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A theoretical analysis of the strength of composite gels with rigid filler particles.


Gel systems are commonly used in the food industry. In order to control the mechanical performance (including strength) of these systems, filler fill·er 1  
One that fills, as:
a. Something added to augment weight or size or fill space.

b. A composition, especially a semisolid that hardens on drying, used to fill pores, cracks, or holes in wood, plaster,
 particles may be dispersed dis·perse  
v. dis·persed, dis·pers·ing, dis·pers·es
a. To drive off or scatter in different directions: The police dispersed the crowd.

 in the gel matrix to form composites. When analyzing the strength of composite gels, their micro structure must be taken into account. If the volume fraction of filler is small, i.e., the distances between the particles are much larger than the size of the particles, the interaction between particles can be neglected. In this case the composites be considered to be equivalent to a single particle embedded Inserted into. See embedded system.  in an infinite medium. If the embedded particle is a sphere and the interface is completely bonded, the stress concentration can be found from Goodier's work (1). Gao, et al. (2) have analyzed an·a·lyze  
tr.v. an·a·lyzed, an·a·lyz·ing, an·a·lyz·es
1. To examine methodically by separating into parts and studying their interrelations.

2. Chemistry To make a chemical analysis of.

 the situation where filler and matrix are incompressible in·com·press·i·ble  
Impossible to compress; resisting compression: mounds of incompressible garbage.

, and the interface is smoothly connected or perfectly bonded.

When the volume fraction of filler is large, the interaction between particles is significant and therefore the strength analysis is difficult. In order to consider the influence of the volume fraction of filler on the behavior of composites, Nielsen (3) gave a simplified model for a filled polymer. For the case of a perfectly bonded interface, Nielsen (3) found:

[[Epsilon 1. (language) EPSILON - A macro language with high level features including strings and lists, developed by A.P. Ershov at Novosibirsk in 1967. EPSILON was used to implement ALGOL 68 on the M-220. ].sub.B](filled)/[[Epsilon].sub.B](unfilled) = 1 - [[Phi].sub.F.sup.1/3] (1)

In which [[Epsilon].sub.B] denotes the strain at break, [[Phi].sub.F] the volume fraction of filler.

For the case of no adhesion adhesion /ad·he·sion/ (ad-he´zhun)
1. the property of remaining in close proximity.

2. the stable joining of parts to one another, which may occur abnormally.

 at the interface, Nielsen (3) gave:

[[Sigma].sub.B](filled)/[[Sigma].sub.B](unfilled) = (1 - [[Phi].sub.F.sup.2/3])S (2)

where [Sigma] denotes the stress at break, S is a coefficient coefficient /co·ef·fi·cient/ (ko?ah-fish´int)
1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities.

 (0.5 [is less than] S [is less than] 1). Based on a similar concept, Ross-Murphy and Todd (4) developed a formula to predict the relative stress at break:

[[Sigma].sub.B](filled)/[[Sigma].sub.B](unfilled) = 1 - [[Phi].sub.F.sup.1/3]/(1 - [[Phi].sub.F).sup.5/2] (3)

Equations 1 to 3 give qualitative evaluations of the effect of filler particles. The purpose of the present paper is to give a quantitative analysis Quantitative Analysis

A security analysis that uses financial information derived from company annual reports and income statements to evaluate an investment decision.

 of the stress distribution in the gel matrix.


Self-Consistent Model

We consider a composite specimen under uniaxial uniaxial /uni·ax·i·al/ (u?ne-ak´se-al)
1. having only one axis.

2. developing in an axial direction only.


1. having only one axis.

2. developed in an axial direction only.
 load as shown in Fig. 1. It is assumed that all of the particles are spheres with the same radius R, but their locations are randomly distributed. For such a structural model, it is still impossible to analyze the exact distribution of stress in the whole composite. But if our attention is focused at a typical particle and its immediate surroundings, the problem can be simplified. We divide the whole composite into three parts: (1) a typical filler particle, (2) its immediate vicinity (matrix gel), (3) the remaining domain of the composite. Based on these three parts, we construct a model to simulate simulate - simulation  a local region of the composite as shown in Fig. 2. This model also contains three parts: (1) the kernel The nucleus of an operating system. It is the closest part to the machine level and may activate the hardware directly or interface to another software layer that drives the hardware.  sphere (filler particle), (2) the spherical spher·i·cal
Having the shape of or approximating a sphere; globular.
 shell with outer radius [R.sub.o] (matrix gel), (3) the surrounding infinite domain (imaginary medium with proper elastic elastic

Of or relating to the demand for a good or service when the quantity purchased varies significantly in response to price changes in the good or service.
 moduli In theoretical physics, moduli are scalar fields whose different values are equally good (each one such scalar field is called a modulus). The reason is that the potential energy for moduli is constant, which can be guaranteed, for example, by supersymmetry (with  that can reflect the nature of the composites). The kernel, together with the shell, is called a cell of the composite. In order for the model to represent real composites, it is required that the ratio of R to [R.sub.o] is related to the volume fraction of filler

R/[R.sub.o] = [[Phi].sub.F.sup.1/3] (4)

Further, the elastic moduli of the imaginary medium must be properly specified by some conditions that are called self-consistent conditions, as will be given later.

Basic Equations

Under uniaxial loading, the local model will be deformed de·formed
Distorted in form.
 axisymmetrically. If the spherical coordinate system spherical coordinate system

In geometry, a coordinate system in which any point in three-dimensional space is specified by its angle with respect to a polar axis and angle of rotation with respect to a prime meridian on a sphere of a given radius.
 (r, [Theta], [Phi]), as shown in Fig. 3, is taken to describe the deformation deformation /de·for·ma·tion/ (de?for-ma´shun)
1. in dysmorphology, a type of structural defect characterized by the abnormal form or position of a body part, caused by a nondisruptive mechanical force.

, then we have the geometric relations,

[MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]

in which [u.sub.r], [u.sub.[Theta]] denote de·note  
tr.v. de·not·ed, de·not·ing, de·notes
1. To mark; indicate: a frown that denoted increasing impatience.

 the displacements, [[Epsilon].sub.r], [[Epsilon].sub.[Theta]], [[Epsilon].sub.[Phi]], [[Epsilon].sub.r[Theta]] denote the strain.

For gel materials, the compressibility com·press·i·ble  
That can be compressed: compressible packing materials; a compressible box.

 can be neglected (5) (at least for uniaxial load), then

[[Epsilon].sub.r] + [[Epsilon].sub.[Theta]] + [[Epsilon].sub.[Phi]] = 0 (6)

Further, the stress can be expressed as


where [Sigma], is a hydrostatic hy·dro·stat·ic or hy·dro·stat·i·cal
Of or relating to fluids at rest or under pressure.


pertaining to a liquid in a state of equilibrium or the pressure exerted by a stationary fluid.
 stress to be determined. G stands for shear modulus shear modulus

See under modulus of elasticity.
 of the material.

The equilibrium equation is


General Solution

We search for the solution to Eqs 5 to 8 in the following form




then Eq 16 is automatically satisfied. Substituting Eqs 9 and 10 into Eqs 5, 7, and 8 we eventually obtain

[L + (k - 1)k][L + (k+1)(k+2)]V([Theta]) = 0 (11)


L = [d.sup.2]/[d[[Theta].sup.2] + cot [Theta] d/d[Theta] - 1/[sin.sup.2][Theta] (12)

The General solution to Eq 11 is the associated Legendre polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a , see Arfken (6),

V([Theta]) = [V.sub.k-1]([Theta]) or V([Theta]) = [V.sub.k+1]([Theta]) (13)

in which

[V.sub.m]([Theta]) = sin [Theta][P'.sub.m] (cos [Theta]) (14)

where [P.sub.m]([Xi]) denotes the Legendre polynomial, i.e.


Evidently if k is replaced by -(k+1), Eq 11 still holds. Therefore besides Eq 9 we have the following solution


in which V is still given by Eq 13, but U* is

U* = 1/k - 1 (dV/d[Theta] + V cot [Theta]) (17)

From Eqs 9, 10, 13, 16, and 17 the general form of displacements can be written as


where [a.sub.n], [b.sub.n], [d.sub.n] are constants.

Using Eqs 5, 7, 8 and 18 we can obtain


in which [[Sigma].sub.o], is a constant.

For uniaxial loading, only the terms with [V.sub.2] ([Theta]) are needed, then we have


It should be emphasized that the solution, Eqs 20 to 22, is valid for all of the three domains: kernel, shell, and the imaginary surrounding medium, but that the constants G, a, b, c, d, and [[Sigma].sub.o], have different values. In order to distinguish one from another, the quantities related to the kernel and the surrounding medium will be capped by a bar and wave line respectively. For example, for the kernel and surrounding medium we have


The stress for the kernel and the surrounding medium can be stated in a similar manner, for the sake of brevity Brevity
Adonis’ garden

of short life. [Br. Lit.: I Henry IV]


symbolic of transitoriness of life. [Art: Hall, 54]

cherry fair

cherry orchards where fruit was briefly sold; symbolic of transience.
 the equations are not given.

Perfectly Bonded Interface

Now we consider the case where there is perfect adhesion between the filler particle and the gel matrix. Therefore the displacements and stresses must be continuous at r = R and r = [R.sub.o],


From Eqs 25 and 26 we can obtain


Further, at r = O and r = [Infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ] we have the supplementary boundary conditions boundary condition
n. Mathematics
The set of conditions specified for behavior of the solution to a set of differential equations at the boundary of its domain.


From Eq 29 we obtain


Using Eq 30 we obtain


in which

[Beta] = G/G G/G Gotta Go
G/G Ground/Ground (air traffic management)
G/G Ground-to-Ground
 - 1 (34)

From Eqs 27 and 31 to 33 we can obtain the constants c, d, a, b, but the expression is rather complicated. In order to give a clear discussion on the stress concentration, we only consider an extreme case where the filler particle is absolutely rigid, i.e. G = [Infinity], then we have the following result

c = d = 0 (35)



in which



[Delta] = R/[R.sub.o] (38)

Equations 32 to 38 give the complete solution of the problem but there is an unknown constant [Beta] to be determined from the self-consistent condition. This condition requires that G must take a proper value so that the average value of [[Sigma].sub.z] in the cell equals the nominal stress [[Sigma].sub.[Infinity]]. Then we have


Using Eqs 39 and 24, and the basic equations, we obtain


From Eqs 32 and 40 we have b = 0 therefore [[Delta].sub.2] = 0, then


Considering [[Delta].sub.2] = 0, we can reduce [Delta] and [[Delta].sub.1] as


The important problem is the stress concentration in The gel matrix. It can be shown from Eqs 21 and 22 that the severest stress concentration is near the interface between the particle and matrix, where the maximum shear stress shear stress
See shear.

shear stress

A form of stress that subjects an object to which force is applied to skew, tending to cause shear strain.
 is [[Sigma].sub.r[Theta]. Making use of Eqs 21 and 32 to 37, it follows


The stress concentration coefficient is defined as

[S.sub.c] = 2 [[Sigma].sub.r[Theta]]/[[Sigma].sub.[Infinity]] (44)

Using Eqs 43 and 44 we can calculate the function [S.sub.c]([[Phi].sub.F]) as shown in Fig. 4. The relative strength of the composites is defined as [R.sub.s] = = 1/[S.sub.c], this is shown in Fig. 5 by a solid curve. The calculated result from the formula of Ref. 4 is also shown in Fig. 5 by a dotted line.

Smoothly Connected Interface

Now we consider the case where the interface between the particle and matrix is unbonded but connected. The friction governing slipping between the two composite phases can be neglected. Correspondingly we assume that at r = [R.sub.o], the interface is smoothly connected, but at r = R the interface is completely bonded. For simplicity, we still assume that G = [Infinity], then we have

The boundary conditions for the shell at r = R become

[u.sub.r] = 0, [[Sigma].sub.r[Theta]] = 0 (46)

From Eqs, 46, 20 and 21 we obtain

a = 1/3 [R.sup.3]c, b = 2/5 [R.sup.7]d (47)

The boundary conditions at r = [R.sub.o] are still given by Eq 26. So we still have the same result as Eq 33. Substituting Eq 33 into Eq 47, we formally obtain the same form of expression as Eq 36 but the [delta], [[delta].sub.1], and [[delta].sub.2] are


Using a similar procedure to that used previously, we can give the self-consistent condition that yields b = o ([[delta].sub.2] = 0) from which we have

[Beta] = 0 or

[Beta] = - 5/8 (8[[Delta].sup.10] - [[Delta].sup.3] - 7)(2[[Delta].sup.10] - 5[[Delta].sup.3] - [2).sup.-1] (49)

when [Delta] [right arrow] 0, it is required that [Beta] [right arrow] 0, therefore only the root [Beta] ?? 0 is reasonable. Noting Eq 34 we have

G = G (50)

Equation 50 means that the surrounding medium possesses the same shear modulus as the shell (gel matrix), so the local model becomes a simpler one that only contains a particle embedded in an infinite gel matrix.

Further noting [Beta] = 0, from Eqs 36, 48, 32, and 47 we obtain


From Eqs 21, 22, and 51 it follows


when [Theta] = 0, r = R the stress concentration reaches the maximum value [[Sigma].sub.r]/[[Sigma].sub.[Infinity]] = 3. This value is independent of [Delta] (=[[Theta].sub.F.sup.1/3]).

It should be pointed out that the above self-consistent analysis is only valid for relatively low concentrations of filler. It only predicts some tendencies concerning the influence of the volume fraction of filler. When the volume fraction of filler is high the self-consistent model loses its meaning. In reality, the volume fraction of filler cannot exceed that of close packing for which [[Theta].sub.F] = 0.697.


For most solid material, failure is caused by cracking. Either crack initiation or crack growth must obey some rules (so called criterion) in which both the stress state and the material nature are involved. On the other hand, even a simple specimen of composite is actually a complex structure in which the stress distribution and material nature are not homogeneous The same. Contrast with heterogeneous.

homogeneous - (Or "homogenous") Of uniform nature, similar in kind.

1. In the context of distributed systems, middleware makes heterogeneous systems appear as a homogeneous entity. For example see: interoperable network.
 so that the cracking process becomes tremendously complicated. For simpler cases, i.e. fiber-reinforced materials and reinforced concrete reinforced concrete

Concrete in which steel is embedded in such a manner that the two materials act together in resisting forces. The reinforcing steel—rods, bars, or mesh—absorbs the tensile, shear, and sometimes the compressive stresses in a concrete
, the criterion for crack growth have been given by Gao (7) and Gao, et al (8). As for particle composites, when the interaction between the cracks and particles is considered, the criterion of crack growth has not been reported. But despite the mathematical complexity it is possible to give some qualitative discussion on the failure mechanism of a particulate par·tic·u·late
Of or occurring in the form of fine particles.

A particulate substance.


composed of separate particles.
 composite specimen. In this discussion, the exact details of crack growth will be neglected, and the argument is based on crack initiation caused by stress concentration.

Criterion of Crack Growth

Gels are not like other engineering material (concrete, ceramic) which always contain pre-existing micro cracks. Generally, the cracks in a gel matrix are initiated by stress. The commonly used criterion of crack initiation is that when a certain critical value of the tensile stress tensile stress

See under axial stress.
 is reached, the crack will initiate perpendicular to the stress direction. For gel materials we must consider their intrinsic structure, they contain polymer chains and water. Basically, polymer chains sustain tensile tensile,
adj having a degree of elasticity; having the ability to be extended or stretched.
 forces while water only bears a hydrostatic stress. Further we can presume pre·sume  
v. pre·sumed, pre·sum·ing, pre·sumes
1. To take for granted as being true in the absence of proof to the contrary: We presumed she was innocent.
 that for many gels the crack initiation is caused by the breakage of chains, and the occurrence of breakage depends upon the tensile force in chains but not upon the hydrostatic stress. Based on this consideration we propose the following criterion for crack initiation: when the difference [[Sigma].sub.max] - [[Sigma].sub.min] reaches a certain critical value [[Sigma]], the crack will initiate along the surface on which [[Sigma].sub.max] acts. [[Sigma].sub.max] and [[Sigma].sub.min] denote the maximum and minimum principal stresses, [[Sigma]] is a material constant. This criterion looks like the commonly used maximum shear stress criterion as in engineering but possesses a different meaning, the latter means that the failure happens in the plane of maximum shear stress.

Completely Bonded Interface

For simplicity, we only consider the case of low concentration of filler, i.e. [[Delta] [arrow right] 0, then from Eqs 21, 22 and 32 to 38 it follows that


The maximum difference of principal stresses is located at the latitude latitude, angular distance of any point on the surface of the earth north or south of the equator. The equator is latitude 0°, and the North Pole and South Pole are latitudes 90°N and 90°S, respectively.  circle of [Theta] = [Pi]/4 and [Theta] = 3[Pi]/4. If we transfer to the cyclindrical coordinate system coordinate system

Arrangement of reference lines or curves used to identify the location of points in space. In two dimensions, the most common system is the Cartesian (after René Descartes) system.
 according to according to
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.



then at [Theta] = [Pi]/4 and [Theta] = 3[Pi]/4 we have


(a) Under tension, [[Sigma].sub.[Infinity]] [is greater than] 0

In this case, [[Sigma].sub.z] [is greater than] 0, [[Sigma].sub.z] - [[Sigma].sub.p] = 2.5[[Sigma].sub.[Infinity]]. When [[Sigma].sub.[Infinity]] reaches 0.4[[Sigma]] the crack will initiate from point [P.sub.a] along the horizontal direction as shown in Fig. 6a. If the gel matrix is brittle (jargon) brittle - Said of software that is functional but easily broken by changes in operating environment or configuration, or by any minor tweak to the software itself. Also, any system that responds inappropriately and disastrously to abnormal but expected external stimuli; e.  enough, once a crack is initiated it will grow continuously because it is driven by tension. The crack growth will cause the specimen to fail. Therefore the strength of the specimen may be decreased by 0.4 times that of the plain gel. Of course, if a crack encountered a second-filler particle, it will kink but the specimen also will be broken.

(b) Under compression, [[Sigma].sub.[Infinity]] [is less than] 0

In this case, [[Sigma].sub.p] [is greater than] 0, [[Sigma].sub.p] - [[Sigma].sub.z] = -2.5[[Sigma].sub.[Infinity]]. When [[Sigma].sub.[Infinity]]. reaches - 0.4[[Sigma]] the crack will initiate from a point [P.sub.b], along the vertical direction as shown in Fig. 6b. But this crack cannot grow to a long distance because the tensile stress [[Sigma].sub.p] will be damped to zero when the crack tip is far away from the particle. Therefore, even when a crack is initiated the strength of the specimen may not be decreased as much as in the tension case.

Smoothly Connected Interface

As mentioned before, Eq 52 is valid for all of the values of [Delta], then we obtain


The maximum difference of principal stresses is located at [Theta] = 0 and [Theta] = [Pi], where


a) Under tension, [[Sigma].sub.[Infinity] [is greater than] 0

If the interface is really kept connected, then at [Theta] = 0 (or [Pi]), [[Sigma].sub.r] - [[Sigma].sub.[Theta]] = 3[[Sigma].sub.[Infinity]]. When [[Sigma].sub.[Infinity] reaches [[Sigma]]/3, the crack will initiate from point [P*.sub.a] as shown in Fig. 7a. But the smoothly connected interface is just a simplified model for a non-adhering interface. The model is only approximately valid for that part under compressive stress Compressive stress is the stress applied to materials resulting in their compaction (decrease of volume). When a material is subjected to compressive stress, then this material is under compression. Usually, compressive stress applied to bars, columns, etc. leads to shortening. . When the interface sustains tensile stress, the filler and matrix will separate from each other so that Eq 57 is not valid. In order to reflect the reality, instead of [Theta] = 0 (or [Pi]), we consider the equator [Theta] = [Pi]/2, where


Then we have [[Sigma].sub.[Theta]] - [[Sigma].sub.r] = 2 [[Sigma].sub.[Infinity]] reaches [[Sigma]]/2, the crack will initiate from [P.sub.a] along the horizontal direction as shown in Fig. 7a. Therefore, the strength of the specimen will be decreased to 0.5 times that of a pure gel.

b) Under compression, [[Sigma].sub.[Infinity]] [is less than] 0

From Eq 57 we can see, at [Theta] = 0 (or [Pi]) [[Sigma].sub.[Theta]] - [[Sigma].sub.r] = -3[[Sigma].sub.[Infinity]]. When [Sigma].sub.[Infinity]] reaches - [[Sigma]]/3, the crack will initiate from point [P.sub.b] along the vertical direction as shown in Fig. 7b. But this crack cannot grow because the energy release rate at the crack tip is zero. On the other hand, this kind of crack must have some influence on the strength of the specimen. For this case we cannot give a simple prediction.


1) The stress concentration in the gel matrix near a particle has been calculated based on a self-consistent model. For the completely bonded interface the maximum stress concentration factor depends on the volume fraction of filler. When [[Phi].sub.F] [arrow right] 0 the maximum factor is 2 [[Sigma].sub.r[Theta]/[[Sigma].sub.[Infinity]] = 2.5, when [[Phi].sub.F] [arrow right] 1 the factor is zero.

For the smoothly connected interface, the stress concentration is independent of the volume fraction of the filler. The maximum stress concentration is at [Theta] = 0, r = R, where [[Sigma].sub.r = 3[[Sigma].sub.[Infinity]].

2) The influence of stress concentration on the strength of a specimen depends on the loading condition. Under tension the cracks are easily initiated and propagated. Under compression, the tensile stress is relatively small, so cracks are not easy to initiate. Even when a crack exists along the vertical direction, it still cannot propagate prop·a·gate
1. To cause an organism to multiply or breed.

2. To breed offspring.

3. To transmit characteristics from one generation to another.

 to long distances. The estimation of the real strength of a composite specimen still remains a complicated problem, especially for the compression case.


[1.] J.N. Goodier, J. Appl. Mech., 55, 39 (1933).

[2.] Y. C. Gao, J. Lelievre, and J. Tang tang, in zoology
tang: see butterfly fish.
, "A Theoretical Analysis of Stress Concentrations in Gels Containing Low Concentration of Spherical Filler Particles," to be published.

[3.] L. E. Nielsen, J. Appl. Polym. Sci, 10, 97 (1966).

[4.] S. B. Ross-Murphy and S. Todd, Polymer, 24, 481 (1983).

[5.] D. D. Hamann, Structure Failure in Solid Foods, in: Physical Properties of Foods, p. 532, M. Peleg and E. B. Bagley, eds., AVI (Audio Video Interleaved) A Windows multimedia video format from Microsoft. It interleaves standard waveform audio and digital video frames (bitmaps) to provide reduced animation at 15 fps at 160x120x8 resolution. Audio is 11,025Hz, 8-bit samples.  Publishing Co., Westport, Conn., (1983).

[6.] G. Arfken, Mathematical Methods for Physicists Below is a list of famous physicists. Many of these from the 20th and 21st centuries are found on the list of recipients of the Nobel Prize in physics. A
  • Ernst Karl Abbe — Germany (1840–1905)
  • Derek Abbott — Australia (1960- )
, pp. 417-76 Academic Press, New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of

[7.] Y. C. Gao, Theor. Appl. Fracture fracture, breaking of a bone. A simple fracture is one in which there is no contact of the broken bone with the outer air, i.e., the overlying tissues are intact. In a comminuted fracture the bone is splintered.  Mech., 11, 147 (1989).

[8.] Y. C. Gao, Y. C. Loo, and D. G. Montgomery, Theor. Appl. Fracture Mech, 17, 121 (1992).
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Author:Gao, Y.C.; Lelievre, J.
Publication:Polymer Engineering and Science
Date:Sep 1, 1994
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