PSI - Issue 21

Tuncay Yalçinkaya et al. / Procedia Structural Integrity 21 (2019) 52–60 Yalc¸inkaya et al. / Structural Integrity Procedia 00 (2019) 000–000

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there is not a unique form of the traction-separation law and countless number of them have been suggested in the literature. Mainly divided into initially rigid and initially elastic laws, bilinear, polynomial and exponential forms exist (see Park and Paulino (2011) for an overview). The underlying physical mechanism for the initiation and propagation of ductile cracks in metallic materials is the nucleation, growth and coalescence of micro-voids. Voids nucleate at inclusions and second phase particles due to particle-matrix interface decohesion or particle cracking. Then, voids grow due to plactic deformation and coalesce by localization of plasticity between closely spaced voids. This microscopic physical mechanism can be observed as macroscopic cracks causing failure of the material. This phenomenon has been researched extensively and im plemented as constitiuve porous plasticity and creep models (e.g. Gurson (1977), Tvergaard and Needleman (1984), Gologanu et al. (1993), Cocks (1989)). The idea here is to bridge the information obtained from the physical fracture mechanism due to void growth to a traction-separation law in order to obtain a physically motivated relation to be implemented in crack propagation simulations. The derivation and the numerical anaylsis of a traction-separation law based on the growth of pores is discussed. We start by considering an array of cylindrical representative volume elements with initial cylindrical pores. Under normal and shear displacements, traction separation relations are obtained by aplying upper bound theorem for mode-I and mixed-mode repectively. Then, the e ff ect of micromechanical parameters appearing in the traction separation-law readily such as initial pore fraction and initial pore height are investigated through numerical simulations of a compact-tension (CT) specimen under mode-I loading. The paper is organized as follows. First, in Section 2 derivations of the models and equations are summarized for physics based traction separation laws for mode-I and mixed-mode loading. Then, in Section 3 numerical results are presented, where the capability of the model under mode-I loading is discussed. Finally, in Section 4 the conclusions and the future remarks are given. 2. Model formulation In this section, the main steps for obtaining the traction-separation equations are summarized. Please see Yalcinkaya and Cocks (2015) and Yalc¸inkaya and Cocks (2016) for a more detailed derivation. 2.1. Traction-separation relation for Mode-I loading

2a

r

h

2l

Fig. 1. Idealization of pores within a plane as cylinders (left), and the response to normal traction (right).

Within a plane, imagine an array of pores, idealized as cylinders (see Fig. 1). Take one cell as the representative volume element (RVE) and assume all cells are the same. Assume that the in plane macroscopic strain is zero, i.e. u = 0 at r = l , u being radial displacement in the plane. Let the radius of pores be a , the mean spacing 2 l and the height of the cylinder be h . Initially, we consider the response to normal traction T n only (see Fig. 1). RVE is fully constrained so that l is constant. In polar coordinates, using the incompressibility condition ˙ ε r + ˙ ε θ + ˙ ε z = 0, and small strain-displacement