In fracture mechanics design, the growth of a crack from an intrinsic flaw or pre-existing crack is considered using strain energy release rate (SERR) as the basis. The relationship between the rate of crack growth and G is shown schematically in Figure 2. Two extreme levels of SERR are identified: a high catastrophic or critical level Gc , at which rapid fracture occurs and a low threshold level Gth , below which crack growth does not occur. In between these extremes, the crack growth behaviour tends to follow a power law (.e. is of the form da/dN = AGB , where A and B are constants).
Figure 2: Schematic plot of G versus da/dN
For the particular materials, joint geometry and loading of interest, design analysis using fracture mechanics follows the steps illustrated in Figure 3 as follows.
An analysis of the joint to determine the variation in strain energy release rate with crack length (G vs. a ). It will be necessary to evaluate the contributions from the various modes (i.e. GI , GII and sometimes GIII ).
A materials model that represents the rate of joint failure as a function of strain energy release rate, i.e.
For fracture, the critical strain energy release rate Gc
For fatigue, the rate of crack growth per cycle da/dN as a function of G and Gth
The tests should include the effects of adhesive layer thickness, mode ratio and any relevant environmental effects that need to be accounted for in the analysis such as the rate of loading, temperature, humidity and long-term ageing.
Calculation of the failure conditions from ‘1’ and ‘2’ in terms of maximum load for fracture or cycles to failure for fatigue. For fracture the limiting load is when G exceeds Gc . For fatigue, the number of cycle to failure requires an incremental crack growth calculation since G will in general vary with crack length. This is illustrated in Figure 3 for constant cyclic load conditions. The number of cycles DN for each increment of crack growth (Da) are obtained from the current G (from G vs. a ) and corresponding da/dN (from da/dN vs. G ). These incremental cycles are summed up to the final crack length to obtain the number of cycles to failure Nf . Repeating he calculation at various load levels enables the fatigue life to be obtained as a function of the load level to give the equivalent of a load-life curve. A similar incremental calculation can be carried out when the load levels is also varied to predict the total crack growth due to a spectrum of fatigue loads. In designing against fatigue, it may be necessary to ensure that for all load cases the maximum G level is less than the threshold level Gth fulfilling a no growth design criterion.
The main advantage of the fracture mechanics approach to design analysis is that geometry independent results are generated from the laboratory testing. Strain energy release rate G is measured from the test pieces during cyclic fatigue testing, together with the associated crack growth rates da/dN . Crack growth testing is performed over a range of G values including the threshold value, the G value below which negligible crack growth would ever be expected, typically associated with rates between 10-6 and 10-7 mm per cycle. The threshold value can itself be used during joint design as a limiting criterion, whereby the designer limits the maximum G value within the joint to, or below the measured threshold value. This would be classed as a damage intolerant design. If a damage tolerant system is acceptable then the designer can predict the expected life of the component by applying the measured crack growth model. Allowing a known, acceptable, amount of crack growth within the joint will therefore extend its service capabilities significantly.
For designs where there is no requirement to consider the presence of initial cracks or damage tolerance, this approach is inherently conservative, in that any additional contributions to fatigue life due to any crack initiation processes are neglected.
Figure 3: Steps in fracture mechanics design analysis
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