PSI - Issue 33

R.F.P. Resende et al. / Procedia Structural Integrity 33 (2021) 126–137 Resende et al. / Structural Integrity Procedia 00 (2019) 000 – 000

131

6

( , F

)

( ) 

' 2 J k

(1)

 

= =

.

Therefore, it can be related to  y , as

3

(

)

( )   y

(2)

'   ' 2 ij ij

, F  

0,

=

−

=

where  ij ’ corresponds to the deviatoric stress tensor. Upon mathematical manipulation of equation (2), the resulting equation is the known formulation from the equivalent stress or von Mises stress. Further details of how this criterion is implemented into FEM and into meshless methods can be found in the literature (Owen and Hinton 1980, Crisfield 1997, Dinis et al. 2009). The von Mises yield criterion, however, does not consider the hydrostatic pressure. Because of this, its yield surface describes a cylinder. However, as described before, the yield onset on some adhesives also depends on the hydrostatic pressure, p , as described by Dean et al. (2004). According to this reference, the Exponent Drucker-Prager (EDP) yield criterion provides a closer description to experimental adhesive yielding in confined spaces, as the adhesive joints. 3.3. Exponent Drucker-Prager The Exponent Drucker-Prager (EDP) yield criterion was found to be suitable to describe the yielding of rubber like materials as the adhesives (Dean et al. 2004). Unlike the VM yield criterion, the EDP is sensitive to the hydrostatic pressure in an exponential manner. Therefore, the yield surface is described as a paraboloid in the deviatoric-hydrostatic plane (  e -  H ), as follows ( ) ( ) DP DP t 0. b F a q p p  = − − = (3) being a DP , b DP , and p t material specific constants (Dean et al. 2004); also, p corresponds to the hydrostatic pressure and q to the equivalent stress, i.e., von Mises stress. The constant p t can be derived from the other constants and the yield strength (  y ) obtained from a uniaxial tensile test, as follows

3 

( )

y

(4)

b

.

p a =

+



DP

t

DP y

From equation (3), p is calculated as the average of the first invariant of the stress tensor,  . Thus, the mean stress,  m , is defined as p I , where I is the identity matrix. Then, from equations (3) and (4), the yield surface can be described by

3 

( ) ( ) DP  =

( )

y

(5)

DP a q p a − − b

b

0.

F

− =



DP

DP y

In addition, a DP can be determined as a function of the material yield strength in tensile (  y ) and shear (  y ) at the deviatoric plane, i.e., when  m =0 and considering an exponent b DP =2 (Dean et al. 2004). For clarity, a new constant, λ DP , was added. λ DP is defined as the squared ratio of  y over  y . Hence, a DP is



1 .

y

(6)

a

=

=

(

)

(

)

DP

3

1

 

−

2   − 2

3

y DP

y

y

Consequently, with the introduction of λ DP , equation (5) takes the form