PSI - Issue 13

Fuzuli Ağrı Akçay / Procedia Structural Integrity 13 (2018) 1695 – 1701 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 3 Now, consider an isotropic continuous medium in rectangular cartesian coordinate system subjected to uniform state of stress before fracture. The total work increment for the unfractured medium, , can be written in terms of principal stresses and strains as ( ) I II III I I II II III III dW l l l de de de    = + + (2) where ≥ ≥ . Also, , , and represent the dimensions of the volume element in the direction of principal stresses/strains. On the other hand, total work increment for the fractured system, ∗ , is ( ) * * * * * * * I II III I I II II III III dW l l l de de de    = + + (3) with superscript star denoting the corresponding quantities for the fractured system. 1697

Fig. 1. Representation of tensile mode potential fracture plane.

Tensile mode fracture corresponds to a crack formation with the plane of fracture having a normal in the direction of maximum principal stress. Representation of the potential fracture plane is shown in Fig. 1. In this case, the area of the fracture plane equals to = × . Applying the fracture criterion by equating the rate of energy change of the un-fractured and fractured systems yields ( ) * I I dW A de dW =   + (4) where Γ I represents necessary energy per unit area (per unit increment) to create new (fracture) surfaces, that is, critical effective energy release rate during the formation of new surfaces. This energy is concentrated in the fracture zone such that the increment in strain, , is concentrated in this very small zone as well (Karr & Akçay, 2016). In the case of pure brittle fracture, Γ I = 2 as two (new) surfaces are created during fracture process. Here, represents surface energy per unit area. Let divide both sides with the area of the fracture plane and the increment in strain. Simplifying it yields (5) The fracture is presumed to render a complete loss of the axial component of work, that is, ∗ ∗ = 0 . Applying the consistent boundary conditions, i.e., = ∗ ∗ and = ∗ ∗ , yields the critical state equation * I I l dW dW A de de  −   I I    = 