0 ) should be zero if energy is conserved, and nonzero otherwise. Table 1 Several possible forms of /(A) and Fn.

Linear Hertz-type Power

Fig. 3. Contact feature in corner/corner contact.

/(A)

/0 (A)

kFnk

knA

kn knA1/2

knbw knA1/2bw

knAm1

knAm1bw

3=2 2 3 kn A knAm/m

Fig. 5. Two polygons in corner/corner contact: energy conservation check.

Fig. 4. Smooth transition of the normal direction in corner/corner contact.

174


@A @A ¼ ; @xp0 @xp0

@A @A ¼ : @hp @hp

ð45Þ

Thus this contact situation can be covered by the corner–corner model. (3) Through-penetrating corners: For cases where the two bodies penetrate through each other, such as the case illustrated in Fig. 7(b), one can decompose these complex contact situations into several corner-corner situations, and then apply the corner-corner model to each case. For each body, there are two boundary segments involved in the contact. The normal contact force acting on body I consists of two forces each directed along the inward normal direction and at the middle point of each segment. Then the normal force is the resultant force of the two:

Fn ¼ /0 ðAÞðb1 n1 þ b2 n2 Þ

where n1 and b1, and n2 and b2 are respectively, the unit outward normals and the lengths of the two segments of body I. The line of action of Fn can also be obtained. Similarly, the normal force acting on body II can be computed as

Fig. 6. The x-components of Fn for three different choice of Fn.

Fig. 6 depicts the values of the x-component of the force for the three choices when the body moves from O to O0 . The (signed) area formed by each curve with the x-axis is the total work. The calculation confirms that W(O ? O0 ) = 0 for the first two choices, while for the third choice, W(O ? O0 ) = 0.188 which is about 34% of the total positive work, a significant amount of un-balanced energy. 4.3. Extensions to other cases The contact interaction model developed above for two polygons based on a corner/corner contact configuration can be extended to more general contact situations. Several such contact cases are discussed below: (1) Corner-edge contact: the corner-edge contact can be considered as the case that the two penetrating edges of one body are co-linear. It can be concluded that the normal direction is perpendicular to the edge intersected, as one would expect. (2) Multiple penetrating corners: When more than one corner is penetrating into the other body, as shown in Fig. 7(a), the overlap area A can be obtained as

A ¼ A A1 A2 ;

ð46Þ

ð44Þ

where A is the whole area enclosed by the polyline p0 g q0 h p0 . As A1 and A2 are fixed, i.e. independent of xp0 and hp,

F0n ¼ /0 ðAÞðb3 n3 þ b4 n4 Þ:

ð47Þ

It is not difficult to show that both Fn and F0n are equal in magnitude, opposite in direction and act along the same line of action. Thus any point on the line can be chosen as the contact point. The above procedure can be extended to other contact scenarios to form a unified approach. The procedure involves the determination of all the geometric features of the overlap region of the two contacting polygons, which itself must be a polygon, followed by the well established approach described in [10]. After all the vertices, edge directions, widths and the overlap area (if required) are defined, a force fi is applied at the middle point of the i-th segment:

f i ¼ /0 ðAÞbi ni ;

ð48Þ

where bi is the width of the segment and ni is the unit outward normal. Then the normal force acting on one body will be the resultant force of all the individual forces on the segments of the same body:

FnI;II ¼

X

fi:

ð49Þ

i2I;II

Again, the two normal forces will be in equilibrium, and one contact point can be identified along the same line of action of the two forces.

Fig. 7. Additional contact situations.

175


5. Normal contact models for polyhedra

where

The extension of the energy-based contact model developed above to polyhedra is presented in this section. To highlight the essential features of the model for polyhedra and to simplify the mathematical derivation, two special shaped polyhedra are used for discussion. More general cases can be dealt with in a similar fashion. Consider two rigid polyhedra I and II overlapping to form a typical polyhedron to polyhedron contact situation with two penetrating vertices p and q chosen as the reference points of the two bodies, as shown in Fig. 8. Suppose that there are only three edges and three surfaces attached to each penetrating vertex p or q and two sets of the unit vectors of the three edges connected to p or q are p1, p2, p3, and q1, q2, q3, respectively. The two polyhedra overlap to form a 6-sided polyhedron in general, and its volume is defined as the overlap volume V. Also assume that the outward normal unit vectors and the areas of the surfaces of the overlapping volume are respectively, npi and Api , and nqi and Aqi ði ¼ 1; 2; 3Þ. As is proposed in [8], the potential / is chosen to be a monotonically increasing function of the overlap volume V, /(V). Then the normal contact force, Fn, exerted on body I is defined as

n¼

Fn ¼ rxp /ðVÞ ¼

@/ðVÞ d /ðVÞ @V x ¼ ¼ /0 ðVÞrxp V ¼ F n n; @xp dV @xp ð50Þ

rxp V krxp Vk

with rxp V ¼

@V ; @xp

ð51Þ

F n ¼ kFn k ¼ /0 ðVÞkrxp Vk;

ð52Þ

in which n defines the normal direction, and Fn is the magnitude determined by both /0 (V) and krxp Vk. The ‘contact’ moment, Mph , associated with the rotational movement of body I about p is defined as

Mhp ¼

@/ðVÞ d/ðVÞ @V ¼ ¼ /0 ðVÞrhp V; @hp dV @hp

ð53Þ

where

rhp V ¼

@V @hp

and hp is an arbitrary rotational vector about the point p. When Fn and Mhp are given, the normal contact line can be determined based on (10). In what follows, the explicit expressions for rxp V; rph V; rxq V, and rqh V, which are the key to computing both Fn and Mhp , are established in detail. 5.1. Normal direction and normal contact forces Referring to Fig. 8, the three unit outward normals to the surfaces of the overlap volume associated with body I are defined by

np1 ¼

p2 p3 ; jp2 p3 j

np2 ¼

p3 p1 ; jp3 p1 j

np3 ¼

p1 p2 : jp1 p2 j

Assume that body I is moved along the direction p1 by a distance Dp1 as shown in Fig. 9a. Then the change of the overlap volume, DV, is

DV ¼ Ap1 ðnp1 p1 ÞDp1 thus

dV ¼ Ap1 np1 p1 : dp1 As

Fig. 8. Two polyhedra in corner/corner contact.

dV ¼ rxp V p1 dp1

Fig. 9. Volume change due to: (a) translation along p1 and (b) rotation about p1.

176


then

Similarly

rxp V p1 ¼ Ap1 np1 p1 :

ð54Þ

Similarly

ð55Þ ð56Þ

rxp V ¼ Ap1 np1 þ Ap2 np2 þ Ap3 np3 :

ð57Þ

npi pj ¼ dij npi pi

p

Apj dji

ði ¼ 1; 2; 3Þ;

ð64Þ

j¼1

p dij ¼ rpj npj pi :

Mhp ¼ /0 ðVÞ Ap1 rp1 np1 þ Ap2 rp2 np2 þ Ap3 rp3 np3 :

which are (54)–(56), therefore (57) is proved. The establishment of the expression (57) provides a feasible means to determine the normal direction n

n ¼ Ap1 np1 þ Ap2 np2 þ Ap3 =Ap1 np1 þ Ap2 np2 þ Ap3 and thus the normal force Fn:

Fn ¼ /0 ðVÞ Ap1 np1 þ Ap2 np2 þ Ap3 np3 :

ð58Þ

Similarly, rxq V can be defined as

rxq V ¼ Aq1 nq1 þ Aq2 nq2 þ Aq3 nq3 and thus the normal force

F0n

ð59Þ is

¼ / ðVÞ Aq1 nq1 þ Aq2 nq2 þ Aq3 nq3 : 0

ð60Þ

þ

þ

Aq1 nq1

þ

Aq2 nq2

þ

Aq3 nq3

ð66Þ

Then following (10), the normal contact line can be determined. In contrast to the polygon/polygon contact where all the contact features, including the normal/tangential directions, contact line, and contact width, can be conveniently defined from the line joining the two intersection points, it appears that no such simple geometric features emerge from the overlap region that can be used to define the contact characteristics of polyhedra. In practice, the contact (tangential) plane can be defined as the plane with n as its normal and passing through the mass centre (with coordinates xm) of the overlap volume. Then the contact point will be the intersection point of the line (10) with the contact plane, i.e. the coordinates xc satisfy the following additional condition

ðxc xm Þ n ¼ 0: Further discussion about how to extend the above procedure to more complex contact scenarios similar to the cases in Section 4.3 will be omitted.

It is not difficult to prove that

Ap3 np3

ð65Þ

thus the moment Mhp can be determined by

rxp V pi ¼ Ap1 np1 þ Ap2 np2 þ Ap3 np3 pi ¼ Api npi pi ði ¼ 1; 2; 3Þ

þ

3 X

rhp V ¼ Ap1 rp1 np1 þ Ap2 rp2 np2 þ Ap3 rp3 np3

ði; j ¼ 1; 2; 3Þ

it follows immediately that

Ap2 np2

rhp V pi ¼

It is obvious that rhp V should be of a form

In fact, since

Ap1 np1

ð63Þ

where

It can be proved that rxp V should have the following form

ð62Þ

rhp V p3 ¼ Ap1 dp13 þ Ap2 dp23 þ Ap3 dp33 ; or collectively

rxp V p2 ¼ Ap2 np2 p2 ; rxp V p3 ¼ Ap3 np3 p3 :

F0n

rhp V p2 ¼ Ap1 dp12 þ Ap2 dp22 þ Ap3 dp32 ;

¼0

i.e.

rxp V þ rxq V ¼ 0

6. Normal contact models for two arbitrarily shaped bodies

which leads to Fn þ F0n ¼ 0, as required.

Since any 2D/3D body can be approximately represented by a polygon/polyhedron with an infinite number of sides, the approaches described in the preceding two sections for handling contact situations between two polygons/polyhedra can be readily extended to any arbitrarily shaped 2D/3D bodies, thereby leading to the establishment of a generic contact interaction model for any two bodies in contact. Note that similar (geometric) features of the contact models established for polygon or polyhedron contact may also be applicable to general cases. For instance, for a simple contact of two convex (2D) bodies as shown in Fig. 1, the normal direction is also perpendicular to the line joining the two intersection points, and the magnitude of the two normal forces is proportional to the contact width, i.e. the distance of the two intersection points, and the contact point can be chosen as the middle point of the line. As a matter of fact, one can provide an alternative interpretation to the normal contact model without a rigorous proof. It states that the current choice of the potential / as the function of the overlap volume (or area) is equivalent to a normal contact model where an evenly distributed pressure, p = /0 (V)n (where n is the unit outward normal), is applied on the whole boundary of the overlap region. Then the normal force Fn on one body is the integral of the pressure on the part of the boundary belonging to the body, XI:

5.2. Contact point and contact plane By rotating body I around the edge p1 by an angle Dhp1 , the volume change DV can be computed as (referring to Fig. 9b)

p p p DV ¼ Ap1 d11 þ Ap2 d21 þ Ap3 d31 Dhp1 ; p

p

p

where d11 ; d21 and d31 are respectively, the algebraic distances of the three surface centres, cp1 ; cp2 and cp3 , to the edge p1, which can be obtained by p

di1 ¼ ðri ni Þ p1

ði ¼ 1; 2; 3Þ:

Thus

dV ¼ Ap1 rp1 np1 þ Ap2 rp2 np2 þ Ap3 rp3 np3 p1 : dhp1 As

dV ¼ rhp V p1 dhp1 then

rhp V p1 ¼

3 X j¼1

! rpj npj

p1 :

ð61Þ

Fn ¼

Z XI

p dS ¼

Z XI

/0 ðAÞ ndS ¼ /0 ðVÞ

Z XI

dS

ð67Þ


which indicates that the boundary of the overlap region dictates the normal contact direction. Furthermore, the normal contact line is given by

R xc ¼

XI

x dS SI

þk

Fn ; kFn k

ð68Þ

where SI = jXIj is the total area of the boundary surface XI, and k is a parameter. The position of the contact point, and thus the contact line/plane, can be determined from practical considerations. In theory, both (67) and (68) can be applied to any shapes, smooth or with corners, convex or concave, and any contact scenarios including simple penetration or passing through. It is easy to verify that the contact models for polygons/polyhedra with any complex contact scenario, as discussed in the previous Section 4.3, are the special cases of this general contact model. Furthermore, the model can also be applied to disks or spheres, resulting in alternatives but very similar versions to the existing contact models discussed in Section 3. Clearly, the actual shapes of the contacting bodies away from the contact region play no part in determining the contact forces. 7. Concluding remarks The current work has proposed unified theoretical framework for developing normal contact models for arbitrarily shaped 2D/ 3D bodies in discrete element modelling solely based on the energy-conservation principle for elastic contact. It has established that the normal force must be a potential field vector associated with a potential function. By constructing an appropriate potential function, a complete normal contact model, including the magnitude and direction of the normal force and the normal contact line, can be unambiguously derived for a pair of any shaped bodies. The model in its final form is simple and has a clear geometric perspective for 2D bodies. It can also recover some existing models for bodies with simple geometric shapes.

177

The effectiveness of the current model for the simulation of practical phenomena depends to a certain degree on how close the potential function selected can approximate actual physical situations. Numerical experiments may need to be undertaken to evaluate the efficiency and robustness of the model developed. The main computational cost associated with the model is the determination of all the geometric features/properties of the overlap region of the two contact bodies, and therefore will be shape dependent. An optimal implementation may be achieved for each particular pair of geometric entities. These issues are discussed in [7,8] for polygons or polyhedra. References [1] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1) (1979) 47–65. [2] P. Wriggers, Computational Contact Mechanics, second ed., Springer-Verlag, Berlin, Germany, 2006. [3] C. Kane, E.A. Repetto, M. Ortiz, J.E. Marsden, Finite element analysis of nonsmooth contact, Comput. Methods Appl. Mech. Engrg. 180 (1999) 1–26. [4] A. Munjiza, The Combined Finite-Discrete Element Method, Wiley & Sons, England, 2004. [5] P.A. Cundall, Formulation of a three-dimensional distinct element model – Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks, Int. J. Rock Mech. Min. Sci. Geomech. 25 (1988) 107–116. [6] G.-H. Shi, Discontinuous Deformation Analysis – A New Model for the Statics and Dynamics of Block Systems, Ph.D Thesis, University of California, Berkeley, 1988. [7] Y.T. Feng, D.R.J. Owen, A 2D polygon/polygon contact model: algorithmic aspects, Int. J. Engrg. Comput. 21 (2004) 265–277. [8] Y.T. Feng, K. Han, D.R.J. Owen, An energy based polyhedron-to-polyhedron contact model, in: Proceeding of 3rd M.I.T. Conference of Computational Fluid and Solid Mechanics, MIT, USA, 2005, pp. 210–214. [9] K.J. Johnson, Contact Mechanics, Cambridge University Press, 1985. [10] F.P. Preparata, M.I. Shamos, Computational Geometry, An Introduction, Springer-Verlag, Berlin, Germany, 1985. [11] Y.T. Feng, K. Han, D.R.J. Owen, J. Loughran, On upscaling of discrete element modelling for particle systems: similarity principles, Eng. Comput. 26 (6) (2009) 599–609. [12] J. Marsden, A. Weinstein, Calculus III, second ed., Springer, 1985, ISBN 978-0387-90985-1.