a fi

In the above equations, n is the total number of coupons tested under the i th stress level, oai, r is the number of failed coupons under that stress level, and NSi is the number of cycles after which the test was stopped.

The values of the number of cycles to failure for every coupon at each stress level are subsequently normalized by the corresponding estimated characteristic number, Ni. Thus, the following normalized data set is formed:


It is assumed that this set of data also follows a two-parameter Weibull distribution:

The parameters of the distribution of Eq. (1.28) are estimated by:

i=1 j=1 i=1 1 i 1 -----> >lnX!; - — = 0, m ri m r Z—/ ¿—/ ij a j

The following notation was used:

Ni i=i where rT is the total number of failed coupons.

The value of X0 has to be unity for a perfect fit. If X0 takes any value other than unity, the characteristic number of cycles for each stress level can be adjusted to produce X0 = 1. In particular:

The slope of the S-N curve, 1 Ik, and the y intercept, a0, can be determined by fitting log a, versus log ;V0, to a straight line. With cro, k, and a/ already determined, the S -N curve at any specified level of reliability can be calculated by:

1.10 EXPERIMENTAL PROCEDURE 1.10.1 Material and test Coupons

A comprehensive experimental program was realized consisting of static and fatigue tests of straight-edge coupons cut from a multidirectional laminate. The stacking sequence of the plate consists of four layers, 2 x UD, unidirectional lamina of 100% aligned warp fibers, with a weight of 700 g/m2 and 2 x stitched, ±45°, of 450 g/m2, 225 g/m2 in each off-axis angle. The material used was E-glass/polyester, E-glass from AHLSTROM GLASSFIBRE, while the polyester resin was CHEMPOL 80 THIX by INTERCHEM. This resin is a thixotropic unsaturated polyester and was mixed with 0.4% cobalt naphthenate solution (6% Co), accelerator, and 1.5% methyl ethyl ketone peroxide (MEKP, 50% solution), catalyst. Rectangular plates were fabricated by hand lay-up technique and cured at room temperature. Considering as 0° direction that of the UD layer fibers, the lay-up can be encoded as [0/(±45)2/0]r. Coupons were cut, by a diamond saw wheel, at 0°, on-axis, and 15°, 30°, 45°, 60°, 75°, and 90° off-axis orientations. All data from cyclic loading were used to characterize anisotropic mechanical properties of the material, for the verification of theoretical predictions from FTPF strength criterion and for the study of stiffness variation during life.

The coupons were prepared according to ASTM 3039-76 standard, and aluminum tabs were glued at their ends. Coupon edges were trimmed with sandpaper. The coupons were 250 mm long and had a width of 25 mm. Their nominal thickness was 2.6 mm. The length of the tabs, with a thickness of 2 mm, was 45 mm leaving a gage length of 160 mm.

Static and fatigue tests were performed. The number of coupons tested, 307 in total, was partitioned as follows: 50 coupons for static tests to provide baseline data, both in tension and compression, while 257 coupons were tested under uniaxial cyclic stress for the determination of 17 S-N curves at various off-axis directions and loading conditions.

1.10.2 Test Program and Results Static and Fatigue Strength

Static tests were performed in tension and compression on an MTS machine of 250 kN capacity under displacement control at a speed of 1 mm/min. The coupons used for static compression tests had a gage length of 30 mm to avoid buckling. Fatigue tests, of sinusoidal constant amplitude waveform, were also carried out on the same MTS machine. In total, 17 S -N curves were determined experimentally, under 4 different stress ratios, namely, R = 10 (C-C), R = —1 (T-C), R = 0.1, and R = 0.5 (T-T). The frequency was kept constant at 10 Hz for all the tests, which were continued until coupon ultimate failure or 106 cycles, whichever occurred first. In particular, for the on-axis coupons, 0°, under reversed loading, R = -1, tests were continued for up to 5 x 106 cycles. For stress ratios comprising compression, the antibuckling jig of Fig. 1.1 was used. Its geometry and operational characteristics are in essence those described in [41].

Uniaxial tests on coupons cut off-axis from principal material directions were performed to induce complex stress states in the principal coordinate system (PCS). Denoting by 0i,i = 1, 2, 6, the in-plane stress tensor components in the

Clamping pla

Spei pla

Clamping pla

Spei pla

FIGURE 1.1 Sketch of antibuckling device.

PTFE protective liner

FIGURE 1.1 Sketch of antibuckling device.

FIGURE 1.2 Complex stress state in principal material system of off-axis loaded coupon.

FIGURE 1.2 Complex stress state in principal material system of off-axis loaded coupon.

PCS of the multidirectional laminate, (see Fig. 1.2), and by ox, the applied normal stress at an off-axis angle 0, the following transformation relations are valid:

a1 = ox cos2 0, o2 = ox sin2 0, o6 = ox sin 0 cos 0, (1.33)

The biaxiality ratios a2/a1 and a6/a1 as a function of 0 take values that are proportional to tan2 0 and tan 0, respectively.

With respect to material properties that need to be determined experimentally in order to use the FTPF criterion, Eq. (1.17), it is clear from Fig. 1.2 as well, that X(N) and Y(N) are the S-N curves determined from on-axis and 90° off-axis coupon tests. And S(N) is determined by fitting Eq. (1.17) to the experimental fatigue strength data from any off-axis orientation. By substituting relations (1.33) into Eq. (1.17) and solving for S(N) one has:

sin2 0 cos2 0

1 sin2 0 cos2 0 cos4 0 sin4 0

0 0

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