Issue 74

E.V. Feklistova et alii, Fracture and Structural Integrity, 74 (2025) 55-72; DOI: 10.3221/IGF-ESIS.74.05

    r t r



1 ,

  ,



K r t

(2)



B

Since the strains in the damaged subregions can be much greater than in undamaged ones, the using of the Lagrangian strain tensor is preferrable. On the other hand, it is supposed that the deformation of the subregion does not lead to a significant change of its shape, size and configuration, so the Cauchy stress tensor can be used instead of Piola–Kirchhoff stress tensor. The equations of equilibrium (no mass forces are considered), the strain-displacement equations and the constitutive law (using generalized Hooke’s law, taking into account the scheme of material properties alternation after damage) are represented as:                               , , , , , , 0 1 , , , , , 2 , , , , ; ij j ij i j j i k i k j ij ijkl kl N z z r t r t u r t u r t u r t u r t r t C r t r t C r t C t r           



ijkl

ijkl



z

1

  

    K r 

init ijkl







C

t

,

:max ,

1



  z

V

    K r 

    z ijkl

        z z z z     im jn kp lq mnpq C dam

    ' z ijkl 

init ijkl



  

1 '   

:  



C t

t

C

C

,

:max ,

fract     

  z

V

    K r 

    ' z ijkl 

dam

1 '   





:  

C

t

C

C

,

:max ,

ijkl

ijkl



  z

V



E

E





init ijkl





,        ik jl il jk



ij kl  

C

G

G

,

  1 1 2  



 2 1





 



dam mnpq C C 

init

dam dam dam dam dam C C C C C     

, but

0

mnpq

1111

1122

1133

1313

1212













      

      

  z

  z

  z

x x

x x

x x

cos ,

cos ,

cos ,

1 1

1 2

1 3

 

 

 

 

 

 

  z im

  z

  z

  z





x x

x x

x x

cos ,

cos ,

cos ,

(3)

2 1

2 2

2 3

  z

  z

  z

x x

x x

x x

cos ,

cos ,

cos ,

3 1

3 2

3 3

fract



C

0

ijkl

Here σ ij is the stress tensor; ε ij is the Lagrangian strain tensor; u i is the displacement vector; C ijkl is the elastic constants tensor represented as piecewise-constant function that specifies the elastic properties   z ijkl C of the subregion indexed ( z ); init ijkl C is the elastic properties tensor of the undamaged (initial) material; E ,  ,  , G are the elastic constants of the isotropic material: Young’s modulus, Poisson’s ratio, Lamé moduli, respectively;  ij is the Kronecker delta; dam ijkl C is the elastic properties tensor of the damaged material with partial bearing capacity loss (i.e. the failure criterion was implemented once), the elastic properties are reduced in direction x 1 ( z ) ; ( x 1 , x 2 , x 3 ) is the global coordinate system; ( x 1 ( z ) , x 2 ( z ) , x 3 ( z ) ) is the local coordinate system of the orthotropic damaged material in the subregion indexed ( z ), associated with the direction of the principal stresses at the moment of partial bearing capacity loss (similar to the global coordinate system for the undamaged material);   z im  is the rotation matrix from the local coordinate system to the global one; fract ijkl C is the elastic properties tensor of material after complete bearing capacity loss. The problem is supplemented by the displacement boundary conditions and the traction boundary conditions:

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