13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 2. A laminated composite DCB and its loading conditions. (a) General description. (b) Details of the crack influence region aΔ . 2.1.1. Classical beam partition theory Using the constitutive relation in classical laminated composite beam theory, the ERR at the crack tip at location B, G is ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + − + + − − = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ B BM N B BM N B BM N A N A N A N D M D M D M b G B B B B B B B B B B B B 2 2 2 2 1 2 2 2 2 1 1 1 1 2 2 2 2 1 2 1 2 2 2 2 1 2 1 2 (1) where subscript ‘B’ indicates loads at the crack tip at location B, for example, B M1 is the bending moment on the top sub-laminate at the crack tip. These loads are shown in Fig. 2 (b). Other quantities in Eq. (1) are i i i iA A B D 2 = − ∗ , i i i iB B AD = − ∗ 2 , i i i iD D B A 2 = − ∗ (2) The range of subscript i is 1 and 2, which again refers to the upper and lower sub-laminates respectively. For the intact laminate, the subscript i is dropped. A, B and D are the equivalent extensional, coupling and bending stiffness of the DCB respectively. A novel methodology to partition mixed-mode ERR G in Eq. (1) arises from the fact that G is of quadratic form and non-negative definite in terms of the crack tip bending moments B M1 and B M2 , and the crack tip axial forces B N1 and B N2 . An analogy of this is the positive definite kinetic energy of a vibrating structure, to which individual modal energies are attributed by using modal analysis from orthogonal natural vibration modes. A hypothesis is then made that the total ERR in a mixed-mode delamination can be partitioned into pure mode components by using orthogonal pure modes. There are two sets of fundamental orthogonal pure modes. The first set corresponds to zero relative shearing displacement just behind the crack tip (mode I) and zero crack tip opening force ahead of the crack tip (mode II). The second set corresponds to zero relative opening displacement just behind the crack tip (mode II) and zero crack tip shearing force (mode I). It is simple to derive the zero relative displacement modes first and then to find the zero force modes by applying orthogonality through Eq. (1). An alternative and more complex derivation considers the interface stresses. If the mode vector form is { }T B B B B M M N N2 1 2 1 , , , , then the first set of fundamental orthogonal pure modes, referred to as the { }β θ, set, are found to be { } ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = 0 0 1 1 1 θ ϕθ , { } ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = 0 0 1 2 2 θ ϕθ , { } ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = 3 0 0 1 3 θ ϕθ , { } ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = 0 0 1 1 1 β ϕβ , { } ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = 0 0 1 2 2 β ϕβ , { } ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = 3 0 0 1 3 β ϕβ (3) with
RkJQdWJsaXNoZXIy MjM0NDE=