PSI - Issue 47

Mikhail Perelmuter / Procedia Structural Integrity 47 (2023) 545–551 M. Perelmuter / Structural Integrity Procedia 00 (2023) 000–000

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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 researches 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 in the last decade. 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 composite materials self-healing. 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 pro cesses. We consider problem of a crack bridged zone formation with increases ligaments sti ff ness over time. Bonds restoration is considered on the basis of the thermal 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. In fracture mechanics the crack bridging model is used to analyze fracture toughness and cracks growth assuming of bridged zone destruction as cracks advance. To model cracks self-healing we here consider an inverse process which is the crack bridged zone formation during, for example, a crack filler polymerization or an oxidation reaction inside cracks in ceramics. The formed bridged zone will prevent the crack growth. The restoration of bonds inside of cracks (cracks self-healing) can be considered on the basis of kinetic model. To model self-healing of bonds, we assume that the increase in the density of bonds between the crack faces over time n h ( x , t ) is governed by a first-order kinetic equation, see Khawam and Flanagan (2006) dn h ( x , t ) dt = n 0 − n h ( x , t ) τ h ( x ,σ ) (1) Here n 0 is the maximum of bonds density between crack faces (bonds density in undamaged material), σ is the bond tension at the distance x from the crack tip, τ h is the characteristic time of bonds healing ( 1 /τ h is the healing rate constant) defines by the Arrhenius type relation τ h ( x ,σ ) = ψ ( x ,σ ) A ( T ) , ψ ( x ,σ ) = u 2 x ( x , t ) + u 2 y ( x , t ) H , A ( T ) = τ 0 exp U h RT , (2) where dimensionless function ψ ( x ,σ ) is introduced into (2) to define the dependence of the bond healing time on the distance x to the crack tip, u y , x ( x , t ) are the projections of the crack opening on the coordinate axes (Fig. 1), H is a linear parameter related to the thickness of the formed bridged zone, τ 0 = α h / kT is the characteristic time, h is Planck’s constant, k is the Boltzmann constant, T is absolute temperature, α is a dimensionless coe ffi cient depending on the material type (polymer, metal, or ceramics), U h is the energy of bonds healing, R is the gas constant. The solution of equation (1) (at the initial condition n ( x , 0) = 0) gives the time variation of density of bonds formed between the crack faces 2. Model of cracks healing - bonds restoration inside of cracks

t τ h ( x ,σ )

Z ( x , t ,σ ) = 1 − exp −

n h ( x , t ) = n 0 Z ( x , t ,σ ) ,

(3)