PSI - Issue 5

N.A. Kosheleva et al. / Procedia Structural Integrity 5 (2017) 99–106 V.P. Matveenko et al. / Structural Int grity Procedia 00 (2017) 0 0 – 000

102 4

( ( Г k Г k

1) ( 1) ( k       a k

( ( Г k Г k

1) ( 1) ( k       a k

1) 1)

1) 1)

Here p = 1 –  ; g m , g a are the angels of the composite wedges;

—

,









m

m

m

a

m

a

3 4 k    , i = { m , a };  m ,  a are Poisson’s rations of materials

combined parameters of elastic material constants; and adhesive; Γ = G a / G m ; G m , G a are the shear modulus. i

i

В

g 

g  

B

a

g

t

m

R

C

a g

  g A

A

g m

Fig. 2. Shapes of the free surface of the spew fillet. It follows from Eq. (1) that the eigenvalues depend on the angles g m and g a and on the mechanical characteristics ν m , ν a , and Γ = G a / G m . In this case, the values of  k with the real part 0 < Re  k < 1 correspond to the singular solutions and determine the nature of the stress changes in the vicinity of the singular points. As an example, consider the following parameters of adhesive joint [11]. The elasticity moduli and Poisson ’ s ratios of the plate material and adhesive have the following values: E m = 1.567 · 10 10 Pa, ν m = 0.46, E a = 3.81 · 10 9 Pa,  a = 0.48, plate thickness t = 2.54 mm, thickness of adhesive layer t = 0.762 mm. The adhesive joint is under the influence of the stretching force P = 445 N. Calculations were carried out for different angles g a = g B and g a = g A . In the examples considered that g m at the point A is equal to π, and at the point B is equal to π/2. The numerical analysis results show that for angles g m , g a , where singular solutions are occurred (values satisfying the condition Re  k < 1), the stress concentration takes place nearby the singular points. In this case, the stresses in the vicinity of a singular point, where singular solutions take place, are determined by the degree of mesh refinement. While the dimensions of the finite elements are decreasing, the stresses at the singular point asymptotically tend to infinity, which in the numerical solution indicates as the existence of a singular solution. In the absence of singular solutions (all values of  k satisfy the condition Re  k > 1), the stresses at the singular point tend asymptotically to zero as the size of the finite elements decreases. At a minimum value of Re  k , which is equal to one, the stress at the singular point has a finite value. It should be noted that: the lower the value of Re  k < 1, the brighter the picture of stress concentration nearby the singular points. It should be noted that in the works [6 – 8, 11] for determination of the best version of the spew fillet geometry, the comparison was done using the stresses at the point A or B obtained by the finite element method. In the presence of a stress singularity, such analysis is not completely correct, since in this case the stress values that were obtained at singular points, are determined by the degree of mesh refinement. It was shown in [14, 15] that, in the vicinity of singular points, optimal geometries in terms of reduction of stress concentrations, have a common property: the parameters of optimal geometries in the vicinity of singular points (angles g m , g a ) and mechanical characteristics of the material ( ν m , ν a , Γ = G a / G m ) define the boundary between the solutions with and without the singularity. We use this property of optimal geometries to choose an optimal variant of the circle arc connecting the points A and B . In this case, the consideration of a limited class of surfaces in for choosing the optimal geometry is not critical and can be justified by the technology of adhesive joint manufacturing. The analysis of the eigenvalues in Eq. (1) for g m = 180° shows that eigenvalues with 0 < Re λ 1 < 1 are obtained for all g А > 0. According to the property of optimal geometries in the vicinity of singular points, the angle of coupling of the free surface of the adhesive layer with the adherend surface at the point A should be zero for the optimal choice of the circle arc. At the singular point B , for the considered materials, all the eigenvalues have real parts greater than unity at g В < 63°, and eigenvalues with 0 < Re λ 1 < 1corresponding to singular solutions appear at g В > 63°. Consequently, the optimal angle at the point B is g В ≤ 63°. There are no singular solutions at the point C only at g m = g a = 180°, but this variant does not make sense in