A theoretical analysis of the strength of composite gels with rigid filler particles.
INTRODUCTION
Gel systems are commonly used in the food industry. In order to control the mechanical performance (including strength) of these systems, filler particles may be dispersed 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 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 the situation where filler and matrix are incompressible, 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].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 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 (0.5 [is less than] S [is less than] 1). Based on a similar concept, RossMurphy 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 of the stress distribution in the gel matrix. STRESS ANALYSIS SelfConsistent Model We consider a composite specimen under uniaxial 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 a local region of the composite as shown in Fig. 2. This model also contains three parts: (1) the kernel sphere (filler particle), (2) the spherical shell with outer radius [R.sub.o] (matrix gel), (3) the surrounding infinite domain (imaginary medium with proper elastic moduli 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 selfconsistent conditions, as will be given later. Basic Equations Under uniaxial loading, the local model will be deformed axisymmetrically. If the spherical coordinate system (r, [Theta], [Phi]), as shown in Fig. 3, is taken to describe the deformation, then we have the geometric relations, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in which [u.sub.r], [u.sub.[Theta]] denote the displacements, [[Epsilon].sub.r], [[Epsilon].sub.[Theta]], [[Epsilon].sub.[Phi]], [[Epsilon].sub.r[Theta]] denote the strain. For gel materials, the compressibility 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [Sigma], is a hydrostatic stress to be determined. G stands for shear modulus of the material. The equilibrium equation is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] General Solution We search for the solution to Eqs 5 to 8 in the following form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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) where 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, see Arfken (6), V([Theta]) = [V.sub.k1]([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. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Evidently if k is replaced by (k+1), Eq 11 still holds. Therefore besides Eq 9 we have the following solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [a.sub.n], [b.sub.n], [d.sub.n] are constants. Using Eqs 5, 7, 8 and 18 we can obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in which [[Sigma].sub.o], is a constant. For uniaxial loading, only the terms with [V.sub.2] ([Theta]) are needed, then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The stress for the kernel and the surrounding medium can be stated in a similar manner, for the sake of brevity 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], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From Eqs 25 and 26 we can obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Further, at r = O and r = [Infinity] we have the supplementary boundary conditions: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From Eq 29 we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Using Eq 30 we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in which [Beta] = G/G  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) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [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 selfconsistent 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Using Eqs 39 and 24, and the basic equations, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From Eqs 32 and 40 we have b = 0 therefore [[Delta].sub.2] = 0, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Considering [[Delta].sub.2] = 0, we can reduce [Delta] and [[Delta].sub.1] as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 is [[Sigma].sub.r[Theta]. Making use of Eqs 21 and 32 to 37, it follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Using a similar procedure to that used previously, we can give the selfconsistent 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From Eqs 21, 22, and 51 it follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 selfconsistent 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 selfconsistent model loses its meaning. In reality, the volume fraction of filler cannot exceed that of close packing for which [[Theta].sub.F] = 0.697. STRENGTH AND FAILURE MECHANISM 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 so that the cracking process becomes tremendously complicated. For simpler cases, i.e. fiberreinforced materials and reinforced 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 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 preexisting 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 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 forces while water only bears a hydrostatic stress. Further we can presume 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].sub.cr], 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].sub.cr] 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The maximum difference of principal stresses is located at the latitude circle of [Theta] = [Pi]/4 and [Theta] = 3[Pi]/4. If we transfer to the cyclindrical coordinate system according to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then at [Theta] = [Pi]/4 and [Theta] = 3[Pi]/4 we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (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].sub.cr] the crack will initiate from point [P.sub.a] along the horizontal direction as shown in Fig. 6a. If the gel matrix is brittle 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 secondfiller 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].sub.cr] 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The maximum difference of principal stresses is located at [Theta] = 0 and [Theta] = [Pi], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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].sub.cr]/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 nonadhering interface. The model is only approximately valid for that part under compressive stress. 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then we have [[Sigma].sub.[Theta]]  [[Sigma].sub.r] = 2 [[Sigma].sub.[Infinity]] reaches [[Sigma].sub.cr]/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].sub.cr]/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. CONCLUSION 1) The stress concentration in the gel matrix near a particle has been calculated based on a selfconsistent 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 to long distances. The estimation of the real strength of a composite specimen still remains a complicated problem, especially for the compression case. REFERENCES [1.] J.N. Goodier, J. Appl. Mech., 55, 39 (1933). [2.] Y. C. Gao, J. Lelievre, and J. Tang, "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. RossMurphy 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 Publishing Co., Westport, Conn., (1983). [6.] G. Arfken, Mathematical Methods for Physicists, pp. 41776 Academic Press, New York (1966). [7.] Y. C. Gao, Theor. Appl. Fracture 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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