PSI - Issue 15

Mikhail Perelmuter / Procedia Structural Integrity 15 (2019) 60–66

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M.N. Perelmuter / Structural Integrity Procedia 00 (2019) 000–000

fluid”, which provide self-healing of the material during cracks formation, see White et al. (2001). High e ffi ciency is also demonstrated by the technique of introducing fibers with shape memory properties (nitinol, for example) into a polymer composite material. In this case fibers, due to the reverse phase transformation, lead to crack closure and healing, see Burton et al. (2006). Note also, that the self-healing processes in the stent materials should occur at human body temperature, and the ”healing fluid” should be harmless to health. Analytical and numerical estimations of various factors influence on materials self-healing processes are necessary for increasing of these procedures e ffi ciency. We note, also, that in many cases the experimental investigations can be extremely long and labor-consuming. It is possible to mark out three basic stages of materials process self-healing: 1) formation and growth flaws / cracks under external loading; 2) activation of self-healing mechanisms; 3) healing of flaws / cracks with partial or total restoration of bonds between crack surfaces. Experimental results have been obtained on each of these stages, but physical-mechanical models and numerical techniques have only started to develop. Known techniques are based, mostly, on the using of cohesive models in the framework of finite elements method, see Ozaki et al. (2016); Alsheghri and Al-Rub (2016); Ponnusami et al. (2018). In this paper we use the bridged crack model (in this model it is assumed that the stress intensity factors do not vanish at the crack tip) for the e ffi ciency evaluation of self-healing materials for stents application. The bridged crack model allows to analyze fracture toughness of heterogeneous materials on the basis of microme chanical properties of bonds, see Goldstein and Perelmuter (1999); Perelmuter (2014). The extension of this model with accounting of time-variation in the mechanical properties the crack bridged zone allows one to estimate the long-term strength and variation over time of materials resistance characteristics, Goldstein and Perelmuter (2012); Perelmuter (2013). In the present paper the bridged crack model is used for modelling of cracks self-healing. We consider quasistatic problem for a bridged crack with increases ligaments sti ff ness over time. Bonds restoration is considered on the basis of the thermal fluctuation kinetic model together with the crack bridged zone model. The main target of the modelling is the computational analysis of the bridged stresses distribution and the com puting of the stress intensity factors which are the main characteristics of self-healing e ffi ciency. The mathematical background of the stresses problem solution is based on the singular integral-di ff erential equations method. Some results of self-healing processes analysis are presented. The main measure of the self-healing e ffi ciency is the value of the stress intensity factor at the crack tip. We use previously proposed by Goldstein and Perelmuter (1999) the bridged interface crack model with the as sumption that the crack surfaces interact in some zones starting from the crack tips. Let us consider the planar elasticity problem on a crack ( | x | ≤ ) at the interface of two dissimilar joint half-planes, see Fig. 1. Assume that the uniform loads normal to the material interface are acted at infinity. The crack surface interaction is supposed to be existing in the bridged zones, − d ≤ | x | ≤ . The size of the interaction zone d = d ( t ) depends on time due to the possibility the time changing of the bond properties (bonds degradation or bonds healing). As a simple mathematical model of the crack surfaces interaction we will assume that the linearly elastic bonds act through the crack bridged zones. Denote by Σ ( x , t ) the bond stresses, occurring under the external loads action, also depend on time Σ ( x , t ) = σ yy ( x , t ) − i σ xy ( x , t ) , i 2 = − 1; σ ( x , t ) = σ 2 yy ( x , t ) + σ 2 xy ( x , t ) (1) where σ yy , xy ( x , t ) are the normal and shear components of the stress, respectively, σ ( x , t ) is the stress vector modulus. The crack opening, u ( x , t ) at − d ≤ | x | < is determined as follows u ( x , t ) = u y ( x , t ) − i u x ( x , t ) = c ( x , t ) ( σ yy ( x , t ) − i σ xy ( x , t )) , c ( x , t ) = ϕ ( x , t ) H E b (2) where u y , x ( x , t ) are the projections of the crack opening on the coordinate axes (Fig. 2), c ( x , t ) is bonds compliance, H is a linear scale related to the thickness of the intermediate layer adjacent to the interface, E b is the e ff ective elasticity modulus of the bond and ϕ ( x , t ) is dimensionless function. By incorporating linearity of the problem one can represent the crack opening u ( x , t ) as follows u ( x , t ) = u ∞ ( x , t ) − u b ( x , t ) , (3) 2. Bridged crack model