International Journal of Engineeringweb.ics.purdue.edu/~gonza226/papers/AgarwalGonzalez2018.pdf · 2018. 12. 8. · 28 A. Agarwal, M. Gonzalez / International Journal of Engineering

International Journal of Engineering Science 133 (2018) 26–46

Contents lists available at ScienceDirect

International Journal of Engineering Science

journal homepage: www.elsevier.com/locate/ijengsci

Contact radius and curvature corrections to the nonlocal

contact formulation accounting for multi-particle interactions

in elastic confined granular systems

Ankit Agarwal a , Marcial Gonzalez a , b , ∗

a School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA b Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA

a r t i c l e i n f o

Article history:

Received 13 July 2018

Revised 14 August 2018

Accepted 15 August 2018

Available online 3 September 2018

Keywords:

Contact mechanics

Granular systems

Nonlocal contact formulation

Multi-particle interactions

Hertz theory

Powder compaction

a b s t r a c t

We present contact radius and curvature corrections to the nonlocal contact formulation

that account for multi-particle interactions in elastic confined granular systems. The nonlo-

cal contact formulation removes the classical assumption of independent contacts by tak-

ing into account the interplay of deformations due to multiple contact forces acting on a

single particle. The contact radius correction considers the components of these deforma-

tions that contribute to the inter-particle contact area. The curvature correction improves

the description of contacting surface profiles by including higher order terms in their Tay-

lor series expansions. To validate the corrected formulation, we restrict attention to rubber

spheres under different loading conditions, in the absence of gravitational forces, adhesion

or friction. Specifically, we show that the predictions of contact force and radius are in re-

markable agreement with finite-element simulations and experimental observations up to

levels of deformation at which contact impingement occurs, which was not possible with

the original elastic nonlocal contact formulation. Convergence of the curvature corrected

formulation is observed at a four-term correction.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

The extensive applications of powder compaction, especially in manufacturing processes of critical industries like phar-

maceutics, ceramic, energy, food, and metallurgy, make it a subject of intense research in the scientific community. Devel-

opment of predictive and computationally efficient models that could accurately describe the behavior of granular media

during compaction would directly impact optimality in manufacturing, waste reduction, and price and quality of the end

product.

Macroscopic behavior of confined granular systems has been conventionally described by continuum-based models , which

consider granular media as a continuous system and hence have a minimal emphasis on the behavior at particle scale.

Many of these models were originally developed for analyzing the behavior of geological materials, such as Drucker and

Prager (1952) , Cam-Clay plasticity and Cap plasticity models. More recently, the Drucker-Prager Cap (DPC) plasticity model

( DiMaggio & Sandler, 1971 ), where a cap yield surface is added to the Drucker and Prager model to allow for material

∗ Corresponding author. E-mail addresses: [email protected] (A. Agarwal), [email protected] (M. Gonzalez).

URL: http://www.marcialgonzalez.net (M. Gonzalez)

https://doi.org/10.1016/j.ijengsci.2018.08.006

0020-7225/© 2018 Elsevier Ltd. All rights reserved.

https://doi.org/10.1016/j.ijengsci.2018.08.006http://www.ScienceDirect.comhttp://www.elsevier.com/locate/ijengscihttp://crossmark.crossref.org/dialog/?doi=10.1016/j.ijengsci.2018.08.006&domain=pdfmailto:[email protected]:[email protected]://www.marcialgonzalez.nethttps://doi.org/10.1016/j.ijengsci.2018.08.006

A. Agarwal, M. Gonzalez / International Journal of Engineering Science 133 (2018) 26–46 27

hardening and dilatancy control during inelastic deformation, has been used for analysis of metal, ceramic and pharmaceu-

tical powder compaction. Although requiring an elaborate mechanical testing procedure for calibration of model parameters

( Cunningham, Sinka, & Zavaliangos, 2004 ), the DPC model is widely used due to its adaptability to finite element method

( Sinka, Cunningham, & Zavaliangos, 2004 ). However, the accuracy of model response, especially during the decompression

(unloading) phase, relies heavily on the design of calibration experiments ( Sinha, Curtis, Hancock, & Wassgren, 2010 ) and

proper elastic constitutive modeling ( Han et al., 2008 ). In order to incorporate microstructural properties of the granular

system into its global behavior, discrete models have been proposed, where contact behavior of individual particles is taken

into account. Numerical methods in this category, such as dynamic discrete element methods ( Cundall & Strack, 1979; Zhu,

Zhou, Yang, & Yu, 2008 ) and quasi-static particle mechanics approaches ( Gonzalez & Cuitiño, 2016; Gonzalez, Poorsolhjouy,

Thomas, Liu, & Balakrishnan, 2018; Yohannes et al., 2016 ), are used in combination with a suitable contact formulation to

predict the macroscopic behavior of compacted granular systems and, thus, predictability is heavily dependent on the con-

tact law involved. The Hertz (1882) contact law for linear-elastic materials and similarity solution by Storåkers, Biwa, and

Larsson (1997) for viscous-plastic power law hardening materials are fairly predictable at small deformations and low rela-

tive densities of powder compacts. However, due to the occurrence of contact interactions at higher deformations, as pointed

out by Mesarovic and Fleck (20 0 0) in their study of elasto-plastic spheres, their predictions become increasingly deviant due

to the assumption of independent contacts. This was partially overcome by introducing a local relative density parameter

in contact laws curve-fitted to finite element simulations of small three-dimensional packings ( Harthong, Jérier, Dorémus,

Imbault, & Donzé, 2009 ). Finally, a systematic and mechanistic connection between macroscopic and particle level behaviors,

using continuum and discrete models respectively, was recently proposed ( Poorsolhjouy & Gonzalez, 2018 ) to capture the

anisotropic evolution of die-compacted systems.

Several efforts towards experimental characterization of confined granular systems have also been made to understand

the deformation behavior at granular scale and to provide an efficient validation tool for the analytical contact formulations.

Of particular interest is the mechanical response of single particles under confined conditions, most commonly studied using

uniaxial compression experiments ( Liu, Williams, & Briscoe, 1998; Lu, Tung, Hung, Shiau, & Hwang, 2001; Shima, Tatara, Iio,

Shu, & Lucero, 1993; Tatara, Shima, & Lucero, 1991; Topuz & Okay, 2009; Zhang, Kristiansen, & Liu, 2007 ). Recently, an

apparatus has been developed for triaxial testing of single particles ( Jonsson, Gråsjö, Nordström, Johansson, & Frenning,

2015 ), providing a more realistic insight into the behavior of individual particles during powder compaction.

For elastic confined granular systems, relaxing the underlying assumptions of the Hertz contact theory that limit its

applicability to small deformations could be the key to achieving predictability at moderate to large deformations. Signif-

icant effort s in this direction were made by Zhupanska (2011) , who relaxed the small-strain Hertz assumption of consid-

ering contacting surfaces as elastic half spaces by proposing an analytical solution to the boundary value problem of an

elastic sphere subject to contact stresses on a finite region of its surface and supported at its center. The results showed

that the Hertz pressure distribution remained accurate for relatively large contact areas. Recently, Argatov, Kachanov, and

Mishuris (2017) have explored the concept of far points in Hertz contact problems, emphasizing the limitations of the ”local

character” of Hertz predictions. Another major contribution in this regard is the nonlocal contact formulation for confined

granular systems by Gonzalez and Cuitiño (2012) , which provides an accurate and mechanistic description of the force-

deformation behavior at contacts of a linear-elastic spherical particle subject to multiple contact forces, a typical config-

uration in particulate systems compressed to high relative densities. It follows the work of Tatara (1989) and relaxes the

classical contact mechanics assumption of independent contacts by invoking the principle of superposition to express the

deformation at a particular contact as a sum of local (i.e., Hertzian) deformation and nonlocal deformations generated by

other contact forces acting on the same particle. The nonlocal contact formulation has recently been employed success-

fully to study the die-compaction of large frictionless noncohesive granular systems comprising weightless elastic spherical

particles ( Gonzalez & Cuitiño, 2016 ).

A complete description of the inter-particle contact behavior in confined granular systems includes determination of both

contact force and area with respect to particle deformation. While critical macroscopic quantities like compaction pressure

and the reaction from die walls are directly related to inter-particle contact forces, the prediction of contact area is needed to

estimate strength formation in the compacted solid ( Gonzalez, 2018 ). In addition, the evolution of contact area is associated

with contact impingement, i.e., with the merger of neighboring contacts. Since the assumption of circular contacts no longer

remains valid after contact impingement, the predictions of a contact formulation may not be representative of real contact

behavior beyond the occurrence of this phenomenon, making an accurate determination of contact areas ever so important.

In the context of the nonlocal contact formulation ( Gonzalez & Cuitiño, 2012 ), nonlocal mesoscopic deformations are

derived from the Boussinesq solution ( Johnson, 1987; Timoshenko & Goodier, 1970 ) of an elastic half-space under a con-

centrated force. The components of these deformations normal to the contact surface constitute the nonlocal contribution

to the contact displacement, for which a closed-form solution has been obtained ( Gonzalez & Cuitiño, 2012 ). However, the

derivation of an analytical solution for nonlocal components radial to the contact center, that contribute to the evolution of

contact radius, remains an open problem. Therefore, part of the work presented in this paper is concerned with the devel-

opment of an analytical framework for predicting nonlocal effects in the evolution of inter-particle contact area. The analysis

presented is in the spirit of Tatara ’s (1991) work on expanded contact radius during uniaxial compression of rubber spheres.

An important aspect of this nonlocal contact formulation is that it is a direct extension of the classical Hertz contact

theory. When nonlocal effects are neglected, the formulation reduces to Hertz theory. Therefore, any correction introduced

in the Hertz solution should, in turn, improve the accuracy of the formulation. The second analysis presented in this paper

28 A. Agarwal, M. Gonzalez / International Journal of Engineering Science 133 (2018) 26–46

r

z

p(r)

r

z

θ1

θ2

θiθNP

P1

P2

Pi

PNP

R

2a 2a

P1

P2

Pi

PNP

p (r )

Fig. 1. Depiction of deformation of an elastic sphere under the action of a general configuration of contact forces, due to a distributed surface pressure p

acting on its surface. The figure on the left denotes the configuration before deformation (reference configuration) while the figure on the right denotes

the deformed configuration.

is concerned with the improvement of the formulation using contact pressures of higher accuracy obtained by correcting

the description of the profiles of contacting surfaces through higher order terms in the Taylor series expansion of the profile

functions. This methodology, termed curvature correction, is similar to the one reported by Cattaneo (1947) for solids of

revolution and Luo (1958) for general solids. Closed form solutions of contact force and radius in terms of displacement are

obtained for two-, three- and four-term curvature corrections.

Finally, a validation of the two proposed corrections is performed by comparison of the analytical predictions of contact

force and radius versus deformation for compression of rubber spheres under various loading configurations with finite

element simulations and experimental measurements.

The paper is organized as follows. The deformation of an elastic sphere in a confined granular system under the action of

multiple contact forces is discussed in Section 2 . The analytical formulation for predicting nonlocal effects in the evolution of

contact radius is then presented in Section 3 . In Section 4 , a curvature corrected nonlocal contact formulation is discussed.

Section 5 presents a numerical and experimental validation of contact radius and curvature corrections. Finally, a summary

and concluding remarks are presented in Section 6 .

2. Deformation of an elastic sphere in a confined granular system under the action of a distributed surface pressure

We consider the deformation of a linear elastic sphere with radius R , Young’s modulus E and Poisson’s ration ν , un-der the action of a distributed pressure and a general loading configuration of multiple contact forces. By approximating

the deformation with an axially symmetric field, a cylindrical coordinate system defined in Fig. 1 is adopted. The reference

coordinates are denoted by X : ( r, z ), while the deformed coordinates are denoted by x : ( r ′ , z ′ ). The general loading config-uration is represented by ( P i , θ i ), where P i are the forces acting on the sphere’s surface and θ i are their angular distanceswith respect to z or z ′ axis.

A spherical cap of base radius a deforms to a flat circular surface of radius a ′ under the action of an ellipsoidally dis-tributed pressure p proposed by Hertz (1882) and given by

p(r) = p̄ m √

1 − r 2

a 2 (1)

where p̄ m = 3 P/ 2 πa 2 is the maximum value of the pressure and P is the effective contact force. The displacement of theelastic cap is represented by a deformation mapping ϕ(X) , given by

{r ′ z ′

}= ϕ( X ) =

⎧ ⎪ ⎨ ⎪ ⎩

r + u (r) + N P ∑ i =1

u P i (r)

R − √ R 2 − r 2 + w (r) −N P ∑ i =1

w P i (r)

⎫ ⎪ ⎬ ⎪ ⎭ (2)


Fig. 2. Depiction of nonlocal effects on the contact area between two dissimilar particles being compressed by a general loading configuration. The figure

shows a schematic of the loading configuration, and the contact configuration without (long dashed curve) and with (solid curve) nonlocal correction. The

contact with effective nonlocal correction is depicted by a short dashed curve, with radius a ′ i j

.

where z = R −√

R 2 − r 2 is used, thus enabling the deformation mapping of surface points solely in terms of r . Quantitiesw ( r ) and u ( r ) are the vertical and radial displacements of the cap’s surface points due to local pressure p ( r ), which can be

approximated by means of Boussinesq solution ( Johnson, 1987; Timoshenko & Goodier, 1970 ), i.e., by

w (r) = 3 P (1 − ν2 )

8 a 3 E ( 2 a 2 − r 2 ) (3)

and, according to Tatara (1991) , by

u (r) = (1 + ν) P 4 πEr

⎡ ⎣ r √

2 R

√ 1 +

√ 1 −

(r

R

)2 − 2( 1 − 2 ν)

⎛ ⎝ 1 − 1 √

2

√ 1 −

√ 1 −

(r

R

)2 ⎞ ⎠ ⎤ ⎦ (4)

Quantities u P i (r) and w P i (r) are the nonlocal contributions to radial and vertical displacements induced by concentrated

forces P i acting on the surface of the sphere. As a valid approximation, Gonzalez and Cuitiño (2012) have represented w P i (r)

by the value of w P i at the contact center, i.e., by

w P i (r) � w P i (0) � (1 + ν) P i

4 πRE

[−2(1 − ν) − 2(1 − 2 ν) sin (θi / 2) + (7 − 8 ν) sin 2 (θi / 2)

sin (θi / 2)

](5)

Determination of nonlocal radial displacements u P i (r) is addressed in the next section and it is a major contribution of this

work. It is worth noting that if both local and nonlocal radial displacements are assumed negligible, the nonlocal contact

formulation proposed by Gonzalez and Cuitiño (2012) is recovered.

From the above particle deformation analysis, an important conclusion can be drawn. Although physical quantities, such

as pressure and contact radius, compatibility conditions and equilibrium are defined in the deformed configuration, it is

more convenient to work in the reference configuration. Physical quantities can be easily converted to spacial quantities

by using the deformation mapping ϕ(X) and push-forward operations. This understanding will be used in the analysis per-formed in subsequent sections.

3. Nonlocal effects in the evolution of inter-particle contact area

We now consider the evolution of contact surface between two elastic spherical particles i and j of radius R i and R j , and

material properties E i , ν i and E j , ν j respectively, being pressed together in a general configuration of particles simulating aconfined granular system ( Fig. 2 ). We propose that the particles deform to accommodate a flat contact surface of effective

radius a ′ i j , given by

a ′ i j = a i j + 1

2

[ P i j

m L i j (a i j )

+ ∑

h ∈N i ,h � = j

P ih

m NL jih

(x j , x i , x h ) +

∑ k ∈N j ,k � = i

P jk

m NL i jk

(x i , x j , x k )

] (6)


where a ij is the contact radius in the reference configuration, x i and N i are the position and neighbors of particle i , and P ij is the effective contact force between particles i and j . The factor of 1/2 signifies an average of the radial deformations at

the contact due to particles i and j and their respective neighbors.

Term m L i j

corresponds to the local contribution to radial deformation of the contact boundary. Using Eq. (4) for r = a i j , itcan be expressed as

1

m L i j

= ∑ l= i, j

1 + νl 4 πE l R l

⎡ ⎣ 1 √

2

√ 1 +

√ 1 −

(a i j

R l

)2 − 2(1 − 2 νl )

(R l a i j

)⎡ ⎣ 1 − 1 √ 2

√ 1 −

√ 1 −

(a i j

R l

)2 ⎤ ⎦ ⎤ ⎦ (7)

Term m NL jih

corresponds to the nonlocal contribution to the radial deformation of the contact boundary, given by

1

m NL jih

= 1 2 π

∫ 2 π0

d φ

m NL jih,Q

(x j , x i , x h , φ) (8)

where m NL jih,Q

corresponds to a particular point Q on the contact boundary, denoted by its angular position φ with respect

to the plane T given by the equation [(x i − x j ) × (x i − x h )

] · (x − x i ) = 0 (the plane shown in Fig. 2 ). Specifically, m NL jih,Q ismathematically represented by

1

m NL jih,Q

= 1 + νi 2 πE i R i

[ sin θ jih cos φ

(sin (θ jih / 2) − sin (β jih,Q / 2)

)(sin (θ jih / 2) sin (β jih,Q / 2) − 2 + 2 νi

)2 sin (θ jih / 2) sin (β jih,Q / 2)

+ √

1 − sin 2 θ jih cos 2 φ[

cos (β jih,Q / 2) − cos (θ jih / 2) 2

−(1 − 2 νi ) (

1 − sin (β jih,Q / 2) sin β jih,Q

− cos (θ jih / 2) 2 sin (θ jih / 2)(1 + sin (θ jih / 2))

)]] (9)

where β jih,Q = ̂ x i j,Q x i x h is the angle between point Q and position coordinates of particles i and h , given by

β jih,Q = cos −1 [

cos

∣∣∣θ jih − sin −1 (a i j R i )∣∣∣− (a i j

R i

)(1 − cos φ) sin θ jih

] (10)

and θ jih = ̂ x j x i x h is the angle between position coordinates of particles j, i and h . A detailed derivation of m NL jih,Q is presentedin Appendix A . It essentially entails calculation of the radial displacement of point Q on the boundary of the deforming

surface due to one of the forces P i depicted in Fig. 1 .

In Eq. (8) , m NL jih

is calculated by taking an average of the radial displacements due to nonlocal forces exerted by particles

h on particle i across the contact boundary between particles i and j . Such averaging is necessary due to the fact that the

nonlocal radial contribution of a particular force is asymmetric over the contact boundary due to dependence on variable

angle φ. By averaging over φ, the assumed symmetry in the deformation field is recovered, allowing the deformed contactsurface to be approximated by a circular surface, although resulting in a shift in the position of its center which will be

neglected in this work (see insert in Fig. 2 and Appendix A , Eq. (A.10) ). It was found that the integral in Eq. (8) does not

have an exact closed-form solution; however, its numerical integration is convergent and is used in further analysis. We next

proceed to identify a closed-form approximate solution for m NL jih

amenable to a computationally tractable implementation.

To this end, we first carry out a Taylor series expansion of 1 / m NL jih,Q

about a i j /R i = 0 to obtain

1

2 π

∫ 2 π0

d φ

m NL jih,Q

(x j , x i , x h , φ) = 1 + νi

16 πE i R i

[ cos 2

(θ jih / 2

)(4 νi − 3 − cos θ jih )

sin (θ jih / 2

)] (

a i j

R i

)+ O

((a i j

R i

)2 )(11)

Similarly, we carry out a Taylor expansion of 1 / m NL jih,Q

about θ jih = π to obtain

1

2 π

∫ 2 π0

d φ

m NL jih,Q

(x j , x i , x h , φ) = 1

m NL jih,π

− (1 + νi )(1 + 2 νi ) 16 πE i R i

(π − θ jih ) + O ((π − θ jih ) 2

)(12)

where m NL jih,π

corresponds to the analytical solution of Eq. (8) for θ jih = π, that is

1

m NL jih,π

= 1 + νi 4 πE i R i

⎡ ⎣ 1 √

2

√ 1 −

√ 1 −

(a i j

R i

)2 − 2(1 − 2 νi )

(R i a i j

)⎡ ⎣ 1 − 1 √ 2

√ 1 +

√ 1 −

(a i j

R i

)2 ⎤ ⎦ ⎤ ⎦ (13)

Fig. 3 presents a comparison of the two approximate analytical solutions obtained from truncated Taylor expansions at

a i j /R i = 0 ( Eq. (11) ) and θ jih = π ( Eq. (12) ) with the numerical solution of Eq. (8) , by plotting the dimensionless quantity


Fig. 3. Plots of E i R i / m NL jih

obtained by truncated Taylor expansion at a i j /R i = 0 ( Eq. (11) , solid curves) and θ jih = π ( Eq. (12) , dashed curves) versus angular distance θ jih , compared with numerical solution of Eq. (8) (dotted curves). Graphs for three different values of ν i equal to (a). 0.15, (b).

0.3 and (c). 0.45 are presented, each having plots evaluated at three different values of a ij / R i ( red curves for a i j /R i = 0 . 25 , green curves for a i j /R i = 0 . 5 and blue curves for a i j /R i = 0 . 75 ). The lower bound of θ jih = sin −1 (a i j /R i ) (dashed-dotted lines) for evaluation of nonlocal contributions has also been delineated in each of the graphs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

E i R i / m NL jih

with respect to θ jih for all possible values of θ jih ∈ (sin −1

(a i j /R i

), π

]. To gain a better insight into the non-linear

dependency of the expressions on a ij / R i and ν i , graphs for three different values of ν i ( = 0.15, 0.3 and 0.45) are shown, witheach graph having plots evaluated at three different values of a ij / R i ( = 0.25, 0.5 and 0.75). Analysis of the plots suggeststhat the numerical solution is fairly represented by Eq. (11) until it shifts to positive values near θ jih = 3 π/ 4 . Thereafter, thenumerical solution is well represented by Eq. (12) . Accordingly, we propose a piecewise continuous function to represent

m NL jih

, given by

1

m NL jih

�

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

1 + νi 16 πE i R i

[cos 2 (θ jih / 2)(4 νi − 3 − cos θ jih )

sin (θ jih / 2) −

cos 2 (θ c jih

/ 2)(4 νi − 3 − cos θ c jih ) sin (θ c

jih / 2)

](a i j

R i

)θ jih ≤ θ c jih

1

m NL jih,π

− (1 + νi )(1 + 2 νi ) 16 πE i R i

(π − θ jih ) θ jih > θ c jih

(14)

where θ c jih

is the critical value of angular distance at which 1 /m NL jih

= 0 , given by

θ c jih = π −16 πE i R i

(1 + νi )(1 + 2 νi ) m NL jih,π(15)

Fig. 4 shows a comparison of the numerical solution of 1 /m NL jih

and the proposed piecewise continuous approximate solution,

using the same values of a ij / R i and ν i as in Fig. 3 . The accuracy of the closed-form approximate solution is acceptable andthe computational tractability of the formulation is attained.

4. Curvature correction to the nonlocal contact formulation

We again consider a configuration of two elastic spheres i and j being pressed together along the direction z i j = (x i −x j ) / || x i − x j || by a general configuration of concentrated forces, as depicted in Fig. 2 . According to the nonlocal contactformulation ( Gonzalez & Cuitiño, 2012 ), any point within the contact area satisfies the following compatibility equation

γi j = R i −√

R 2 i

− r 2 i j

+ R j −√

R 2 j − r 2

i j + w i (r i j ) + w j (r i j ) − γ NL i j (16)


Fig. 4. Plots of E i R i / m NL jih

obtained by piecewise approximate solution Eq. (14) (bold solid curves) versus angular distance θ jih , compared with numerical

solution of Eq. (8) (dotted curves). The values of ν i and a ij / R i are the same as in Fig. 3 .

where γ ij is the relative displacement of x i and x j along the direction z ij , r ij is the radial coordinate from axis z ij of surfacepoints in the reference configuration, and w i ( r ij ) and w j ( r ij ) are the vertical displacements of surface points located at r ij on

spheres i and j . The last term in the equation, γ NL i j

, is the total nonlocal contribution to vertical displacements induced by

all neighbors of the two particles, i.e.,

γ NL i j = ∑

h ∈N i ,h � = j

P ih

n NL jih

(x j , x i , x h ) +

∑ k ∈N j ,k � = i

P jk

n NL i jk

(x i , x j , x k ) (17)

with n NL jih

derived from Eq. (5) ( Gonzalez & Cuitiño, 2012 )

1

n NL jih

= (1 + νi ) 4 πR i E i

[−2(1 − νi ) − 2(1 − 2 νi ) sin (θ jih / 2) + (7 − 8 νi ) sin 2 (θ jih / 2) sin (θ jih / 2)

](18)

An important assumption of the Hertz theory is that the profile of the undeformed spherical contact surface is replaced by

the first term of its Taylor series expansion, i.e.

R i −√

R 2 i

− r 2 i j

= R i [

r 2 i j

2 R 2 i

+ O (

a 4 i j

R 4 i

)]�

r 2 i j

2 R i (19)

This approximation largely deviates from the exact solution for moderate to high mesoscopic deformations. However, the

error can be controlled by including more terms in the series to further correct the profile curvature; for example, the first

four terms are given by

R i −√

R 2 i

− r 2 i j

= R i [

r 2 i j

2 R 2 i

+ r 4

i j

8 R 4 i

+ r 6

i j

16 R 6 i

+ O (

a 8 i j

R 8 i

)]�

r 2 i j

2 R i +

r 4 i j

8 R 3 i

+ r 6

i j

16 R 5 i

+ 5 r 8

i j

128 R 7 i

(20)

Therefore, by adopting a four-term curvature correction, Eq. (16) simplifies to

w i (r i j ) + w j (r i j ) = (γi j + γ NL i j ) −r 2

i j A i j

2 −

r 4 i j B i j

8 −

r 6 i j C i j

16 −

5 r 8 i j D i j

128 (21)

where

A i j = 1

R i + 1

R j B i j =

1

R 3 i

+ 1 R 3

j

C i j = 1

R 5 i

+ 1 R 5

j

D i j = 1

R 7 i

+ 1 R 7

j


Fig. 5. Depiction of the circular contact region Q i j of contact radius a ij under the action of a distributed contact pressure, with B ( q ij , ω ij ) being an elemental region on which the pressure distribution is considered.

Next, we determine the pressure distribution p ij over the circular contact region Q i j of contact radius a ij in the referenceconfiguration, depicted in Fig. 5 , which is compatible with Eq. (21) . Specifically, for a pressure p ij ( q ij , ω ij ) acting over anelemental region B of area q ij dq ij d ω ij , the vertical displacement field according to the theory of elasticity ( Johnson, 1987 ,pg. 53) is given by

w i (r i j ) + w j (r i j ) = (

1 − ν2 i

πE i +

1 − ν2 j

πE j

)∫ ∫ Q i j

p i j (q i j , ω i j ) d q i j d ω i j (22)

and thus, using Eq. (22) in Eq. (21) , we get (1 − ν2

i

πE i +

1 − ν2 j

πE j

)∫ ∫ Q i j

p i j (q i j , ω i j ) dq i j dω i j = (γi j + γ NL i j ) −r 2

i j A i j

2 −

r 4 i j B i j

8 −

r 6 i j C i j

16 −

5 r 8 i j D i j

128 (23)

The pressure distribution p ij is readily available by following the method reported by Luo (1958) , which involves taking an

approximate form of p ij to solve the integral, followed by comparing coefficients of like powers of r ij on both sides of the

equation. For reference, the solution method is described in Appendix B . Using the pressure distribution functions, contact

radius-force-displacement relationships for two-term, three-term and four-term curvature corrections are derived in the

following sub-sections.

4.1. Two-term curvature correction

Pressure distribution ( Section B.1 , Eq. (B.20) ):

p i j (r i j ) = 2 a i j

π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (1 −

r 2 i j

a 2 i j

)1 / 2 [A i j +

2 a 2 i j B i j

9

(1 + 2

r 2 i j

a 2 i j

)](24)

Contact radius-displacement relationship : according to ( Appendix B.1 , Eq. (B.19) )

a 4 i j

(B i j 3

)+ a 2 i j A i j − (γi j + γ NL i j ) = 0 (25)

which yields only one real and positive solution given by

a i j = [

3

2 B i j

( √ 4 B i j

3 (γi j + γ NL i j ) + A 2 i j − A i j

) ] 1 / 2 (26)


Contact force-radius-displacement relationship :

P i j = ∫ a i j

0

p i j (r i j )2 π r i j dr i j = 4 E ∗

i j

3 a i j (γi j + γ NL i j ) 3

[A i j +

2 a i j (γi j + γ NL i j ) 2 B i j 5

](27)

where

1

E ∗i j

= 1 − ν2 i

E i +

1 − ν2 j

E j

and a i j (γi j + γ NL i j ) is given by Eq. (26) .

4.2. Three-term curvature correction


p i j (r i j ) = 2 a i j

π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (1 −

r 2 i j

a 2 i j

)1 / 2 [A i j +

2 a 2 i j B i j

9

(1 + 2

r 2 i j

a 2 i j

)+

a 4 i j C i j

25

(3 + 4

r 2 i j

a 2 i j

+ 8 r 4

i j

a 4 i j

)](28)

Contact radius-displacement relationship : according to ( Section B.2 , Eq. (B.29) )

γi j + γ NL i j = a 2 i j A i j + a 4

i j

3 B i j +

a 6 i j

5 C i j (29)


a i j = [ [

(Q 2 i j

+ 4 R 3 i j ) 1 / 2 + Q i j

]1 / 3 9(2) 1 / 3 C i j

− (2) 1 / 3 R i j

9 C i j [(Q 2

i j + 4 R 3

i j ) 1 / 2 + Q i j

]1 / 3 − 5 B i j 9 C i j ] 1 / 2

(30)

where

Q i j = 2025 A i j B i j C i j − 250 B 3 i j + 3645 C 2 i j (γi j + γ NL i j ) R i j = 135 A i j C i j − 25 B 2 i j


P i j = 4 E ∗

i j

3 a i j (γi j + γ NL i j ) 3

[A i j +

2 a i j (γi j + γ NL i j ) 2 B i j 5

+ 9 a i j (γi j + γ NL i j ) 4 C i j

35

](31)

where a i j (γi j + γ NL i j ) is given by Eq. (30) .

4.3. Four-term curvature correction


p i j (r i j ) = 2 a i j

π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (1 −

r 2 i j

a 2 i j

)1 / 2 [A i j +

2 a 2 i j B i j

9

(1 + 2

r 2 i j

a 2 i j

)+

a 4 i j C i j

25

(3 + 4

r 2 i j

a 2 i j

+ 8 r 4

i j

a 4 i j

)

+ 4 a 6

i j D i j

245

(5 + 6

r 2 i j

a 2 i j

+ 8 r 4

i j

a 4 i j

+ 16 r 6

i j

a 6 i j

)] (32)

Contact radius-displacement relationship : according to ( Section B.3 , Eq. (B.41) )


i j

3 B i j +

a 6 i j

5 C i j +

a 8 i j

7 D i j (33)


a i j = [

1

2

√ 147 C 2

i j

100 D 2 i j

− 14 B i j 3 D i j

− S i j −1

4 √

S i j

(196 B i j C i j

15 D 2 i j

−343 C 3

i j

125 D 3 i j

− 56 A i j D i j

)−√

S i j

2 − 7 C i j

20 D i j

] 1 / 2 (34)


Fig. 6. Schematic of the loading configurations considered for validation of contact radius and curvature corrections to the nonlocal contact formulation.

(a). Simple Compression, (b). Die Compression, (c). Hydrostatic Compaction and (d). Die compression with two additional walls at an angle of 37.5 ° from lateral walls in the y-direction. For each load case, views in x-z and y-z plane are provided for clarity.

Fig. 7. Finite-element mesh of one-eighth of a sphere created in ABAQUS. The depicted mesh is coarser than the final converged mesh and consists of

108,0 0 0 hexahedral elements and 113,521 nodes. (a). Initial undeformed mesh. (b). Deformed mesh for the fourth loading configuration of die compression

with two additional oblique walls.


Fig. 8. Impingement of two contacts of radii a ′ 1 and a ′ 2 at distances (R − γ1

2

)and

(R − γ2

2

)respectively from the center of a particle of radius R, separated

by an angular distance θ12 .

where

S i j = 7(2) 1 / 3 S (1)

i j

9 D i j

[S (2)

i j + √

S (2) i j

2 − 171500 S (1) i j

3 ]1 / 3 +

[S (2)

i j + √

S (2) i j

2 − 171500 S (1) i j

3 ]1 / 3

45(2) 1 / 3 D i j +

49 C 2 i j

100 D 2 i j

− 14 B i j 9 D i j

S (1) i j

= 35 B 2 i j − 189 A i j C i j − 540 D i j (γi j + γ NL i j ) S (2)

i j = 4465125 A 2 i j D i j − 694575 A i j B i j C i j + 85750 B 3 i j + (39690 0 0 B i j D i j − 1250235 C 2 i j )(γi j + γ NL i j )


P i j = 4 E ∗

i j

3 a i j (γi j + γ NL i j ) 3

[A i j +

2 a i j (γi j + γ NL i j ) 2 B i j 5

+ 9 a i j (γi j + γ NL i j ) 4 C i j

35 +

4 a i j (γi j + γ NL i j ) 6 D i j 21

](35)

where a i j (γi j + γ NL i j ) is given by Eq. (34) .

5. Validation of contact radius and curvature corrections

Validation of contact radius and curvature corrections to the nonlocal contact formulation was performed by considering

four types of loading configurations ( Fig. 6 ), three of which are symmetric configurations that simulate the compaction of

particles in a simple cubic lattice, namely simple compression (sphere pressed between two rigid plates, Fig. 6 a), die com-

paction (sphere compressed between rigid plates and constrained laterally by rigid walls, Fig. 6 b) and hydrostatic compaction

(sphere compressed triaxially by rigid plates, Fig. 6 c). The fourth configuration is asymmetric, consisting of two additional

walls perpendicular to the y-z plane between the plates and lateral walls of a die compaction configuration ( Fig. 6 d). Analyt-

ical predictions for all the configurations were compared with detailed finite element simulations, analytical results obtained

from the original nonlocal contact formulation ( Gonzalez & Cuitiño, 2012 ), and the classical Hertz predictions. For simple

compression, predictions were also compared with experimental measurements ( Tatara, 1989; Tatara et al., 1991 ).

The finite element simulations were performed in ABAQUS on one-eighth of a sphere, owing to geometric and loading

symmetries. Finite deformations were considered and the material was characterized as compressible neo -Hookean with an

energy density of the form

W (λ1 , λ2 , λ3 ) = μ[J −2 / 3 (λ2 1 + λ2 2 + λ2 3 ) − 3

]+ λ (J − 1) 2 (36)

2 2


Fig. 9. Load - deformation (a) and contact radius - deformation (b) curves for simple compression of a rubber sphere. Hertz theory predictions (black-

dashed curves), nonlocal contact formulation results without contact radius and curvature corrections (black curves), with contact radius and two- (red

curves), three- (green curves) and four- (blue curves) term corrections, finite element solution (grey curves), and experimental measurements ( Tatara, 1989;

Tatara et al., 1991 ) (five-pointed stars and triangles) are presented. The convergence of predictions at a four-term correction is shown in the inserts. (For

interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

where λ1 , λ2 and λ3 are the principle material stretches, J = det (F ) = λ1 λ2 λ3 and μ and λ are the Lamé constants. InABAQUS, C 10 = μ2 and D 1 = 2 λ are the input paramters. The elastic constants used in the simulations correspond to thevalues of E = 1 . 85 MPa and ν = 0 . 48 reported for rubber ( Tatara, 1989; Tatara et al., 1991 ). Following a mesh convergencestudy, the mesh comprising of 50 0,0 0 0 elements of type C3D8R (8-node linear hexahedral) and 515,201 nodes was chosen.

For illustration purposes, a course mesh of 108,0 0 0 elements is depicted in Fig. 7 .

One of the most important factors to be considered while comparing analytical predictions with numerical simulations,

other than the evolution of contact force and contact radius, is the occurrence of contact impingement. This is a phe-

nomenon that causes the contacts to no longer remain circular and, thereby, restricts the applicability of the nonlocal con-

tact formulation. Fig. 8 provides a geometrical description of the impingement of two contacts in a spherical particle under

a general loading configuration. From the figure, the angular distance θ12 between the two contacts at the inception ofimpingement is given by

θ12 = tan −1 (

a ′ 1 R − (γ1 / 2)

)+ tan −1

(a ′ 2

R − (γ2 / 2)

)(37)

Figs. 9 , 10 , 11 , 12 present the validation results in form of load-versus-deformation ( F / πR 2 versus γ /2 R ) and contactradius-versus-deformation ( a ′ / R versus γ /2 R ) plots. The figures also show the values of deformation at which contact im-pingement occurs for each finite-element simulation (except simple compression where impingement does not occur)—this

information is extracted by using the contact radius and displacement values obtained from the simulations in Eq. (36) .

It is interesting to note that for all loading configurations except simple compression, the analytical predictions show an

apparent strain-hardening or stiffening at various deformation levels. This is an artifact of the formulation, which possibly

results from various modeling assumptions made for evaluation of γ NL ( Gonzalez & Cuitiño, 2012 ), that limits the range of


Fig. 10. Load - deformation (a) and contact radius - deformation (b) curves for die compression of a rubber sphere. Predictions for vertical loaded contacts

are given by solid curves while predictions for lateral constrained contacts are given by dashed-dotted curves. Hertz theory predictions (black-dashed

curves), nonlocal contact formulation results without contact radius and curvature corrections (black curves), with contact radius and two- (red curves),

three- (green curves) and four- (blue curves) term corrections, and finite element solution (grey curves) are presented. The deformation at geometrical

contact impingement of vertical and lateral contacts is marked by a grey dotted line. The convergence of predictions at a four-term correction is shown in

the inserts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

applicability of the nonlocal contact formulation. Such limited range of applicability exists for simple compression as well,

albeit at deformations greater than 80% that are beyond the scope of this study.

Analysis of the plots suggests that the proposed contact radius and curvature corrections increase the range of applicabil-

ity of the nonlocal contact formulation, thereby enabling predictions at higher levels of deformation. Further improvement

is observed with increase in the order of curvature correction until convergence, which is achieved with a four-term cor-

rection. Quantitatively, the range of applicability of the formulation is increased by roughly 5% for die compaction, 2% for

hydrostatic compaction and 6% for die compaction with oblique walls. Additionally, the corrections enable predictions closer

to the geometric contact impingement, marked by dotted straight lines in the graphs. This is well represented in the graphs

for die compaction, where predictions continue right until the impingement of vertical and lateral contacts, and for die

compaction with additional walls, where predictions continue until the impingement of vertical and lateral contacts in the

x-direction. Interestingly, analytical predictions for the fourth configuration are remarkably accurate even after the impinge-

ment of vertical, oblique and lateral (y-direction) contacts (grey dotted line), indicating that predictions of the extended

nonlocal contact formulation are accurate until impingement of all particle contacts. As for the case of simple compression,

an overall improvement in the representation of experimental measurements for both contact force and radius is observed,

which again converges at the four-term curvature correction.

6. Summary and discussion

We have developed the analytical framework for contact radius correction to the nonlocal contact formulation, which

accounts for local and nonlocal contributions to the radial deformation of contact boundaries due to multiple contact forces

acting on a single particle. Furthermore, we have provided a method of curvature correction to relax the traditional as-

sumption of one-term Taylor series representation of underformed contacting surfaces. For definiteness, we have restricted


Fig. 11. Load - deformation (a) and contact radius - deformation (b) curves for hydrostatic compression of a rubber sphere. Hertz theory predictions

(black-dashed curves), nonlocal contact formulation results without contact radius and curvature corrections (black curves), with contact radius and two-

(red curves), three- (green curves) and four- (blue curves) term corrections, and finite element solution (grey curves)) are presented. The deformation at

geometrical contact impingement of contacts is marked by a grey dotted line. The convergence of predictions at a four-term correction is shown in the

inserts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

attention to elastic spheres in the absence of gravitational forces, adhesion or friction. Hence, a notable feature of the con-

tact formulation presented here is that, when no contact radius and curvature corrections are accounted for, it reduces to

the nonlocal contact formulation presented by Gonzalez and Cuitiño (2012) and thus, when all nonlocal effects and cor-

rections are neglected, it reduces to Hertz theory. Another salient feature of the proposed formulation is that it increases

the range of applicability of the nonlocal contact formulation ( Gonzalez & Cuitiño, 2012 ); consequently, it enables accurate

predictions of contact behavior until contact impingement for confined loading configurations. Specifically, we have investi-

gated four different loading conditions (namely simple compression, die compression within four walls and within six walls,

and hydrostatic compaction) and have successfully compared the predictions of the proposed nonlocal formulation with

experimental and detailed finite-element simulation results of rubber particles.

We close by pointing out future research directions and possible approaches for extension of the formulation.

First, the work presented in this paper together with the nonlocal contact formulation by Gonzalez and

Cuitiño (2012) serve as the foundation for conceiving an analytical elasto-plastic nonlocal contact formulation. Similar to

elastic particles, the assumption of independent contacts is not valid for particles deforming predominantly plastically in

the range of moderate to high mesoscopic deformations ( Mesarovic & Fleck, 20 0 0 ). Recently, finite element simulations of

linear elastic-perfectly plastic particles ( Tsigginos, Strong, & Zavaliangos, 2015 ) have shown contact interactions at moder-

ate deformations physically manifesting as coalescence of local plastically deforming zones around individual contacts, with

further deformation leading to the development of a low-compressibility regime when the contact deformations become

predominantly elastic. The systematic investigation of these interaction effects is a worthwhile direction of future research.

Second, although the work presented in this paper is a contribution to predictive modeling of confined granular systems,

considerable effort is required for modeling of contact behavior beyond contact impingement. Finite element simulations

presented in this paper show an almost linear increase in contact force with deformation after impingement of all contacts,


Fig. 12. Load - deformation (a) and contact radius - deformation (b) curves for a rubber sphere under an asymmetric loading configuration of die com-

pression with oblique walls. Predictions are represented by solid curves for vertical loaded contacts, dashed-dotted curves for lateral constrained contacts

in x-direction, dashed curves for lateral constrained contacts in y-direction, and dotted curves for oblique contacts. Hertz theory predictions (bold black-

dashed curves), nonlocal contact formulation results without contact radius and curvature corrections (black curves), with contact radius and two- (red

curves), three- (green curves) and four- (blue curves) term corrections, and finite element solution (grey curves) are presented. The deformation at geo-

metrical contact impingement of vertical, oblique and lateral (y-direction) contacts is marked by a grey dotted line, and that of vertical contacts and lateral

contacts in x-direction by a black dotted line. The convergence of predictions at a four-term correction is shown in the inserts. (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of this article.)

indicating a linear force-displacement relationship in this deformation regime. However, rigorous analysis of this linear cor-

relation is desirable, if beyond the scope of this paper.

Acknowledgments

The authors gratefully acknowledge the support received from the National Science Foundation grant number CMMI-

1538861 , from Purdue University’s startup funds, and from the Network for Computational Nanotechnology (NCN) and

nanoHUB.org. Ankit Agarwal also acknowledges the Frederick N. Andrews Fellowship from Purdue University.

Appendix A. Determination of radial displacement of a contact boundary point due to a single force

Fig. A.13 depicts a three dimensional view of a linear-elastic spherical particle under the action of a concentrated force P i applied at the origin A of cylindrical coordinates ( z, r ). We consider the deformation of a spherical cap of base radius a and

center C , situated at an angular distance θ i from the force P i , due to an ellipsoidally distributed pressure given by Eq. (1) inSection 2 . The pressure distribution is approximated by an effective force P applied at C . A point Q on the cap boundary is

situated at angle φ from the plane defined by points A, O and C . Using vector algebra, the angle AOQ , denoted by βQ , canbe expressed as

βQ = cos −1 [

cos

∣∣∣θi − sin −1 ( a R

)∣∣∣− ( a R

)(1 − cos φ) sin θi

] (A.1)

https://doi.org/10.13039/100000001


Fig. A1. A linear-elastic sphere of radius R under the action of an ellipsodially distributed pressure, approximated by an effective force P on a spherical cap

of radius a and center C , with concentrated force P i acting normally on one of its surface points.

The above equation implies that the angular distance between a contact force and any point on the cap boundary can be

represented in terms of a constant reference angle θ i and a variable angle φ. The dependency on angle φ vanishes for thetwo extreme values of θ i , i.e. 0 and π . For θi = 0 , which corresponds to the case when P i = P, Eq. (A.1) reduces to

βQ | θi =0 = sin −1 (

a

R

)(A.2)

the above result restates the geometrical fact that the contact force P is equidistant from any point on the cap boundary,

the angular distance being sin −1 (a/R ) . A similar result is obtained for θi = π, which makes the force P i equidistant fromany point on the cap boundary. The angular distance in this case becomes

βQ | θi = π = π − sin −1 (

a

R

)(A.3)

At point Q with coordinates ( z, r ): (2 R sin 2 ( βQ /2), R sin βQ ), displacements due to force P i along the ( z, r ) axes, denotedby ( W i, Q , U i, Q ), are given by the Boussinesq solution ( Johnson, 1987; Timoshenko & Goodier, 1970 ) and can be expressed in

terms of angle βQ as

W i,Q (βQ ) = (1 + ν) P i

2 πER

[sin ( βQ / 2)

2 + 1 − ν

sin ( βQ / 2)

](A.4)

U i,Q (βQ ) = (1 + ν) P i

4 πER

[cos ( βQ / 2) − ( 1 − 2 ν) cos ( βQ / 2)

sin ( βQ / 2) ( 1 + sin ( βQ / 2) )

](A.5)

Similarly, for the cap center C with coordinates ( z, r ): (2 R sin 2 ( θ i /2), R sin θ i ), displacements due to force P i along the ( z,r ) axes, denoted by ( W i, C , U i, C ), can be expressed in terms of angle θ i as

W i,C (θi ) = (1 + ν) P i

2 πER

[sin ( θi / 2)

2 + 1 − ν

sin ( θi / 2)

](A.6)

U i,C (θi ) = (1 + ν) P i

4 πER

[cos ( θi / 2) −

( 1 − 2 ν) cos ( θi / 2) sin ( θi / 2) ( 1 + sin ( θi / 2) )

](A.7)

The displacement of point Q due to force P i in the radial direction with respect to center C , denoted by u i, Q , can be expressed

as

u i,Q = [(W i,Q − W i,C ) ̂ e z + (U i,Q − U i,C ) ̂ e r

]· ˆ t (A.8)


where ˆ t is a unit vector in the radial direction at Q , as shown in Fig. A.13 . Using vector algebra, it can be proved that the

vector ˆ t can be expressed in terms of ( x, y, z ) coordinates as

ˆ t = − cos θi cos φ ˆ e x − sin φ ˆ e y − sin θi cos φ ˆ e z (A.9) which implies that ˆ t in terms of ( z, r ) coordinates becomes

ˆ t = − sin θi cos φ ˆ e z + (√

1 − sin 2 θi cos 2 φ)

ˆ e r (A.10)

Substituting ˆ t from Eq. (A.10) into Eq. (A.8) , we get

u i,Q = −(W i,Q − W i,C ) sin θi cos φ + (U i,Q − U i,C ) √

1 − sin 2 θi cos 2 φ (A.11) Eq. (A.11) provides a general definition of the radial displacement of a contact boundary point on an elastic spherical

particle due to a concentrated force acting on the surface of the particle. After substituting the values of W i, Q , U i, Q , W i, C and U i, C from Eqs. (A .4) , (A .5), (A .6) and (A .7) respectively into Eq. (A.11) , the resulting final expression for u i, Q in terms of

angles θ i and φ is given by

u i,Q = (1 + ν) P i

2 πER

[sin θi cos φ( sin (θi / 2) − sin (βQ (θi , φ) / 2) ) ( sin (θi / 2) sin (βQ (θi , φ) / 2) − 2 + 2 ν)

2 sin (θi / 2) sin (βQ (θi , φ) / 2)

+ √

1 − sin 2 θi cos 2 φ[

1

2 ( cos (βQ (θi , φ) / 2) − cos (θi / 2) )

−(1 − 2 ν) (

1 − sin (βQ (θi , φ) / 2) sin βQ (θi , φ)

− cos (θi / 2) 2 sin (θi / 2)(1 + sin (θi / 2))

)]] (A.12)

The above expression can be further reduced to the following form

u i,Q = P i

ηi,Q (A.13)

where,

1

ηi,Q = 1 + ν

2 πER

[sin θi cos φ( sin (θi / 2) − sin (βQ (θi , φ) / 2) ) ( sin (θi / 2) sin (βQ (θi , φ) / 2) − 2 + 2 ν)

2 sin (θi / 2) sin (βQ (θi , φ) / 2)

+ √

1 − sin 2 θi cos 2 φ[

1

2 ( cos (βQ (θi , φ) / 2) − cos (θi / 2) )

−(1 − 2 ν) (

1 − sin (βQ (θi , φ) / 2) sin βQ (θi , φ)

− cos (θi / 2) 2 sin (θi / 2)(1 + sin (θi / 2))

)]] (A.14)

Appendix B. Calculation of pressure distribution on the contact surface of an elastic sphere

Considering the contact configuration presented in Fig. 2 in Section 3 , the contact area is circular and depicted in Fig. 5 in

Section 4 . Based on the analysis of Luo (1958) and Cattaneo (1947) , the following approximate form of pressure distribution

can be assumed at a radial distance r ij .

p i j (r i j ) = N ∑

n =1 ρn

(1 −

r 2 i j

a 2 i j

) 2 n −1 2

(B.1)

where N corresponds to the number of Taylor series terms considered in the profile function, and ρn are N unknown func-tion parameters. For an internal point A in the contact region at a radial distance r ij from the contact center, we can write

the following using cosine rule

s 2 i j = r 2 i j + q 2 i j + 2 r i j q i j cos ω i j (B.2) Using Eqs. (B.1) and (B.2) , the pressure distribution at elemental region B ( q ij , ω ij ) can be written as

p i j (q i j , ω i j ) = N ∑

n =1

ρn

a 2 n −1 i j

(a 2 i j − r 2 i j − 2 r i j q i j cos ω i j − q 2 i j ) 2 n −1

2 (B.3)

Let ζi j = √

a 2 i j

− r 2 i j

and δi j = r i j cos ω i j . Then Eq. (B.2) reduces to

p i j (q i j , ω i j ) = N ∑

n =1

ρn

a 2 n −1 i j

(ζ 2 i j − 2 δi j q i j − q 2 i j ) 2 n −1

2 (B.4)


Substituting the expression for p ij ( q ij , ω ij ) given by Eq. (B.4) in Eq. (22) ( Section 4 ), the displacement field inside the circularregion now becomes

w i (r i j ) + w j (r i j ) = ū iz i j (r i j ) + ū jz i j (r i j )

= (

1 − ν2 i

πE i +

1 − ν2 j

πE j

)∫ 2 π0

d ω i j

∫ q 1 0

{ N ∑

n =1

ρn

a 2 n −1 i j

(ζ 2 i j − 2 δi j q i j − q 2 i j ) 2 n −1

2

} d q i j

(B.5)

where q 1 is the positive root of the equation ( Johnson, 1987 , pg. 60)

q 2 i j + 2 δi j q i j − ζ 2 i j = 0 (B.6)given by

q 1 = −δi j + √

ζ 2 i j

+ δ2 i j

(B.7)

We now consider three cases depending upon the number of Taylor series terms ( N ) taken in the profile function. The three

cases correspond to N = 2 , 3 and 4. We shall calculate the pressure distribution for each of these cases. We first calculate the displacement field for N = 4 and then modify the function accordingly for different cases. First, the

internal integrals with respect to dq ij for n = 1 , 2, 3 and 4 are obtained and given by ∫ q 1 0

(ζ 2 i j − 2 δi j q i j − q 2 i j ) 1 / 2 dq i j = −1

2 ζi j δi j +

1

2 (ζ 2 i j + δ2 i j )

{π

2 − tan −1

(δi j ζi j

)}(B.8)

∫ q 1 0


8 ζi j δi j (5 ζ

2 i j + 3 δ2 i j ) +

3

8 (ζ 2 i j + δ2 i j ) 2

{π

2 − tan −1

(δi j ζi j

)}(B.9)

∫ q 1 0


48 ζi j δi j (33 ζ

4 i j + 40 ζ 2 i j δ2 i j + 15 δ4 i j ) +

5

16 (ζ 2 i j + δ2 i j ) 3

{π

2 − tan −1

(δi j ζi j

)}(B.10)

∫ q 1 0


384 ζi j δi j (279 ζ

6 i j + 511 ζ 4 i j δ2 i j + 385 ζ 2 i j δ4 i j + 105 δ6 i j )

+ 35 128

(ζ 2 i j + δ2 i j ) 4 {

π

2 − tan −1

(δi j ζi j

)} (B.11)When the resulting expression from combination of Eqs. (B.8) , (B.9), (B.10) and (B.11) is substituted in Eq. (B.5) , and in-

tegrated with respect to ω ij from 0 to 2 π , the terms containing ζ ij δij and tan −1 (

δi j ζi j

)are eliminated. The final form of

Eq. (B.5) re-expanded in terms of a ij and r ij is given by

w i (r i j ) + w j (r i j ) = ū iz i j (r i j ) + ū jz i j (r i j )

= (

1 − ν2 i

E i +

1 − ν2 j

E j

)(π

4 a i j

)[ a 2 i j

(2 ρ1 + 3

2 ρ2 + 5

4 ρ3 + 35

32 ρ4

)− r 2 i j

(ρ1 + 3

2 ρ2 + 15

8 ρ3

+ 35 16

ρ4

)+ r 4 i j

(9 ρ2

16 a 2 i j

+ 45 ρ3 32 a 2

i j

+ 315 ρ4 128 a 2

i j

)− r 6

(25 ρ3 64 a 4

i j

+ 175 ρ4 128 a 4

i j

)+ r 8 i j

(1225 ρ4 4096 a 6

i j

)] (B.12)

We now consider the three cases individually.

B1. Case I: two terms (N = 2)

For N = 2 , terms with ρ3 and ρ4 are eliminated from Eq. (B.12) . Substituting the modified equation in Eq. (23) ( Section 4 )with two terms, we have (

1 − ν2 i

E i +

1 − ν2 j

E j

)(π

4 a i j

)[a 2 i j

(2 ρ1 + 3

2 ρ2

)− r 2 i j

(ρ1 + 3

2 ρ2

)+ r 4 i j

(9 ρ2

16 a 2 i j

)]

= (γi j + γ NL i j ) −r 2

i j A i j

2 −

r 4 i j B i j

8

(B.13)

In order to satisfy Eq. (B.13) for all points within the circular contact region, coefficients of like powers of r ij on both sides

must be equal. Hence,

−(

1 − ν2 i

E i +

1 − ν2 j

E j

)(9 πρ2 8 a 3

i j

)= B i j (B.14)


(1 − ν2

i

E i +

1 − ν2 j

E j

)(π

2 a i j

)(ρ1 + 3

2 ρ2

)= A i j (B.15)

(1 − ν2

i

E i +

1 − ν2 j

E j

)(πa i j 4

)(2 ρ1 + 3

2 ρ2

)= γi j + γ NL i j (B.16)

Solving Eqs. (B.14) and (B.15) for ρ1 and ρ2 , we get

ρ1 = 2 a i j (3 A i j + 2 a 2 i j B i j )

3 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.17)

ρ2 = −8 a 3

i j B i j

9 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.18)

Substituting the expressions for ρ1 and ρ2 obtained above into Eq. (B.16) , we get an expression for displacement γ ij interms of contact radius a ij


i j

3 B i j (B.19)

Also, by substituting the expressions for ρ1 and ρ2 in Eq. (B.1) and rearranging, we obtain a pressure distribution of theform

p i j (r i j ) = 2 a i j

π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (1 −

r 2 i j

a 2 i j

)1 / 2 [A i j +

2 a 2 i j B i j

9

(1 + 2

r 2 i j

a 2 i j

)](B.20)

B2. Case II: Three terms (N = 3)

For N = 3 , terms with ρ4 are eliminated. Substituting the modified Eq. (B.12) in Eq. (23) with three terms, we have (1 − ν2

i

E i +

1 − ν2 j

E j

)(π

4 a i j

)[a 2 i j

(2 ρ1 + 3

2 ρ2 + 5

4 ρ3

)− r 2 i j

(ρ1 + 3

2 ρ2 + 15

8 ρ3

)+ r 4 i j

(9 ρ2

16 a 2 i j

+ 45 ρ3 32 a 2

i j

)− r 6 i j

(25 ρ3 64 a 4

i j

)]

= (γi j + γ NL i j ) −r 2

i j A i j

2 −

r 4 i j B i j

8 −

r 6 i j C i j

16

(B.21)

Since the coefficients of like powers of r ij on both sides of Eq. (B.21) must be equal, we get (1 − ν2

i

E i +

1 − ν2 j

E j

)(25 πρ3 16 a 5

i j

)= C i j (B.22)

−(

1 − ν2 i

E i +

1 − ν2 j

E j

)(9 π

8 a 3 i j

)(ρ2 + 5 ρ3

2

)= B i j (B.23)

(1 − ν2

i

E i +

1 − ν2 j

E j

)(π

2 a i j

)(ρ1 + 3

2 ρ2 + 15

8 ρ3

)= A i j (B.24)

(1 − ν2

i

E i +

1 − ν2 j

E j

)(πa i j 4

)(2 ρ1 + 3

2 ρ2 + 5

4 ρ3

)= γi j + γ NL i j (B.25)

solving Eqs. (B.22) , (B.23) and (B.24) for ρ1 , ρ2 and ρ3 , we get

ρ1 = 2 a i j (15 A i j + 10 a 2 i j B i j + 9 a 4 i j C i j )

15 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.26)

ρ2 = −8 a 3

i j (5 B i j + 9 a 2 i j C i j )

45 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.27)

ρ3 = 16 a 5

i j C i j

25 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.28)


By substituting the expressions for ρ1 , ρ2 and ρ3 obtained above into Eq. (B.25) , we get


i j

3 B i j +

a 6 i j

5 C i j (B.29)

And by substituting the expressions for ρ1 , ρ2 and ρ3 into Eq. (B.1) and rearranging, we obtain a pressure distribution ofthe form

p i j (r i j ) = 2 a i j

π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (1 −

r 2 i j

a 2 i j

)1 / 2 [A i j +

2 a 2 i j B i j

9

(1 + 2

r 2 i j

a 2 i j

)+

a 4 i j C i j

25

(3 + 4

r 2 i j

a 2 i j

+ 8 r 4

i j

a 4 i j

)](B.30)

B3. Case III: four terms (N = 4)

For N = 4 , we substitute Eq. (B.12) into Eq. (23) to get (1 − ν2

i

E i +

1 − ν2 j

E j

)(π

4 a i j

)[ a 2 i j

(2 ρ1 + 3

2 ρ2 + 5

4 ρ3 + 35

32 ρ4

)− r 2 i j

(ρ1 + 3

2 ρ2 + 15

8 ρ3 + 35

16 ρ4

)

+ r 4 i j (

9 ρ2 16 a 2

i j

+ 45 ρ3 32 a 2

i j

+ 315 ρ4 128 a 2

i j

)− r 6 i j

(25 ρ3 64 a 4

i j

+ 175 ρ4 128 a 4

i j

)+ r 8 i j

(1225 ρ4 4096 a 6

i j

)]

= (γi j + γ NL i j ) −r 2

i j A i j

2 −

r 4 i j B i j

8 −

r 6 i j C i j

16 −

5 r 8 i j D i j

128

(B.31)

Since the coefficients of like powers of r ij on both sides of Eq. (B.31) must be equal, we get

−(

1 − ν2 i

E i +

1 − ν2 j

E j

)(245 πρ4 128 a 7

i j

)= D i j (B.32)

(1 − ν2

i

E i +

1 − ν2 j

E j

)(π

16 a 5 i j

)(25 ρ3 + 175

2 ρ4

)= C i j (B.33)

−(

1 − ν2 i

E i +

1 − ν2 j

E j

)(9 π

8 a 3 i j

)(ρ2 + 5 ρ3

2 + 35

8 ρ4

)= B i j (B.34)

(1 − ν2

i

E i +

1 − ν2 j

E j

)(π

2 a i j

)(ρ1 + 3

2 ρ2 + 15

8 ρ3 + 35

16 ρ4

)= A i j (B.35)(

1 − ν2 i

E i +

1 − ν2 j

E j

)(πa i j 4

)(2 ρ1 + 3

2 ρ2 + 5

4 ρ3 + 35

32 ρ4

)= γi j + γ NL i j (B.36)

Solving Eqs. (B.32) , (B.33), (B.34) and (B.35) for ρ1 , ρ2 , ρ3 and ρ4 , we get

ρ1 = 2 a i j (105 A i j + 70 a 2 i j B i j + 63 a 4 i j C i j + 60 a 6 i j D i j )

105 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.37)

ρ2 = −8 a 3

i j (35 B i j + 63 a 2 i j C i j + 90 a 4 i j D i j )

315 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.38)

ρ3 = 16 a 5

i j (7 C i j + 20 a 2 i j D i j )

175 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.39)

ρ4 = −128 a 7

i j D i j

245 π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (B.40)

By substituting the expressions for ρ1 , ρ2 , ρ3 and ρ4 obtained above into Eq. (B.36) , we get


i j

3 B i j +

a 6 i j

5 C i j +

a 8 i j

7 D i j (B.41)

And by substituting the expressions for ρ1 , ρ2 , ρ3 and ρ4 into Eq. (B.1) and rearranging, we obtain a pressure distributionof the form

p i j (r i j ) = 2 a i j

π

(1 − ν2

i

E i +

1 − ν2 j

E j

)−1 (1 −

r 2 i j

a 2 i j

)1 / 2 [A i j +

2 a 2 i j B i j

9

(1 + 2

r 2 i j

a 2 i j

)+

a 4 i j C i j

25

(3 + 4

r 2 i j

a 2 i j

+ 8 r 4

i j

a 4 i j

)

+ 4 a 6

i j D i j

(5 + 6

r 2 i j

2 + 8

r 4 i j

4 + 16

r 6 i j

6

)] (B.42)

245 a

i j a

i j a

i j


Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.ijengsci.2018.08.

006 .

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Contact radius and curvature corrections to the nonlocal contact formulation accounting for multi-particle interactions in elastic confined granular systems1 Introduction2 Deformation of an elastic sphere in a confined granular system under the action of a distributed surface pressure3 Nonlocal effects in the evolution of inter-particle contact area4 Curvature correction to the nonlocal contact formulation4.1 Two-term curvature correction4.2 Three-term curvature correction4.3 Four-term curvature correction

5 Validation of contact radius and curvature corrections6 Summary and discussion AcknowledgmentsAppendix A Determination of radial displacement of a contact boundary point due to a single forceAppendix B Calculation of pressure distribution on the contact surface of an elastic sphereB1 Case I: two terms (N=2)B2 Case II: Three terms (N=3)B3 Case III: four terms (N=4)

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International Journal of Engineeringweb.ics.purdue.edu/~gonza226/papers/AgarwalGonzalez2018.pdf · 2018. 12. 8. · 28 A. Agarwal, M. Gonzalez / International Journal of Engineering