Jul 14, 1986 - Weibull distribution function is observed to be a good representation of the hybrid break- ing strain. The estimated ultimate breaking strains ...
The Tensile Behavior of Intraply Hybrid Composites II: Micromechanical Model S. J. FARIBORZ
AND
D. G. HARLOW
Department of Mechanical Engineering and Mechanics Lehigh University Bethlehem, PA 18015 (Received July 14, 1986) (Revised November 5, 1986)
ABSTRACT The standard shear-lag model is modified to allow for the determination of the stress concentration factors for any unidirectional intraply hybrid composite with random as well as deterministic arrangements of the filaments. Utilizing this micromechanical model in conjunction with the chain-of-bundles probability model, hybrids with different volume ratios of the constituents are analyzed. The method of analysis is the numerical Monte Carlo simulation technique. The well known "hybrid effect" for strains is confirmed. The Weibull distribution function is observed to be a good representation of the hybrid breaking strain. The estimated ultimate breaking strains show a significant improvement over previously cited results, and they appear to be more characteristic of experimental observations.
1. INTRODUCTION HE EXTENSIVE APPLICATION OF THE INTRAPLY HYBRID COMPOSITE MATE-
Trials
which consists of matrix stiffened by two or more types of filaments is due to their several advantages over metals, polymers, laminated composites and composite materials fabricated with a single kind of fiber reinforcement. Material and manufacturing costs can be drastically reduced with a proper combination of fibers. The strength to weight ratio of composites is substantially higher than metals. The resistance of composites to corrosion allows for the use of less vulnerable components in harsh environments. The greater rigidity and thermal degradation make the composites more durable than polymers. Moreover, from the design standpoint the most important aspect of intraply hybrids is that they offer a broad spectrum of material properties. By changing the volume ratio and the degree of dispersion of the constituent fibers in the matrix, hybrids with various mechanical properties may be produced. The other key factor is the selection of the type of matrix and fibers to be used. A variety of matrices can be chosen from either thermoplastic or thermosetting materials. Common fibers 856
Journal
of COMPOSITE MATERIALS,
Vol.
0021-9983/87/09 0856-20 $4.50/0 © 1987 Technomic Publishing Co., Inc
21-September
1987
857
used in composites are glass, carbon, boron, or kevlar. Given the enormous number of possibilities for manufacturing hybrid composites, many desirable material characteristics can be obtained. Survey articles such as [1-4] should be consulted for the various applications and mechanical properties of hybrids. One of the major obstacles in the wide spread exploitation of hybrid composites is the uncertainty involved in reliability estimations. If the potential use of hybrid composites is to be realized it is imperative to clearly understand the failure mechanism under the applied tractions. The &dquo;hybrid effect&dquo; which may be defined as the increase in failure strain of hybrids with respect to the parent composite having the lowest ultimate strain must be characterized. Unfortunately, a good estimate of the material strength is not obtained from the rule-of-mixtures. In general the problem has two major aspects: the statistical aspect to model the scatter which exists in the ultimate stress/strain of composite materials observed in the experimental data and the micromechanical aspect to model the failure mechanism. With the application of the chain-of-bundles probability model some progress in describing the scatter in hybrid strength has been made. Intraply hybrid composites with the volume ratio of one to one and full dispersion of the filaments were the subject of study in [5]. The main characteristic of the foregoing effort is that once the model has been established the resulting mathematical problem can be solved explicitly. Nevertheless computations become onerous for intraply hybrids with different volume ratios, degrees of dispersion, and typical populations of filaments. To overcome this burden numerical simulations were made in order to approximate the model. Monte Carlo simulation was utilized by Oh [6] and Fukuda and Chou [7] to predict the reliability of a composite material reinforced by only five similar fibers. This obviously was not sufficient to reflect the salient features of the mechanical behavior of a laboratory specimen. In [8] a Monte Carlo simulation was employed, and the probability of failure for intraply hybrid composites subjected to a uniform tensile load for typical laboratory size specimens was estimated. It was demonstrated that even for a rather moderate sample size the Monte Carlo simulation for the reliability of hybrid composites can be estimated by the Weibull cumulative distribution function (c.d. f. ) with a high degree of confidence. For simple tensile loading it suffices to consider the hybrid as a discrete system so that the micromechanical chain-of-bundles model is acceptable. The basic assumption is that the applied load is originally distributed uniformly among the reinforcing filaments. This assumption implies the negligible tensile resistance of the matrix, and it considerably simplifies the problem. However, difficulties arise when the failure of one or more fibers occurs. The redistribution of the load previously carried by the broken fibers among the remaining fibers must be specified. This requires a micromechanical model for the failure mechanism. The pioneering work of Hegepeth [9] is based on the shear-lag model which states that the matrix between the fibers is only capable of transmitting shear force. Also the assumption of the linearly elastic behavior of filaments even to failure was made. Hedgepeth computed the values of stress concentration factors (s.c.f.s) for an intact fiber adjacent to an array of broken fibers in a sheet of
858 material subjected to a tensile load. An extension from two-dimensional to threedimensional models where filaments were arrayed in both square and hexagonal patterns was carried out by Hegepeth and Van Dyke [10]. A plastic failure-mode model was introduced in [11] wherein the constitutive equations of an elastic perfectly plastic material were considered for the matrix. Fitcher [12,13] investigated the stress concentration around single and double cuts by generalizing the standard shear-lag model. The paper by Zweben [14] analyzes a composite material containing a crack in the matrix perpendicular to the fibers. Zweben used an approximate method to obtain s.c.f.s which were lower than the values previously reported in [9]. The model developed by Goree and Gross [15] considered the yielding and splitting of the matrix. Again the shear-lag model was used with the effects of the matrix damage included in the analysis and by assuming that the behavior of the matrix is analogous with the elastic perfectly plastic model in [11]. The resultant equilibrium equation in the axial direction yields a pair of integral equations which are solved numerically. In all of the foregoing studies the composite material consisted of only one type of fiber. Fukuda and Chou [16] computed the s.c.f.s for intraply hybrids composed of glass and graphite fibers with a volume ratio of one to one and full dispersion. They considered a variety of combinations of broken and intact fibers. In the above references, except for [15], the influence-function method was utilized to solve the mixed boundary value problem in the shear-lag analysis. This technique is somewhat complicated. However, it is possible to simplify the mathematical analysis by specifying the strain of the broken fibers, the displacement of the intact fibers at the origin, and the strain far from the origin by utilizing the elasticity assumption of fibers. Using this boundary condition, Zweben [17] determined the s.c.f.s and the ineffective length, i.e., the axial length of a broken fiber over which the fiber is unable to sustain load. The hybrid composite considered by Zweben consisted of two types of fibers arranged in alternating
positions. A quantitatively different micromechanical model to analyze the stress redistribution in any intact fiber adjacent to a series of broken fibers was introduced by Harlow and Phoenix [18,19] for composites with one type of reinforcement and Harlow [5] for the hybrid composites. The local load sharing rule devised by these authors ignored the interactions between the matrix and the fibers. By satisfying the equilibrium between the applied tractions and stresses in the intact filaments in conjunction with the assumption of the elastic behavior of fibers the s.c.f.s were calculated. This model is rather straightforward but the results are overly conservative compared to those obtained from the shear-lag theory. In particular when the number of broken fibers increases this discrepancy is more pronounced. The local load sharing rule was generalized in [8] for intraply hybrid composites with very diverse types of lay-up and degrees of dispersion. It was shown that even though the ultimate strength of the hybrid is slightly underestimated nevertheless the major features of the intraply hybrid tensile failure mechanism for a broad class of hybrids could be captured by the model. Some other attempts to compute the s.c.f.s were carried out by Smith [20] and Fukuda [21]. They assumed that the s.c.f.s have statistical characteristics which
859
spacing of the fibers in the matrix. A Weibull the normalized distances between the fibers. The shear-lag theory was applied for the composites composed of one type of fiber and some efforts were made to acquire a c.d.f., preferably Weibull, for the s.c.f.s. In the present study a simplified shear-lag model is presented to determine the stress redistribution for various arrangements of broken and unbroken filaments. The governing differential equations are more general than those derived in [17]. However, the properties of the ineffective length for broken fibers which considerably simplify the computations are utilized. This allows for the analysis of intraply hybrid composites with any kind of fiber dispersion and lay-up. Moreover, the model is sufficiently flexible so that it can be generalized to hybrids with a random arrangement of filaments. It is demonstrated that the s.c.f.s not only depend upon the types and number of broken fibers, but also they are sensitive to the sequence in which the fibers fail. Thus, they are influenced by the statistical characteristics of the fibers. Having specified a model for the failure mechanism, the chain-of-bundles probability model is used. By means of the Monte Carlo simulation technique the c.d.f. for the ultimate strength of hybrids reinforced by relatively large populations of filaments is estimated. The results are compared to those obtained in [8]. In the next section the statistical model is described briefly, and the stress redistribution model is specified more fully. The third and fourth sections contain numerical results and conclusions, respectively. may be attributed to the random
c.d.f.
was
fitted
to
2. DEVELOPMENT OF THE MODEL
Hybrid Geometry The hybrid, which is assumed to be in the form of tape or thin tube, consists of a large number of filaments equally spaced in a matrix. The fiber material is either low modulus (L) or high modulus (H). Due to the high population of fibers, any boundary irregularities or edge effects may be neglected [22,23]. Both deterministic and random arrangements of fibers are investigated. For the randomly distributed arrangements the volume ratio of H to L fibers is fixed but the lay-up pattern is determined by randomly arranging the fibers according to a Bernoulli probability distribution. Let 100p be the percentage of H-fibers, where 1, then 100(1 p ) is the percentage of L-fibers. Each position in the ran0
