13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Figure 1. Some examples of one-dimensional fracture. and then to partition a mixed mode into pure modes. Delamination propagation criteria can then be established by using the partition together with experimental data. The through-width delamination in a DCB made of isotropic material with rigidly bonded interface can be considered to be the ‘simplest’ one-dimensional delamination. Although it seems to be a straightforward matter to determine the pure modes and to partition a mixed mode, it has been proved to be an extremely complex and sophisticated problem. There has been a lot of confusion on the matter during the last 25 years. Ref. [1] may be the earliest work on the ‘simplest’ problem by Williams. A mixed-mode partition theory [1] was developed based on classical beam theory. Ref. [2] reported a combined numerical and analytical theory by Schapery and Davidson based on combined classical beam theory and 2D elasticity. It disagrees with the Williams’ theory [1] and concludes that classical beam theory does not provide quite enough information to obtain an analytical decomposition of the mixed-mode ERR into its opening and shearing mode components. Hutchinson-Suo reported their work in Ref. [3] in which the mixed-mode ERR is calculated based on the classical beam theory but the partition of ERR is calculated based on stress intensity factors from 2D elasticity. Their theory [3] agrees well with the theory in Ref. [2] and claims that Williams’ theory [1] contains conceptual errors. To respond to this claim, Williams reported some experimental work in Ref. [4] showing that Williams’ theory [1] is in a better agreement with the test results than Hutchinson-Suo theory [3]. This has caused a lot of confusion, which has affected many academic researchers and design engineers until today. A great deal of research effort has been made during the last two decades to resolve the confusion. Among many others, the following significant works are referenced here. Ref. [5] reported a mixed-mode partition theory for laminated composite beams with rigidly bonded interface based on first-order shear-deformable beam theory, which gives different mixed-mode partitions to those from Williams’ theory [1] and the Hutchinson-Suo theory [3]. The same theory as that in Ref. [5] was derived in Refs. [6, 7] but these are based on classical beam theory, which caused yet more confusion. Recently, the authors have developed analytical theories for one-dimensional delamination in laminated composite beams and plates by using a novel methodology [8–13]. All the confusion is explained. This paper reports some of the major results in Refs. [8–13]. 2. Partition of mixed-mode fracture in laminated composite beams and plates with rigid interfaces The mechanics of delamination depend on the mechanical properties of lamina interfaces. A lamina interface is considered to be a rigid interface when the interface separation is negligible before an existing delamination propagates. Otherwise, it is considered to be a non-rigid interface or as it often called, a cohesively bonded interface. Bare-bonded interfaces in the conventional manufacturing process from glass or carbon fiber epoxy pre-pregs are typical rigid interfaces because of their brittleness. While cohesively bonded interfaces are typical non-rigid interfaces which are achieved by adding adhesive layers between bare plies when manufacturing components. 2.1. Laminated composite DCBs A laminated composite DCB with a delamination of length is shown in Fig. 2 (a). The interface stresses in Fig. 2 (b) only show the sign convention rather than any representative distribution.
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