PSI - Issue 68

Roman Vodička et al. / Procedia Structural Integrity 68 (2025) 212 – 218 Roman Vodicˇka / Structural Integrity Procedia 00 (2024) 000–000

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Both damaging processes, in material and along interfaces, are unidirectional. Thus, both internal variables may only increase by satisfying the conditions ˙ α ≥ 0 in Ω , and ˙ ζ ≥ 0 in Γ i , where ’dot’ means the time derivative. Additionally, there are other nonlinear processes which dissipate the energy, e. g. plastic deformations. Not taking them into account, the fracture energy still may be di ff erent if the crack tends to propagate in a shear mode. That was anticipated in (1), where the superscript ’I’ for G c appeared to emphasise the crack mode I, opening. In the crack mode II, the fracture energy is distinguished as G II c , supposing G II c ≥ G I c both inside material or along the interface. Therefore, a crack mode dependent fracture energy is considered, and in the energy formulation it is introduced in a dissipation (pseudo)potential R ( u ; ˙ α, ˙ ζ ) = i Ω i 3 8 D i ( u ) − G I i c ˙ α d Ω + jk Γ jk D jk u − G I jk c ˙ ζ d Γ , ˙ α ≥ 0 in Ω i ˙ ζ ≥ 0 in Γ jk . (2) Not being satisfied the constraints, the value of R is set to infinity. The mode mixity character can be introduced by the aforementioned splits of the strain energy: Inside subdomains, the orthogonal split of the strain tensor used for the strain energy based on the formulae of Vodicˇka (2023). From the series analysis of the crack tip strain relation follows that in front of the crack in mode I there holds | dev e | = 0, and similarly in front of the crack in mode II | sph e | = 0 is valid. At the interface crack tip, in mode I there holds [[ u ]] s = 0, and in mode II [[ u ]] n = 0 is valid. Thus, the functions D are supposed in the following form: Additionally, the functions introduce the di ff erence between tension and compression so that if the state is close to uniform compression, no crack will appear. Both options introduce a new parameter ω which a ff ects the shear in compression. Small ω clearly increases the denominator containing G II c , thus crack changes are suppressed. Taking the inertia into account, the kinetic energy is also a vital part of the energy balance. It is given as K (˙ u ) = i Ω i 1 2 ρ | ˙ u | 2 d Ω , (4) where ρ denotes mass density of the material. Finally, the external forces also contribute to energy balance, here, represented by boundary forces f . Their contribution is F ( t ; u ) = i Γ i N f ( t ) · u d Γ . (5) The relations which govern the evolution in a deformable structure with a regularised and interface cracks can be obtained from the Hamilton variational principle extended to a dissipative system, see Kruzˇ´ık and Roub´ıcˇek (2019), and they can be written in a form of nonlinear variational equations or inclusions along with the initial conditions for the variables determining the trajectory of the solution: K (¨ u ) + E u ( t ; u ,α,ζ ) = F ( t ; u ) , u (0) = u 0 , ˙ u (0) = v 0 ∂ ˙ α R ( u ; ˙ α, ˙ ζ ) + ∂ α E ( t ; u ,α,ζ ) 0, α (0) = α 0 , ∂ ˙ ζ R ( u ; ˙ α, ˙ ζ ) + ∂ ζ E ( t ; u ,α,ζ ) 0, ζ (0) = ζ 0 , (6) where ∂ denotes a partial subdi ff erential as the functionals does not have to be smooth, e. g. R jumps from zero to infinity at zeros of the rate arguments. For smooth functionals, the (Gateaux) di ff erentials (the prime) are used. The initial conditions usually correspond to an intact state: α 0 = 0, ζ 0 = 0. D i ( u ) = K p sph K p | sph G I c + e ( u ) 2 + µ | dev e ( u ) | 2 + e ( u ) | 2 + µ | dev e ( u ) | 2 G II c 1 + ω K p | sph − e ( u ) | 2 G I c ¸ and D jk ( u ) = κ n [[ u ]] + n ) 2 G I c + n + 2 + κ s [[ u ]] s 2 κ s ([ u ] s ) 2 κ G ( [ u ] − n ) 2 κ n ([ u ] G II c 1 + 1 ω G I c (3) where in the pure crack modes holds: in the mode I, the value of D is G I c , and in the mode II, the value of D is G II c .