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

Fig. 1. Crack with two bridged zones at the materials interface.

Fig. 2. Normal u y and shear u x crack opening at the bridged zone edge.

where 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. Dimensionless function ψ ( x , σ ) is introduced into (9) to define the dependence of the bond healing time on the distance to the crack tip and the external load. The solution of equation (8) (at the initial condition n ( x , 0) = 0) gives the time variation of density of bonds formed between the crack faces

t τ h ( x , σ )

Z ( x , t , σ ) = 1 − exp −

n h ( x , t ) = n 0 Z ( x , t , σ ) ,

(10)

The increasing in the bond density over time leads to increasing the bonds sti ff ness in the crack bridged zone. Let k s ( x ) denote the rigidity of a single bond formed between crack faces. Then the e ff ective rigidity of bonds per unit area of the crack bridged zone is k ( x , t ) = k s ( x ) n ( x , t ) = k b Z ( x , t , σ ) (11) where k b ( x ) = k s ( x ) n 0 is the final bond rigidity per unit area of the crack bridged zone. It follows from (11) that the bond compliance in the crack bridged zone can be represented as the decreasing function over time where c h 0 is the bonds compliance in the crack bridged zone after the ending of the self-healing process (bridged zone formation), dimensional function ϕ h ( x ) defines compliance variation over the bridged zone. This relation enables us to model of the crack bridged zone formation by means of bonds compliance variation over time. The system of SIDE is solved by the time-steps scheme as in Goldstein and Perelmuter (2012). The bond compliances at each time step depend on the density of bonds according to relation (12). For example, see White et al. (2001), if the initial crack reaches su ffi cient size in the polymer composite with the healing microcapsules and / or with shape-memory alloy wires then the self-healing process is started. On this stage the problem of bonds restoration for the crack bridged zone is solved. The healing time is dependent on the chemical reaction rate of the healing agent, wires properties, crack size and the external loads. The decreasing of the stress intensity factors is used as the measure of the healing e ff ect. The model can be used for the evaluation of healed composite materials durability. The computation modelling results of bonds healing and interfacial crack repairing are presented below. c ( x , t ) = c h 0 ( x ) Z ( x , t , σ ) , c h 0 ( x ) = c 0 ϕ h ( x ) E b (12)

4. Modelling results

The proposed above approach for cracks self-healing was used for several problems analysis. A crack at the in terface between di ff erent materials was considered. It can be treated as a model of interface delamination in polymer