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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On the contrary, the model proposed by Pugno and Carpinteri assumes that the joint is also subjected to σ y stresses, whilst the stresses through the adhesive thickness are assumed constant. The scheme to derive the formulations is shown in Fig. 3.

Fig. 3. Tubular lap joint: nomenclature for the Pugno & Carpinteri model. Adapted from (Pugno and Carpinteri 2003).

σ y and τ xy stresses in the adhesive layer are calculated as follows:

(1 ) − 

E

a a E



( ) x

( ) x

x  ( ( ) x

( )), x

(4)

a a



=

r 

+

+

θ 

y

a  (1 )(1 2 ) + − a 

a  (1 )(1 2 ) + − a 

( x x N C e C e     ) −

− 

( ) x

,



= −

1

2

(5)

xy

2

R

where E a is the adhesive’s Young’s modulus and  a its Poisson ratio, whilst  x ,   ,  r are strains in the axial, transverse, and radial directions, respectively. The strains are calculated as follows:

  

  

( ) (1 ( )) − f x

N f x

( ) x

,

=

+



x

2

1 1 E A E A f x  2 2 ( ( ) 1) −

  

  

( )

f x

1 

( )

,

= x N

−

θ 

2

(6)

2

E A

1 1 E A

2 2

  

  

(1 ( )) − f x

( )

f x



1 

( )

,

= x NR h

−

r 

2

E A

1 1 E A

2 2

where N is the axial force imposed to the joint, R is the inner tube’s outer radius, h the adhesive layer’s thickness, E 1 and E 2 are respective outer and inner tubes’ Young’s moduli. Similarly, A 1 , A 2 are the respective cross-sectional areas,  1 and  2 their respective Poisson ratio. In addition, f(x) is a function calculates as shown in Equation (7).

2

1 1 RG E A E A h E A E A + 2 2 1 1 2 2 ( a

)

1 1 E A



( ) f x C e C e  = + x

x

,  

,

,

−



+

=

=



1

2

1 1 E A E A + 2 2

(7)

c

c e e  −

c

c

c e e − −

c  

−



−





e

e

,

,

C

C

=

+

=

+





1

2

2

2

2

2

2 e e  − c

2

2 e e  − c

2

c

c

c

c

c

c

−





−





−



−



e

e

e

e

−

−

From Equation (7), c corresponds to half L O .

2.5. Numerical models The numerical work was done within ABAQUS (Abaqus 2017, Dassault Systèmes INC. RI, USA). The joint geometries correspond to that described in Section 2.1. Furthermore, due to the axisymmetric nature of the tubular joint, this approach was followed because it allows saving computational resources. The numerical modelling followed two approaches. The first approach used cohesive elements in the adhesive layer, and it was used to calculate