PSI - Issue 33

I.J. Sánchez-Arce et al. / Procedia Structural Integrity 33 (2021) 149–158 Sánchez-Arce et al. / Structural Integrity Procedia 00 (2019) 000–000

152 4

deformability matrix, H , and then, using Galerkin’s weak form, establish the stiffness matrix: . Afterwards, the equilibrium system of equations can be written: Ku = F , being F the force vector and u the displacement field. Here, the method has been briefly described; nevertheless, it is described in detail in (Belinha, 2015; Dinis et al., 2009). In this work, the meshless methodology was implemented by means of custom-written MATLAB scripts (MATLAB R2018a, The MathWorks, Inc.). 3. Hyper-elastic models The Neo-Hookean hyper-elastic model defines the strain-energy ( W ) as a function of the shear modulus ( G ) and the first invariant of the Cauchy-Green tensor; it can also be represented using the principal stretches, as in Equation (5) (Treloar, 1943).   2 2 2 1 2 3 1 3 . 2 W G        (5) The Mooney-Rivlin model considers both the first and second invariants of the Cauchy-Green tensor, which can also be expressed in terms of the principal stretches (Equation (6)). Moreover, the double of the sum of its two constants, C 1 and C 2 , is equal to G (Equation (7)) (Ogden, 1972). T  K H cH d  

  

 

2    2 2 1 1 1









(6)

2    2 2

1 2 3 , ,   

3    

3 , 

W

C

C



  

1 1

2

3

2

1

2

3

(7)

2( ) . G C C  

10

01

The Ogden model, however, considers the strain-energy as a power function of the principal invariants (Equation (8)). Moreover, its constants  p and its powers  p can also be related to G , as shown in Equation (9), where N OGD is the number of terms on the strain-energy function, being N OGD =3 a typical value.   1 2 3 1 3 , OGD p p p N p p p W               (8)

1    1 2 OGD N p

.

G

p p  

(9)

The hyper-elastic model chosen to be implemented is the Ogden model. This model also allows analysing cases with Neo-Hookean and Mooney-Rivlin material behaviours (de Souza Neto et al., 1995; Ogden, 1982). These equations are expressed in terms of the principal stretches, which is convenient for the following stages. At each increment of the non-linear solution, the deformation gradient ( F ) was calculated with the aid of the partial derivatives of  i ( x I ), as follows:

1 d x X x X x X x X d d / d d d d / / / d 1 1

1 d d d d / /

x X x X

 

     

     

2

n

2

1

2

2

2

n

F

,

(10)







  

/ d d d d x X x X /

/ d d

x X

1

2

m

m

m n

T  B FF (11) The principal stretches had to be calculated from the polar decomposition of the left Cauchy-Green tensor ( B ); this procedure is detailed by de Souza Neto et al., (2008) and Baaser, (2010). Subsequently, the derivative of W with