Mixed-mode partition theories for one-dimensional fracture

Many practical cases of fracture can be considered as one-dimensional, that is, propagating in one dimension and characterised by opening (mode I) and shearing (mode II) action only with no tearing (mode III) action. A double cantilever beam (DCB) represents the most fundamental one-dimensional fracture problem. There has however been considerable confusion in calculating its mixed-mode energy release rate (ERR) partition. In this work, new and completely analytical mixed-mode partition theories are developed for one-dimensional fractures in isotropic homogeneous and laminated composite DCBs, based on linear elastic fracture mechanics (LEFM) and using the Euler and Timoshenko beam theories. They are extended to isotropic homogeneous and laminated composite straight beam structures and isotropic homogeneous plates based on the Kirchhoff-Love and Mindlin-Reissner plate theories. They are also extended to non-rigid elastic interfaces for isotropic homogeneous DCBs. A new approach is used, based on orthogonal pure fracture modes. Two sets of orthogonal pairs of pure modes are found. They are distinct from each other in the present Euler beam and Kirchhoff-Love plate partition theories and coincide on the first set in the present Timoshenko beam and Mindlin-Reissner plate partition theories. After the two sets of pure modes are shown to be unique and orthogonal, they are used to partition mixed modes. Interaction is found between the mode I and mode II modes of the first set in the present Euler beam and Kirchhoff-Love plate partition theories. This alters the ERR partition but does not affect the total ERR. There is no interaction in the present Timoshenko beam or Mindlin-Reissner plate partition theories. The theories distinguish between local and global ERR partitions. Local pureness is defined with respect to the crack tip. Global pureness is defined with respect to the entire region mechanically affected by the crack. It is shown that the global ERR partition using any of the present partition theories or two-dimensional elasticity is given by the present Euler beam or Kirchhoff-Love plate partition theories. The present partition theories are extensively validated using the finite element method (FEM). The present beam and plate partition theories are in excellent agreement with results from the corresponding FEM simulations. Approximate 'averaged partition rules' are also established, based on the average of the two present beam or plate partition theories. They give close approximations to the partitions from two-dimensional elasticity. The propagation of mixed-mode interlaminar fractures in laminated composite beams is investigated using experimental results from the literature and various partition theories. The present Euler beam partition theory offers the best and most simple explanation for all the experimental observations. It is in excellent agreement with the linear failure locus and is significantly closer than other partition theories. It is concluded that its excellent performance is either due to the failure of materials generally being based on global partitions or due to the through-thickness shear effect being negligibly small for the specimens tested. The present partition theories provide an excellent tool for studying interfacial fracture and delamination. They are readily applicable to a wide-range of engineering structures and will be a valuable analytical tool for many practical applications

Mixed mode partition theories for one dimensional fractures

The crack in a double cantilever beam is the most fundamental one-dimensional fracture problem. It has caused considerable confusions due to its in-depth subtleness and complex entanglement with different theories and numerical simulations. The present paper presents completely analytical theories based on Euler and Timoshenko beam theories using a brand new approach which reveals the hidden mechanics of the problem. Orthogonal pairs of pure modes are found and used to partition mixed modes. The developed theories are extensively validated against numerical simulations using finite element methods. Moreover, the fracture mode partition space is thoroughly investigated and crack tip running contact is found which results in a region of pure mode II. The theories are finally applied to general one-dimensional fracture in beams and axisymmetric plates

A theory of one-dimensional fracture

A completely analytical theory is developed for the mixed mode partition of one-dimensional fracture in laminated composite beams and plates. Two sets of orthogonal pure modes are determined first. It is found that they are distinct from each other in Euler beam or plate theory and coincide at the Wang-Harvey set in Timoshenko beam or plate theory. After the Wang- Harvey set is proved to form a unique complete orthogonal pure mode basis within the contexts of both Euler and Timoshenko beam or plate theories, it is used to partition a mixed mode. Stealthy interactions are found between the Wang-Harvey pure mode I modes and mode II modes in Euler beam or plate theory, which alter the partitions of a mixed mode. The finite element method is developed to validate the analytical theories