PSI - Issue 13

Ondřej Krepl et al. / Procedia Structural Integrity 13 (2018) 1279 – 1284 Ond ř ej Krepl & Jan Klusák / Structural Integrity Procedia 00 (2018) 000–000

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materials such as concrete (an aggregate in cement paste), the end points of layers or fibres embedded with matrix in composites etc. Study of the order of singularity of SMI can be found e.g. in Pageau et al. (1994) and Paggi and Carpinteri (2008). Only few studies regarding the calculation of stress series terms can be found (Yang and Munz (1995)). Establishment of failure condition of bi-material closed corner composed of CFRP and epoxy was conducted in Barroso et al. (2012). However, to the authors’ best knowledge no study that would in general describe crack initiation direction or a stability condition formulation in SMI exists. The geometry of a bi-material junction as shown in Figure 1 is characterized by angles  0 ,  1 and  2 . Complementary opening angle 2  is defined. The joint has two interfaces and no free surface. The material is considered as linear elastic and fully described by Young’s moduli and Poisson’s ratios in terms of elasticity. The material characteristics change by step at each interface. Perfect bonding is assumed at the interfaces, thus displacements and tractions are assumed to be continuous. The solution mostly presupposes the approximation of plane strain or plane stress. In this paper, the bi-material junction tip is presumed to be sharp. The stress distribution in the case of a bi-material junction is given by the asymptotic expansion Pageau et al. (1994):       1 1 1 1 2 1 1 2 2 3 3 3 ij ij ij ij H r f H r f H r f               (1) where the indices ( i , j )  ( r ,  ) are polar coordinates. The symbol H k stands for the generalized stress intensity factor (GSIF). The exponents of the singularity are p k = 1 − λ k . Generally, the eigenvalue λ k and corresponding H k are complex numbers. Nevertheless since the majority of bi-material junction cases have leading terms real, we assume the stress series (1) in its real form. For λ k satisfying 0 < λ k < 1, the corresponding k -th stress term is considered singular. For λ k where 1 < λ k the corresponding k -th stress term is considered non-singular. An effective tool to describe problems of isotropic plane elasticity is the Muskhelishvili’s complex potential theory, England (2003), applied to multi-material domain problems in Paggi and Carpinteri (2008). The eigenvalues λ k are gained from a numerical solution of the characteristic equation depending on the geometry and material properties of a bi-material junction, see Pageau et al. (1994), Paggi and Carpinteri (2008) and Yang and Munz (1995). The characteristic equation originates from equations enforcing stress and displacement continuity at the i th interface  i . For determination of the GSIFs of particular stress terms the overdeterministic method (Ayatollahi and Nejati, 2011) is an appropriate method to be used. An alternative method is the path independent  -integral where both methods give consistent results.

Fig. 1. Bi-material junction as a model for a sharp material inclusion

Detailed description of stress field reconstruction can be found in Krepl and Klusák (2016) and Krepl and Klusák (2017a). The latter paper showed that description of stress field by singular stress terms only can be insufficient in some cases. It was shown that for the concave inclusion stiffer than matrix the higher order terms are indispensable. For this case even if we decrease the radial distance to the nanometers, the singular terms still does not describe the stress field well.