PSI - Issue 42

Roman Vodička et al. / Procedia Structural Integrity 42 (2022) 927 – 934 R. Vodicˇka / Structural Integrity Procedia 00 (2019) 000–000 R ( u ; ˙ α, ˙ ζ ) = −  η = A , B 3 8 ϵ  Ω η D η ( u ) ˙ α η d Ω −  Γ i D i (  u  ) ˙ ζ d Γ ,

930

4

(3)

provided that the constraining inequalities are satisfied. The functions manifesting fracture mode dependence in the last relation may take the following form:

arctan 

2    2 π

n  

+ e ( w )    2 + e ( w ) | 2 p ( G iII c − G

p    sph K η p | sph iII c + µ

+ µ η | dev e ( w ) | 2

K η

G iII c G iI

w s w +

D i ( w ) = G iI

η ( w ) = G iII c

− G iI

c tan

1 · arctan

c , (4)

 , D

c −

+ µ η | dev e ( w ) | 2

c / ( G

iI c ) )

η / K η

G iI

which contains G II c and G iII c as the fracture energies for material and interface, respectively, in the shearing mode. The formulae are set so that for an opening crack the energy release per unit area is G I c , G iI the case of the shearing crack. The quasi-static evolution in deformable domains with cracks can be expressed in a form of nonlinear variational inclusions with initial conditions ∂ u E ( t ; u , α, ζ ) ∋ 0, ∂ ˙ α R ( ˙ α, ˙ ζ ) + ∂ α E ( t ; u , α, ζ ) ∋ 0, ∂ ˙ ζ R ( ˙ α, ˙ ζ ) + ∂ ζ E ( t ; u , α, ζ ) ∋ 0, u η (0 , · ) = u η 0 , α η (0 , · ) = α η 0 = 1 in Ω η , ζ (0 , · ) = ζ 0 = 1 on Γ c . (5) where ∂ denotes the partial subdi ff erential (e. g. R may jump for zero damage rates). It is also supposed that the energy functional (with respect to separated variables of state) is separately convex (assumptions for degradation functions). The initial values for damage parameters pertain to a non-degraded state. In the computational implementation, all state variables are approximated by an adequate finite element mesh and interpreted within an own MATLAB code of FEM analysis which uses simple implementation in the MATLAB environment based on Alberty et al. (2002). This is allowed by the computational scheme for the time discretisation which uses a staggered algorithm, as it was described in Vodicˇka (2019) for a procedure with a fixed time step size τ (proposed in Roub´ıcˇek et al. (2015)). This computational implementation leads to separation of deformation quantities from the damage ones. The separation of variables in the staggered approach provides a variational structure to the discretisation of the solved problem in Eq. (5). In each time step, two minimisations have to be resolved recursively. Based on Eq. (1), it is generally admitted that with respect to displacement a quadratic functional is obtained for whose minimisation quadratic programming (QP) algorithms are applied, based on Dosta´l (2009). The other restriction generally leads to a convex functional (based on the aforementioned assumptions on the functions Φ ( α ) and ϕ ( ζ )) for which the QP algorithm is applied sequentially. c . There remain just G II c , G iII c in 3. Notes on the computational algorithms Crack propagation as a competition between interface and bulk cracks is demonstrated in a domain with two inclusions. The scheme for the problem is shown in Fig. 2. Two slightly unsymmetrically placed inclusions within a square domain are considered. The initial elastic properties (introduced in Eq. 1 for an undamaged material) are: K A p = 52 GPa, µ A = 29 GPa (inclusions), K B p = 3 GPa, µ B = 1 GPa (matrix), and κ =  1 0 0 0 . 5  PPam − 1 , κ G = 1 EPam − 1 at the interfaces. The mesh size (min.) in Fig. 2 is h = 0 . 25 mm. The fracture energy in all domains is G c = 1 kJm − 2 , while that of the interfaces is G i c = 1 Jm − 2 . The hard-device loading g 2 ( t ) = v 0 t is increasing at the velocity v 0 = 1 mms − 1 , and with a refined time step of 0 . 1 ms. The PFM degradation function Φ is chosen in a simple quadratic form: Φ ( α ) =  α 2 + δ 2  , where the parameter δ as mentioned above adjusts the residual sti ff ness after total damage to avoid material degeneration in the numerical solution. The interface degradation function ϕ is chosen to obey a bilinear stress-strain relation as defined in Vodicˇka and Manticˇ (2017). It is determined by the function ϕ ( ζ ) = βζ 1 − ζ + β , with β = 0 . 1. 4. Examples