PSI - Issue 13

Lucie Malíková et al. / Procedia Structural Integrity 13 (2018) 1798–1803 Lucie Malíková & Jan Klusák / Structural Integrity Procedia 00 (2018) 000 – 000

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component of the stress tensor is utilized, see Eq. 1, the critical value of the GSIF must be calculated from the fracture criteria. Stress distribution around a tip of a crack at bi-material interface for polar coordinates r ,  : = √2 ∙ ( , , ) (1) In Eq. 1, f ij ( p,  ) is known function of bi-material parameters  and  , see Lin and Mar (1976), and p = 1 -  is the stress singularity exponent. Parameters  and  express the elastic mismatch and can be understood as elastic constants of bi-materials (depending on the elastic modulus and Poisson’s ratio of the material before and behind the interface) . For plane strain conditions and the interface between the MTX and ITZ are given as: = M IT T Z X ∙ 1 1 + + M IT T Z X −1 4(1− MTX ) , = MTX ITZ ∙ 1− I 2 TZ 1− M2 TX . (2) Eigenvalue  can be determined (for a crack perpendicular to an interface) from characteristic equation, see Lin and Mar (1976): 2 (−4 2 + 4 ) + 2 2 − 2 + 2 − + 1 + (−2 2 + 2 − 2 + 2 )cos( ) = 0 . (3) The value of the critical load, F crit , is then obtained from the formula: = ( ) , (4) where F appl is applied force defined in the finite element model of the three-point bending test and H I is the GSIF corresponding to this value. 2.1. Mean tangential stress value criterion According to this criterion, the critical value of the GSIF can be calculated as: = 2 −1⁄2 2− + , (5) Where K IC is fracture toughness of the material behind the interface, d is a length parameter (often referred to as critical distance) that should be connected to the material behind the interface (the material where the crack will propagate) and g R is a known function of the parameters  and  , see for instance Knésl et al. (1998). Similarly, the critical value of the GSIF can be derived by means of the generalized strain energy density factor: = ∙ −1⁄2 ( 1−2 (1− ) 2 [4(1−2 )+( − ) 2 ] ) 1⁄2 . (6) The meaning of the symbols is the same as in the previous equations, the elastic properties values and the value of the fracture toughness, K IC , belong to the material behind the interface. 2.2. Generalized strain energy density factor criterion Because of the symmetry, only one half of the three-point bending cracked specimen can be modeled in the finite element software ANSYS, see ANSYS (2016). The schema of the numerical model can be seen in Fig. 1. In the MTX a circular AGG with a very thin ITZ layer at its surface was modelled. The crack length a = W /2 and its tip was modelled at the MTX/ITZ interface. Several parameters of the model were kept constant whereas other ones were varied to study their influence on the fracture response of the system. In Fig. 1, L = 80 mm represents the half specimen length, S = 60 mm represents the half span between the supports, W = 40 mm represents the specimen width and B = 40 mm represents the specimen thickness. The value of the loading force applied to the 2D numerical model was F = 1 kN, i.e. the real force corresponding to the total specimen thickness would be 40 kN. Elastic Young’s 3. Geometry and material properties of the numerical model