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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If the di ff erentiation in (6) is performed, the first relation provides the equation of motion at time t together with the boundary and interface conditions ρ ¨ u − div σ = 0, σ = Φ ( α ) 2 K p sph + e ( u ) + 2 µ dev e ( u ) + 2 K p sph − e ( u ), in each Ω i , u = g ( t ), on each Γ i D , p = f ( t ), with p = σ n , on each Γ i N , p = − cu , on each Γ i R , p j + p k = 0 , p n = φ ( ζ ) κ n u + n + κ G u − n , p s = φ ( ζ ) κ s u s , with p n = p k · n k , p s = p k · s k on each Γ jk . (7) Di ff erentiation in the second relation of (6), including the subdi ff erential provides the relations in a form of com plementarity inequalities in the domain Ω i as follows 0 ≤ ˙ α , 1 ≥ α , 0 ≤ λ , 0 = λ ( α − 1), 0 ∗ ≤ Φ ( α ) · K p sph + e ( u ) 2 + µ | dev e ( u ) | 2 + λ + 3 8 − G I c 2 ∆ α + D i ( u ) , 0 = ˙ α Φ ( α ) · K p sph + e ( u ) 2 + µ | dev e ( u ) | 2 + λ + 3 8 − G I c 2 ∆ α + D i ( u ) , ∂ α ∂ n = 0, on Γ i , (8) where also the boundary condition for the phase-field variable α was set in the last equation. It should also be noted that a Lagrange multiplier λ was introduced in order to guarantee the constraint on α . These relations represent the flow rule for the evolution of the phase-field variable. Similar relations are provided for the interface damage using the third relation of (6). On Γ jk , they read 0 ≤ ˙ ζ , 1 ≥ ζ , 0 ≤ λ , 0 = λ ( ζ − 1) , 0 ∗ ≤ 1 2 c ε A stress criteria for both the phase-field damage and interface damage triggering can be obtained, if pertinent relation from (3) is substituted into the ∗ -inequality of (8) and (9), respectively. At respective instants of damage initiation, there holds α = 0 (with ∆ α = 0, λ = 0), and ζ = 0 (with ∆ s ζ = 0, λ = 0) sph σ = 2 K p sph e ( u ), dev σ = 2 µ dev e ( u ), see (7), and the conditions for a related critical stress σ crit in Ω i , and for p crit on Γ jk read sph + σ crit 2 4 K p G I c + | dev σ crit | 2 4 µ G II c 1 + ω | sph − σ crit | 2 4 K p G I c = 3 8 ( − Φ (0)) , (10) p + ncrit 2 κ n G I c + p 2 scrit κ s G II c 1 + 1 ω ( p − ncrit ) 2 κ G G I c = 2 ( − φ (0)) . (11) As the triggering of the phase-field damage is related to the stress state by the relation (10), the curves corre sponding to that relation are shown in Figure 1 scaled to dimensionless variables, restricting to the in-plane relation | sph σ | 2 = (tr σ ) 2 2 . Using the scaled variables, the domain pertaining to the positive stress trace is the same quarter circle which defines a safe zone where damage is not initiated. For the negative stress traces the safe zone varies depending on the selected parameter ω and on some material characteristics. The phase-field version was discussed in Vicentini et al. (2024), which provides the same set of critical curves. In the compression, each curve (a hyperbola) has an asymptote which does not enable occurring of damage below it (geometrically), i. e. if the ratio between the norm of the deviatoric and the spherical stress inside a subdomian, or the ratio between tangential and (compressive) normal stress along an interface, is small enough. The same triggering curves are obtained for the interface damage, though di ff erent scaling to dimensionless quan tities is used. In Figure 1, this refers to the second labels used for both axes. φ ( ζ ) κ n u n 2 jk u − G I + κ s u 2 i ∆ s ζ , 2 + λ + D jk u − G I 2 i ∆ s ζ , 0 = ˙ ζ 1 2 φ ( ζ ) κ n u n 2 + κ s u s 2 + λ + D c ε ∂ ζ ∂ s = 0, on ∂ Γ jk . (9)