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On the relative motions of two rigid bodies at a compliant contact: Application to granular media
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 32 (2005) 463–480 www.elsevier.com/locate/mechrescom
On the relative motions of two rigid bodies at a compliant contact: Application to granular media Matthew R. Kuhn a
a,*
, Katalin Bagi
b
Department of Civil Engineering, School of Engineering, University of Portland, 5000 N. Wllamette Blvd., Portland, OR 97203, USA b Research Group for Computational Structural Mechanics, Hungarian Academy of Sciences, Department of Structural Mechanics, Technical University of Budapest, Mu¨egyetem rkp. 3, K.mf. 35, H-1521, Budapest, Hungary Available online 10 February 2005
Abstract We consider the relative motions of a pair of discrete three-dimensional particles that share a common contact point. The contact interactions are characterized by three independent motions: rigid movement, translational deformation, and rotational (rolling and twisting) deformation. The rigid motions are non-objective but satisfy a weakened objectivity condition; whereas, the two types of deformation are objective. Three types of rolling motions are discussed: all are objective and two are independent of the choice of the reference points at which the particle motions are measured. To aid in implementation, tensor and matrix descriptions are provided for each type of motion. 2005 Elsevier Ltd. All rights reserved. Keywords: Granular media; Kinematic; Rolling; Contact; Rigid bodies
1. Introduction Granular materials are large assemblies of particles that interact in a pair-wise manner. These materials include powders, soils, pharmaceutical pills, cereal grains, and piecemeal objects stored in bulk. Often, the interactions between pairs of grains occur only when grains are touching, and the resulting surface indentations are much smaller than the grains themselves. For these two conditions, a micromechanical analysis or simulation requires understanding the relationship between the short-range
*
Corresponding author. Fax: +1 503 943 7316. E-mail addresses: [email protected] (M.R. Kuhn), [email protected] (K. Bagi).
0093-6413/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2005.01.006
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contact force and the relative motion of the particle pair. Our purpose is not to develop a specific contact force-movement law, as these have received much attention, starting with the work of Hertz (1882), and have evolved into complex constitutive relations. We instead consider the general dependence of contact force on the particle movements, especially in the presence of particle rotations. The translations and rotations of two particles are called particle motions, and these motions of the two individuals must, in some way, be resolved into contact motions associated with the pair. Certain contact motions are admissible in a local, contact constitutive relation; others are not. An admissible contact motion must satisfy the principle of material frame indifference (objectivity), which we intend to clarify and apply to the combined motions of two particles. As an example, translational contact deformation is the relative translation of material points on either side of the contact, taking the forms of contact indentation and sliding. This particular form of contact motion is objective and has been extensively used in standard numerical simulations, such as the Discrete Element Method (DEM), to compute the inter-particle forces in large assemblies of particles (Cundall and Strack, 1979; Molenkamp, 1984; Koenders, 1987; Rothenburg and Bathurst, 1988; Chang and Misra, 1989). We show, however, that translational contact deformations, by themselves, do not exhaust the full range of admissible, objective contact motions. Our first contribution is to complete the subspace of objective motions by adding alternative forms of contact rolling motion, which we will also refer to as rotational contact deformation. Various forms of rolling have already been suggested in the literature. In their study of a two-dimensional (2D) model granular material, Oda et al. (1982) developed an experimental means of distinguishing a particular form of rolling from sequenced photographs. Molenkamp (1984), working with the same data, developed expressions for the sliding motion and a combined rolling-rigid motion of 2D grains. Bardet (1994) used a similar technique to analyze simulations of 2D circular disks. In each program, the investigators were able to characterize the motions of a particle pair as being ‘‘predominantly rolling’’ or ‘‘predominantly sliding.’’ Iwashita and Oda (1998) presented expressions for quantifying a rolling rate between 2D circular disks, and they introduced a force-movement mechanism that included a contact moment that depended upon the rolling motion. The authors have derived two other definitions of rolling between three-dimensional (3D) objects, which we will summarize and place into the context of a unified description of the relative motions of a particle pair (Bagi and Kuhn, 2004; Kuhn and Bagi, 2004). From this survey, we see that rolling can conjure many meanings, but we take a broad view toward its definition and provide the rationales for three alternative descriptions. In the broader sense, we view rolling as any form of objective contact motion that is independent of translational contact deformation and that, together with contact deformation, exhausts the full vector space of objective contact motions. The three alternative forms of rolling are applicable to three dimensional (3D) particles of arbitrary smooth shape, and details of their computational forms are provided in the Appendix A. Our second contribution is a framework for decomposing the motions of two particles into their objective and non-objective parts, with the latter referred to as the rigid motion of the pair. We present a definition of rigid motion that will satisfy a weakened objectivity condition. In short, we clarify the notion of objectivity when applied to the motions of a particle pair; we define an admissible, objective range of contact motions that can be used in numerical models to govern the interactions of the pair; we define several forms of admissible rolling motion; and we define the non-objective, rigid motion of the pair. The Appendix details the computational forms of each motion, which are readily applicable to numerical simulations. Notation conventions are as follows. Scalars are represented in normal weight fonts (rp, xi). Secondorder tensors are written in upper-case bold fonts (A), and vectors are written in lower-case bold fonts (v, x). Matrices are written with brackets, e.g. [v] and [A], with lower case fonts denoting column vectors. The inner and cross products of a vector pair are written with the ‘‘ Æ ’’ and ‘‘ · ’’ operators. The ‘‘ · ’’ symbol is also used in designating the sizes of matrices (e.g., 2 · 5).
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2. Analysis Two particles, p and q, are touching at the contact surface c, but because the contact surface is assumed to be small, it will be treated as a point in our analysis (Fig. 1). We identify two reference points, xp and xq, attached to the particles, perhaps at their centers. The motions of the particles are known through the translational velocities of these two points, vp and vq, and through their rates of rotation, xp and xq. Vectors rp and rq connect the points xp and xq to the contact point c; whereas the branch vector l connects point xp to point xq, l xq xp ¼ rp rq ;
ð1Þ
with length ‘ jlj. Particle motions are referred to global Cartesian axes with the unit basis vectors 1e, 2e, and 3e (Fig. 1). When formulating the constitutive behavior of a contact, it might be more convenient to refer the motions to local Cartesian axes having rotated basis vectors, perhaps ne, t1 e, and t2 e, with one axis normal to the contact surface and two tangential axes. The results in the paper can be changed to these local axes by properly transforming the underlying data—the components of vectors xp, xq, vp, vq, xq, rp, rq, and l—and, with these transformations, the labels 1, 2, and 3 can then be changed to n, t1, and t2. We note, however, that certain 3-tuple (column vectors) derived from this data may not represent first-order tensors, since they may not transform as tensors with a change in basis (frame). We will identify such 3-tuples of scalars. We arrange the motions of points xp and xq into a 12 · 1 partitioned column vector, written as ð2Þ where the sub-vectors ½vp and ½vq contain the stacked motions of the two particles:
:
Fig. 1. Particle and contact positions.
ð3Þ
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The paper identifies certain generalized contact motions [^v] and evaluates an N · 12 transformation matrix [A] that yields these N generalized motions: :
ð4Þ
The generalized motions include translational contact deformation (indentation and sliding), rotational contact deformation (rolling and twisting), and rigid motion of the particle pair. Although only 12 generalized motions are required for an invertible mapping in Eq. (4), we allow for more than 12 generalized motions, so that the extended size of [^v] can accommodate redundant or alternative descriptions of the rotational deformations. We are particularly interested in a collection of generalized contact motions [^v] in which the N elements ^vi are separated into two types: (1) generalized motions that are objective and (2) generalized motions that are not. Objective contact motions are admissible in a constitutive relation for the contact; they can alternatively be applied as internal constraints to the inter-particle motions; and they can be properly included as forms of internal displacement in a virtual work formulation. Besides identifying objective contact motions, we are also interested in finding those objective motions whose values are independent of the choice of the two reference points xp and xq assigned to the particles, in the event that they are not placed at the particle centers (Fig. 1). 2.1. Objectivity A constitutive relation will typically be assigned to a contact, giving the contact force and moment, or their co-rotational rates, as functions of certain contact motions ^v1 ; ^v1 ; . . . ; ^vn . In general, the response of a material (or of a contact) must be independent of the observer, even if the observer is moving (e.g. Truesdell and Toupin (1960, Section 293)). This requirement places a constraint upon admissible contact motions, and this constraint can be separated into two objectivity conditions. A ‘‘g’’ generalized contact motion can be either a vector ^vg or a scalar ^vg , and the objectivity conditions apply differently to these two objects. The conditions are as follows: Condition 1. We consider two observers, A and B, who are stationary but have different locations and frames. Observer A is located at the origin of its frame (1e, 2e, 3e); whereas observer B is located at a finite offset c relative to A and with a rotated frame. The two observers would record different locations xO;A and xO;B of a material point O: xO;B ¼ c þ QxO;A ;
ð5Þ
where Q effects a finite rotation of their frames. A contact motion g is objective only if its observations by A and B are related by ^vg;B ¼ Q^vg;A ; g;B
^v
g;A
¼ ^v ;
for vectors; for scalars:
ð6Þ
That is, with a change in observer frame, vector motions must transform as vectors, scalar motions must be invariant (e.g., Eringen (1967, 2.10)). Condition 2. Suppose that two observers are moving with different translational and rotational velocities but they briefly share the same frame (i.e., the same location and orientation). The translation and rotation velocities of a material point ‘‘O’’ that are observed by B are vO;B ¼ vO;A ðvBA þ xBA xO Þ;
ð7Þ
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xO;B ¼ xO;A þ xBA ; O;A
O;B
O;B
ð8Þ BA
O;B
BA
where v ; x v and x are the velocities of point O as observed by V and B; A and by x are the velocities of observer B as seen by observer A; and xO is the position of O relative to both A and B. g A contact motion g is objective only if the same motion, bv or ^vg , is observed by both A and B. Together, the two conditions assure that the observed motions are related by (6) even if the observers have different frames and are moving. We consider Condition 2 as the more restrictive test, and the paper is primarily directed toward its requirement. We will defer the application of Condition 1, after having first applied Condition 2. If observers A and B are located at the contact point c, then xO corresponds to rp for the point xp; whereas, xO corresponds to rq for the point xq. By separately substituting rp and rq into expressions (7) and (8), the difference in the observed motions of p and q is given by ð9Þ
:
Both of the observed motions on the left are 12 · 1 stacked vectors in the form of Eqs. (2) and (3); the vecand xBA in the stacked form (3); and [T] is the 12 · 6 tor ½vBA is an arbitrary 6 · 1 vector with contents vBA i i matrix
ð10Þ
with [Wp] and [Wq] given by 2 3 0 rp3 rp2 6 7 0 rp1 5; ½Wp ¼ 4 rp3 rp2 rp1 0
2
0 6 q q ½W ¼ 4 r3 rq2
rq3 0 rq1
3 rq2 7 rq1 5;
ð11Þ
0
and [I] and [0] representing the 3 · 3 identity and zero matrices. We now return to the generalized motions [^v] in Eq. (4) and test the objectivity of the kth generalized motion bv k . If objectivity Condition 2 is met, the two observers would measure the same generalized motion: bv Ak 9bv Bk :
ð12Þ
This test can be applied to the individual rows of transformation matrix [A]: ð13Þ 9½0:
ð14Þ
The kth generalized motion bv k in matrix expression (4) is objective only if the kth row of [A] is orthogonal to the column space of [T], that is, only if all columns of [T] are orthogonal to row k of [A]. If so, the kth number on the right of Eq. (14) will indeed be zero for arbitrary observer motions ½vBA , and the same generalized motion bv k would always be assigned by the two observers. The union of all objective motions is the left nullspace of [T], or
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½Aobjective ½T126 ¼ ½0
ð15Þ
and the objective part of the motion [vp =vq ] is given by ð16Þ
:
Because the six columns of [T] are linearly independent, the system (15) admits only six linearly independent objective motions, and these motions correspond to six linearly independent rows of [Aobjective]. Any six such motions (rows) can serve as a basis of all objective motions in the sense that any objective motion can be expressed as a linear combination of the six selected basis motions. We will later formalize objective motions of two types: those that produce a translational deformation of the contact and those that produce, in part, a rotational deformation of the contact. These two types of generalized objective motions will be def rot collected into the two vectors [bv ] and [bv ]. Also of interest are non-objective motions that are purely non-objective—motions that are orthogonal to rigid all objective motions—and these generalized motions will be gathered into a separate vector [bv ]. Our p q purpose is to define a proper choice of matrix [A] that will transform the actual motions [v =v ] of Eqs. (2) and (3) into generalized motions that give the separate effects of translational deformation, rotational deformation, and rigid motions:
:
ð17Þ
The union of the row spaces of [Adef] and [Arot] is the six-dimensional basis of all objective motions and is the left null space of [T]. The sub-matrices [Adef] and [Arot] that are associated with translational and rotational deformation are stipulated in Sections 2.3 and 2.4. The six rigid motions [^vrigid ] are defined in the following section. 2.2. Rigid motions The mixing (or sum) of an objective and a non-objective motion gives a non-objective motion, and it may be of advantage to entirely separate these motions into two distinct and orthogonal subspaces. Toward this end, we place the following three conditions on the matrix [Anon-objective] in Eq. (17), with the result being a new matrix [Arigid] that characterizes the rigid motions of a particle pair. (1) The row space[Arigid] should equal the column space of [T]. With this restriction, the row spaces of the two matrices [Aobjective] and [Arigid] are orthogonal and any particle motion [vp =vq ] can be represented as a combination of its objective contact motion, [Aobjectlve] [vp =vq ], and its purely non-objective, rigid motion, ½Arigid ½vp =vq . This restriction on [Arigid] limits its representation to a product ½Arigid 612 ¼ ½Z66 ½TT612 ;
ð18Þ
where matrix [Z] is non-singular. (2) We further restrict [Arigid] by considering two observers A and B, who are momentarily located at the contact c and share the same frame, but who have different velocities, as in (7) and (8). The two A; rigid B; rigid observers would assign the following generalized ‘‘^’’ rigid motions, ½bv and ½bv , to the particle pair:
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;
ð19Þ
:
ð20Þ
We require that the difference in the generalized motions (19) and (20) equal the difference in the motions that would be observed if the two particles were stationary and only the observers were moving. That is, the difference in the observed motions ½^vA; rigid and ½^vB; rigid should equal the difference in the observer motions, A;rigid
½bv
B;rigid
½bv
¼ ½vBA
ð21Þ
for arbitrary observer motions. Condition (21) is, of course, weaker than the objectivity condition (14). Substituting expressions (19), (20), and (7) into condition (21) gives the restriction ½Arigid ½T½vBA ¼ ½vBA
ð22Þ
for arbitrary observer motions [v
BA
], or
½Arigid 612 ½T126 ¼ ½I66 :
ð23Þ
(3) The rigid motions should be represented by translational and rotational parts, and each part should satisfy the objectivity Condition 1 given in (6). That is, the rigid motions should transform as vector objects. The two conditions (18) and (23) are uniquely satisfied by the Moore-Penrose pseudo-inverse of [T]: T
1
T
½Arigid ¼ ð½T ½TÞ ½T ;
ð24Þ
where the matrix [Z] in Eq. (18) appears as the 6 · 6 inverse ([T]T[T])1. This inverse will always exist and be non-singular, since the columns of [T] are linearly independent. With definition (24), the product of [A] and [T] becomes:
ð25Þ
in which we allow for alternative definitions of rotational deformation (hence, the dimension 3+ P 3). The product of [Arigid] and the particle motions ½vp =vq yields the six generalized rigid motions:
:
ð26Þ
The contents of the 6 · 12 matrix [Arigid] are detailed in Appendix A as equation (A.10). In Appendix A, the ; ^vrigid ; ^vrigid T are the rigid translational velocities of the particle pair; the second set first three motions ½^vrigid 1 2 3 T of motions ½^vrigid ; ^vrigid ; ^vrigid are the rigid rotational velocities. These rigid motions have the following 5 4 6 properties:
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• Motions ^vrigid are non-objective—their values depend upon the observers motion. The rigid motions do i not conform to the objectivity condition (12), but instead satisfy the weaker condition (21). T T • Each of the two 3-tuples ½^vrigid ; ^vrigid ; ^vrigid and ½^vrigid ; ^vrigid ; ^vrigid is a vector: each 3-tuple can be 5 1 2 3 4 6 translation i rotation i expressed as the inner products ^v e and ^v e of a single vector and the three unit basis vectors. That is, each of the two 3-tuples is a vector object, satisfying the third requirement of the rigid motions (i.e., the objectivity Condition 1). This property is evident in the matrix form (A.10) given in Appendix A, since each 3 · 3 sub-matrix within matrix (A.10) transforms as a second-order tensor under frame rotations. • Matrix [Arigid] has rank 6, since both ([T]T[T]1) and [T]T are of rank 6 (Eq. (24)). That is, the non-objective motions are fully described by the six generalized velocities ^vrigid . These characteristics are summai rized in the first row of Table 1.
2.3. Translational contact deformation Translational contact deformation results from a relative translation of the particles p and q at the contact point c, and this form of contact motion produces indentation and sliding. The translations of material on either side of the contact are given by vp;c ¼ vp þ xp rp ;
ð27Þ
vq;c ¼ vq þ xq rq ;
ð28Þ
p,c
q,c
where v and v are the velocities of material in grains p and q near their common contact. The difference vq,c vp,c is the generalized translational deformation velocity of q relative to p, and this deformation velocity is responsible for any sliding or indentation at c: ^vdef vq;c vp;c
ð29Þ
¼ ðvq vp Þ þ ðxq rq xp rp Þ
ð30Þ
or in matrix form, ð31Þ
:
The translational deformation (28) has long been used in the analysis of granular materials and in DEM simulations (e.g., Cundall and Strack, 1979; Molenkamp, 1984; Koenders, 1987; Chang and Misra, 1989).
Table 1 Properties of the generalized motions Motion
Objective?a
Independent of xp,xq?
Rank of [A(Æ)]
Vector or scalar?
Eq.
Eq. [A(Æ)]
½^vrigid ^vdef ^vroll;1 ½^vroll;2 ^vroll;3 ^vtwist
Nob Yes Yes Yes Yes Yes
No Yes Yes No Yes Yes
6 3 2d 3 2d 1
Vectorc Vector Vector 3 Scalars Vector Scalar
(24) (30) (33) (43) (44) (34)
(A.10) (A.16) (A.17) (A.18) (A.28) (A.29)
a b c d
Motions that satisfy objectivity Conditions 1 and 2. Rigid motions satisfy the weaker conditions of Section 2.2. Each set of three components ½^vrigid ; ^vrigid ; ^vrigid and ½^vrigid ; ^vrigid ; ^vrigid is a 3-vector. 1 2 3 1 2 3 Together, [Aroll,(Æ)] and [Atwist]1·12 have rank 3.
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The motions ½^vdef satisfy objectivity Condition 2, as the product (13) for [Adef] is zero for an arbitrary vector ½vBA . The contents of [Adef] are given in the Appendix A (Eq. (A.16)), and the deformation motion has the following characteristics, which are summarized in Table 1: • The components ^vdef are independent of the choice of reference points xp and xq at which the particle i motions are measured (Fig. 1). • The object ^vdef is a vector and will transform as a vector under frame rotations, since each object on the i right of Eqs. (29) and (30) is also a vector. The deformation ^vdef satisfies objectivity Condition 1 as well i as Condition 2. • Matrix [Adef] is rank 3.
2.4. Rotational contact deformation Although the translational deformation motions ½^vdef can be unambiguously defined, numerous definitions are possible for the rotational deformation motions ½^vrot , and we summarize three definitions of rolling (with twisting) that have been proposed by the authors. The definitions can be collected into a partitioned matrix [Arot],
ð32Þ
with the superscripts denoting the alternative descriptions. The fourth, ‘‘twisting’’ type of rotational motion in Eq. (32) is included to augment (and complete) the first and third forms of rolling, as will be seen in Sections 2.4.1 and 2.4.3. Any of these rotational motions could be used in a constitutive description of the contact behavior. A fourth type of rolling (a ‘‘Type 4 rolling’’) has been proposed elsewhere, such that the rolling motions ^vroll;4 are orthogonal both to the rigid motions and to the translational contact deformations (Kuhn and Bagi, 2004). 2.4.1. Type 1 rolling and contact twisting One definition of rolling, termed ‘‘Type 1 rolling,’’ is simply the tangential component of the difference between the rotations of particles p and q: ^vroll;1 ðxq xp Þ ½ðxq xp Þ nn
ð33Þ
as in (Bojta´r, 1989; Bagi, 1993; Zhou et al., 1999; Satake, 2001). The motion ^vroll;1 has been applied in 2D simulations of granular assemblies, a situation in which (33) contains no out-of-plane motions of the particles and in which the bracketed term on the right of (33) is zero. The contents of [Aroll, 1] are given in the Appendix A (Eq. (A.17)). This form of rolling clearly satisfies objectivity Condition 2 by meeting the test (13,14) for arbitrary ½vBA . The generalized rolling ½^vroll;1 has the following additional properties, which are summarized in Table 1: • The movements v^i roll;1 are independent of the choice of reference points xp and xq at which the particle motions are measured (Fig. 1), although the signs of the v^i roll;1 depend on the assignment of the particle labels ‘‘p’’ and ‘‘q’’ (if this characteristic is undesirable, the alternative (xq xp) · n can be used instead).
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• The objects in Eq. (33) are written without matrix brackets to emphasize that ^vroll;1 is a vector: its three (Cartesian) components are the inner products ^vroll;1 i e of a single vector quantity and the unit basis vectors. That is, ^vroll;1 transforms as a vector under frame rotations, and it satisfies the objectivity Condition 1 as well as Condition 2. • The rolling vector ^vroll;1 lies in the tangent plane of the contact, so that the matrix [Aroll,1] is of rank 2. The six rows of the combined matrices [Aroll,1] and [Adef] are likewise not linearly independent, but are of rank 5. Motions ½^vdef and ½^vroll;1 must be augmented by a separate motion in order to span the full six-dimensional subspace of objective motions. In this regard, we could augment ½^vroll;1 with any one of the three rolling quantities in Section 2.4.2 or 2.4.3. A more reasonable choice, however, is a vector that is orthogonal to ½^vroll;1 as would be the case with the particles relative rotation about the contact normal n. We will refer to this rotation as a ‘‘twisting’’ motion having the magnitude ^vtwist : ^vtwist ðxq xp Þ n:
ð34Þ roll,1
twist
The four rows of the combined matrices [A ] and [A ] are rank 3. The contents of the corresponding row vector [Atwist]1·12 are given in the Appendix A (Eq. (A.29)).
2.4.2. Type 2 rolling Iwashita and Oda (1998) considered the relative motions of two 2D circular disks (Fig. 2) and arrived at the following expression for rolling: 1 ^vIO ¼ ½rp ðxp3 rq xq3 Þ ðrp rq Þb_ 3 ð35Þ 2 1 ¼ ½rp ðxp3 b_ 3 Þ rq ðxq3 b_ 3 Þ: ð36Þ 2 Both rp and rq are (positive) radii, and b_ 3 is the rate at which the center of particle q rotates around the center of particle p: b_ 3 ¼ ½ðvq vp Þ t=ðrp þ rq Þ:
ð37Þ
The unit vector t is tangent to the contact and is directed counter-clockwise around p (Fig. 2). The rotation b_ 3 will be referred to as the ‘‘pair-spin’’, which is subtracted from the individual particle spins to calculate the Iwashita–Oda rolling of Eq. (36). When both vp and vq are zero, ^vIO is the average of the tangential velocities vp,c Æ t and vq,c Æ t of two points, one attached to each of the particles p and q at the contact (Eqs. (27) and (28), Fig. 1). The authors have extended the Iwashita–Oda rolling to 3D particles of arbitrary shape (Bagi and Kuhn, 2004). We briefly review this general type of rolling, referred to as ‘‘Type 2 rolling,’’ using the global frame
Fig. 2. Iwashita–Oda rolling between circular disks.
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(1e, 2e, 3e) instead of the local frame (n, t1, t2) as in reference (Kuhn and Bagi, 2004). The rolling ½^vroll;2 has three elements, each of which is an averaged translational velocity of two material points, one attached to each of the particles at their contact. The element ^vroll;2 connotes a translational velocity in the ie direction, i just as the Iwashita–Oda rolling velocity occurs in the tangential t direction and has 1e and 2e components. To define the single element ^vroll;2 of Type 2 rolling, we construct a vector il that is the orthogonal projection i p q of the branch vector l = r r onto a plane perpendicular to direction ie (Fig. 3 and Eq. 1): i
l l ðl i eÞi e
ð38Þ i
with the associated unit vector k, i
k i l=i ‘;
ð39Þ
where i‘ is the length of vector il, or i‘ jilj. The Type 2 rolling associated with the ie direction is attributed to material motions in the vicinity of the contact that occur about an axis iz that is perpendicular to both the unit vector ie and the unit vector ik (Fig. 3): i
z i k i e:
ð40Þ i
i
i
Together, the three vectors form a right-hand orthogonal triad ( e, z, k), but each of the three basis directions ie (i = 1,2,3) will have a different associated triad. In the event that l and ie are aligned (i.e. l · ie = 0, cf. Eq. (39), then ik in (39) is ill-defined, but the right-hand triad of basis vectors, (ie, je, ke), can then serve in place of (ie, iz, ik). Type 2 rolling in the ie direction is the difference of two parts, which we label ^vroll;2a and ^vroll;2b . The first i i i part is due to particle rotations about the z-axis (Fig. 3): 1 ^vroll;2a ½ðxp i zÞðrp i kÞ þ ðxq i zÞðrq i kÞ: ð41Þ i 2 The terms in this expression can be thought to correspond to the rx3 products in Eq. (35). The product xp Æ iz is the rotation of particle p about the iz axis, and rp Æ ik is the corresponding rotation arm that is perpendicular to both ie and iz. The products in Eq. (41) yield a velocity in the ie direction. The pair-spin about the iz axis is ðvqi vpi Þ=i ‘, and the second part of rolling ^vroll;2 is produced by this spin: i
Fig. 3. Vectors for computing the rolling motion ^vroll;2 of particle q relative to particle p. This motion is referred to the 1e direction. The 1 unit vectors 1k,1z, and 1e are orthogonal. Similar sets of unit vectors are required for computing ^vroll;2 and ^vroll;2 . 2 3
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^vroll;2b i
1 vqi vpi p q i ðr þ r Þ k ; i‘ 2
where i‘ = l Æ ik is the length of vector il in equation (38) (Fig. 3). The difference ^vroll;2a ^vroll;2b defines the rolling rate ^vroll;2 of q relative to p: i i i q p 1 p i vi vi vqi vpi roll;2 p i q i q i ^vi ¼ ðr kÞ x z i þ ðr kÞ x z i : 2 ‘ ‘
ð42Þ
ð43Þ
In the event that l and ie are aligned, the length i‘ is zero and the corresponding component ^vroll;2 does not i exist. This form of rolling satisfies the objectivity Condition 2 by meeting the test (14) for an arbitrary observer motion ½vBA . The contents of matrix [Aroll,2] are detailed in Appendix A (Eq. (A.18)). The generalized rolling ½^vroll;2 has the following additional properties, which are summarized in Table 1: • The rolling motions ^vroll;2 can be computed for particles of arbitrary shape. i • The values of motions ^vroll;2 depend upon the choice of the reference points xp and xq at which the pari ticle motions are measured. • The motions ^vroll;2 are scalars, which can be assembled into the 3-tuple ð^vroll;2 ; ^vroll;2 ; ^vroll;2 Þ. These motions 1 2 3 i roll;2 are not the Cartesian components of a single vector, since the three values ^vi in Eq. (43) can not be expressed as the inner products ^vroll;2 i e of a single vector ^vroll;2 . • The scalar motions satisfy objectivity Condition 1 as well as Condition 2. Each term in (43) is either a norm or an inner product of two vectors, so the scalar ^vroll;2 is invariant under changes in the observer i frame, provided that both observers agree to the same ie directions. • The three rows of [Aroll,2] are linearly independent so that the matrix is rank 3.
2.4.3. Type 3 rolling The authors have defined a third type of rolling, which has its origins in robotics applications (Montana, 1988; Kuhn and Bagi, 2004). This ‘‘Type 3 rolling’’ is the average of the rates at which the two points of contact (one on each particle surface) move across the two rolling surfaces. This rolling motion is distinct from Type 2 rolling: for the rotations shown in Fig. 2, material points near the contact move in direction t (Type 2 rolling); whereas, the points of touching move in the opposite direction (Type 3 rolling). Like Type 1 rolling, the rolling motion ^vroll;3 is a vector, but its direction and speed depend upon the local surface shapes of the two particles at their contact. The local shapes are described by curvature tensors Kp and Kq, and the rolling velocity is defined as 1 ^vroll;3 ðKp þ Kq Þ1 ð^vroll;1 n Kqvdef Þ vdef ; ð44Þ 2 where n is the unit normal vector at the contact, directed outward from particle p (Kuhn and Bagi, 2004). Vector ^vroll;3 lies in the tangent plane of the contact and is orthogonal to n. The deformation rate vdef in Eq. (44) is the projection of the deformation ^vdef in Eq. (30) onto the tangent plane of the contact: vdef ^vdef ð^vdef nÞn:
ð45Þ
The rolling in Eq. (44) can be expressed in matrix form as ð46Þ and the contents of matrix [Aroll,3] are detailed in Appendix A (Eq. (A.28)). The motion (44) satisfies objectivity Condition 2, since it is a linear combination of the two objective motions ^vroll;1 and ^vdef . The Type 3 rolling motion has the following additional characteristics, which are summarized in Table 1:
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• The rolling motion ^vroll;3 can be computed for particles of arbitrary shape, provided that the particle surfaces are sufficiently smooth to admit their curvatures K. • Like ^vroll;1 and ^vdef (but unlike ^vroll;2 ), the rolling motion ^vroll;3 is independent of the choice of reference i p q points x and x at which the particle motions are measured. • Unlike the motions ^vroll;2 , the rolling ^vroll;3 is a vector: its Cartesian components ^vroll;3 are the inner prodi i ucts of the single vector ^vroll;3 and the unit vectors ie. Because it transforms as a vector, ^vroll;3 satisfies i objectivity Condition 1 as well as Condition 2. • Like the Type 1 rolling in Section 2.4.1, the rolling vector ^vroll;3 lies in the tangent plane of the contact, so that the matrix [Aroll,3] is of rank 2, and the six rows of the combined matrices [Aroll,3] and [Adef] are rank 5. Motions ½^vdef and ½^vroll;3 must be augmented by a separate motion, such as the twist ^vtwist of Eq. (34), in order to span the full six-dimensional subspace of objective motions.
3. Example We consider two two-dimensional (2D) disks of the same radius r arranged as shown in Fig. 4. To simplify the calculations and conserve space, we only consider motions in the 1e 2e plane. As such, the 12 T kinematic variables in Eq. (2) are reduced to the six motions ½vp1 ; vp2 ; xp3 ; vq1 ; vq2 ; xq3 . The various [A] matrices that are associated with generalized motions are also reduced in size: with 6 columns instead of 12, and with fewer rows as well. To conserve space, we have collected the relevant data into Table 2, which displays all of the quantities defined in the Appendix A. For example, the 3 · 3 matrix [Wp] in Eq. (11) is reduced to the 2 · 1 matrix [0, r]T, since it is only multiplied by the x3 rotations to produce a 2-component translation in the 1e 2e plane. The resulting [A] matrices for deformation, rolling, rigid translation, and rigid rotation are as follows:
:
ð47Þ
Types 2 and 3 rolling are both 2-component translation vectors that appear as pairs of rows in Eq. (47), but Type 3 rolling is a single component of rotation in the 3e direction. Because Types 2 and 3 rolling are associated with the contacts tangential direction, the two components are not linearly independent. The matrix in (47) contains six independent rows that correspond to six canonical motions for the pair of particles. T Each of these six motions is produce by setting the corresponding motion vector ½vp1 ; vp2 ; xp3 ; vq1 ; vq2 ; xq3 equal to its corresponding row in [A]: (1) a stretching of the two particles in the normal n = 1e direction, as in the first row, (2) a tangential shearing of the particles in the 2e direction, as in the second row. The particles simultaneously translate and rotate to produce this canonical motion,
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Fig. 4. Example of two equal-size disks.
Table 2 Data for an example of two disks of equal size (Fig. 4) Data values
Equations
rigid
]3·6 [rp]2·1 = [r,0]T, [rq]2·1 = [r,0]T, [n]2·1 = [1,0]T
(A.10) –
[l]2·1 = [2r,0]T, [s]2·1 = [0,0]T
(1)–(A.3)
l
‘ = 0, 2‘ = 2r, 3‘ = 2r G = 4r2 + 4, ‘ = 2r, a = 0 [Wp]2·1 = [0,r]T, [Wq]2·1 = [0,r]T 0 0 ½B22 ¼ , ½C21 ¼ ½0; 0T , ½D11 ¼ D33 ¼ 0 0 0 [Adef]3·6
(A.4)–(A.6) (A.7), (A.8), and (A.12) (11)
[Aroll,1]1·6
(A.17)
[A
1 ¼ 0
½M22 roll,2
[A
]2·6
½E22
0 0
0 0 ¼ , ½FP 21 ¼ ½0; r=2T , ½Fq 21 ¼ ½0; r=2T 0 0
[Aroll,3]2·6
0 0 , ½N21 ¼ ½0; 1T ½P22 ¼ 0 1 0 0 ½Kp 22 ¼ ½Kq 22 ¼ 0 1=r 0 0 p q þ ½K þ K 22 ¼ 0 r=2
(3) (4) (5) (6)
a a a a
(A.13)–(A.15) A.16)
(A.17) (A.18) (A.19)–(A.23) (A.28) (A.25), (A.27) – –
rolling motion in either of the rows 3, 5, or 7, rigid-body horizontal translation (row 8), rigid-body vertical translation (row 9), and rigid-body rotation of the particles about the central point (r, 0), as in row 10.
The twisting motion [Atwist] does not apply, of course, to these in-plane motions. 4. Conclusions The discrete motions of two particles can be unambiguously separated into objective and non-objective parts. The non-objective motions can be defined as the rigid motions of the pair, and these motions are the
M.R. Kuhn, K. Bagi / Mechanics Research Communications 32 (2005) 463–480
477
orthogonal projection of the actual motions onto the vector space of all non-objective motions. The objective motions can be separated into motions that produce a translational deformation at the contact and motions that produce a rotational (rolling or twisting) deformation. The rolling motions can be expressed in at least three different forms. Table 1 summarizes the properties of the rigid motion, the translational deformation, and the various forms of rotational deformation. Details of each motion are given in Appendix A.
Acknowledgements This work was funded in part by the Arthur Butine Faculty Development Fund of the University of Portland. The support of the OTKA 31889 grant is also acknowledged.
Appendix A This appendix gives the contents of the six sub-matrices [Arigid], [Adef], [Aroll, 1], [Aroll, 2], [Aroll,3], and [A ]. These sub-matrices can be assembled, as in Eqs. (17) and (32), to create the full matrix [A] for computing the generalized motions of a particle pair: twist
ðA:1Þ
where relative twisting has been added to augment the two-dimensional (and incomplete) vector sub-space of Type 3 rolling. The expressions in this Appendix can be readily applied in numerical simulations to the the various contact motions. Expressions for the six sub-matrices are greatly condensed by using the following intermediate terms: s rp þ rq ;
ðA:2Þ
l rp rq ¼ 1 e‘1 þ 2 e‘2 þ 3 e‘3 ; with components ‘1 ; ‘2 ; ‘3 ;
ðA:3Þ
2
2 1=2
;
ðA:4Þ
2
2 1=2
;
ðA:5Þ
2
2 1=2
;
ðA:6Þ
1
‘ l 1 k ¼ ½ð‘2 Þ þ ð‘3 Þ
2
‘ l 2 k ¼ ½ð‘1 Þ þ ð‘3 Þ
3
‘ l 3 k ¼ ½ð‘1 Þ þ ð‘2 Þ 2
G ð‘Þ þ 4;
ðA:7Þ
‘ j l j;
ðA:8Þ
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M.R. Kuhn, K. Bagi / Mechanics Research Communications 32 (2005) 463–480
½nT ½n1 ; n2 ; n3 :
ðA:9Þ rigid
Rigid motions: The 6 · 12 matrix [A tively, matrix [Arigid] is explicitly
] can be computed directly from expressions (10) and (24). Alterna-
;
;
ðA:10Þ
ðA:11Þ
where [I] and [0] are the 3 · 3 identity and zero matrices, [Wp] and [Wq] are given in Eq. (11), and scalar a and the 3 · 3 matrices [B], [C], and [D] are defined as a ¼ l s;
ðA:12Þ
Bij ¼ ‘i sj ;
ðA:13Þ
C ij ¼ i e ðrp rq Þ‘j ;
ðA:14Þ
Dij ¼ ‘i ‘j :
ðA:15Þ
Translational deformations: The 3 · 12 matrix [Adef] is defined by expression (30), or ðA:16Þ where matrices [Wp] and [Wp] are given in Eq. (11). Type 1 rolling: The 3 · 12 matrix [Aroll,1] in Eq. (33) is ðA:17Þ where Mij = ninj. Type 2 rolling: The 3 · 12 matrix [Aroll,2] is defined by (40), (39), and (43), or by ðA:18Þ where
2s 16 6 ½E ¼ 6 24
2 ‘2 þs3 ‘3 ð1 ‘Þ2
0
0
0
s1 ‘1 þs3 ‘3 ð2 ‘Þ2
0
0
0
s1 ‘1 þs2 ‘2 ð3 ‘Þ2
2
0
16 ðÞ ½FðÞ ¼ 6 ‘ R 24 3 2 ðÞ ‘2 R3 with
ðÞ
‘ 3 R1 0
ðÞ ‘1 R3
3 7 7 7; 5 ðÞ
‘2 R1
ðA:19Þ
3
7 ðÞ ‘ 1 R2 7 5; 0
ðA:20Þ
M.R. Kuhn, K. Bagi / Mechanics Research Communications 32 (2005) 463–480 ðÞ
ðÞ
R1 ¼
ðÞ
r2 ‘2 þ r3 ‘3 ð1 ‘Þ
2
ðÞ
ðÞ
R2 ¼
;
ðA:21Þ
;
ðA:22Þ
:
ðA:23Þ
ðÞ
r1 ‘1 þ r3 ‘3 ð2 ‘Þ
2
ðÞ
ðÞ
R3 ¼
479
ðÞ
r1 ‘1 þ r2 ‘2 3
ð ‘Þ
2
Type 3 rolling: The vector ^vroll;3 in Eq. (44) is a linear combination of the motions ^vroll;1 and ^vdef . Vector vdef in Eq. (45) can be expressed as the matrix product ½vdef ¼ ½P½^vdef ;
ðA:24Þ
where the projection matrix [P] is related to the contact normal vector n, as ½P ¼ ½I ½M ¼ ½I ½n½nT : roll
The cross product ^v roll;1
½^v with
n is 1
n ¼ ½N½^vroll; ; 2
0 6 ½N ¼ 4 n3 n2
ðA:25Þ
n3 0 n1
3 n2 7 n1 5 :
ðA:26Þ
ðA:27Þ
0
In standard presentations of differential geometry, the surface curvatures Kp and Kq are usually referred to a two-dimensional basis of vectors that are tangent to the surfaces. We have modified the standard approach by referring Kp and Kq to the three-dimensional basis of vectors 1e, 2e, and 3e (Kuhn and Bagi, 2004). In this approach, the 3 · 3 matrix sum [Kp] + [Kq] is of rank 2, and the pseudo-inverse (‘‘+’’ inverse) is used in place of (Kp + Kq)1. By substituting expressions (A.24) through (A.27) into (44), the matrix [Aroll,3] can be expressed as ðA:28Þ Twisting: The 1 · 12 matrix [Atwist] is defined by Eq. (34), or ðA:29Þ where n is the unit normal vector (Eq. (A.9)).
References Bagi, K., 1993. Mech. Mater. 16, 101. Bagi, K., Kuhn, M.R., 2004. J. Appl. Mech. 71, 493. Bardet, J.P., 1994. Mech. Mater. 18, 159. Bojta´r, I., 1989. Epites-Epiteszettudomany 75. Chang, C.S., Misra, A., 1989. Comput. Geotech. 7, 269.
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Cundall, P.A., Strack, O.D.L., 1979. Ge´otechnique 29, 47. Eringen, A.C., 1967. Mechanics of Continua. John Wiley and Sons, New York. Hertz, H., 1882. J. Reine Angewandte Math. 92, 156. Iwashita, K., Oda, M., 1998. J. Engrg. Mech. 124, 285. Koenders, M.A., 1987. Acta Mech. 70, 31. Kuhn, M.R., Bagi, K., 2004a. J. Engrg. Mech. 130, 826. Kuhn, M.R., Bagi, K., 2004b. Int. J. Solids Struct. 41, 5793. Molenkamp, F., 1984. Soil Found 24, 17. Montana, D.J., 1988. Int. J. Robotics Res. 7, 17. Oda, M., Konishi, J., Nemat-Nasser, S., 1982. Mech. Mater. 1, 269. Rothenburg, L., Bathurst, R., 1988. J. Appl. Mech. 55, 17. Satake, M., 2001. In: Kishino, Y. (Ed.), Powders and Grains. A.A. Balkema, Lisse, pp. 149–152. Truesdell, C., Toupin, R.A., 1960. In: Flu¨gge, S. (Ed.), Encyclopedia of Physics, Vol. III/l. Springer-Verlag, Berlin, pp. 226–902. Zhou, Y.C., Wright, B.D., Yang, R.Y., Xu, B.H., Yu, A.B., 1999. Physica A 269, 536.
Mechanics Research Communications 32 (2005) 463–480 www.elsevier.com/locate/mechrescom
On the relative motions of two rigid bodies at a compliant contact: Application to granular media Matthew R. Kuhn a
a,*
, Katalin Bagi
b
Department of Civil Engineering, School of Engineering, University of Portland, 5000 N. Wllamette Blvd., Portland, OR 97203, USA b Research Group for Computational Structural Mechanics, Hungarian Academy of Sciences, Department of Structural Mechanics, Technical University of Budapest, Mu¨egyetem rkp. 3, K.mf. 35, H-1521, Budapest, Hungary Available online 10 February 2005
Abstract We consider the relative motions of a pair of discrete three-dimensional particles that share a common contact point. The contact interactions are characterized by three independent motions: rigid movement, translational deformation, and rotational (rolling and twisting) deformation. The rigid motions are non-objective but satisfy a weakened objectivity condition; whereas, the two types of deformation are objective. Three types of rolling motions are discussed: all are objective and two are independent of the choice of the reference points at which the particle motions are measured. To aid in implementation, tensor and matrix descriptions are provided for each type of motion. 2005 Elsevier Ltd. All rights reserved. Keywords: Granular media; Kinematic; Rolling; Contact; Rigid bodies
1. Introduction Granular materials are large assemblies of particles that interact in a pair-wise manner. These materials include powders, soils, pharmaceutical pills, cereal grains, and piecemeal objects stored in bulk. Often, the interactions between pairs of grains occur only when grains are touching, and the resulting surface indentations are much smaller than the grains themselves. For these two conditions, a micromechanical analysis or simulation requires understanding the relationship between the short-range
*
Corresponding author. Fax: +1 503 943 7316. E-mail addresses: [email protected] (M.R. Kuhn), [email protected] (K. Bagi).
0093-6413/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2005.01.006
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contact force and the relative motion of the particle pair. Our purpose is not to develop a specific contact force-movement law, as these have received much attention, starting with the work of Hertz (1882), and have evolved into complex constitutive relations. We instead consider the general dependence of contact force on the particle movements, especially in the presence of particle rotations. The translations and rotations of two particles are called particle motions, and these motions of the two individuals must, in some way, be resolved into contact motions associated with the pair. Certain contact motions are admissible in a local, contact constitutive relation; others are not. An admissible contact motion must satisfy the principle of material frame indifference (objectivity), which we intend to clarify and apply to the combined motions of two particles. As an example, translational contact deformation is the relative translation of material points on either side of the contact, taking the forms of contact indentation and sliding. This particular form of contact motion is objective and has been extensively used in standard numerical simulations, such as the Discrete Element Method (DEM), to compute the inter-particle forces in large assemblies of particles (Cundall and Strack, 1979; Molenkamp, 1984; Koenders, 1987; Rothenburg and Bathurst, 1988; Chang and Misra, 1989). We show, however, that translational contact deformations, by themselves, do not exhaust the full range of admissible, objective contact motions. Our first contribution is to complete the subspace of objective motions by adding alternative forms of contact rolling motion, which we will also refer to as rotational contact deformation. Various forms of rolling have already been suggested in the literature. In their study of a two-dimensional (2D) model granular material, Oda et al. (1982) developed an experimental means of distinguishing a particular form of rolling from sequenced photographs. Molenkamp (1984), working with the same data, developed expressions for the sliding motion and a combined rolling-rigid motion of 2D grains. Bardet (1994) used a similar technique to analyze simulations of 2D circular disks. In each program, the investigators were able to characterize the motions of a particle pair as being ‘‘predominantly rolling’’ or ‘‘predominantly sliding.’’ Iwashita and Oda (1998) presented expressions for quantifying a rolling rate between 2D circular disks, and they introduced a force-movement mechanism that included a contact moment that depended upon the rolling motion. The authors have derived two other definitions of rolling between three-dimensional (3D) objects, which we will summarize and place into the context of a unified description of the relative motions of a particle pair (Bagi and Kuhn, 2004; Kuhn and Bagi, 2004). From this survey, we see that rolling can conjure many meanings, but we take a broad view toward its definition and provide the rationales for three alternative descriptions. In the broader sense, we view rolling as any form of objective contact motion that is independent of translational contact deformation and that, together with contact deformation, exhausts the full vector space of objective contact motions. The three alternative forms of rolling are applicable to three dimensional (3D) particles of arbitrary smooth shape, and details of their computational forms are provided in the Appendix A. Our second contribution is a framework for decomposing the motions of two particles into their objective and non-objective parts, with the latter referred to as the rigid motion of the pair. We present a definition of rigid motion that will satisfy a weakened objectivity condition. In short, we clarify the notion of objectivity when applied to the motions of a particle pair; we define an admissible, objective range of contact motions that can be used in numerical models to govern the interactions of the pair; we define several forms of admissible rolling motion; and we define the non-objective, rigid motion of the pair. The Appendix details the computational forms of each motion, which are readily applicable to numerical simulations. Notation conventions are as follows. Scalars are represented in normal weight fonts (rp, xi). Secondorder tensors are written in upper-case bold fonts (A), and vectors are written in lower-case bold fonts (v, x). Matrices are written with brackets, e.g. [v] and [A], with lower case fonts denoting column vectors. The inner and cross products of a vector pair are written with the ‘‘ Æ ’’ and ‘‘ · ’’ operators. The ‘‘ · ’’ symbol is also used in designating the sizes of matrices (e.g., 2 · 5).
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2. Analysis Two particles, p and q, are touching at the contact surface c, but because the contact surface is assumed to be small, it will be treated as a point in our analysis (Fig. 1). We identify two reference points, xp and xq, attached to the particles, perhaps at their centers. The motions of the particles are known through the translational velocities of these two points, vp and vq, and through their rates of rotation, xp and xq. Vectors rp and rq connect the points xp and xq to the contact point c; whereas the branch vector l connects point xp to point xq, l xq xp ¼ rp rq ;
ð1Þ
with length ‘ jlj. Particle motions are referred to global Cartesian axes with the unit basis vectors 1e, 2e, and 3e (Fig. 1). When formulating the constitutive behavior of a contact, it might be more convenient to refer the motions to local Cartesian axes having rotated basis vectors, perhaps ne, t1 e, and t2 e, with one axis normal to the contact surface and two tangential axes. The results in the paper can be changed to these local axes by properly transforming the underlying data—the components of vectors xp, xq, vp, vq, xq, rp, rq, and l—and, with these transformations, the labels 1, 2, and 3 can then be changed to n, t1, and t2. We note, however, that certain 3-tuple (column vectors) derived from this data may not represent first-order tensors, since they may not transform as tensors with a change in basis (frame). We will identify such 3-tuples of scalars. We arrange the motions of points xp and xq into a 12 · 1 partitioned column vector, written as ð2Þ where the sub-vectors ½vp and ½vq contain the stacked motions of the two particles:
:
Fig. 1. Particle and contact positions.
ð3Þ
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M.R. Kuhn, K. Bagi / Mechanics Research Communications 32 (2005) 463–480
The paper identifies certain generalized contact motions [^v] and evaluates an N · 12 transformation matrix [A] that yields these N generalized motions: :
ð4Þ
The generalized motions include translational contact deformation (indentation and sliding), rotational contact deformation (rolling and twisting), and rigid motion of the particle pair. Although only 12 generalized motions are required for an invertible mapping in Eq. (4), we allow for more than 12 generalized motions, so that the extended size of [^v] can accommodate redundant or alternative descriptions of the rotational deformations. We are particularly interested in a collection of generalized contact motions [^v] in which the N elements ^vi are separated into two types: (1) generalized motions that are objective and (2) generalized motions that are not. Objective contact motions are admissible in a constitutive relation for the contact; they can alternatively be applied as internal constraints to the inter-particle motions; and they can be properly included as forms of internal displacement in a virtual work formulation. Besides identifying objective contact motions, we are also interested in finding those objective motions whose values are independent of the choice of the two reference points xp and xq assigned to the particles, in the event that they are not placed at the particle centers (Fig. 1). 2.1. Objectivity A constitutive relation will typically be assigned to a contact, giving the contact force and moment, or their co-rotational rates, as functions of certain contact motions ^v1 ; ^v1 ; . . . ; ^vn . In general, the response of a material (or of a contact) must be independent of the observer, even if the observer is moving (e.g. Truesdell and Toupin (1960, Section 293)). This requirement places a constraint upon admissible contact motions, and this constraint can be separated into two objectivity conditions. A ‘‘g’’ generalized contact motion can be either a vector ^vg or a scalar ^vg , and the objectivity conditions apply differently to these two objects. The conditions are as follows: Condition 1. We consider two observers, A and B, who are stationary but have different locations and frames. Observer A is located at the origin of its frame (1e, 2e, 3e); whereas observer B is located at a finite offset c relative to A and with a rotated frame. The two observers would record different locations xO;A and xO;B of a material point O: xO;B ¼ c þ QxO;A ;
ð5Þ
where Q effects a finite rotation of their frames. A contact motion g is objective only if its observations by A and B are related by ^vg;B ¼ Q^vg;A ; g;B
^v
g;A
¼ ^v ;
for vectors; for scalars:
ð6Þ
That is, with a change in observer frame, vector motions must transform as vectors, scalar motions must be invariant (e.g., Eringen (1967, 2.10)). Condition 2. Suppose that two observers are moving with different translational and rotational velocities but they briefly share the same frame (i.e., the same location and orientation). The translation and rotation velocities of a material point ‘‘O’’ that are observed by B are vO;B ¼ vO;A ðvBA þ xBA xO Þ;
ð7Þ
M.R. Kuhn, K. Bagi / Mechanics Research Communications 32 (2005) 463–480
467
xO;B ¼ xO;A þ xBA ; O;A
O;B
O;B
ð8Þ BA
O;B
BA
where v ; x v and x are the velocities of point O as observed by V and B; A and by x are the velocities of observer B as seen by observer A; and xO is the position of O relative to both A and B. g A contact motion g is objective only if the same motion, bv or ^vg , is observed by both A and B. Together, the two conditions assure that the observed motions are related by (6) even if the observers have different frames and are moving. We consider Condition 2 as the more restrictive test, and the paper is primarily directed toward its requirement. We will defer the application of Condition 1, after having first applied Condition 2. If observers A and B are located at the contact point c, then xO corresponds to rp for the point xp; whereas, xO corresponds to rq for the point xq. By separately substituting rp and rq into expressions (7) and (8), the difference in the observed motions of p and q is given by ð9Þ
:
Both of the observed motions on the left are 12 · 1 stacked vectors in the form of Eqs. (2) and (3); the vecand xBA in the stacked form (3); and [T] is the 12 · 6 tor ½vBA is an arbitrary 6 · 1 vector with contents vBA i i matrix
ð10Þ
with [Wp] and [Wq] given by 2 3 0 rp3 rp2 6 7 0 rp1 5; ½Wp ¼ 4 rp3 rp2 rp1 0
2
0 6 q q ½W ¼ 4 r3 rq2
rq3 0 rq1
3 rq2 7 rq1 5;
ð11Þ
0
and [I] and [0] representing the 3 · 3 identity and zero matrices. We now return to the generalized motions [^v] in Eq. (4) and test the objectivity of the kth generalized motion bv k . If objectivity Condition 2 is met, the two observers would measure the same generalized motion: bv Ak 9bv Bk :
ð12Þ
This test can be applied to the individual rows of transformation matrix [A]: ð13Þ 9½0:
ð14Þ
The kth generalized motion bv k in matrix expression (4) is objective only if the kth row of [A] is orthogonal to the column space of [T], that is, only if all columns of [T] are orthogonal to row k of [A]. If so, the kth number on the right of Eq. (14) will indeed be zero for arbitrary observer motions ½vBA , and the same generalized motion bv k would always be assigned by the two observers. The union of all objective motions is the left nullspace of [T], or
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½Aobjective ½T126 ¼ ½0
ð15Þ
and the objective part of the motion [vp =vq ] is given by ð16Þ
:
Because the six columns of [T] are linearly independent, the system (15) admits only six linearly independent objective motions, and these motions correspond to six linearly independent rows of [Aobjective]. Any six such motions (rows) can serve as a basis of all objective motions in the sense that any objective motion can be expressed as a linear combination of the six selected basis motions. We will later formalize objective motions of two types: those that produce a translational deformation of the contact and those that produce, in part, a rotational deformation of the contact. These two types of generalized objective motions will be def rot collected into the two vectors [bv ] and [bv ]. Also of interest are non-objective motions that are purely non-objective—motions that are orthogonal to rigid all objective motions—and these generalized motions will be gathered into a separate vector [bv ]. Our p q purpose is to define a proper choice of matrix [A] that will transform the actual motions [v =v ] of Eqs. (2) and (3) into generalized motions that give the separate effects of translational deformation, rotational deformation, and rigid motions:
:
ð17Þ
The union of the row spaces of [Adef] and [Arot] is the six-dimensional basis of all objective motions and is the left null space of [T]. The sub-matrices [Adef] and [Arot] that are associated with translational and rotational deformation are stipulated in Sections 2.3 and 2.4. The six rigid motions [^vrigid ] are defined in the following section. 2.2. Rigid motions The mixing (or sum) of an objective and a non-objective motion gives a non-objective motion, and it may be of advantage to entirely separate these motions into two distinct and orthogonal subspaces. Toward this end, we place the following three conditions on the matrix [Anon-objective] in Eq. (17), with the result being a new matrix [Arigid] that characterizes the rigid motions of a particle pair. (1) The row space[Arigid] should equal the column space of [T]. With this restriction, the row spaces of the two matrices [Aobjective] and [Arigid] are orthogonal and any particle motion [vp =vq ] can be represented as a combination of its objective contact motion, [Aobjectlve] [vp =vq ], and its purely non-objective, rigid motion, ½Arigid ½vp =vq . This restriction on [Arigid] limits its representation to a product ½Arigid 612 ¼ ½Z66 ½TT612 ;
ð18Þ
where matrix [Z] is non-singular. (2) We further restrict [Arigid] by considering two observers A and B, who are momentarily located at the contact c and share the same frame, but who have different velocities, as in (7) and (8). The two A; rigid B; rigid observers would assign the following generalized ‘‘^’’ rigid motions, ½bv and ½bv , to the particle pair:
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;
ð19Þ
:
ð20Þ
We require that the difference in the generalized motions (19) and (20) equal the difference in the motions that would be observed if the two particles were stationary and only the observers were moving. That is, the difference in the observed motions ½^vA; rigid and ½^vB; rigid should equal the difference in the observer motions, A;rigid
½bv
B;rigid
½bv
¼ ½vBA
ð21Þ
for arbitrary observer motions. Condition (21) is, of course, weaker than the objectivity condition (14). Substituting expressions (19), (20), and (7) into condition (21) gives the restriction ½Arigid ½T½vBA ¼ ½vBA
ð22Þ
for arbitrary observer motions [v
BA
], or
½Arigid 612 ½T126 ¼ ½I66 :
ð23Þ
(3) The rigid motions should be represented by translational and rotational parts, and each part should satisfy the objectivity Condition 1 given in (6). That is, the rigid motions should transform as vector objects. The two conditions (18) and (23) are uniquely satisfied by the Moore-Penrose pseudo-inverse of [T]: T
1
T
½Arigid ¼ ð½T ½TÞ ½T ;
ð24Þ
where the matrix [Z] in Eq. (18) appears as the 6 · 6 inverse ([T]T[T])1. This inverse will always exist and be non-singular, since the columns of [T] are linearly independent. With definition (24), the product of [A] and [T] becomes:
ð25Þ
in which we allow for alternative definitions of rotational deformation (hence, the dimension 3+ P 3). The product of [Arigid] and the particle motions ½vp =vq yields the six generalized rigid motions:
:
ð26Þ
The contents of the 6 · 12 matrix [Arigid] are detailed in Appendix A as equation (A.10). In Appendix A, the ; ^vrigid ; ^vrigid T are the rigid translational velocities of the particle pair; the second set first three motions ½^vrigid 1 2 3 T of motions ½^vrigid ; ^vrigid ; ^vrigid are the rigid rotational velocities. These rigid motions have the following 5 4 6 properties:
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• Motions ^vrigid are non-objective—their values depend upon the observers motion. The rigid motions do i not conform to the objectivity condition (12), but instead satisfy the weaker condition (21). T T • Each of the two 3-tuples ½^vrigid ; ^vrigid ; ^vrigid and ½^vrigid ; ^vrigid ; ^vrigid is a vector: each 3-tuple can be 5 1 2 3 4 6 translation i rotation i expressed as the inner products ^v e and ^v e of a single vector and the three unit basis vectors. That is, each of the two 3-tuples is a vector object, satisfying the third requirement of the rigid motions (i.e., the objectivity Condition 1). This property is evident in the matrix form (A.10) given in Appendix A, since each 3 · 3 sub-matrix within matrix (A.10) transforms as a second-order tensor under frame rotations. • Matrix [Arigid] has rank 6, since both ([T]T[T]1) and [T]T are of rank 6 (Eq. (24)). That is, the non-objective motions are fully described by the six generalized velocities ^vrigid . These characteristics are summai rized in the first row of Table 1.
2.3. Translational contact deformation Translational contact deformation results from a relative translation of the particles p and q at the contact point c, and this form of contact motion produces indentation and sliding. The translations of material on either side of the contact are given by vp;c ¼ vp þ xp rp ;
ð27Þ
vq;c ¼ vq þ xq rq ;
ð28Þ
p,c
q,c
where v and v are the velocities of material in grains p and q near their common contact. The difference vq,c vp,c is the generalized translational deformation velocity of q relative to p, and this deformation velocity is responsible for any sliding or indentation at c: ^vdef vq;c vp;c
ð29Þ
¼ ðvq vp Þ þ ðxq rq xp rp Þ
ð30Þ
or in matrix form, ð31Þ
:
The translational deformation (28) has long been used in the analysis of granular materials and in DEM simulations (e.g., Cundall and Strack, 1979; Molenkamp, 1984; Koenders, 1987; Chang and Misra, 1989).
Table 1 Properties of the generalized motions Motion
Objective?a
Independent of xp,xq?
Rank of [A(Æ)]
Vector or scalar?
Eq.
Eq. [A(Æ)]
½^vrigid ^vdef ^vroll;1 ½^vroll;2 ^vroll;3 ^vtwist
Nob Yes Yes Yes Yes Yes
No Yes Yes No Yes Yes
6 3 2d 3 2d 1
Vectorc Vector Vector 3 Scalars Vector Scalar
(24) (30) (33) (43) (44) (34)
(A.10) (A.16) (A.17) (A.18) (A.28) (A.29)
a b c d
Motions that satisfy objectivity Conditions 1 and 2. Rigid motions satisfy the weaker conditions of Section 2.2. Each set of three components ½^vrigid ; ^vrigid ; ^vrigid and ½^vrigid ; ^vrigid ; ^vrigid is a 3-vector. 1 2 3 1 2 3 Together, [Aroll,(Æ)] and [Atwist]1·12 have rank 3.
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The motions ½^vdef satisfy objectivity Condition 2, as the product (13) for [Adef] is zero for an arbitrary vector ½vBA . The contents of [Adef] are given in the Appendix A (Eq. (A.16)), and the deformation motion has the following characteristics, which are summarized in Table 1: • The components ^vdef are independent of the choice of reference points xp and xq at which the particle i motions are measured (Fig. 1). • The object ^vdef is a vector and will transform as a vector under frame rotations, since each object on the i right of Eqs. (29) and (30) is also a vector. The deformation ^vdef satisfies objectivity Condition 1 as well i as Condition 2. • Matrix [Adef] is rank 3.
2.4. Rotational contact deformation Although the translational deformation motions ½^vdef can be unambiguously defined, numerous definitions are possible for the rotational deformation motions ½^vrot , and we summarize three definitions of rolling (with twisting) that have been proposed by the authors. The definitions can be collected into a partitioned matrix [Arot],
ð32Þ
with the superscripts denoting the alternative descriptions. The fourth, ‘‘twisting’’ type of rotational motion in Eq. (32) is included to augment (and complete) the first and third forms of rolling, as will be seen in Sections 2.4.1 and 2.4.3. Any of these rotational motions could be used in a constitutive description of the contact behavior. A fourth type of rolling (a ‘‘Type 4 rolling’’) has been proposed elsewhere, such that the rolling motions ^vroll;4 are orthogonal both to the rigid motions and to the translational contact deformations (Kuhn and Bagi, 2004). 2.4.1. Type 1 rolling and contact twisting One definition of rolling, termed ‘‘Type 1 rolling,’’ is simply the tangential component of the difference between the rotations of particles p and q: ^vroll;1 ðxq xp Þ ½ðxq xp Þ nn
ð33Þ
as in (Bojta´r, 1989; Bagi, 1993; Zhou et al., 1999; Satake, 2001). The motion ^vroll;1 has been applied in 2D simulations of granular assemblies, a situation in which (33) contains no out-of-plane motions of the particles and in which the bracketed term on the right of (33) is zero. The contents of [Aroll, 1] are given in the Appendix A (Eq. (A.17)). This form of rolling clearly satisfies objectivity Condition 2 by meeting the test (13,14) for arbitrary ½vBA . The generalized rolling ½^vroll;1 has the following additional properties, which are summarized in Table 1: • The movements v^i roll;1 are independent of the choice of reference points xp and xq at which the particle motions are measured (Fig. 1), although the signs of the v^i roll;1 depend on the assignment of the particle labels ‘‘p’’ and ‘‘q’’ (if this characteristic is undesirable, the alternative (xq xp) · n can be used instead).
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• The objects in Eq. (33) are written without matrix brackets to emphasize that ^vroll;1 is a vector: its three (Cartesian) components are the inner products ^vroll;1 i e of a single vector quantity and the unit basis vectors. That is, ^vroll;1 transforms as a vector under frame rotations, and it satisfies the objectivity Condition 1 as well as Condition 2. • The rolling vector ^vroll;1 lies in the tangent plane of the contact, so that the matrix [Aroll,1] is of rank 2. The six rows of the combined matrices [Aroll,1] and [Adef] are likewise not linearly independent, but are of rank 5. Motions ½^vdef and ½^vroll;1 must be augmented by a separate motion in order to span the full six-dimensional subspace of objective motions. In this regard, we could augment ½^vroll;1 with any one of the three rolling quantities in Section 2.4.2 or 2.4.3. A more reasonable choice, however, is a vector that is orthogonal to ½^vroll;1 as would be the case with the particles relative rotation about the contact normal n. We will refer to this rotation as a ‘‘twisting’’ motion having the magnitude ^vtwist : ^vtwist ðxq xp Þ n:
ð34Þ roll,1
twist
The four rows of the combined matrices [A ] and [A ] are rank 3. The contents of the corresponding row vector [Atwist]1·12 are given in the Appendix A (Eq. (A.29)).
2.4.2. Type 2 rolling Iwashita and Oda (1998) considered the relative motions of two 2D circular disks (Fig. 2) and arrived at the following expression for rolling: 1 ^vIO ¼ ½rp ðxp3 rq xq3 Þ ðrp rq Þb_ 3 ð35Þ 2 1 ¼ ½rp ðxp3 b_ 3 Þ rq ðxq3 b_ 3 Þ: ð36Þ 2 Both rp and rq are (positive) radii, and b_ 3 is the rate at which the center of particle q rotates around the center of particle p: b_ 3 ¼ ½ðvq vp Þ t=ðrp þ rq Þ:
ð37Þ
The unit vector t is tangent to the contact and is directed counter-clockwise around p (Fig. 2). The rotation b_ 3 will be referred to as the ‘‘pair-spin’’, which is subtracted from the individual particle spins to calculate the Iwashita–Oda rolling of Eq. (36). When both vp and vq are zero, ^vIO is the average of the tangential velocities vp,c Æ t and vq,c Æ t of two points, one attached to each of the particles p and q at the contact (Eqs. (27) and (28), Fig. 1). The authors have extended the Iwashita–Oda rolling to 3D particles of arbitrary shape (Bagi and Kuhn, 2004). We briefly review this general type of rolling, referred to as ‘‘Type 2 rolling,’’ using the global frame
Fig. 2. Iwashita–Oda rolling between circular disks.
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(1e, 2e, 3e) instead of the local frame (n, t1, t2) as in reference (Kuhn and Bagi, 2004). The rolling ½^vroll;2 has three elements, each of which is an averaged translational velocity of two material points, one attached to each of the particles at their contact. The element ^vroll;2 connotes a translational velocity in the ie direction, i just as the Iwashita–Oda rolling velocity occurs in the tangential t direction and has 1e and 2e components. To define the single element ^vroll;2 of Type 2 rolling, we construct a vector il that is the orthogonal projection i p q of the branch vector l = r r onto a plane perpendicular to direction ie (Fig. 3 and Eq. 1): i
l l ðl i eÞi e
ð38Þ i
with the associated unit vector k, i
k i l=i ‘;
ð39Þ
where i‘ is the length of vector il, or i‘ jilj. The Type 2 rolling associated with the ie direction is attributed to material motions in the vicinity of the contact that occur about an axis iz that is perpendicular to both the unit vector ie and the unit vector ik (Fig. 3): i
z i k i e:
ð40Þ i
i
i
Together, the three vectors form a right-hand orthogonal triad ( e, z, k), but each of the three basis directions ie (i = 1,2,3) will have a different associated triad. In the event that l and ie are aligned (i.e. l · ie = 0, cf. Eq. (39), then ik in (39) is ill-defined, but the right-hand triad of basis vectors, (ie, je, ke), can then serve in place of (ie, iz, ik). Type 2 rolling in the ie direction is the difference of two parts, which we label ^vroll;2a and ^vroll;2b . The first i i i part is due to particle rotations about the z-axis (Fig. 3): 1 ^vroll;2a ½ðxp i zÞðrp i kÞ þ ðxq i zÞðrq i kÞ: ð41Þ i 2 The terms in this expression can be thought to correspond to the rx3 products in Eq. (35). The product xp Æ iz is the rotation of particle p about the iz axis, and rp Æ ik is the corresponding rotation arm that is perpendicular to both ie and iz. The products in Eq. (41) yield a velocity in the ie direction. The pair-spin about the iz axis is ðvqi vpi Þ=i ‘, and the second part of rolling ^vroll;2 is produced by this spin: i
Fig. 3. Vectors for computing the rolling motion ^vroll;2 of particle q relative to particle p. This motion is referred to the 1e direction. The 1 unit vectors 1k,1z, and 1e are orthogonal. Similar sets of unit vectors are required for computing ^vroll;2 and ^vroll;2 . 2 3
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^vroll;2b i
1 vqi vpi p q i ðr þ r Þ k ; i‘ 2
where i‘ = l Æ ik is the length of vector il in equation (38) (Fig. 3). The difference ^vroll;2a ^vroll;2b defines the rolling rate ^vroll;2 of q relative to p: i i i q p 1 p i vi vi vqi vpi roll;2 p i q i q i ^vi ¼ ðr kÞ x z i þ ðr kÞ x z i : 2 ‘ ‘
ð42Þ
ð43Þ
In the event that l and ie are aligned, the length i‘ is zero and the corresponding component ^vroll;2 does not i exist. This form of rolling satisfies the objectivity Condition 2 by meeting the test (14) for an arbitrary observer motion ½vBA . The contents of matrix [Aroll,2] are detailed in Appendix A (Eq. (A.18)). The generalized rolling ½^vroll;2 has the following additional properties, which are summarized in Table 1: • The rolling motions ^vroll;2 can be computed for particles of arbitrary shape. i • The values of motions ^vroll;2 depend upon the choice of the reference points xp and xq at which the pari ticle motions are measured. • The motions ^vroll;2 are scalars, which can be assembled into the 3-tuple ð^vroll;2 ; ^vroll;2 ; ^vroll;2 Þ. These motions 1 2 3 i roll;2 are not the Cartesian components of a single vector, since the three values ^vi in Eq. (43) can not be expressed as the inner products ^vroll;2 i e of a single vector ^vroll;2 . • The scalar motions satisfy objectivity Condition 1 as well as Condition 2. Each term in (43) is either a norm or an inner product of two vectors, so the scalar ^vroll;2 is invariant under changes in the observer i frame, provided that both observers agree to the same ie directions. • The three rows of [Aroll,2] are linearly independent so that the matrix is rank 3.
2.4.3. Type 3 rolling The authors have defined a third type of rolling, which has its origins in robotics applications (Montana, 1988; Kuhn and Bagi, 2004). This ‘‘Type 3 rolling’’ is the average of the rates at which the two points of contact (one on each particle surface) move across the two rolling surfaces. This rolling motion is distinct from Type 2 rolling: for the rotations shown in Fig. 2, material points near the contact move in direction t (Type 2 rolling); whereas, the points of touching move in the opposite direction (Type 3 rolling). Like Type 1 rolling, the rolling motion ^vroll;3 is a vector, but its direction and speed depend upon the local surface shapes of the two particles at their contact. The local shapes are described by curvature tensors Kp and Kq, and the rolling velocity is defined as 1 ^vroll;3 ðKp þ Kq Þ1 ð^vroll;1 n Kqvdef Þ vdef ; ð44Þ 2 where n is the unit normal vector at the contact, directed outward from particle p (Kuhn and Bagi, 2004). Vector ^vroll;3 lies in the tangent plane of the contact and is orthogonal to n. The deformation rate vdef in Eq. (44) is the projection of the deformation ^vdef in Eq. (30) onto the tangent plane of the contact: vdef ^vdef ð^vdef nÞn:
ð45Þ
The rolling in Eq. (44) can be expressed in matrix form as ð46Þ and the contents of matrix [Aroll,3] are detailed in Appendix A (Eq. (A.28)). The motion (44) satisfies objectivity Condition 2, since it is a linear combination of the two objective motions ^vroll;1 and ^vdef . The Type 3 rolling motion has the following additional characteristics, which are summarized in Table 1:
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• The rolling motion ^vroll;3 can be computed for particles of arbitrary shape, provided that the particle surfaces are sufficiently smooth to admit their curvatures K. • Like ^vroll;1 and ^vdef (but unlike ^vroll;2 ), the rolling motion ^vroll;3 is independent of the choice of reference i p q points x and x at which the particle motions are measured. • Unlike the motions ^vroll;2 , the rolling ^vroll;3 is a vector: its Cartesian components ^vroll;3 are the inner prodi i ucts of the single vector ^vroll;3 and the unit vectors ie. Because it transforms as a vector, ^vroll;3 satisfies i objectivity Condition 1 as well as Condition 2. • Like the Type 1 rolling in Section 2.4.1, the rolling vector ^vroll;3 lies in the tangent plane of the contact, so that the matrix [Aroll,3] is of rank 2, and the six rows of the combined matrices [Aroll,3] and [Adef] are rank 5. Motions ½^vdef and ½^vroll;3 must be augmented by a separate motion, such as the twist ^vtwist of Eq. (34), in order to span the full six-dimensional subspace of objective motions.
3. Example We consider two two-dimensional (2D) disks of the same radius r arranged as shown in Fig. 4. To simplify the calculations and conserve space, we only consider motions in the 1e 2e plane. As such, the 12 T kinematic variables in Eq. (2) are reduced to the six motions ½vp1 ; vp2 ; xp3 ; vq1 ; vq2 ; xq3 . The various [A] matrices that are associated with generalized motions are also reduced in size: with 6 columns instead of 12, and with fewer rows as well. To conserve space, we have collected the relevant data into Table 2, which displays all of the quantities defined in the Appendix A. For example, the 3 · 3 matrix [Wp] in Eq. (11) is reduced to the 2 · 1 matrix [0, r]T, since it is only multiplied by the x3 rotations to produce a 2-component translation in the 1e 2e plane. The resulting [A] matrices for deformation, rolling, rigid translation, and rigid rotation are as follows:
:
ð47Þ
Types 2 and 3 rolling are both 2-component translation vectors that appear as pairs of rows in Eq. (47), but Type 3 rolling is a single component of rotation in the 3e direction. Because Types 2 and 3 rolling are associated with the contacts tangential direction, the two components are not linearly independent. The matrix in (47) contains six independent rows that correspond to six canonical motions for the pair of particles. T Each of these six motions is produce by setting the corresponding motion vector ½vp1 ; vp2 ; xp3 ; vq1 ; vq2 ; xq3 equal to its corresponding row in [A]: (1) a stretching of the two particles in the normal n = 1e direction, as in the first row, (2) a tangential shearing of the particles in the 2e direction, as in the second row. The particles simultaneously translate and rotate to produce this canonical motion,
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Fig. 4. Example of two equal-size disks.
Table 2 Data for an example of two disks of equal size (Fig. 4) Data values
Equations
rigid
]3·6 [rp]2·1 = [r,0]T, [rq]2·1 = [r,0]T, [n]2·1 = [1,0]T
(A.10) –
[l]2·1 = [2r,0]T, [s]2·1 = [0,0]T
(1)–(A.3)
l
‘ = 0, 2‘ = 2r, 3‘ = 2r G = 4r2 + 4, ‘ = 2r, a = 0 [Wp]2·1 = [0,r]T, [Wq]2·1 = [0,r]T 0 0 ½B22 ¼ , ½C21 ¼ ½0; 0T , ½D11 ¼ D33 ¼ 0 0 0 [Adef]3·6
(A.4)–(A.6) (A.7), (A.8), and (A.12) (11)
[Aroll,1]1·6
(A.17)
[A
1 ¼ 0
½M22 roll,2
[A
]2·6
½E22
0 0
0 0 ¼ , ½FP 21 ¼ ½0; r=2T , ½Fq 21 ¼ ½0; r=2T 0 0
[Aroll,3]2·6
0 0 , ½N21 ¼ ½0; 1T ½P22 ¼ 0 1 0 0 ½Kp 22 ¼ ½Kq 22 ¼ 0 1=r 0 0 p q þ ½K þ K 22 ¼ 0 r=2
(3) (4) (5) (6)
a a a a
(A.13)–(A.15) A.16)
(A.17) (A.18) (A.19)–(A.23) (A.28) (A.25), (A.27) – –
rolling motion in either of the rows 3, 5, or 7, rigid-body horizontal translation (row 8), rigid-body vertical translation (row 9), and rigid-body rotation of the particles about the central point (r, 0), as in row 10.
The twisting motion [Atwist] does not apply, of course, to these in-plane motions. 4. Conclusions The discrete motions of two particles can be unambiguously separated into objective and non-objective parts. The non-objective motions can be defined as the rigid motions of the pair, and these motions are the
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477
orthogonal projection of the actual motions onto the vector space of all non-objective motions. The objective motions can be separated into motions that produce a translational deformation at the contact and motions that produce a rotational (rolling or twisting) deformation. The rolling motions can be expressed in at least three different forms. Table 1 summarizes the properties of the rigid motion, the translational deformation, and the various forms of rotational deformation. Details of each motion are given in Appendix A.
Acknowledgements This work was funded in part by the Arthur Butine Faculty Development Fund of the University of Portland. The support of the OTKA 31889 grant is also acknowledged.
Appendix A This appendix gives the contents of the six sub-matrices [Arigid], [Adef], [Aroll, 1], [Aroll, 2], [Aroll,3], and [A ]. These sub-matrices can be assembled, as in Eqs. (17) and (32), to create the full matrix [A] for computing the generalized motions of a particle pair: twist
ðA:1Þ
where relative twisting has been added to augment the two-dimensional (and incomplete) vector sub-space of Type 3 rolling. The expressions in this Appendix can be readily applied in numerical simulations to the the various contact motions. Expressions for the six sub-matrices are greatly condensed by using the following intermediate terms: s rp þ rq ;
ðA:2Þ
l rp rq ¼ 1 e‘1 þ 2 e‘2 þ 3 e‘3 ; with components ‘1 ; ‘2 ; ‘3 ;
ðA:3Þ
2
2 1=2
;
ðA:4Þ
2
2 1=2
;
ðA:5Þ
2
2 1=2
;
ðA:6Þ
1
‘ l 1 k ¼ ½ð‘2 Þ þ ð‘3 Þ
2
‘ l 2 k ¼ ½ð‘1 Þ þ ð‘3 Þ
3
‘ l 3 k ¼ ½ð‘1 Þ þ ð‘2 Þ 2
G ð‘Þ þ 4;
ðA:7Þ
‘ j l j;
ðA:8Þ
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½nT ½n1 ; n2 ; n3 :
ðA:9Þ rigid
Rigid motions: The 6 · 12 matrix [A tively, matrix [Arigid] is explicitly
] can be computed directly from expressions (10) and (24). Alterna-
;
;
ðA:10Þ
ðA:11Þ
where [I] and [0] are the 3 · 3 identity and zero matrices, [Wp] and [Wq] are given in Eq. (11), and scalar a and the 3 · 3 matrices [B], [C], and [D] are defined as a ¼ l s;
ðA:12Þ
Bij ¼ ‘i sj ;
ðA:13Þ
C ij ¼ i e ðrp rq Þ‘j ;
ðA:14Þ
Dij ¼ ‘i ‘j :
ðA:15Þ
Translational deformations: The 3 · 12 matrix [Adef] is defined by expression (30), or ðA:16Þ where matrices [Wp] and [Wp] are given in Eq. (11). Type 1 rolling: The 3 · 12 matrix [Aroll,1] in Eq. (33) is ðA:17Þ where Mij = ninj. Type 2 rolling: The 3 · 12 matrix [Aroll,2] is defined by (40), (39), and (43), or by ðA:18Þ where
2s 16 6 ½E ¼ 6 24
2 ‘2 þs3 ‘3 ð1 ‘Þ2
0
0
0
s1 ‘1 þs3 ‘3 ð2 ‘Þ2
0
0
0
s1 ‘1 þs2 ‘2 ð3 ‘Þ2
2
0
16 ðÞ ½FðÞ ¼ 6 ‘ R 24 3 2 ðÞ ‘2 R3 with
ðÞ
‘ 3 R1 0
ðÞ ‘1 R3
3 7 7 7; 5 ðÞ
‘2 R1
ðA:19Þ
3
7 ðÞ ‘ 1 R2 7 5; 0
ðA:20Þ
M.R. Kuhn, K. Bagi / Mechanics Research Communications 32 (2005) 463–480 ðÞ
ðÞ
R1 ¼
ðÞ
r2 ‘2 þ r3 ‘3 ð1 ‘Þ
2
ðÞ
ðÞ
R2 ¼
;
ðA:21Þ
;
ðA:22Þ
:
ðA:23Þ
ðÞ
r1 ‘1 þ r3 ‘3 ð2 ‘Þ
2
ðÞ
ðÞ
R3 ¼
479
ðÞ
r1 ‘1 þ r2 ‘2 3
ð ‘Þ
2
Type 3 rolling: The vector ^vroll;3 in Eq. (44) is a linear combination of the motions ^vroll;1 and ^vdef . Vector vdef in Eq. (45) can be expressed as the matrix product ½vdef ¼ ½P½^vdef ;
ðA:24Þ
where the projection matrix [P] is related to the contact normal vector n, as ½P ¼ ½I ½M ¼ ½I ½n½nT : roll
The cross product ^v roll;1
½^v with
n is 1
n ¼ ½N½^vroll; ; 2
0 6 ½N ¼ 4 n3 n2
ðA:25Þ
n3 0 n1
3 n2 7 n1 5 :
ðA:26Þ
ðA:27Þ
0
In standard presentations of differential geometry, the surface curvatures Kp and Kq are usually referred to a two-dimensional basis of vectors that are tangent to the surfaces. We have modified the standard approach by referring Kp and Kq to the three-dimensional basis of vectors 1e, 2e, and 3e (Kuhn and Bagi, 2004). In this approach, the 3 · 3 matrix sum [Kp] + [Kq] is of rank 2, and the pseudo-inverse (‘‘+’’ inverse) is used in place of (Kp + Kq)1. By substituting expressions (A.24) through (A.27) into (44), the matrix [Aroll,3] can be expressed as ðA:28Þ Twisting: The 1 · 12 matrix [Atwist] is defined by Eq. (34), or ðA:29Þ where n is the unit normal vector (Eq. (A.9)).
References Bagi, K., 1993. Mech. Mater. 16, 101. Bagi, K., Kuhn, M.R., 2004. J. Appl. Mech. 71, 493. Bardet, J.P., 1994. Mech. Mater. 18, 159. Bojta´r, I., 1989. Epites-Epiteszettudomany 75. Chang, C.S., Misra, A., 1989. Comput. Geotech. 7, 269.
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Cundall, P.A., Strack, O.D.L., 1979. Ge´otechnique 29, 47. Eringen, A.C., 1967. Mechanics of Continua. John Wiley and Sons, New York. Hertz, H., 1882. J. Reine Angewandte Math. 92, 156. Iwashita, K., Oda, M., 1998. J. Engrg. Mech. 124, 285. Koenders, M.A., 1987. Acta Mech. 70, 31. Kuhn, M.R., Bagi, K., 2004a. J. Engrg. Mech. 130, 826. Kuhn, M.R., Bagi, K., 2004b. Int. J. Solids Struct. 41, 5793. Molenkamp, F., 1984. Soil Found 24, 17. Montana, D.J., 1988. Int. J. Robotics Res. 7, 17. Oda, M., Konishi, J., Nemat-Nasser, S., 1982. Mech. Mater. 1, 269. Rothenburg, L., Bathurst, R., 1988. J. Appl. Mech. 55, 17. Satake, M., 2001. In: Kishino, Y. (Ed.), Powders and Grains. A.A. Balkema, Lisse, pp. 149–152. Truesdell, C., Toupin, R.A., 1960. In: Flu¨gge, S. (Ed.), Encyclopedia of Physics, Vol. III/l. Springer-Verlag, Berlin, pp. 226–902. Zhou, Y.C., Wright, B.D., Yang, R.Y., Xu, B.H., Yu, A.B., 1999. Physica A 269, 536.