13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- The current study uses a two-parameter method to quantify the constraint of a variety of specimens and also to predict failure of specimens of biaxial loads. The J-A2 method has been applied to experimental data published by the Fraunhofer Institute for Mechanics. Fraunhofer conducted fracture toughness testing on 3PB and cruciform specimens, with the cruciform specimens subjected to biaxial loads. Such experimental data can be used to determine if the J-A2 method can be used as a tool to i) predict failure of mechanical components of different constraints and ii) quantify the constraint due to loading and geometry. 2. J-A2 Two Parameter Method The stress distribution ahead of the crack tip described by the well-known HRR solution [3, 4] is ( )n I r J ij n n ij , ~1 1 0 0 0 θ σ σ αε σ σ + = , (1) where σ0 is a reference (or yield) stress, ε0 = σ0 / E is a reference (or yield) strain, E is Young’s modulus, α is a material constant, and n is the strain-hardening exponent. The integration constant In and the dimensionless angular functions of stresses ijσ ~ depend on the strain-hardening exponent. Equation 1 is the first term of a series solution. Further, the HRR solution described by Equation 1 is defined by only one parameter, the J-integral, which is related to the energy release rate around the crack tip during loading. As shown in Figure 1, the stress field as described by the HRR solution is valid as r →0. In theory, the stress approaching the crack tip increases asymptotically to infinity, meaning a component subjected to such stress would fail under very small loads. In practice, this is typically not the case, especially with materials exhibiting elastic-plastic behavior. The HRR solution does not consider large deformations or blunting of the crack and is only valid for small strains as r →0. An example of the actual stress field near the crack tip is also shown in Figure 1. Ritchie, et al. [5] determined that cleavage fracture actually occurs at some critical distance (rc) when the stress at that location exceeds the critical stress (σc) and this failure criterion is often referred to as the RKR model. Clearly, the single term HRR solution can overestimate the actual stress ahead of the crack tip. As a result, various procedures have been developed to include additional terms from the series expansion. Yang et al. [6, 7] and Chao et al. [8] developed an asymptotic crack-tip solution using three terms of the series solution from which the HRR solution is derived. The J-A2 procedure uses two parameters to better define the stress ahead of the crack tip. The J-integral is representative of the magnitude of the applied loading, while the A2 term is used to describe the constraint at the crack tip based on the loading and specimen geometry. The procedure begins by using the Ramberg-Osgood power-law relationship for a strain-hardening material for a Mode I crack under plane strain conditions. The Ramberg-Osgood equation relating uniaxial strain ε to the uniaxial stress σ in tension is n = + 0 0 0 σ σ α σ σ ε ε . (2) Eq. 2 can be rewritten in general terms by applying the J2 deformation theory of plasticity
RkJQdWJsaXNoZXIy MjM0NDE=