Title:

Brittle mixedmode cracks between linear elastic layers

Original analytical theories are developed for partitioning mixedmode fractures on rigid interfaces in laminated orthotropic double cantilever beams (DCBs) based on 2D elasticity by using some novel methods. Note that although the DCB represents a simplified case, it provides a deep understanding and predictive capability for real applications and does not restrict the analysis to a simple class of fracture problems. The developed theories are generally applicable to socalled 1D fracture consisting of opening (mode I) and shearing (mode II) action only with no tearing (mode III) action, for example, straight edge cracks, circular blisters in plates and shells, etc. A salient point of the methods is to first derive one loading condition that causes one pure fracture mode. It is conveniently called the first pure mode. Then, all other pure fracture modes can be determined by using this pure mode and the property of orthogonality between pure mode I modes and pure mode II modes. Finally, these 2Delasticitybased pure modes are used to partition mixedmode fractures into contributions from the mode I and mode II fracture modes by considering a mixedmode fracture as the superposition of pure mode I and mode II fractures. The partition is made in terms of the energy release rate (ERR) or the stress intensity factor (SIF). An analytical partition theory is developed first for a DCB composed of two identical linear elastic layers. The first pure mode is obtained by introducing correction factors into the beamtheorybased mechanical conditions. The property of orthogonality is then used to determine all other pure modes in the absence of throughthicknessshear forces. To accommodate throughthickness shear forces, first two pure throughthicknessshearforce pure modes (one pure mode I and one pure mode II) are discovered by extending a Timoshenko beam partition theory. Partition of mixedmode fractures under pure throughthickness shear forces is then achieved by using these two pure modes in conjunction with two thicknessratiodependent correction factors: (1) a shear correction factor, and (2) a puremodeII ERR correction factor. Both correction factors closely follow a normal distribution around a symmetric DCB geometry. The property of orthogonality between all pure mode I and all pure mode II fracture modes is then used to complete the mixedmode fracture partition theory for a DCB with bending moments, axial forces and throughthickness shear forces. Fracture on bimaterial interfaces is an important consideration in the design and application of composite materials and structures. It has, however, proved an extremely challenging problem for many decades to obtain an analytical solution for the complex SIFs and the crack extension sizedependent ERRs, based on 2D elasticity. Such an analytical solution for a brittle interfacial crack between two dissimilar elastic layers is obtained in two stages. In the first stage the bimaterial DCB is under tip bending moments and axial forces and has a mismatch in Young s modulus; however, the Poisson s ratios of the top and bottom layers are the same. The solution is achieved by developing two types of pure fracture modes and two powerful mathematical techniques. The two types of pure fracture modes are a SIFtype and a loadtype. The two mathematical techniques are a shifting technique and an orthogonal pure mode technique. In the second stage, the theory is extended to accommodate a Poisson s ratio mismatch. Equivalent material properties are derived for each layer, namely, an equivalent elastic modulus and an equivalent Poisson s ratio, such that both the total ERR and the bimaterial mismatch coefficient are maintained in an alternative equivalent case. Cases for which no analytical solution for the SIFs and ERRs currently exist can therefore be transformed into cases for which the analytical solution does exist. It is now possible to use a completely analytical 2Delasticitybased theory to calculate the complex SIFs and crack extension sizedependent ERRs. The original partition theories presented have been validated by comparison with numerical simulations. Excellent agreement has been observed. Moreover, one partition theory is further extended to consider the blister test and the adhesion energy of mono and multilayered graphene membranes on a silicon oxide substrate. Use of the partition theory presented in this work allows the correct critical mode I and mode II adhesion energy to be obtained and all the experimentally observed behaviour is explained.
