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Often fibre composites experience fibre bridging during cracking. This can be beneficial from the point of view of crack resistance; fibre bridging leads to increasing crack growth resistance (R-curve behaviour). The bridging length can be large in comparison with specimen geometry. The large scale bridging (LSB) occurs. Under LSB the fracture process cannot properly be characterised by GIc, which is a small process zone concept. Instead, fibre bridging can be described in terms of a bridging law (a basic solid mechanics concept); the relationship between the local crack opening d and the resulting bridging stress σ ; σ = σ(δ). A maximum opening δ0 exists, beyond which the bridging stress vanishes.
A basic fracture mechanics tool, the J integral provides a relationship between global energy release rate Jext and the energy uptake at the crack tip Jtip and in the crack wake,
where σ(δ) is the bridging law and d* is the end-opening of the bridging zone. By definition Jext = JR during crack growth. Initially, the crack is unbridged. Thus, crack growth initiates when Jext = Jtip, which corresponds to J0 at the R-curve. As the crack grows, JR increases. When the end-opening of the bridging zone, δ*, reaches δ0, the overall R-curve attains a steady-state value Jss. Differentiation of the equation above gives
σ(δ*)=dJR/dδ*
Thus, by recording JR and the end-opening of the bridging zone, δ*, the bridging law can be determined.
Under LSB Jext (and thus JR) cannot easily be determined for most specimens. One exception is the double cantilever beam (DCB) specimens loaded with pure bending moments (Sørensen et al., 1996; Sørensen and Jacobsen, 1998).
A test specimen suitable for measuring JR and thus bridging law σ(δ) during large scale bridging (LSB).
Large scale bridging (due to fibre cross over bridging) was studied in a carbon/epoxy composite. R-curves and bridging laws were determined for DCB-specimens loaded with pure bending moments. R-curves were found to depend strongly on specimen geometry. In contrast, the bridging law was identical for all geometries, suggesting that the bridging law is a material property under LSB (Sørensen and Jacobsen, 1998).
The concept of bridging laws (or cohesive laws) can also be used for characterising other failure types, where large scale process zones develop, such as failure of adhesive joints (Sørensen, 2002).
Measured crack growth resistance curves: (a) JR as a function of crack extension and (b) JR as a function of end-opening of the bridging zone
Selected publications
Sørensen, B. F., Brethe, P. and Skov-Hansen, P., 1996, "Controlled Crack Growth in Ceramics: The DCB-Specimen Loaded with Pure Moments", J. Euro. Ceram. Soc., Vol. 16, pp. 1021-5.
Sørensen, B. F. and Jacobsen, T. K., 1998, "Large Scale Bridging in Composites: R-curve and Bridging Laws", Composites part A, vol. 29A, pp. 1443-51.
Sørensen, B. F., Gamstedt, E. K., and Jacobsen. T. K., 2000, "Equivalence of J Integral and Stress Intensity Factor Approach for Large Scale Bridging Problems", Int. J. Fracture, Vol. 104, pp. L31-6.
Jacobsen, T. K., and Sørensen, B. F., 2001, "Mode I Intra-laminar Crack Growth in Composites - Modelling of R-curves from Measured Bridging Laws", Composites part A, Vol. 32, pp 1-11.
Sørensen, B. F., 2002, "Cohesive Law and Notch Sensitivity of Adhesive Joints", Acta Mater., Vol. 50, pp. 1053-61.
Sørensen, B. F., and Kirkegaard, P., 2006, "Determination of mixed mode cohesive laws", Engineering Fracture Mechanics, Vol. 73, pp. 2642-61.
Sørensen, B. F., Jørgensen, K., Jacobsen, T. K., and Østergaard, R. C., 2006, "DCB-specimen loaded with uneven bending moments", International Journal of Fracture, Vol. 141, pp. 163-76. |