PSI - Issue 60

A.K. Dwivedi et al. / Procedia Structural Integrity 60 (2024) 286–297 A.K.Dwivedi / Structural Integrity Procedia 00 (2019) 000 – 000

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Numerical calculations are performed under pure Mode-I loading, which corresponds to tensile or crack opening mode. Displacement increments are prescribed at the outer boundary, see Fig. 1a, based on the elastic K I field which controls the crack tip deformation. For small-scale yielding condition, the outer radius r of the MBL model should be sufficiently larger than the maximum size of the plastic zone, thus, requiring a sufficiently large ratio of r/X 0 . In our numerical calculations, this ratio is set to 4500. Due to the assumed symmetrical distribution of voids and crack extension, only one-half of the model is analyzed. Numerical values of the prescribed displacements ( u i ) is calculated using the following equations (Andersen et al., 2019). 1 = 2 √ 2 cos( 2 )[3−4ν−1+2 2 ( 2 )] (3a) 2 = 2 √ 2 sin( 2 ) [3 − 4ν + 1 − 2 2 ( 2 ) ] (3b) Here, is the applied stress intensity factor under mode-I loading, is the shear modulus, ν is the Poisson’s ratio, and θ is the angle in the far field measured with respect to the x 1 axis. The intensity of far-field loading under pure mode-I can also be expressed in terms of the J -integral using the following equation, = 2 ( 1−ν 2 ) (4) All the numerical calculations are carried out using a rate-independent J 2 isotropic hardening response for the elastic – plastic matrix between the two populations of voids. The matrix response is characterized by the following representation in uniaxial tension, ={ , ≤ 0 0 ( 0 ) , > 0 (5) Here, E is Young’s modulus, σ 0 is the initial yield stress, and n is the strain-hardening exponent. In most of our studies, ⁄ = 0.004 , ν=0.333 , and = 0.1. Sufficiently fine meshes are used to discretize the region ahead of crack tip, see Fig. 2. To free the mesh from artificial constraint/locking associated with plastic flow, computations are carried out using reduced integration (CPE8R). 3. Results We present here, numerical results providing some details of the growth and coalescence of the two-scale voids and the influence of this complex void interaction occurring ahead of the crack tip on ductile crack propagation. As mentioned earlier, only mode-I loading is considered. In this study, the material decohesion in the ligament of voids resulting from crack propagation is not modelled. The measurement of crack extension is carried out rather in an approximate manner in terms of the active softening zone resulting from strain localization (Hutter, 2015). As the imposed deformation increases, the strains get concentrated in the ligament between the neighboring voids. For in-plane crack growth, the crack extension Δ a is calculated in terms of the normalized width of each intervoid ligament defined as χ = ⁄ . Here, and denote the width of the i th ligament in the initial and in the current configuration, respectively. The crack extension is evaluated using the following equation (Hutter, 2015). ∆a= 0 ∑ (1−χ ) =1 (6)