PSI - Issue 2_A

Mikhail Perelmuter et al. / Procedia Structural Integrity 2 (2016) 2030–2037 M. Perelmuter / Structural Integrity Procedia 00 (2016) 000–000

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If u ( ℓ ) 0 has been chosen in the crack model, then it is assumed that the crack bridged zone consists of two parts of sizes d 1 and d 2 with di ff erent deformation laws and G m is the density of energy dissipated in the zone d 1 . In our approach it will be assumed that d 1 ≪ d 2 , the thickness H of the interface layer ahead of the crack tip (for adhesion junction) is infinitely thin, H /ℓ → 0, and, therefore, the integral for G m in (9) is omitted and the contribution of all ligaments near the crack tip is regarded as G Ic , see (1).

The two types of material junction can be considered in the frames of the conditions (4): a) composite materials with fibers and b) adhesion joining two di ff erent materials without fibers. In these cases the parameter G Ic is defined as G Ic = { 2 c m γ m + 2 c f γ f ( a ) η G b ( b )

(10)

where in the case a) c m = 1 − c f and c f is the specific concentration of fibres in the composite, γ m and γ f are the specific face energy of the matrix and fibers materials; in the case b) the parameter η ≥ 0 defines a relative intrinsic toughness of the material junction and it can be considered as the interface weakness measure ahead of the crack tip. We will assume that η = 1 corresponds to the similar deformation law of bonds in the bridged zone and bonds ahead of the crack tip, if η > 1 then the adhesion junction is hard and the case 0 ≤ η < 1 corresponds to a weak interface junction. The value η = 0 corresponds to ’an ideal weak’ interface - it is the case of cohesive models and it was shown in (Perelmuter, 2007) that criterion (4) is degenerated for η = 0 to the fracture condition in Barenblatt Dugdale-Panasyuk model, see (Barenblatt, 1959; Dugdale, 1960; Panasyuk, 1971) Expression (9) can be written accounting to relations (2) and u ( ℓ ) = 0 , G m = 0 as (see details in (Perelmuter, 2007))

ℓ ∫ ℓ − d (

q x ( u ) ) dx + G Ic − G b

∂ u y ( x ) ∂ ℓ

∂ u x ( x ) ∂ ℓ

G bond ( d , ℓ ) =

q y ( u ) +

(11)

Note, that in the crack limit equilibrium state the quantity G b in (11) is the strain energy density released during bonds rupture in the bridged zone trailing edge x = ℓ − d . If we assume that η = 1 (the rate of the energy released at the trailing edge of the bridged zone is equal to the rate of the energy consumed by newly deformed bonds during the crack tip advancing) then in the case of a homogeneous material or an infinite thin adhesive layer joining di ff erent materials

δ cr ∫ 0

G Ic = G b =

σ ( u ) du

(12)

Fulfilment of the necessary and su ffi cient conditions (4) corresponds to the limit equilibrium state of the crack tip and the trailing edge of the crack bridged zone. The parameter δ cr is defined by the properties of the bonds in the crack bridged zone and can also depends on the scale of the crack (for example, when the type of bonds changes as the crack grows). From the simultaneous solution of Eqs. (4) it is possible to determine the size of the bridged zone d cr and the critical external stress σ cr in the crack limit equilibrium state. The deformation energy consumption rate obtained from this solution is the energy characteristic of the adhesive fracture toughness, G cr = G bond ( d cr , ℓ ), and the quantity G cr does not remain constant when the crack length changes. After the critical external load has been determined, the critical stress intensity factor and the energy flux to the crack tip, due to the external load σ cr , can also be determined.

3. Results

In this section the results of the analysis of the SIF and the fracture parameters for the problem of uniaxial tension of the plate with straight crack of length 2 ℓ on the interface of two dissimilar elastic half-planes are presented. The following material data were used: E 1 = 135 GPa, E 2 = 25 GPa, ν 1 = ν 2 = 0 . 35 (Cu-epoxy polymer junction), E b = E 2 , ϕ 1 , 2 = 1 (linear-elastic bonds). For these data β < 0 and | β | = 0 . 0509313, see relations (6). Initial size of the half crack and the critical crack opening were assigned to ℓ = 0 . 5 · 10 − 3 m and δ cr = 2 · 10 − 7 m , see details in (Goldstein and Perelmuter, 1999). The crack opening and the tractions distributions along the interfacial crack bridged zone are