International Journal of Fracture 58:R51-R55, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands.

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FRACTURE STRENGTH OF BRI'I'I'LE POROUS MATERIALS

M. Zheng Physics Department, Xidian University Xi' an, 710071, People' s Republic of China X. Zheng and Z.J. Luo Materials Science and Engineering Department Northwestern Polytechnical University Xi' an, 710072, People's Republic of China In the present study, the expression reflecting the influence of porosity on the fracture strength of brittle porous materials is established based on recent studies of the elastic behavior of porous materials. Moreover, three-point-bend experiments of rectangular specimens of phenolic plastics with different porosity are carded out to test the proposed formula. It is found that the calculated results of the present theory agree with those of the experiments very well. It enriches the brittle fracture theory extremely. The brittle fracture of materials is not only one of the most important parts of mechanical behavior but the basis of the modern fracture theory as well. The philosophy of the brittle fracture theory is still based on the energy balance concept of crack propagation due to Griffith [1]. The statistical theory of fracture strength of brittle materials has been fully studied since the pioneering work of Weibull [2]. However, there may exist initial voids randomly distributed in real materials instead of a single flaw. Therefore, the fracture strength of materials may undoubtedly be influenced by voids. The computer simulation in [3] and [4] indicated that the effects of voids on the fracture strength of brittle porous materials is nonlinear and that there exists a percolation failure phenomena for the fracture strength of brittle porous materials. However, the problems mentioned above have not yet been studied comprehensively. In the present report, a formula reflecting the influence of voids on fracture strength of brittle porous solids is derived based on recent achievements in the study of elastic behavior of porous materials. As stated in the previous section there may exist initial voids randomly distributed in real materials; practical materials may also be thought of as porous materials in a sense. As to porous materials, the probability distribution of voids size can be written as follows [5],

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(1)

F(1) = f . e -I't,

where F(1) is the probability distribution function of voids size; l is the size of voids; f i s a parameter related to the volume fraction of voids, i.e., the porosity. For a material with defined porosity,f is a constant. It is clear from (1) that the voids size distribution function F(1) is normalized, i.e.,

- F(l).dl

=

£-f

.e q~.dl

= 1,

(2)

therefore, the average value of voids size 7 for a material with a given porosity can be written as i =

(3)

l . f . e - I t . dl = 1If

Equation (3) indicates thatfis in fact a parameter characterizing the average value of the voids size. According to the percolation theory [6], the average value of voids size in porous materials can be written as, (4)

.10

i=

where lo is the unit of the voids size; v is the scaling exponent for 3-dimensional solids, v = 0.85; ¢ is the porosity of porous materials; ¢c is the elastic percolation failure threshold which depends on Poisson's ratio u of materials [7]. According to the research achievements in [7], ¢o can be written as ¢c = 1 - ((1 + u)/(3(1 - u))) = 2(1 - 2u)/(3(1 - u))

(5)

In general, for metals, u 0.3, ~o = 0.381; for ceramics, u 0.2, ~o = 0.5; for polymers, u 0.33, ¢o = 0.338. The average value of l -la can be calculated as follows,

fo

Int Jouan~ of Fracture 58 (1992)

[.Tot,f o-,o yl)1

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R53

therefore, according to the brittle fracture theory proposed by Griffith early in 1920, the fracture strength of porous materials with porosity t) can be written as [1]. V i

°°=J0 --Ie-' f

l ,c

~O~ • K i c

(7)

where K~ois the fracture toughness of porous materials; ot is a coefficient concerning stress state. It is proved that the influence of porosity on fracture toughness of porous materials can be written as [8], K,c =

K,<..

(8)

• (1 - Oa3)J

where K~ is the fracture toughness of voids-free materials. Substituting (8) into (7), one obtains (9)

go Suppose o~, = ct.K,J¢~o is the fracture strength of voids-free materials, then the following is easily obtained,

ojOo =kt

)

"

(1

-Ow3)J

(io)

Equation (10) is the general expression reflecting the influence of voids on fracture strength of porous solids. For polymers, ~o = 0.338, (10) becomes, (ii)

Oc/C~o = (1 - 2.956t)) °'925 • (1 - t~w3)°'5 In order to check the validity of (11), the fracture strength of phenolic plastics with various porosity is experimentally measured by using three-pointbend rectangular specimens of 100 X 15 X 10(mm). The experiments were conducted on ZMG 1 250 three-point-bend equipment. The fracture strength % is calculated according to the following formula,

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3Pc • L ~c - 2 b ~ 2 where Pc is the peak load (or fracture load); L is the length of specimen; b and h are the width and the height of the specimens, respectively. The experimental measurement results of fracture strength of porous phenolic plastics are listed in Table 1. Also listed in Table 1 are the values of the fracture strength of the phenolic plastics predictd by using (11) and taking c~o = 60.83 N/mm 2, which is the average value of the experimental data for the phenolic plastics with ~ = 0. As may be seen, the predicted values of fracture strength of phenolic plastics agree well with the experimental results. Therefore, it can be thought that (11) expresses the nonlinear influence of the porosity on the fracture strength of brittle porous materials and the existence of the percolation failure threshold of fracture strength, which is the same as that of the elastic percolation failure. The following conclusions can be drawn: 1) The formula reflecting the influence of porosity on the fracture strength of brittle porous materials, i.e. (11), is developed and checked by the experimental results of fracture strength of phenolic plastics. Good agreement is obtained. 2) The effects of porosity on fracture strength of brittle porous solids is nonlinear. 3) The percolation threshold of fracture strength of brittle porous materials exists, and is the same as that of the threshold of elastic percolation failure. Acknowledgements: We gratefully acknowledge Senior Engineer Guojun Liu and his colleague for their great help in conducting the experiments. REFERENCES [1] H. Zhang and Y. Chen, Deformation and Fracture of Solids, Advanced Educational Press, Beijing (1989) 10, in Chinese. [2] B.R. Lawn and T.R. Wilshaw, Fracture of Brittle Solids, Cambridge University Press, Cambridge (1975). [3] Paul D. Beale and David J. Srolovitz, Physics Reviews B37(10), (1988) 5500-5507. [4] P. Ray and B.K. Chakrabarti, Solids State Communications 53(5), (1985) 477-479. [5] A. Nordgren and A. Melander, Powder Metallurgy 31(3), (1988) 189-200. [6] R. Zallen, The Physics of Amorphous Solids, Wiley-Interscience (1985). [7] M. Zheng and X. Zheng, Metallurgical Transactions 22A (1991) 507-511.

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[8] M. Zheng, "Ductile Damage and Mechanical Behavior of Materials Containing Damage", Doctorial dissertation, Northwestern Polytechnical University, Xi'an, China (1992) in Chinese. 20 August 1992

Table 1. Fracture strength of porous phenolic plastics

~--'f

0

0.067

0.015 I

51.57J 51 .oi

~, ~. 54 ~ .

~4]4~.oo 43.72 45.39.

,.'

0.121

44.1o

35.~o

__] . . . . . . . . . . . . . . . . . . . . .

~/ote: 6~ ezpr~5~e~ e,xperlmen*al

results.

Int J o u ~

of Fracture 58 (1992)