Mechanics of Materials 37 (2005) 641–662 www.elsevier.com/locate/mechmat
A comparison of micro-mechanical modeling of asphalt materials using finite elements and doublet mechanics Martin H. Sadd *, Qingli Dai Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island, 92 Upper College Road, Kingston, RI 02881, USA Received 11 July 2003; received in revised form 27 May 2004
Abstract A comparative study is given between two micro-mechanical models that have been developed to simulate the behavior of cemented particulate materials. The first model is a discrete analytical approach called doublet mechanics that represents a solid as an array of particles. This scheme develops analytical expressions for the micro-deformation and stress fields between particle pairs (a doublet). The second approach is a numerical finite element method that establishes a network of elements between neighboring cemented particles. Each element has been developed to model the local load transfer between particles. While the two modeling schemes come from very different beginnings, they have a fundamental similarity. However, they also have some basic differences. In order to pursue these similarities and differences, three example problems are investigated using each modeling approach. Even with the differences, the two model predictions of the micro-stress distributions for each example compared quite closely. These results also indicated significant micro-structural effects that differ from continuum elasticity theory and could lead to better explanations of observed failures of these types of materials. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Finite element modeling; Doublet mechanics; Asphalt concrete; Micro-mechanical modeling; Cemented particulate materials; Material micro-structure
1. Introduction
*
Corresponding author. Tel.: +1 401 874 2425; fax: +1 401 874 2355. E-mail address:
[email protected] (M.H. Sadd).
The mechanical behavior of heterogeneous solids is commonly approached from two general viewpoints depending on whether the material phases are distributed as either continuous or discrete. Under continuous distribution, theories are
0167-6636/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2004.06.004
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based on continuum mechanics and have provided many useful solutions to problems of engineering interest. However, these theories generally develop models that do not contain scaling effects and this is normally regarded as a limitation to predicting micro-mechanical material behavior. In order to overcome this limitation, various elegant modifications to continuum mechanics have been made to incorporate scale and micro-structural features into the theory. On the other hand, discrete modeling develops particular micro-force–deformation relations between the various material phases within the solid. Thus the micro-forces are distributed at specific points and along directions inherently connected to the discrete micro-structural geometry. From a fundamental point of view, such distributions are not directly representable by traditional tensor variables used in continuum mechanics. Discrete models normally lead to theories with one or more length scales resulting in non-local behavior where the stress at a point will depend on the deformation in a neighborhood about the point. These length scales may represent the sizes and/or separations of particles, dimensions of internal cells, characteristic ranges of particle or phase interactions, etc. The particular material of interest in the current work is asphalt concrete, a complex heterogeneous material generally composed of aggregates, binder/ cement, and void space. Because of these features, the material has particular micro-structures that affect the load carrying behavior. Typically these micro-structures are related to aggregate geometry such as particle size or spacing and occur at length scales in the range 0.1–20 mm. Recently a considerable amount of research has been conducted on developing micro-mechanics models to predict the behavior of asphalt, concrete, ceramics, rock and other cemented granular materials. This previous work can be divided into analytical and computational modeling. Past analytical work has incorporated several types of micro-mechanical continuum mechanics theories. Examples include Cosserat/micro-polar theories which introduce an additional kinematic rotational degree of freedom; see review articles by Eringen (1968, 1999) and Kunin (1983). These theories have been applied to granular materials
by Chang and Liao (1990) and Chang and Ma (1991, 1992). Continuum theories using higher order displacement gradients have also been used to develop micro-mechanical models; Bardenhagen and Trianfyllidis (1994) for elastic lattice models, and Chang and Gao (1995) for granular materials. A large volume of work has used fabric tensor theories to characterize the material microstructure and relate particular fabric tensors to the materialÕs constitutive stress–strain response; e.g. Nemat-Nasser and Mehrabadi (1983), Konshi and Naruse (1988) and Bathurst and Rothenburg (1988). Another area of analytical modeling has used distributed body theory whereby porous/ multiphase micro-structure is accounted for using an additional independent volume distribution function. The general theory for elastic materials was established by Cowin and Nunziato (1983), and this was followed by many application papers; e.g. Cowin (1984). Some work has approached the problem using statistical methods to develop models with random variation in micro-mechanical properties; e.g. Ostoja-Starzewski and Wang (1989). One particular theory that has recently been applied to granular and asphalt materials is the doublet mechanics model. This approach originally developed by Granik (1978), has been applied to granular materials by Granik and Ferrari (1993) and Ferrari et al. (1997). Recently Wang et al. (2003) presented a micro-mechanical study of top–down cracking of asphalt materials using some results from this theory. Doublet mechanics is a micro-mechanical discrete model whereby solids are represented as arrays of points, particles or nodes at finite distances. A particle pair is referred to as a doublet, and the particle spacing introduces length scales into the micro-structural theory. The model develops micro-stress–strain constitutive laws for the extensional (axial), shear and torsional deformations between the particles in each doublet. The theory has shown promise is predicting observed behaviors that are not predictable using continuum mechanics. These behaviors include the so-called Flamant paradox (Ferrari et al., 1997), where in a half-space under compressive boundary loading, continuum theory predicts a completely compressive stress field but observa-
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tions indicate regions of tensile stress. Other anomalous behaviors include dispersive wave propagation. Another general area of micro-mechanical modeling has taken the numerical computation approach using finite element (FEM), boundary element (BEM) and discrete element (DEM) methods. Examples of finite element modeling include Liao and Chang (1992) for granular materials, Stankowski (1990) on cemented particulate composites, and Sepehr et al. (1994) for a study on asphalt pavement layers. Soares et al. (2003) used cohesive zone elements to develop micro-mechanical fracture model of asphalt materials. A common FEM approach to simulate particulate and heterogeneous materials has used an equivalent lattice network system to represent the inter-particle load transfer behavior. This type of micro-structural modeling has been used previously; Bazant et al. (1990), Mora (1992), Sadd and Gao (1998) and Budhu et al. (1997). Along similar lines, Guddati et al. (2002) recently presented a random truss lattice model to simulate micro-damage in asphalt concrete, and Chang et al. (2002) used a lattice micro-structure to develop a FEM model for concrete failure. Bahia et al. (1999) have also used finite elements to model the aggregate-binder response of asphalt materials. Boundary element applications for asphalt modeling have been presented by Birgisson et al. (2002). Discrete element methods simulate particulate systems by modeling the translational and rotational behaviors of each particle using NewtonÕs laws. DEM studies on cemented particulate materials include the work by Rothenburg et al. (1992), Chang and Meegoda (1993), Trent and Margolin (1994), Sadd et al. (1992), Sadd and Gao (1997), Buttlar and You (2001), and Ullidtz (2001). Of special interest with respect to the current study is the micro-mechanical finite element model previously developed by the authors, Sadd et al. (2004a,b). Similar to the doublet mechanics approach this computational model uses basic load transfer mechanics between particles to establish a numerical scheme to simulate asphalt material behavior. This is accomplished by first developing a frame-type element to simulate the micro-load carrying response between cemented aggregates.
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The overall asphalt behavior is then modeled by a network of these micro-finite elements. This scheme has shown good success in several applications and has compared reasonably well with experimental results. The purpose of this work is to present a comparison of modeling results from doublet mechanics and the above mentioned finite element model. We pursue such a comparison because of the interesting similarities and differences between doublet mechanics and the finite element approach. Each model comes from very different beginnings, doublet mechanics from a discrete analytical scheme, and finite elements based on the usual numerical element equation model. Both schemes are based on fundamental micro-mechanics between particle pairs that make up the media, and each theory will lead to the inclusion of one or more length scales. However, there are also some important differences in each approach that warrant investigation. After briefly reviewing the basics of the two models, we apply each method to solve three basic example problems. The first two examples will incorporate a simplified doublet mechanics solution that has no length scale, while the final example will use a more complete solution with a single length scale. Each of these doublet mechanics solutions is compared with a corresponding finite element simulation of the equivalent problem.
2. Doublet mechanics Originally developed by Granik (1978), doublet mechanics (DM) is a micro-mechanical theory based on a discrete material model whereby solids are represented as arrays of points or nodes at finite distances. A pair of such nodes is referred to as a doublet, and the nodal spacing distances introduce length scales into the micro-structural theory. Current applications of the theory have normally used regular arrays of nodal spacing thus generating a Bravais lattice geometry. Each node in the array is allowed to have a translation and rotation, and increments of these variables are expanded in a Taylor series about the nodal point. The order at which the series is truncated defines the degree of approximation employed. The lowest
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order case using only a single term in the series will not contain any length scales, while using more than one term will produce a multilength scale theory. This allowable kinematics develops microstrains of elongation, shear and torsion (about the doublet axis). Through appropriate constitutive assumptions, these micro-strains can be related to corresponding elongational, shear and torsional micro-stresses. Applications of this theory to geomechanics problems have been given by Granik and Ferrari (1993) and Ferrari et al. (1997). For these applications, a granular interpretation of doublet mechanics has been employed, in which the material is viewed as an assembly of circular or spherical particles. A pair of such particles represents a doublet as shown in Fig. 1. Corresponding to the doublet (A, B) there exists a doublet or branch vector fa connecting the adjacent particle centers and defining the doublet axis a. The magnitude of this vector ga = jfaj is simply the particle diameter for particles in contact. However, in general the particles need not be in contact, and for this case the length scale ga could be used to represent a more general micro-structural feature. As mentioned the kinematics allow relative elongational, shearing and torsional motions between the particles, and this is used to develop an elongational micro-stress pa, shear micro-stress ta, and torsional micro-stress ma as shown in Fig. 1. It should be pointed out that these micro-stresses are not second order tensors in the usual continuum mechanics sense. Rather, they are vector quantities that represent the elastic micro-forces and micro-couples of interaction between doublet particles. Their directions are dependent on the doublet axes
which are determined by the material micro-structure. These micro-stresses are not continuously distributed but rather exist only at particular points in the medium being simulated by DM theory. If u(x, t) is the displacement field coinciding with a particle displacement, then the increment function at x = xA (xA is the position vector of the particle A) is written as Dua ¼ uðx þ fa ; tÞ uðx; tÞ
ð1Þ
Here, a = 1, . . ., n, while n is referred to as the valence of the Bravais lattice. Under the assumption that the doublet interactions are symmetric, the shear and torsional micro-deformations and micro-stresses vanish, and thus only extensional strains and stresses will exist. The extensional micro-strain scalar measure ea, representing the axial deformation of the doublet vector, is defined q Dua ea ¼ a ð2Þ ga where qa = fa/ga is the unit vector in the a-direction. The increment function (1) can be expanded in a Taylor series as m M X ðga Þ m Mþ1 ðqa rÞ uðx; tÞ þ O jga j Dua ¼ ð3Þ m! m¼1 Using this result into relation (2) develops the series expansion for the extensional or axial micro-strain ea ¼ qai
M X ðga Þm1 om ui qak1 . . . qakm m! oxk1 . . . oxkm m¼1
ð4Þ
where qak are the cosines of the angles between the directions of micro-stress and the coordinates.
Fig. 1. Basic doublet geometry and micro-stress definitions.
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As mentioned, the number of terms used in the series expansion of the local deformation field determines the order of the approximation in DM theory. For the first order case (m = 1), it has been shown that the scaling parameter ga will drop from the formulation, and the axial microstrain is reduced to ea ¼ qai qaj eij
ð5Þ
where eij is the usual continuum strain tensor given by eij = 1/2(ui,j + uj,i). For this case, the elastic DM solution can be calculated directly from the corresponding isotropic continuum elasticity solution through the relation rij ¼
n X
ð6Þ
qai qaj pa
a¼1
and this can be expressed in matrix form 1
frg ¼ ½Qfpg ) fpg ¼ ½Q frg
ð7Þ
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where {r} is the continuum elastic stress vector in rectangular Cartesian coordinates, {p} is the micro-stress vector, and [Q] is a transformation matrix. For plane problems, this transformation matrix can be written as 2 3 2 2 2 ðq11 Þ ðq21 Þ ðq31 Þ 6 7 ½Q ¼ 4 ðq12 Þ2 ðq22 Þ2 ðq32 Þ2 5 ð8Þ q11 q12
q21 q22
q31 q32
where qij are the cosines of the angles between the micro-stresses and Cartesian coordinates. This result allows a straightforward development of first order DM solutions for many problems of engineering interest, and will be used to generate DM solutions for comparative example problems. Specific applications of doublet mechanics have been developed for two-dimensional problems with regular particle packing micro-structures. One case that has been studied is the two-dimensional hexagonal packing as shown in Fig. 2. This geometrical micro-structure establishes three
Fig. 2. Two-dimensional hexagonal particle packing geometry.
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doublet axes at 60° angles as shown. For this case using only the first order approximation, the shear and torsional micro-stresses vanish, leaving only the elongational micro-stress components (p1, p2, p3) as shown. Positive elongational components correspond to tensile forces between particles. For the second order approximation (m = 2) case, the axial micro-strain term is now given by g ea ¼ qai qaj ui;j þ a qak ui;jk 2!
ð9Þ
Choose the simplifying case that all doublets originating from a common node have the same magnitudes; i.e. ga = g (a = 1, . . ., n), and also assume that the interactions are purely axial (no shear or torsional micro-stresses). For homogeneous interaction, there will be only one micro-modulus C0, and the constitutive relationship between elongation micro-stress and micro-strain is expressed by pa = C0ea. The displacement field for the second order approximation can be written u ¼ v þ gz þ g2 w
ð10Þ
where v represents the solution without the length scale, and corresponds to the classical elasticity solution, z and w are first order and second order displacement fields, respectively. By applying displacement equilibrium equations and length-scale independent boundary conditions, it follows that z 0. The governing equation for the second
order displacement field w has been previously given as n n X 1 X qai qaj qak qal wj;kl ¼ q q q q q q vj;klpq 12 a¼1 ai aj ak al ap aq a¼1 ð11Þ The solutions of w can be found by using Papkovich–Neuber displacement potentials. Having determined the displacements v and w, the microstress can then be evaluated by g pa ¼ C 0 qaj qak vj;k þ qal vj;kl þ g2 wj;k ð12Þ 2 This second order theory will be used later in a particular application/comparison problem.
3. Micro-mechanical finite element model The basics of our micro-mechanical finite element model have been presented in previous studies (Sadd and Dai, 2001; Sadd et al., 2004a,b). Here we will only briefly review some of the basic model developments for the elastic case. Bituminous asphalt can be described as a multiphase material containing aggregate, binder cement (including mastic and fine particles) and air voids (see Fig. 3(a)). The load transfer between the aggregates plays a primary role in determining the load carrying capacity and failure of such complex materials. In order to develop a micromechanical model of this behavior, proper simulation of the load transfer between the aggregates
Fig. 3. Micro-mechanical finite element modeling. (a) Asphalt material micro-structure, (b) resultant load transfer and (c) micro-frame finite element.
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must be accomplished. The aggregate material is normally much stiffer than the binder, and thus aggregates are taken as rigid particles. On the other hand, the binder cement is a compliant material with elastic, inelastic, and time-dependent behaviors. In order to properly account for the load transfer between aggregates, it is assumed that there is an effective binder zone between neighboring particles. It is through this zone that the micro-mechanical load transfer occurs between each aggregate pair. For the two-dimensional case, this loading can be reduced to resultant normal and tangential forces and a moment as shown in Fig. 3(b). The similarities and differences in the inter-particle load transfer for doublet mechanics (Fig. 1) and the micro-finite element model (Fig. 3(b)) should be noted. While the normal and tangential micro-forces are similar, the moment loadings are not. DM theory formulates a torsional loading about the in-plane doublet axis, while the FEM model postulates a moment loading along the out-of-plane direction. In order to model the inter-particle load transfer behavior, some simplifying assumptions must be made about allowable aggregate shape and binder geometry. Aggregate geometry is commonly quantified in terms of particle size, shape, angularity and texture. However, for the present modeling only size and shape are considered. In general, asphalt concrete contains aggregate of very irregular geometry as shown in Fig. 4(a). Our approach is to allow variable size and shape using an idealized elliptical aggregate model as represented in Fig. 4(b). Simplifying the shape allows a straight-forward determination of binder geometry necessary to calculate particular finite element properties. The finite element model then uses an equivalent lattice network approach, whereby the inter-parti-
647
cle load transfer is simulated by a network of specially created frame-type finite elements connected at particle centers as shown in Fig. 4(c). From granular materials research, the material microstructure or fabric can be characterized to some extent by the distribution of branch vectors which are the line segments drawn from adjacent particle mass centers. Note that the finite element network coincides with the branch vector distribution. Cementation between neighboring particles was generated using a scheme shown in Fig. 3(c), whereby the cementation was asymmetrically distributed parallel to the branch vector. The cementation geometry parameters are shown in Fig. 3(c). Since this scheme allows arbitrary non-symmetric cementation, an eccentricity variable is defined by e = (w2w1)/2. The current network model uses a specially developed, two-dimensional frame-type finite element to simulate the inter-particle load transfer. These two-noded elements have the usual three degrees-of-freedom (two displacements and a rotation) at each node and the element equation can thus be written in general form as 2
K 11 6 : 6 6 6 : 6 6 : 6 6 4 : :
K 12 K 22
K 13 K 23
K 14 K 24
K 15 K 25
: :
K 33 :
K 34 K 44
K 35 K 45
: :
: :
: :
K 55 :
9 8 9 38 K 16 > U 1 > > F n1 > > > > > > > > > >V1> > > > > K 26 7 F t1 > > > > > 7> > > > > > > > > 7< = < K 36 7 h1 M1 = 7 ¼ > K 46 7 F n2 > > > > U2 > > 7> > > > > > > > 7> > > > > > > >V2> > F t2 > K 56 5> > > > > > ; : ; : K 66 h2 M2
ð13Þ
where Ui, Vi and hi are the nodal displacements and rotations, and F.. and M. are the nodal forces and moments. The usual scheme of using bar and/ or beam elements to determine the stiffness terms is not appropriate for the current applications, and
Fig. 4. Asphalt modeling concept. (a) Typical asphalt material, (b) model asphalt system and (c) network finite element model.
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center to the cementation boundary, w1 and w2 are left and right width of cementation. Each binder element stiffness matrix will be different depending on the two-particle layout and size, and binder geometry. This procedure establishes the elastic stiffness matrix, which is a function of the material micro-structure and binder moduli. We now develop doublet mechanics and finite element solutions to three example problems. Comparisons of the resulting micro-stress solutions will be made. Fig. 5. Cementation between two adjacent particles.
4. Comparisons of DM analysis and FEM simulation therefore these terms were determined using an approximate elasticity solution from Dvorkin et al. (1994) for the stress distribution in a cement layer between two particles. The two-dimensional model geometry (uniform thickness case) is shown in Fig. 5. The stresses rx, rz and sxz within the cementation layer can be calculated for normal, tangential and rotational particle motion cases. These stresses can then be integrated to determine the total load transfer within the cement binder, thus leading to the calculation of the various stiffness terms needed in the element equation. Details of this process have been previously reported by Sadd and Dai (2001), and the final result is given by
2
K nn 6 6 6 6 ½K ¼ 6 6 6 6 4
0 K tt
K nn e K tt r1
4.1. Surface compression loading of a semi-infinite mass Surface loading of a semi-infinite body represents an important application problem in asphalt concrete research related to roadway performance. For example, top–down cracking is a type of failure that initiates at or near the pavement surface and is typically generated by surface compression loading. As pointed out by Wang et al. (2003), this problem is still not completely understood. Some of the conflicting issues are related to the stress distribution under concentrated and distributed surface loadings as
K nn 0
K tt r21 þ K3nn w22 w1 w2 þ w21 K nn e
0 K tt K tt r1
K nn
0 K tt
where K nn ¼ ðk þ 2lÞw= h, K tt ¼ lw= h, k and l are the usual elastic moduli, w and h are the cementation width and average thickness, r1 and r2 are the radial dimensions from each aggregate
3 K nn e 7 K tt r2 2 7 K nn 2 7 K tt r1 r2 3 w2 w1 w2 þ w1 7 7 7 K nn e 7 7 5 K tt r2 K nn 2 2 2 K tt r2 þ 3 w2 w1 w2 þ w1
ð14Þ
shown in Fig. 6(a) and (b). The elastic stress distribution in a semi-infinite solid under concentrated loading (Fig. 6(a)) is given by the classical Flamant solution
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649
Fig. 6. Surface compression problem. (a) Flamant problem, (b) integrated flamant problem and (c) computation model.
rx ¼ 2Px2 y=pðx2 þ y 2 Þ2 2
sxy ¼ 2Pxy 2 =pðx2 þ y 2 Þ
ð15Þ
2
ry ¼ 2Py 3 =pðx2 þ y 2 Þ
This continuum mechanics solution specifies that the stresses are everywhere compressive in the region below the surface loading, and this would also be true for the case shown in Fig. 6(b) where the surface loading acts over a distributed area. However as pointed out in the literature, there exists considerable experimental evidence that a granular medium under similar surface loading will exhibit tensile openings. Ferrari et al. (1997) refer to this issue as FlamantÕs paradox. It would thus appear that a micro-mechanical model is needed to resolve this paradox, and Ferrari et al. (1997) and Wang et al. (2003) have ap-
plied doublet mechanics to investigate this issue. For the Flamant problem using the transformation given by (7) and (8), Ferrari et al. (1997) have developed the micro-stresses for the first order, non-scale case for a medium with hexagonal packing as shown in Fig. 2. pffiffiffi 2 p1 ¼ 4Py 2 3x þ y =3pðx2 þ y 2 Þ pffiffiffi 2 p2 ¼ 4Py 2 3x y =3pðx2 þ y 2 Þ
ð16Þ
p3 ¼ 2Pyð3x2 y 2 Þ=3pðx2 þ y 2 Þ2 These micro-stress directions are defined for a granular micro-structure as shown in Fig. 6(c). These micro-stresses can be integrated to generate the more useful solution to the problem shown in Fig. 6(b). For a uniformly distributed line load q over the range a 6 x 6 a, the micro-stresses at
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location (x,y) can be obtained by integration over the coordinate origin variable u as
P1
¼ ¼
P2
¼ ¼
P3
¼ ¼
mal axial force of each micro-frame element was computed from the code. In order to compare fi-
pffiffiffi 2 1 2 2 3ðx uÞ þ y 3p ðx uÞ þ y p1 ðx uÞ du ¼ 4qy du a a h pffiffi i pffiffi 2qy 3yxa 3yxþa ðxaÞ 1y tan1 xþa þ 1y tan1 xa 2 y y 3p ðxþaÞ2 þy 2 þy 2 Z a Z a pffiffiffi 2 1 p2 ðx uÞ du ¼ 4qy 2 3ðx uÞ y 3p ðx uÞ2 þ y 2 du a a h pffiffi i pffiffi 3yþxþa 3yþxa 1 1 xþa 1 1 xa þ tan tan 2qy 2 2 2 2 3p ðxþaÞ þy y y y y ðxaÞ þy Z a Z a 2 1 2 2 p3 ðx uÞ du ¼ 2qyð3ðx uÞ y 2 Þ 3p ðx uÞ þ y 2 du a a h i 2qy 2xþ2a 2x2a 1 1 xþa 1 1 xa ðxaÞ tan tan þ 2 2 y y y y 3p ðxþaÞ2 þy 2 þy Z
a
Z
a
2
Although these DM micro-stresses actually exist only at discrete points and directions in the domain, we will use these results to make continuous contour and x–y plots over the domain under study. A similar statement would also apply for the finite element micro-forces to be developed next. In order to generate a similar model for finite element simulation, a 2D-hexagonal Bravais lattice structure was generated using a MATLAB Material Generator Code. This model had 1296 circular particles and 3709 micro-frame elements as shown in Fig. 6(c). Chosen model parameters include: k = 0.58 MPa, l = 0.38 MPa, w1 = w2 = 4 mm, h0 = 1 mm, and particle size D = 10 mm. For the integrated Flamant problem, three central particles on the boundary had prescribed vertical compressive loading and zero horizontal displacement. Particles on the bottom layer were supported with a very stiff vertical spring foundation (compared with the asphalt binder stiffness). Boundary particles on the vertical sides were unloaded and not constrained. To avoid boundary effects present in the FEM model, only the central portion of the domain (indicated by box in Fig. 6(c)) was used to compare with the doublet mechanics results. Micro-structural finite element simulation was then conducted on this model, and the elastic nor-
ð17Þ
nite element results with the corresponding doublet mechanics predictions, the DM microstresses from Eq. (17) were calculated at the mid-point of each element. It should be noted that the directions of these micro-stresses coincide with the element axial forces. Fig. 7 shows comparisons between DM micro-stress contours and FEM micro-force distributions for this problem. Plus and minus signs indicate regions of tensile and compressive micro-stresses. It is observed that the DM micro-stress and FEM micro-force contours are quite similar. Comparable tensile zones exist for each micro-stress–force distribution, thus indicating possible regions of tensile or mode I fracture behavior. Tensile zones for the P1 and P2 distributions are located adjacent to the surface loading, and these have been observed regions of surface or top–down cracking behavior. It is also evident that a significant zone of horizontal tensile micro-stress (P3), is located directly below the loading. According to each theory, the maximum value of this tensile field appears to be located at a somewhat different location below the loading surface. Clearly this location would be dependent on the fact that the DM solution is for a semi-infinite half space, while the FEM model has used particular dimensions and boundary conditions for the asphalt domain. Since the continuum elasticity case predicts only a compres-
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651
Fig. 7. Comparison of DM analysis and FEM results for integrated flamant problem.
sive stress field, these tensile stress regions are attributable to the micro-mechanical modeling through the inter-particle mechanics based on either doublet mechanics or finite element simulation. Of course material failure could also occur via shearing or mode II fracture at other locations in the model.
in any standard elasticity text and is given in polar coordinates as 2 4 2 rr ¼ r20 1 Rr2 þ r20 1 þ 3Rr4 4Rr2 cos 2h 2 4 rh ¼ r20 1 þ Rr2 r20 1 þ 3Rr4 cos 2h 4 2 ð18Þ srh ¼ r20 1 3Rr4 þ 2Rr2 sin 2h
4.2. Void under uniform compression
where R is the radius of the hole and r and h are the usual polar coordinates. To obtain the DM micro-stresses from the continuum field using transformation relation (7), the polar coordinate stresses need to be first transformed to Cartesian components. The micro-stress directions for the problem (hexagonal Bravais lattice) are shown in Fig. 8(a), and thus the transformation matrix Q can be expressed as
The existence of voids in asphalt pavements resulting from improper compaction or entrapment of spurious material is another important issue related to roadway failure. Voids can raise the local stress field and produce fatigue or fracture failures emanating from the void boundary. We wish to consider the problem of a circular void in an asphalt material under uniform farfield compression loading as shown in Fig. 8(a). The primary goal of this example is to investigate the elevation of micro-stresses around the void, and to look for zones of possible tensile behavior. The continuum elastic stress distribution around a circular hole under uniform far-field compression r0 as shown in Fig. 8(a) may be found
2 6 ½Q ¼ 4
cos2 c
cos2 c
sin2 c
sin2 c
cos c sin c cos c sin c
1
3
7 05
ð19Þ
0
where the doublet structure angle c = 60° for this case. Applying these appropriate transformations, the DM micro-stresses become
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r0 p1 ¼ 3
"
pffiffiffi 1 2 3 sin 4h 1 þ cos 2h þ sin 2h þ 2 2 ! 2
pffiffiffi 3 sin 2hcos2 h
þ 5 cos 2h 4cos2 h cos 2h þ sin2 2h R2 pffiffiffi pffiffiffi pffiffiffi þ 3 sin 2h 3 sin 4h 2 3cos2 h sin 2h 2 r pffiffiffi 3 3 3 sin 4h þ 3cos2 2h sin2 2h þ 2 2 ! # pffiffiffi 2 R4 þ3 3cos h sin 2h 4 r " pffiffiffi r0 1 2 3 2 sin 4h p2 ¼ 1 þ cos 2h þ sin 2h 2 2 3 ! pffiffiffi 2 þ 3 sin 2hcos h þ 5 cos 2h 4cos2 h cos 2h þ sin2 2h R2 pffiffiffi pffiffiffi pffiffiffi 3 sin 2h þ 3 sin 4h þ 2 3cos2 h sin 2h 2 r pffiffiffi 3 3 3 sin 4h þ 3cos2 2h sin2 2h 2 2 ! # pffiffiffi 2 R4 3 3cos h sin 2h 4 r r0 p3 ¼ cos2 2h þ ½4cos2 hð1 2 cos 2hÞ þ 2 cos 2h 3 R2 R4 2 2 ð1 cos 2hÞ 2 þ ð6cos 2h 3sin 2hÞ 4 r r ð20Þ The corresponding finite element model of this problem was again generated using our MATLAB code and the result is shown in Fig. 8(b). This model has the required regular 2D hexagonal Bravais lattice structure with 662 circular particles and 1880 micro-frame elements. The circular void was created in the central area of the model by removing a single particle from the regular packing pattern. Model boundary conditions included uniform compression loading on the top layer while the bottom particles were supported with a
Fig. 8. Void in granular asphalt materials under uniform compression loading. (a) Void problem and (b) computation model.
very stiff vertical spring foundation. The horizontal x-displacements of the top and bottom layers were fixed. Boundary particles on the vertical sides were again unloaded and unconstrained. The overall sample dimensions were chosen to be more than 10 times the size of the void diameter. As in the previous problem to reduce boundary condition effects, only the central subregion (indicated by box in Fig. 8(b)) was used for comparisons with the doublet mechanics results. Finite element simulation was then conducted on the model, and the axial element force distributions were obtained. As before, the corresponding doublet micro-stresses from Eq. (20) were calcu-
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653
Fig. 9. Comparison of DM analysis and FEM results.
Fig. 10. Micro-stress concentration. (a) p1 micro-stress, (b) p2 micro-stress and (c) p3 micro-stress.
lated at the mid-point of each element. Fig. 9 illustrates comparisons between the corresponding DM and FEM distributions for the void example. Again, plus and minus signs indicate regions of tensile and compressive microstresses. As in the previous example, results from each model for the void case are very similar. Results for the p1 and p2 micro-stresses indicate a general compression distribution in the domain except for particular portions on the boundary of the void. The horizontal micro-stress p3 is found to be primarily
tensile in the region except on the boundary of the void. Fig. 10 shows each of the DM micro-stresses as a function of the angular coordinate both on the void surface and in the far field (r ! 1). It can be observed that the micro-stresses p1 and p2 have small tensile zones on the void boundary and totally compressive behavior for the far field. The local maximum micro-stresses on the void surface are elevated when compared with far field values. A second void example was generated by rotating the previous lattice through 90° (CCW). This
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r0 p1 ¼ 3
"
pffiffiffi 1 2 3 1 þ cos 2h þ sin 2h þ sin 4h 2 2 ! 2
pffiffiffi þ 3 sin 2hsin2 h
pffiffiffi þ 3 cos 2h þ 4sin2 h cos 2h þ 2 3sin2 h sin 2h þ sin2 2h þ
R2 pffiffiffi pffiffiffi 3 sin 2h 3 sin 4h 2 r
pffiffiffi þ 3 cos 2h 6sin2 h cos 2h 3 3sin2 h sin 2h ! # pffiffiffi 3 2 3 3 R4 sin 2h þ sin 4h 4 2 r 2 " pffiffiffi r0 1 2 3 2 p2 ¼ 1 þ cos 2h þ sin 2h sin 4h 2 3 2 ! pffiffiffi 2 3 sin 2hsin h pffiffiffi þ 3 cos 2h þ 4sin2 h cos 2h 2 3sin2 h sin 2h þsin2 2h
R2 pffiffiffi pffiffiffi 3 sin 2h þ 3 sin 4h 2 r
pffiffiffi þ 3 cos 2h 6sin2 h cos 2h þ 3 3sin2 h sin 2h
Fig. 11. Rotated hexagonal Bravais lattice structure for void problem. (a) Void problem and (b) computation model.
case and the new doublet axes are shown in Fig. 11(a). Notice that micro-stress p3 is now along the compression loading direction. The new transformation matrix for this case becomes 2 6 ½Q ¼ 4
sin2 c cos2 c
sin2 c cos2 c
sin c cos c
sin c cos c
3 0 7 15
ð21Þ
0
and the doublet structure angle is again c = 60°. Applying the transformations, the DM microstresses now become
! # pffiffiffi 3 2 3 3 R4 sin 2h sin 4h 4 2 r 2 r0 ðcos2 2h þ 2Þ þ ð8sin2 h cos 2h þ 2sin2 2hÞ p3 ¼ 3 R2 R4 2 2 þ ð6cos 2h 3sin 2hÞ ð22Þ r2 r4
The finite element model for this second void problem was generated in analogous fashion and is shown in Fig. 11(b). This model had the rotated hexagonal Bravais lattice structure with 594 circular particles and 1680 micro-frame elements. Boundary conditions for the FEM modeling were identical to the previous case, and boundary effects
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655
Fig. 12. Comparison of DM and FEM analysis for rotated lattice structure.
Fig. 13. Micro-stresses concentration. (a) p1 micro-stress, (b) p2 micro-stress and (c) p3 micro-stress.
were minimized by only considering the domain interior to the box shown in Fig. 11(b). Fig. 12 shows the comparisons between DM and FEM distributions for the rotated void example. As in the previous cases, results from each model are similar. It is observed from the p1 and/or p2 contours that each model generates tensile micro-stress domains under the uniform compressional loading. These tension domains are caused by the presence of the void. If the void is removed, the tension zones dis-
appear and the p1 and p2 fields are totally compressive. It appears that the zone of maximum tensile micro-stress p1 or p2 lies approximately along the direction of the corresponding doublet axis. Since the micro-stress p3 lies along the compressional loading direction, it has only compression action within model domain. Along a horizontal line (h = 0), p3 increases significantly as one approaches the void. As shown in Fig. 13, the micro-stress distributions were again investigated as a function of
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the angular coordinate on the void boundary and at far field. It can be observed that the micro-stresses p1 and p2 have tensile zones on the void boundary while p3 exhibits totally compressive behavior on the void and at far-field. For the void problem shown in Fig. 8, the p3 micro-stress distribution around the void shown in Fig. 10(c) indicates a range of r0/3 6 p3 6 r0, while the far-field variation is given by 0 6 p3 6 r0/ 3. Thus in the horizontal direction (h = 0), the local value of this micro-stress is three times higher than the far-field value. Likewise, for the void problem of Fig. 11, the ranges of p3 are shown in Fig. 13(c) and would be 3r0 6 p3 6 0 around the void and r0 6 p3 6 2r0/3 for the far-field. For this case, the stress concentration factor for the vertical p3 micro-stress is three which matches with classical continuum elasticity result.
4.3. Internal line loading (second order DM model) We now wish to conduct a final comparison study between the FEM model and a second order doublet mechanics solution that will include a length scale. Existing DM solutions that include the second order approximation are very limited, and only the Kelvin problem has been given by Ferrari et al. (1997). We therefore will use this basic problem as our final comparative example. The classic Kelvin problem shown in Fig. 14(a) will be superimposed (integrated) to generate the case of an internal line loading shown in case (b). The two-dimensional continuum elastic displacement field for KelvinÕs problem is given by xy v1 ¼ c1 2 ; r
x2 v2 ¼ c1 2 log r þ 2 r
ð23Þ
Fig. 14. Internal compression example. (a) Kelvin problem, (b) integral Kelvin problem, (c) computation model and (d) maximum micro-stress locations shown in bold (solid-tension, dotted-compression).
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where r2 = x2 + y2, c1 ¼ 4P =9Ep, and E is the modulus of elasticity. The second order displacement field was formulated by Ferrari et al. (1997) 7 5 3 w1 ¼ c91 y 64 rx10 96 xr8 þ 32 xr6 þ rx4 8 6 4 2 c1 w2 ¼ 18 128 rx10 352 xr8 þ 320 xr6 100 xr4 þ r52 ð24Þ Thus from relation (10) with z = 0, the second order displacement field for the Kelvin problem can be expressed as u1 ¼ v 1 þ g2 w 1 ; u2 ¼ v 2 þ g2 w 2
ð25Þ
The micro-stresses for the second order case of the structure angle c = 60° can be developed from Eq. (12) with C0 = E, h P p1 ¼ g2 12x8 y þ 444x2 y 7 780x4 y 5 6 81pðx2 þ y 2 Þ pffiffiffi pffiffiffi pffiffiffi þ 52x6 y 3 504 3x5 y 4 þ 84 3x7 y 2 102 3xy 8 pffiffiffi pffiffiffi þ 588 3x3 y 6 2 3x9 16y 9 þ g 108xy 9 þ 288x3 y 7 þ 216x5 y 5 36x9 y pffiffiffi 9 3x11 þ 45y 11 þ 189x2 y 9 þ 306x4 y 7 pffiffiffi þ 234x6 y 5 þ 81x8 y 3 þ 9x10 y þ 306 3x5 y 6 pffiffiffi pffiffiffi pffiffiffi þ 189 3x3 y 8 þ 45 3xy 10 þ 234 3x7 y 4 i pffiffiffi þ 81 3x9 y 2 ð26Þ p2 ¼
P 81pðx2 þ y 2 Þ
h g2 12x8 y þ 444x2 y 7 780x4 y 5 6
pffiffiffi pffiffiffi pffiffiffi þ 52x6 y 3 þ 504 3x5 y 4 84 3x7 y 2 þ 102 3xy 8 pffiffiffi pffiffiffi 588 3x3 y 6 þ 2 3x9 16y 9 þ g 108xy 9 288x3 y 7 216x5 y 5 þ 36x9 y pffiffiffi þ 9 3x11 45y 11 189x2 y 9 306x4 y 7 pffiffiffi 234x6 y 5 81x8 y 3 9x10 y þ 306 3x5 y 6 pffiffiffi pffiffiffi pffiffiffi þ 189 3x3 y 8 þ 45 3xy 10 þ 234 3x7 y 4 i pffiffiffi þ 81 3x9 y 2 ð27Þ
p3 ¼
4Py 81pðx2
þ
y 2 Þ6
657
2 8 g 3x 152x6 y 2 þ 390x4 y 4
96x y y 8 þ g 9x9 þ 54x5 y 4 þ72x3 y 6 þ 27xy 8 þ 9x10 27x8 y 2 þ18x4 y 6 18x6 y 4 þ 27x2 y 8 þ 9y 10 2 6
ð28Þ
where the micro-stress directions are defined in Fig. 14(c). We now wish to develop the solution for a uniformly distributed line load q over the range a 6 x 6 a (at y = 0) as shown in Fig. 14(b). This can be accomplished as before by integrating the micro-stresses at location (x, y) over the coordinate origin variable u as P1 ¼
Z
a
p1 ðx uÞ du
a
¼
q
324p ðx þ aÞ2 þ y 2
8 2 8 5 g 12y þ 6ðx þ aÞ
þ64ðx þ aÞy 7 400ðx þ aÞ3 y 5 þ 64ðx þ aÞ5 y 3 246ðx þ aÞ2 y 6 þ 390ðx þ aÞ4 y 4 114ðx þ aÞ6 y 2 þ16ðx þ aÞ7 y þ g 144ðx þ aÞ2 y 7 þ144ðx þ aÞ6 y 3 þ 72ðx þ aÞ8 y 72y 9 þ 108y 10 þ 108ðx þ aÞ8 y 2 þ 432ðx þ aÞ6 y 4 þ 648ðx þ aÞ4 y 6 þ 432ðx þ aÞ2 y 8 þ 72ðx þ aÞ9 y þ 288ðx þ aÞ7 y 3 þ 432ðx þ aÞ5 y 5 þ 288ðx þ aÞ3 y 7 x þ a 10 þ 72ðx þ aÞy 9 þ 108tan1 y y 27 ln ðx þ aÞ2 þ y 2 ðx þ aÞ10 xþa 27 ln ðx þ aÞ2 þ y 2 y 10 þ 108tan1 y 10 2 ðx þ aÞ 135 ln ðx þ aÞ þ y 2 ðx þ aÞ2 y 8 xþa þ 540tan1 ðx þ aÞ2 y 8 y 135 ln ðx þ aÞ2 þ y 2 ðx þ aÞ8 y 2 1 x þ a þ 1080tan ðx þ aÞ6 y 4 y
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1 x þ a ðx þ aÞ8 y 2 þ 540tan y 270 ln ðx þ aÞ2 þ y 2 ðx þ aÞ6 y 4 270 ln ðx þ aÞ2 þ y 2 ðx þ aÞ4 y 6 xþa ðx þ aÞ4 y 6 þ1080tan1 y q 8 2 8 5 g 12y þ 6ðx aÞ 2 2 324p ðx aÞ þ y þ64ðx aÞy 7 400ðx aÞ3 y 5 þ 64ðx aÞ5 y 3 246ðx aÞ2 y 6 þ 390ðx aÞ4 y 4 114ðx aÞ6 y 2 þ16ðx aÞ7 y þ g 144ðx aÞ2 y 7 þ144ðx aÞ6 y 3 þ 72ðx aÞ8 y 72y 9 þ 108y 10 þ108ðx aÞ8 y 2 þ 432ðx aÞ6 y 4 þ 648ðx aÞ4 y 6 þ 432ðx aÞ2 y 8 þ 72ðx aÞ9 y þ 288ðx aÞ7 y 3 þ 432ðx aÞ5 y 5 þ 288ðx aÞ3 y 7 þ 72ðx aÞy 9 1 x a y 10 27 ln ðx aÞ2 þ y 2 þ 108tan y ðx aÞ10 27 ln ðx aÞ2 þ y 2 y 10 xa ðx aÞ10 135 ln ðx aÞ2 þ 108tan1 y xa ðx aÞ2 y 8 þy 2 ðx aÞ2 y 8 þ 540tan1 y 135 ln ðx aÞ2 þ y 2 ðx aÞ8 y 2 xa ðx aÞ6 y 4 þ 1080tan1 y xa ðx aÞ8 y 2 þ 540tan1 y 270 ln ðx aÞ2 þ y 2 ðx aÞ6 y 4 270 ln ðx aÞ2 þ y 2 ðx aÞ4 y 6 4 6 1 x a ðx aÞ y ð29Þ þ1080tan y
P3 ¼
Z
a
p3 ðx uÞ du a
h 2qy 5 2 2 5 g 58ðx þ aÞ y 2 81p ðx þ aÞ þ y 2 3 7 70ðx þ aÞ y 4 2ðx þ aÞy 6 2ðx þ aÞ 2 6 þ g 9y 8 þ 18ðx þ aÞ y 6 18ðx þ aÞ y 2 8 9 7 9ðx þ aÞ þ 18ðx þ aÞ þ 72ðx þ aÞ y 2
¼
5
3
i
5
3
i
þ108ðx þ aÞ y 4 þ 72ðx þ aÞ y 6 þ 18ðx þ aÞy 8 h 2qy 5 2 2 þ 5 g 58ðx aÞ y 2 81p ðx aÞ þ y 2 3 7 70ðx aÞ y 4 2ðx aÞy 6 2ðx aÞ 2 6 þ g 9y 8 þ 18ðx aÞ y 6 18ðx aÞ y 2 8 9 7 9ðx aÞ þ 18ðx aÞ þ 72ðx aÞ y 2 þ108ðx aÞ y 4 þ 72ðx aÞ y 6 þ 18ðx aÞy 8
ð30Þ Note that micro-stress P2 can be simply obtained from P1 by replacing x with x. To compare a numerical simulation with these DM results, a similar finite element model was generated with a 2D-hexagonal Bravais lattice structure shown in Fig. 14(c). The model had 946 particles and 2715 micro-frame elements, and neighboring particles all had the same separation (diameter plus cement spacing) of 1.1 cm. To simulate the integrated Kelvin problem, four central particles on the mid-line had vertical loading and zero horizontal displacement. Particles on the top and bottom layers were connected to a stiff vertical spring foundation. Boundary particles on the vertical sides were left unconstrained. To avoid boundary effects, a central portion of the model (indicated by box in Fig. 14(c)) was again used. The box had a horizontal dimension of 22 cm and a vertical height of 19 cm. The loading line size 2a = 3.3 cm and this gives a dimensionless length scale (particle separation/a) of 0.67. Fig. 15 shows the comparisons between DM and FEM micro-stress distributions for the integrated Kelvin problem. For the DM analysis, the length
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659
Fig. 15. Comparison of DM analysis and FEM results for integrated Kelvin problem.
scale is taken to be the same as the particle spacing used in the FEM model; i.e. g = 1.1 cm, and the loading-line size was selected to match the FEM model. The thick line contour indicates the separation between compression and tension zones in the DM distribution. Again, results from each model show very similar distributions. P1 and P2 contours are symmetric to each other about the central vertical line, and their compression and tension domains have similar distribution patterns. Maximum values of the P1 and P2 micro-stresses occur at the edges of the loading line as shown in Fig. 14(d), with maximum tensile and compressive values occurring at the doublets indicated by the solid and dotted bold lines. Contours for P3 indicate symmetry about the vertical centerline and skew-symmetry about the horizontal. Results indicate a significant P3 tension zone below the load line and a corresponding compression zone above. It was found that the maximum tensile and compressive doublets were on the loading centerline as indicated in Fig. 14(d). These DM results match very closely with corresponding predictions from the FEM model. Finally it was desired to investigate the effect of model length scale on the maximum DM micro-
Fig. 16. Effect of the length scale on maximum micro-stress (unit compression loading).
stresses. The model geometry and the loading line size were kept fixed as before, and length scale was varied from 0 to a. The loading magnitude q was taken as unity, and this resulted in a constant total loading for each analysis. Fig. 16 shows the trend of the dimensionless maximum micro-stresses versus the dimensionless length scale g/a. It is
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observed from the figure that the maximum microstresses show sizeable increase with length scale, thus indicating significant dependency on this micro-structural parameter. As the length scale decreases, these micro-stresses reduce to the firstorder elastic solution (without length scale). For a unit compression loading, the maximum value of P1 (or P2) for the elastic case was calculated approximately as 0.5, while the maximum of P3 was about 0.14. An FEM investigation on the effect of length scale was also conducted on the computation model shown in Fig. 14(c). Six simulations were conducted on the same model geometry with a loading line size 2a changed to impose the uniform load to 3, 4, 5, 6, 7, 9 particles, respectively. For a given simulation each particle had the same loading force, and this was adjusted to keep the total loading identical for all simulations. FEM results are shown in Fig. 16, and these predictions match reasonably well with those from doublet mechanics. Thus the effects of length scale within each model appear to provide similar behaviors.
5. Summary and conclusions A comparative study has been given between two micro-mechanical models that have been developed to simulate the behavior of cemented particulate materials. Specific applications of these models are made to asphalt concrete. The first model came from a discrete analytical approach called doublet mechanics that represents microstructural solids as an array of particles. This scheme developed analytical expressions for the micro-deformation and stress fields between particle pairs that define a doublet. The second model was based on a numerical finite element technique that establishes a network of elements between neighboring cemented particles. Elements within this model were specially developed in order to simulate the local load transfer between particles. While the two modeling schemes come from very different origins, they have a common similarity of constructing a theory based on local interaction between neighboring particle pairs. Both theories allow for normal and tangential micro-forces be-
tween adjacent particles. However, the local moment interactions are not the same in each theory. Doublet mechanics incorporates an inplane moment load transfer as shown in Fig. 1, while the finite element scheme used a more common out-of-plane moment as illustrated in Fig. 3b. In order to pursue the effects of such similarities and differences, three example problems were investigated using each approach. Comparisons were made between the DM micro-stresses with the FEM element micro-forces. In spite of the differences, model predictions compared favorably for each of the examples. Results also indicated significant micro-structural effects that differ from continuum elasticity theory. For example, results from the first problem dealing with compressive loading of a half-space, indicated sizeable zones of tensile micro-stress. This situation is completely different from continuum elasticity and could lead to material failure through delamination between particles. The second example problem of an infinite medium under uniform compression with a stress-free hole also showed tensile zones that differ from the continuum solution field. The final example investigated an integrated Kelvin problem for the case where the DM solution yields a length scale dependent solution. Micro-structural dependent tensile zones were again found and compared well with the corresponding FEM solution. It was found from both DM and FEM that maximum micro-stresses increased with increasing relative length scale. In general, these comparisons tend to provide additional verification for each theory. The good comparisons between doublet mechanics and finite element modeling would tend to indicate that the extensional response between particle pairs must be the dominant micromechanical mechanism in the problems considered. It would thus be interesting to investigate additional cases where the shear and torsional micro-stresses are present. Unfortunately such DM solutions have not been found in the literature, and it is expected that development of such solutions might be a formidable task. It is anticipated that for these cases, the comparisons would not be as good. The doublet mechanics model offers analytical solutions to simulate micro-mechanical behavior
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of particular mechanics problems. However, for more realistic problems involving irregular particle sizes and packing geometries, it is unlikely that such analytical DM solutions can be found. Of course the finite element model can be used for such problems, and would therefore be the recommended approach. The presented comparisons have been limited to only the elastic response. However, our finite element model has incorporated damage mechanics concepts in order to simulate inelastic and failure of cemented particulate materials (Sadd et al., 2004a,b). Likewise, doublet mechanics theory has also been extended to describe inelastic behavior, and some specific failure criteria have been developed (Ferrari and Granik, 1995). Thus comparisons of the inelastic predictions between the two theories appear to be workable and would be an interesting future study. Since most real particulate materials are threedimensional in nature, two-dimensional modeling is always subject to concern. Two-dimensional limitations on material micro-structure (fabric) and on out-of-plane degrees of freedom will normally not capture all of the micro-mechanics in the particulate system. The usual simple uniform scaling through the thickness can only approximate the actual behavior. In regard to threedimensional modeling, the additional degrees of freedom will add considerable complexity to both the doublet mechanics and micro-mechanical finite element models. It would be expected that constructing three-dimensional DM solutions would be very difficult. We have recently begun research on a three-dimensional micro-finite element modeling scheme. Within this new model each particle is allowed to have 6 degrees of freedom, and this generates a two-noded, micro-finite element with 12 degrees of freedom. We hope to report three-dimensional simulations in the near future.
Acknowledgment The authors would like to acknowledge support from the Transportation Center at the University of Rhode Island under Grants 01-64 and 02-86.
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