PSI - Issue 2_B

Moslem Shahverdi et al. / Procedia Structural Integrity 2 (2016) 1886–1893 Shahverdi et al./ Structural Integrity Procedia 00 (2016) 000–000

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increasing opening displacement the traction across the interface reaches a maximum, at (  , σ max ), then decreases and eventually reaches zero, presenting a complete separation at an opening displacement of δ f , at the end of the fiber bridging length, l br , see Figure 3 left. The area under the σ - δ curve represents the amount of energy, Φ , dissipated during crack propagation in the cohesive zone. The three parameters: Φ , maximum traction, σ max , and maximum opening displacement, δ f , are interdependent and therefore the CZM can be described by two of them assuming an appropriate traction-separation cohesive law model. The exponential laws implemented in ANSYS software were used in this work. The normal and tangential tractions, T n and T t respectively are: 2 ( ) max n t n n e t n n T e e           and 2 ( ) 2 (1 ) max n t n t n n e t t t t n T e e                (13) where  n and  t are the normal and tangential opening displacements along the cohesive zone and n  and t  are the arbitrary normal and tangential opening displacements at maximum traction. The values of  and σ max required by the CZM were estimated by an iterative procedure. The estimated cohesive element model parameters for different lever lengths are listed in Table 2. The following equation represents the amount of energy dissipated in the crack-bridging zone, G br , according to the CZM approach (Sorensen, Botsis et al. 2008). f br br 0 = d G     (14) where  br is the bridging traction and  is the relative opening displacement along the fiber bridging length of the upper and lower arms. The G br can be partitioned into Mode I and Mode II components as: f-n br I br η η 0 ( ) = ( ) d G     and f-t 0 br ζ br II ( ) = ( ) d ζ G     (15) where δ f-n is the maximum normal crack displacement, δ f-t is the maximum tangential crack displacement, and  and η are the local axes, see Figure 1. In Eqs. (20-22) the bridging traction is obtained from the cohesive elements in the FE models along the bridging length. Table 2. Traction-separation cohesive model parameters for different lever lengths Specimen (lever length) σ max (MPa) n  (mm) t  (mm) c=227 mm 0.65 0.37 0.24 c=150 mm 0.85 0.41 0.24 c=100 mm 0.75 0.45 0.23 c=060 mm 0.85 0.48 0.23

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 br,max

Extended global method Finite element method

c =227 mm

4

3

G I /G II (-)

Traction,  br , (MPa)

c =150 mm

2



c =100 mm

1

c =60 mm

 f

50 60 70 80 90 100 110 0



Opening displacement,  , (mm)

Crack length (mm)

Figure 3. Left: Schematic illustration of cohesive traction-separation law. Right: Mode ratio, G I / G II , versus crack length determined by extended global method and finite element modeling The VCCT was used for calculation of the fracture parameters at the crack tip. Bi-material interfaces were present in all specimens. Therefore, the calculated G I-tip and G II-tip components and the calculated mode-mixity, G I / G II , depended on the FE mesh size around the crack tip, Δ a , and did not represent the actual fracture development. In order to diminish the effect of the Δ a a thin “resin” interlayer was inserted which had the average properties of the adjacent layers of the interface. The G I / G II components for a crack with a resin interlayer were independent of the mesh size.