PSI - Issue 41

A.E.S. Pinheiro et al. / Procedia Structural Integrity 41 (2022) 60–71 Pinheiro et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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5

2.4. Analytical models Stress distributions in the adhesive layer can be determined using analytical models. Although this approach is suitable to a limited type of joints, it also allows performing parametric studies faster than other techniques like numerical models (Garcia Momm and Fleming 2021). The analytical models can be linear-elastic, elastic-plastic, and both bi-dimensional or three-dimensional. The choice of model depends on many factors like the type of joint and adherend and adhesive materials. The simplest model consists in calculating the average τ xy stress on the adhesive layer by using the reaction force ( F ) and the adhesive layer cross-section, being O /( ) = F L B  , where B is the out-of plane width (da Silva et al. 2009a). Analytical models to calculate stresses in the bond-line had also been developed, each one with its strengths. However, these models are not fully relevant for tubular joints and, thus, they are not described here, but detailed descriptions of these models are available in the literature, e.g., (da Silva et al. 2009a, da Silva et al. 2009b, Quispe Rodríguez et al. 2012). For tubular joints subjected to an axial load, two analytic models stand out and were developed by Nayeb-Hashemi et al. (1997), and Pugno and Carpinteri (2003). The model of Nayeb Hashemi et al. considers the adhesive to be only subjected to τ xy stresses, and from this assumption the equations are derived using the nomenclature shown in Fig. 2.

Fig. 2. Tubular lap joint: scheme to derive the formulations. Adapted from (Nayeb-Hashemi et al. 1997).

 xy stresses in the adhesive layer are calculated as: ( ) ( ) ( ) 2 1 a xy 3 2 , ln ln  −    = − u u G R R r

(1)

where G a is the adhesive’s transverse elastic modulus, R 2 is the exterior radius of the inner tube, R 3 the inner radius of the outer tube, r is the variable position through the adhesive layer ( R 2 ≤ r ≤ R 3 ), and x is the normalized position along the bond-line, O / = x L  . Finally, u 1 (  ) and u 2 (  ) are the adherend displacements along L O , which are calculated from the solution of differential equations, being:

1

B

 

 

2 1 2 4 C C E R R u  + − − 3 2 ( ) 2 2

(2)

a C e C e   +

−

a  

( ) 

and ( ) u 

(

),

u

C C  +

=

=

−

1

2

3

4

1

2

2 E R R − 1 2 (

2

2

)

a 

1

where R 1 corresponds to the inner radius of the inner tube, R 4 to the outer radius of the outer tube, E 1 and E 2 are Young’s moduli of tube 1 and 2, respectively. The constants C 1 , C 2 , C 3 , and C 4 are calculated from the loading conditions. In addition, B and  a are calculated as:

  

2 E R R 2 4 (

2 −  )

2

2

2

G

a G L

2

L

1

and

.

B

(3)

a

3

=

+

= −

a 

 

2 ln ln R R E R R E R R − − − ( ( ) 2 2

2

2 3 2 1 2 2 R R E E R R R R − − − 1 4 ) ( ( 2 2

2

)

(ln ln )

)

3

2

1 2

1

2 4

3

3