Ai 622

where Nn is the number of fibers per row, ls is the sliding distance, and G represents the matrix shear modulus. Equation 6.22 is obtained considering displacement (strain) compatibility between the fiber and the matrix at the interface (Fig. 6.20). Additionally, Eq. 6.22 acknowledges that all the fibers in the same row are subjected to an equal strain.

An expression for the sliding distance, ls, is given in [98]

where Ec is the Young's modulus of the composite (calculated by the rule of mixtures), Vm is the volume fraction of the matrix, and t is the interfacial shear stress.

Since the COD depends on the value of o\, a numerical iterative method is required for the calculation of o\. Initially a o\ = 0 value is adopted, and then an interim value of o\ is calculated. The iterations are repeated until there is

FIGURE 6.20 Schematic of strain compatibility setup at bridging fiber.

no difference between successive values of a1. Once the friction stresses, acting at the interface, have been determined, the stress at each fiber row, af, can be evaluated [99]:


dV m Em

Equation 6.24 assumes that all the fibers in the same row exhibit the same sliding distance. Since the friction stresses depend on COD, a numerical iterative method between 6.22 and 6.24 is required for the calculation of af. Moreover, Eq. (6.24) is controlled by the strength integrity of the fiber, and therefore an additional equation should be included in the iteration. In general, the average strength of fibers of a given strength distribution and a particular gage length and fiber diameter is calculated by the Weibull function for average strength and is written as [100, 101]:

d/ \ m where m is the Weibull modulus, L is the gage length, afr is the average fiber strength, a0 is the normalizing factor, and V is a tabulated gamma function. A typical behavior of the most commonly used Textron SCS-6 fiber is given in Fig. 6.21.

The length L over which possible fiber failure should be expected is the sliding distance given by Eq. 6.23. In this region, shear tractions are developed at the fiber ends, allowing stress to be transferred from the matrix to the fibers. Such behavior, increases the probability of failure within the sliding distance, while outside of this region fiber strength can be considered invulnerable. Additionally, the matrix plastic displacement, COD, at the fiber, should also be considered in the sliding distance calculations. Even though the COD region is characterized by the absence of interfacial shear stress and consequently defines a different statistical environment, for simplicity reasons, it is assumed that the interfacial shear stress t is also acting along the COD (COD very small compared to ls). Thus, the average strength of the debonded fibers in the sliding region can be evaluated by considering a gage length equal to a sliding distance, L = ls + CODcr (CODcr defines the critical displacement at the time of fiber failure). The evaluation of the CODcr in respect to typical strength data provided by the manufacturer is

FIGURE 6.21 Average strength of SCS-6 fiber for several values of Weibull modulus.

achieved by employing the Masson and Bourgain [101] estimation of the Weibull probability:

0 0

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