PSI - Issue 33

4

Author name / Structural Integrity Procedia 00 (2019) 000–000

Sergey Smirnov et al. / Procedia Structural Integrity 33 (2021) 259–264

262

h h E E

1 1 2 h E h E h E h E     2 2 1 1 2 2 1

1 h G G G   1 h h

h

(3)

,

,





1  

2

2

E

1

2

2

where h 1 , h 2 are the glue and substrate layer thicknesses in the selected microvolume; E 1 , ν 1 , G 1 and E 2 , ν 2 , G 2 are the values of the normal elastic modulus, Poisson’s ratio and the shear modulus of the glue and substrate materials, respectively. Assume here that the ratio between the values of the thicknesses h 1 and h 2 in the selected microvolume is equal to the inverse ratio of the values of the normal elastic moduli of the glue and the substrate, i.e.

(4)

2 1 h h E E  1 2

Substituting (4) into (3), after simple transformations, we arrive at the following formulas for calculating the effective values � , ̅ and ν� for the selected microvolume:

1 E E E E   1 E E

1 G G G G G G G    1

2 E E E E   2 1 2 2

2

1 2  

(5)

E

,

,



2

2





2

2

1

2

2

1

2

1

The following values of the elastic constants are assumed in the calculations by the formulas in Eq. (5): Е 1 = 4518 MPa, ν 1 = 0.35 for the glue (Yu Jia et al. (2011)); E 2 = 69.58 GPa, ν 2 = 0.27 for the 1570 alloy substrate (Bronz et al. (2014)). The values obtained by Eq. (5) are given in Table 2.

Table 2 – The values of the effective characteristics of elastic properties in the selected microvolumes and the limit values of the strain energy density components � (GPa) ̅ (GPa) ν� � ∗ , (kJ/m 3 ) � ∗ , (kJ/m 3 ) 4793 1783 0.349 5.89 5.67

Table 1 shows the values of σ n , σ s , W n , and W s calculated for the averaged values of Р max ; fig. 2 shows graphical interpretations of the data from Table 3. Table 2 presents the limit values of strain energy density � ∗ under pure breakaway (α = 0) and � ∗ under pure shear (α = 90°). The values of � ∗ and � ∗ have been calculated by Eq. (2) for the averaged values of � and � from Table 1, and they are quantitatively similar. For a complex stress state, when the breakaway and shear stresses act simultaneously, ratio between � ∗ and � ∗ in adhesive failure are described by straight lines in the diagrams (Fig. 2, b). The equations of the approximating straight lines in Fig. 2, b can be written as

* s W W W W n n

(6)

1   , s

*

where the fraction denominators are equal to the lengths intercepting the straight lines when crossing the coordinate axes; these lengths are equal to the values of the strain energy density components at failure under breakaway � ∗ and shear � ∗ from Table 2. Equation (6) has the meaning of the adhesive failure criterion under the tension+shear