P-wave induced energy and damage distribution in agglomerated granules

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Modelling Simul. Mater. Sci. Eng. 15 (2007) S435–S448 doi:10.1088/0965-0393/15/4/S09

P-wave induced energy and damage distribution inagglomerated granules

T I Zohdi

Department of Mechanical Engineering, University of California, Berkeley, CA, 94720-1740,USA

Received 2 October 2006, in final form 24 January 2007Published 30 May 2007Online at stacks.iop.org/MSMSE/15/S435

AbstractThe goal of this work is to develop a model and corresponding computationalmethod capable of determining p-wave induced energy and damage distributionin microstructures composed of hard agglomerated granules embedded withina binder of low shear modulus. The damage to the matrix binding the granulesis assumed to be a function of the absorbed energy, which is tracked granuleby granule. We consider cases where the wavelengths of the incident p-waves(10−6 m � λ � 10−5 m) are at least an order of magnitude smaller than thediameter of a granular scatterer (10−4 m � d � 10−3 m), making ray-tracingtechniques applicable. Accordingly, a ray-tracing algorithm is developed todetermine the amount of propagating energy that is reflected and absorbed bymultiple granules, as well as the resulting local damage. Three dimensionalsimulations are given to illustrate the method.

1. Introduction

In many applications, it is important to simulate the properties/behaviour of materialscomprising of hard agglomerated granules, bound together within a binder of low shearmodulus, under shock-type loading. There are a large variety of materials that fall withinthis category either in their solidified gel-like state or in their fluid-like (molten-matrix/particlesuspension) state during processing. See Torquato (2002) or Nemat-Nasser and Hori (1999)for extensive surveys of such materials. Many of the applications of interest arise fromagglomerations of hard suspensions in a fluid-saturated or gel-like matrix. For example,understanding the behaviour of such materials is important in astrophysics, where the study ofcollisions of agglomerations of ‘space dust’ with satellites has become of increasing importancedue to commercialization of space (Brown and Cooke 2001). However, from a non-commercialspace-related aspect, particle accretions are also important in the formation of planetessimals,which are thought to initiate by the agglomeration of dust particles. In this vein, it isimportant to understand how much force such structures can withstand before breaking up.For reviews of these phenomena, see Benz (1994, 2000), Blum and Wurm (2000), Dominikand Tielens (1997), Chokshi et al (1993), Wurm et al (2001), Kokubu and Ida (1996, 2000),

0965-0393/07/040435+14$30.00 © 2007 IOP Publishing Ltd Printed in the UK S435

S436 T I Zohdi

P-WAVE

SURROUNDING FLUID

FLUID-SATURATEDBINDING MATRIX

GRANULES

Figure 1. Sketch of a model material with embedded granules.

Grazier et al (1999, 2000), Supulver and Lin (2000), Tanga et al (1996), Cuzzi et al (1993),Weidenschilling and Cuzzi (1993), Weidenschilling et al (1997), Beckwith et al (2000), Bargeand Sommeria (1995), Pollack et al (1994) and Lissauer (1993). The examples cited areonly a few of the many applications where the study of such agglomerations is important.Further applications include undersea drilling through fluid-saturated rock, shock-loading offluid-saturated concrete, mixing of paints, shampoos and pastes, food processing, chemicalmechanical planarization (CMP)1, etc. For various other applications, see Zohdi (2003, 2004,2006) and Zohdi and Wriggers (2005).

The purpose of the present work is not to concentrate on a specific application but to isolateone aspect of the complex series of events associated with shock-type loading of agglomeratedmicrostructures. We consider a ‘model’ solid material, composed of an agglomeration of hardgranules in (possibly fluid-saturated) low shear modulus binder (figure 1). In order to motivatethe need for numerical simulations, let us first qualitatively estimate the amount of damagein an idealized spherical object for a given energy input. We denote by W the transmittedenergy from a shock wave (figure 1). In order to make a very simple estimate, we assume thatthe energy is distributed evenly throughout the material. The amount of volume encompassedwithin the domain is V . The volume occupied by the granules within a volume is v2V , wherev2 is the volume fraction of granules.2 The number of granules contained in a volume V istherefore

N = v2V

4π

3b3

, (1.1)

where b is the nominal radius of the (spherical) granules. We denote the amount of energyper unit area needed to break a granule free of the binding matrix as γ . Thus, the amount ofenergy needed to break all the bonds fusing one granule to the agglomeration is S = γ 4πb2.The amount of energy needed to dislodge all the granules in a volume V is

S = γ 4πb2N. (1.2)

We eliminate N and equate the energy for debonding to the energy input; this yields the energyrequired to debond all the granules in the volume

W = 3γ v2V

b. (1.3)

1 CMP involves using chemically reacting particles embedded within a fluid (gas or liquid) to ablate rough surfacesflat, etc. For reviews of CMP practice and applications, see Luo and Dornfeld (2001, 2003a, 2003b, 2004).2 The volume fraction of the matrix is v1, where v1 + v2 = 1.

Large-scale P-wave simulation S437

This simple estimate, despite its sweeping assumptions, provides a rough guide for the energyrequired for a material to be degraded via debonding. Namely, it expresses the somewhatobvious conclusion that this requirement increases with increasing bond energy. The mostegregious assumptions are (i) that the energy is uniformly distributed throughout the material,(ii) that the granules are spherical and (iii) that material fragmentation (agglomeration break-up) proceeds by damage to the binding matrix. The goal of this work is to develop acomputational approach capable of addressing the first assumption; in other words we willdetermine the distribution of energy induced by p-waves and estimate the correspondingdamage, within the class of materials considered. The damage to the low shear modulusmatrix that binds the granules is assumed to be a function of the absorbed energy, which istracked, granule by granule. We consider cases where the wavelengths of the incident p-waves (10−6 m � λ � 10−5 m) are at least an order of magnitude smaller than the diameterof a granular scatterer (10−4 m � d � 10−3 m), making ray-tracing techniques applicable.Accordingly, a ray-tracing algorithm is developed to determine the amount of propagatingincident energy that is reflected and absorbed by multiple granules, as well as the resultinglocal damage.

2. A simple model of a class of agglomerated granular materials

2.1. Mathematical notation

Throughout this work, boldface symbols indicate vectors or tensors. The inner product of twovectors u and v is denoted as u · v. At the risk of oversimplification, we ignore the distinctionbetween second order tensors and matrices, since we exclusively employ a Cartesian basis.Hence, if we consider the second order tensor A with its matrix representation [A], then theproduct of two second order tensors A · B is defined by the matrix product [A][B], withcomponents of AijBjk = Cik . The second order inner product of two tensors or matricesis A : B = AijBij = tr([A]T[B]). Finally, the divergence of a vector u is defined by∇ · u = ui,i , whereas for a second order tensor A, ∇ · A describes a contraction to a vectorwith the components Aij,j .

2.2. Model of the binding matrix

In our approach, we model the individual granules as being rigid and the (fluid-saturated)binding material surrounding the granules as being isotropic and having a relatively low shearmodulus. Generally, for an isotropic material, one has the classical relationship between thecomponents of infinitesimal strain (ε) to the Cauchy stress (σ):

σ = IE : ε = 3κtrε

31 + 2µε′, (2.1)

where IE is the elasticity tensor and ε′ is the strain deviator. The corresponding strain energydensity is

W = 1

2ε : IE : ε = 1

2

(9κ

(trε

3

)2

+ 2µε′ : ε′)

. (2.2)

We focus on the dilatational deformation in the low shear modulus matrix surrounding thegranules. This naturally leads to an idealized ‘acoustical’ material approximation, µ ≈ 0.Hence, equation (2.1) collapses to σ = −p1, where the pressure is p = −3κ trε

3 1 and with a

S438 T I Zohdi

corresponding strain energy of W = 12

p2

κ. By inserting the simplified expression of the stress

σ = −p1 into the equation of equilibrium, we obtain

∇ · σ = −∇p = ρu, (2.3)

where u is the displacement. By taking the divergence of both sides and recognizing that∇ · u = −p

κ, we obtain

∇2p = ρ

κp = 1

c2p. (2.4)

If we assume a harmonic solution, we obtain

p = P ej(k·r−ωt) ⇒ p = P jωej(k·r−ωt) ⇒ p = −Pω2ej(k·r−ωt) (2.5)

and

∇p = Pj (kxex + kyey + kzez)ej(k·r−ωt)

⇒ ∇ · ∇p = ∇2p = −P (k2x + k2

y + k2z )︸ ︷︷ ︸

‖k‖2

ej(k ·r−ωt). (2.6)

We insert these relations into equation (2.4), and obtain an expression for the magnitude of thewave number vector:

− P ‖k‖2ej(k·r−ωt) = −ρ

κPω2ej(k·r−ωt) ⇒ ‖k‖ = ω

c. (2.7)

Equation (2.3) (balance of linear momentum) implies

ρu = −∇p = −Pj (kxex + kyey + kzez)ej(k·r−ωt). (2.8)

Now we integrate once, which is equivalent to dividing by −jω, and obtain the velocity

u = Pj

ρω(kxex + kyey + kzez)e

j(k·r−ωt) (2.9)

and do so again for the displacement

u = Pj

ρω2(kxex + kyey + kzez)e

j(k·r−ωt). (2.10)

Thus, we have

‖u‖ = P

cρ. (2.11)

3. Reflection and ray tracing

Now we turn to the problem of determination of the p-wave scattering by large numbers ofrandomly distributed granules comprising a model material.

3.1. Ray tracing

As mentioned previously, we consider cases where the granules are in the range10−4 m � d � 10−3 m and wavelengths in the range 10−6 m � λ � 10−5 m. In such cases,geometric ray tracing can be used to determine the amount of propagating incident energy thatis reflected and the amount that is absorbed by multiple granules. Ray tracing is amenable torapid large-scale computation required to track the scattering of incident p-waves, comprisingmultiple rays, by multiple granules. For the benefit of readers unfamiliar with ray tracing, weremark that it is essentially an approximate solution to the wave equation, based on the eikonalequation, which is the limiting case of wave phenomena as the wave length tends towards zero.Reviews of this classical topic are given by Elmore and Heald (1985).

Large-scale P-wave simulation S439

DEBONDING

ZONE

iI

DAMAGE

rI

Θt

rΘ

Θ i

tI

XY

Figure 2. Control volume for a granule.

3.2. Incidence, reflection and transmission

The reflection of a plane harmonic pressure wave at an interface is given by enforcing continuityof the (acoustical) pressure and disturbance velocity at that location; this yields the ratiobetween the incident and reflected pressures. We use a local coordinate system (figure 2), andrequire that the number of waves per unit length in the x − direction must be the same for theincident, reflected and refracted (transmitted) waves,

ki · ex = kr · ex = kt · ex. (3.1)

From the pressure balance at the interface, we have

Piej(ki ·r−ωt) + Prej(kr ·r−ωt) = Pte

j(kt ·r−ωt), (3.2)

where Pi is the incident pressure ray, Pr is the reflected pressure ray, and Pt is the transmittedpressure ray. This forces a time-invariant relation to hold at all parts on the boundary, becausethe arguments of the exponential must be the same. This leads to (ki = kr )

ki sin θi = kr sin θr ⇒ θi = θr (3.3)

and

ki sin θi = kt sin θt ⇒ ki

kt

= sin θt

sin θi

= ω/ct

ω/ci

= ci

ct

= vi

vt

= nt

ni

. (3.4)

Equations (3.1) and (3.2) imply

Piej(ki ·r) + Prej(kr ·r) = Pte

j(kt ·r). (3.5)

The continuity of the displacement, and hence the velocity

vi + vr = vt , (3.6)

leads to, after use of equation (2.11),

− Pi

ρici

cos θi +Pr

ρrcr

cos θr = − Pt

ρtct

cos θt . (3.7)

S440 T I Zohdi

We solve for the ratio of the reflected and incident pressures to obtain

r = Pr

Pi

= A cos θi − cos θt

A cos θi + cos θt

, (3.8)

where Adef= At

Ai= ρt ct

ρici, ρt is the medium which the ray encounters (transmitted), ct is the

corresponding sound speed in that medium, At is the corresponding acoustical impedance, ρi

is the medium in which the ray was travelling (incident), ci the is corresponding sound speedin that medium and Ai is the corresponding acoustical impedance. The relationship (the lawof refraction) between the incident and the transmitted angles is ct sin θt = ci sin θi . Thus,we may write the Fresnel relation:

r = cA cos θi − (c2 − sin2 θi)12

cA cos θi + (c2 − sin2 θi)12

, (3.9)

where cdef= ci

ct. The reflectance for the (acoustical) energy R = r2 is

R =(

Pr

Pi

)2

=(

A cos θi − cos θt

A cos θi + cos θt

)2

. (3.10)

For the cases where sin θt = sin θi

c> 1, one may rewrite the reflection relation as

r = cA cos θi − j (sin2 θi − c2)12

cA cos θi + j (sin2 θi − c2)12

, (3.11)

where j = √−1. The reflectance is Rdef= rr = 1, where r is the complex conjugate. Thus,

for angles above the critical angle θi � θ∗i , all of the energy is reflected. We note that when

At = Ai and ci = ct , then there is no reflection. Also, when At � Ai or when At Ai , thenr → 1.

Remark. If one considers for a moment an incoming pressure wave (ray), which is incidenton an interface between two general elastic media (µ �= 0), reflected shear waves must begenerated in order to satisfy the continuity of the traction, [|σ · n|] = 0. This is due to the factthat

[|(

3κtrε

31 + 2µε′

)· n|] = 0. (3.12)

For an idealized acoustical medium, µ = 0, no shear waves need to be generated to satisfyequation (3.12).

4. Computational algorithm

We now consider initially coherent p-waves composed of multiple parallel rays (figures 1and 4). Each ray is a vector in the direction of the flow of energy, which, in isotropic media,corresponds to the normal to the wave front. For isotropic media, the rays are parallel to thewave propagation vector (figures 1 and 4). It is of particular interest to describe the break-upof initially highly directional coherent beams which do not spread out into multidirectionalrays unless they encounter multiple scatterers.3 It is convenient to define vectors for a ray byprojecting the pressure onto the unit vector associated with the velocity

Pdef= P

v

‖v‖ , (4.1)

3 In isotropic media, the direction of propagation of pressure and energy is in the same direction as k.

Large-scale P-wave simulation S441

where P = ‖P‖ is the magnitude of the pressure. Because v‖v‖ also defines the direction of

the flow of energy, we may write for the beam

Idef= I

v

‖v‖ , (4.2)

where I is the energy per unit cross-sectional area of the wave front and initially

Iodef= ||I(t = 0)||. Note that the energy is proportional to the square of the amplitude of

the pressure; thus the reflectance, defined in equation (3.10), also determines the amount ofenergy reflected:

R = Ir

Ii

, (4.3)

where Ii is the incident and Ir is the reflected energy per unit cross-sectional area. To obtainthe ray contributions from a beam, we simply partition the initial beam cross-section into equal

rays, prayardef= pab/Nr and I rayar = Iab/Nr , where Nr is the number of rays in the beam

and where ab is the cross-sectional area of the beam. Clearly, pab can be interpreted as thetotal force applied by the beam, because p is the force intensity (the pressure) and Iab can beconsidered the total energy associated with the beam. The parameter ar provides us with a wayof appropriately distributing or ‘lumping’ the amount of energy in the overall wave (beam)into rays.

It is relatively straightforward to track quantities of interest related to the rays that comprisethe p-wave; for example, the average ray position may be defined as

rxdef= 1

Nr

Nr∑q=1

rxq, rydef= 1

Nr

Nr∑q=1

ryq, and rzdef= 1

Nr

Nr∑q=1

rzq . (4.4)

Here rxq , ryq and rzq are the individual ray positions. The primary quantity of interest isthe behaviour of the propagation of the (normalized) acoustical energy retained in the beam,defined similarly:

Ixdef= 1

Io

Nr∑q=1

Iq · ex, Iydef= 1

Io

Nr∑q=1

Iq · ey and Izdef= 1

Io

Nr∑q=1

Iq · ez, (4.5)

where Iq are the individual ray contributions.If we consider damage to be the debonding of a granule from the aggregate, a relatively

straightforward way to track such damage is to determine the amount of energy absorbed byeach individual granule (figure 2). The assumption here is that the amount of energy absorbedis available to break bonds that bind the granules to the matrix. It is reasonable to assume thatthe rate of damage increase is proportional to the rate of energy absorbed. For each granule,one can parametrize damage by determining all rays that come in contact with each granule(at a moment in time)

α ∝rays∑q=1

I absq ar . (4.6)

Here I absq ar is the rate of energy absorbed from a ray, interacting with a granule, and a value of

α(t = 0) = 1 indicates an undamaged granule (perfectly bonded), whereas α(t) = 0 indicatesa completely damaged (debonded) granule. One may simply write

α = CI absar ⇒ α(t + t) = α(t) + CI absar, (4.7)

S442 T I Zohdi

where C represents the damage per unit energy and where the absorbed energy is the differencebetween the incident and the reflected rays, I abs = I

rayi − I

rayr = (1 − R)I

rayi . The

computational algorithm is as follows, starting at t = 0 and ending at t = T :

(1) COMPUTE RAY REFLECTIONS (FRESNEL RELATION(equation(3.9))

(2) COMPUTE ENERGY ABSORBED BY EACH GRANULE :

I abs = (Irayi − I ray

r ) = (1 − R)Irayi

(3) COMPUTE DAMAGE SURROUNDING EACH GRANULE :

α(t + t) = α(t) + C∑

I absar

(4) INCREMENT ALL RAY POSITIONS :

rq(t + t) = rq(t) + tvq(t), q = 1, . . . , RAYS

(5) GO TO (1) AND REPEAT WITH (t = t + t) (4.8)

The time step size t is dictated by the size of the granules. A somewhat ad hoc approach isto scale the time step size according to t ∝ ξb

‖v‖ , where b is the radius of the granules, ‖v‖ isthe magnitude of the velocity of the rays and ξ is a scaling factor, typically 0.05 � ξ � 0.1.This ensures that meaningful interactions are not skipped.

5. Numerical examples

5.1. System parameters

As an example, we consider a group of Np randomly dispersed spherical granules (particles),of equal size, generated within a spherical (aggregate) domain of diameter D = 10−2 m. Theratio of granule diameter, d , to total domain diameter, D, was d/D = 0.05, correspondingto an individual granule diameter of d = 500 µm. The ratio of refractive indices was set to

cdef= ci

ct= 1500/4250, where ci = 1500 m s−1 (water) and ct = 4250 m s−1. We make use

of typical stone data. We have assumed that the fully fluid-saturated binding material hasapproximately the same acoustical impedance as the surrounding fluid (water). The densitieswere set to ρi = 1000 kg m−3 (water) and ρt = 1750 kg m−3 (again for a typical stone). Thisleads to a ratio of acoustical impedances of A = 4.96 and, consequently, a reflectivity, at azero angle of incidence, from equation (3.10), of

R(θ = 0) =(

4.96 − 1

4.96 + 1

)2

= 0.441. (5.1)

Thus, at a zero angle of incidence, the amount of energy transmitted is approximately1 − 0.441 = 0.559.

In order to generate the random granule positions, the classical random sequential addition(RSA) algorithm was used to place nonoverlapping granules into the domain of interest (Widom1966). This algorithm was adequate for the volume fraction range of interest (under 30%),because it can deliver up to a limit of approximately 38%. If higher volume fractions aredesired, particularly if nonspherical particles are involved, more sophisticated algorithms,such as the equilibrium-based Metropolis algorithm, can be used. See Torquato (2002) fora detailed review of such methods. For much higher volume fractions, effectively packing(and ‘jamming’) granules to theoretical limits (approximately 74%), a new novel class ofmethods, based on simultaneous granular flow and growth, has been developed by Torquatoand coworkers (see, for example, Kansaal et al (2002) and Donev et al (2004), (2005a) and

Large-scale P-wave simulation S443

(2005b)). Due to the relatively moderate volume fraction range of interest in the present work,this class of methods was not employed. However, such methods, which are relatively easy toimplement, computationally efficient and robust, are recommended to generate high volumefractions.

5.2. Results and discussion

In the example calculation, we used Np = 1000 granules and Nr = 1600 rays, initiallyarranged in a square (beam cross-section) 40 × 40 pattern (figure 4). This system providedstable results, i.e. increasing the number of rays and/or the number of granules beyond theselevels resulted in negligibly different overall system responses. We note successive refinementof the grid to 100 × 100 rays (10000 total) was found to produce no noticeable differencecompared with the 40 × 40 result. Thus, we can consider, for all practical purposes, thatthe results are independent of the ray-grid density. One can consider the representation of abeam by multiple rays as simply taking a large ‘sampling’ of the diffraction by the beam (wavefront) over the portion of the scatterer where the beam is incident, as opposed to a discretizationtechnique.

The initial energy of the wave was set to I (t = 0) = Io = 102 J m−2. The initial energyfor each ray was calculated as Iab/Nr , where Nr = 40 × 40 = 1600 was the number of raysin the beam and ab = (10−2 m) × (10−2 m) = 10−4 m2 was the cross-sectional area of thebeam. The damage parameter was calibrated by (i) knowing the amount of energy per unitarea needed to debond two materials, denoted as γ , (ii) realizing that equation (4.7) implies

α = C∑

I absar (5.2)

and (iii) defining for a perfectly bonded granule α = 1 and for a completely debonded granuleα = 0. This yields

α = (0 − 1) = Cγ 4πb2 ⇒ C = − 1

γ 4πb2. (5.3)

The chosen value for γ was γ = 102 J m−2, which is in the range of values for lower end (interms of strength) industrial adhesives. The amount of damage incurred in the system scaleslinearly with C (which depends inversely on γ ), thus making it a relatively natural parameterto use in such modelling.

Figure 3 shows the components of the average ray position and the components ofaverage energy versus time. The sequence of frames in figure 4 show the propagation ofthe wave through the system. On the proximal side of the object the damage is clearly greater.Figure 5 indicates the damage distribution, grain by grain, which is controlled by the energyabsorption. Clearly, if repeated wave fronts were to be applied, the subsurface would becomemore damaged, provided that the top surface were to spall away. On an algorithmic level,within the simulation, one could remove the highly damaged (debonded and spalled away)layers of granules after each front passes through the object, leading to a temporal model ofdegradation.

Remark 1. It is important to remark that a relatively natural algorithmic parallelism is possiblewith this computational technique since, within a time step, the rays do not explicitly interact.There are two natural ways to proceed in this direction: (1) assigning each processor its share ofthe rays and checking which particles interact with those rays or (2) assigning each processorits share of particles and checking which rays interact with those particles. After all thevarious contacts between rays and particles have been determined, then post processing canbe performed for the state of the particles (damage).

S444 T I Zohdi

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

AV

ER

AG

E P

OSI

TIO

N (

M)

TIME (MICRO-SEC)

RXRYRZ

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

NO

RM

AL

IZE

D E

NE

RG

Y

TIME (MICRO-SEC)

Ix/||I(0)||Iy/||I(0)||Iz/||I(0)||

||I(t)||/||I(0)||

Figure 3. The components of the average ray position and the components of average energyversus time.

Remark 2. The use of the parameter C (which stems from γ ) to relate the absorbed energyto the extent of damage is a somewhat phenomenological approach. A much more indepthanalysis would involve correlating the extent of debonding with the growth of microscalecracks in the matrix material. The growth, propagation and coalescence of such cracks isnontrivial, and we refer the reader to the extensive works of Kachanov (1980, 1992, 1993).These works analyse similar problems in general solids related to the effective elasticity inmaterials containing defects, such as cracks.

6. Closing statements

The goal of this work was to develop a computational approach that is capable of tracking thedistribution of energy and damage induced by p-waves within a class of materials composedof an agglomeration of hard granules in a softer (fluid-saturated), binding matrix of low shear

Large-scale P-wave simulation S445

0.980.960.940.920.90.880.860.840.820.80.78

0.980.960.940.920.90.880.860.840.820.80.78

0.980.960.940.920.90.880.860.840.820.80.78

0.980.960.940.920.90.880.860.840.820.80.78

0.980.960.940.920.90.880.860.840.820.80.78

0.980.960.940.920.90.880.860.840.820.80.78

Figure 4. From left to right and top to bottom, the progressive movement of rays comprising abeam. The vector lengths indicate the energy associated with the ray and the colour bars indicatethe value of α.

modulus. The damage to the binding matrix was assumed to be a function of the absorbedenergy, which was tracked, granule by granule. Because the wavelength of the p-waves wasconsidered small compared with the granular scatterers, geometric ray-tracing theory wasapplicable. This technique was used to determine the amount of propagating incident energythat was reflected and the amount that was absorbed by each granule. Three-dimensionalexamples were given to illustrate the method. To conclude, there are limitations in the useof ray theory. In particular, for small granules with size on the order of the wavelength,ray theory is inappropriate. Limitations of ray tracing are elaborated upon by Elmore andHeald (1985).

The paper focused only on the propagation of p-waves in the structure. The energy thatwas absorbed was assumed ‘available’ to damage the binding material. Clearly, not all ofthe absorbed energy would be ‘converted’ into damage; however, this serves as a simpleconservative estimate of the effects. The purpose of the present work was only to isolate oneaspect of the complex series of events associated with shock-type loading of agglomerated

S446 T I Zohdi

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

AV

ER

AG

E D

AM

AG

E

TIME (MICRO-SEC)

DAMAGE

0.75

0.8

0.85

0.9

0.95

1

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005

DA

MA

GE

GRAIN DISTANCE FROM CENTER (Meters)

Figure 5. The average damage (α) and the grain by grain damage distribution.

microstructures—mainly the determination of the energy absorbed and the distribution of theaccumulated damage that would result. To illustrate the overall energy tracking process anddamage distribution, the damage was assumed to follow a simple accumulated damage law.The damage was not explicitly introduced in the mechanical behaviour, but was simply post-processed from the energy absorption. It was assumed that the material properties governingthe propagation of the p-waves were negliglibly affected by the damage. However, such effectscould be added in subsequent works. Aside from these considerations, much work needs to bedone to consider more realistic morphologies and possible shear effects (s-waves) in the solid.Finally, we emphasize that the incremental damage to the material surrounding each granule

Large-scale P-wave simulation S447

was scaled from the incremental energy that was absorbed; the constant of proportionality(C) needs to be calibrated experimentally for different kinds of material, once γ is known.A standard peel test can be used to determine γ , which is the amount of energy absorbed inmaking a unit area of crack, which is a standard measure of the strength of an adhesive.

Acknowledgments

The author would like to thank Professor Andrew Szeri for very productive technicaldiscussions during the preparation of this paper.

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