Issue 71

E. Kormanikova et alii, Fracture and Structural Integrity, 71 (2025) 182-192; DOI: 10.3221/IGF-ESIS.71.13

  

  

∫

∫

∫ q O

T T u B EBu

T N p

T N q

(23)

δ δ = Π

−

−

dV

dV

dO

V

V

of which applies ( δ

)

T u Ku f f

− − = p q 0

(24)

where K is the stiffness matrix = ∫ T V dV K B EB

(25)

f p and f q are vectors of volume and surface external forces = ∫ T p V dV f N p = ∫ q T q O dO f N q .

(26)

If the components δ u are independent, we get from equation (24) the linear equation system

= f Ku

= + p q f f f .

(27)

The given equations valid for the finite element have an added index E . We express the internal energy of the finite element

1 2

1

∫

T E

T

T

=

= 2 u B EB u u K u E E E E E V dV

U

(28)

E

E K is the finite element stiffness matrix

where

= ∫

T

E V dV K B EB .

(29)

E

Using the Boolean matrix L E is determined the correct position of each finite element in the stiffness matrix of the entire structure. For the entire structure, the vector of displacements of the finite element is placed into the vector of displacements using the relation = E E u L u . (30) Then we obtain a system of equations by summation over all elements ( ) ( )   = +   ∑ ∑ T iE iE iE iE iEp iEq i i L K L u L f f . (31) After applying the boundary conditions, the matrix K becomes a regular, i.e. positive definite matrix. We used the programs ADINA [24] for the numerical analysis of delamination process of the CFRP composite.

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