Issue 29

G. Gianbanco et alii, Frattura ed Integrità Strutturale, 29 (2014) 150-166; DOI: 10.3221/IGF-ESIS.29.14

deformation tensor is calculated at the material point of the finite element and it is next used to formulate the kinematical boundary conditions to be imposed on the UC associated to this point.

Figure 2 : Mechanical scheme of the CH strategy. In literature the periodic boundary conditions are suggested for periodic materials. In the present work, considering that the UC mechanical response in the non linear stage is strongly affected by the decohesion process occurring at the material interfaces, firstly the linear boundary conditions are imposed to the UC, thus on m m M UC   u D ε (6) where, for 2D applications:

y

    

   

x

0 2



D

x y 

m

0

2

is a matrix that depends on the x-y coordinates of the nodal point once a reference system is fixed. m u are the prescribed values of the displacements for the point of position x located on the boundary UC  of the UC. This choice could be easily modified on the basis of the obtained numerical results. Since the body forces can be neglected and the whole boundary is constrained, the variational form of the UC equilibrium can be expressed as follows: 0 d    (7) and it has to hold in presence of the kinematical constraint on m m UC   u u (8) The kinematical constraint (8) can be incorporated in the variational equality (7) making use of the Lagrange multiplier method, thus   T T T d d d          σ ε r u u r u (9) UC T m m   σ ε







m m

m m

m







UC

UC

UC

 ε the virtual strain tensor of the UC related to the virtual

where r is the vector of Lagrangian multipliers and m displacements through the following compatibility condition:

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