PSI - Issue 42

Roman Vodička et al. / Procedia Structural Integrity 42 (2022) 927 – 934 R. Vodicˇka / Structural Integrity Procedia 00 (2019) 000–000

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whose distribution in material or in interface simulates a crack. The material crack is denoted Γ c , and a crack developed along Γ i is denoted Γ ic . As the final crack may split the body, it is needed to consider only hard-device loading in a quasi-static com putational model. This load prescribes boundary conditions for displacement field u by a time dependent function u ( t ) | Γ D = g ( t ) on a part of the domain boundary Γ B D . The remaining part of the outer boundary is supposed to be load free. Pertinent portion of the boundary is denoted Γ B N in Fig. 1. The current state of the body is described by two kinds of variables: the displacement field u in the interior of the domains and internal parameters of damage. The displacements generate also the gap [[ u ]] along interfaces Γ i . A crack can be identified by a Gri ffi th-like energy expressed by an integral crack G c d Γ , which introduces the fracture energy G c as a characteristic of the cracked material or interface. Nevertheless, integration domain ’crack’ is not known inside materials. For an interface undergoing degradation, it is required to determine the length of the crack along the interface Γ i . In models of adhesive contact, an interface damage parameter ζ is introduced so that the aforementioned integral is replaced by Γ i G iI c (1 − ζ ) d Γ (known integration domain), where the parameter ζ ∈ [0; 1] is defined so that ζ = 1 pertains to the intact interface and ζ = 0 reflects the actual crack. Along with it, the interface is meant as a negligibly thin adhesive layer having a finite sti ff ness κ which is progressively decreased according to a degradation function ϕ obeying the relations ϕ (1) = 1, ϕ (0) = 0, ϕ ′ ( x ) > 0 for all x ∈ [0; 1] (for computational purposes also ϕ ′′ ( x ) > 0). The presence of the adhesive layer however contributes some elastic energy: Γ i 1 2 ϕ ( ζ ) κ [[ u ]] · [[ u ]] d Γ . Similarly, energy representation for bulk cracks may be introduced by another internal parameter, denoted here α available at each point of the material domains as the crack location is not known. In phase-field models, such a substitution is obtained by a regularisation functional, Ambrosio and Tortorelli (1990), which allows for con tinualisation of displacements across a crack, i.e. the displacements are continuous though with high gradients de pending on a length parameter ϵ . Particular expression of the regularisation introduces the energy in the form: Ω 3 8 G I c 1 ϵ (1 − α ) + ϵ ( ∇ α ) 2 d Ω (known integration domain, though it is not a curve). The resulting (continuous) dis placements generate so called smeared cracks exhibiting a finite width determined by ϵ , see Fig. 1. As presented by Tanne´ et al. (2018), the length parameter ϵ can be used to control a stress criterion in damage and crack propagation. The assumptions about cracks provide an expression for the stored energy as follows: E ( t ; u , α, ζ ) = η = A , B Ω η Φ ( α η ) K η p sph + e ( u η ) 2 + µ η | dev e ( u η ) | 2 + K η p sph − e ( u η ) 2 + + G iI (1) G (given by a large number) was added to approximate contact conditions by permitting only a small interpenetration (e.g. caused by surface roughness), and v ± = max(0 , ± v ). In order for both damage ingredients to have the same form, ϵ i is a small number accounting for non-local character of interface damage ζ and allowing for high (surface) gradients ∇ s in ζ distribution. The elastic energy term was written in Eq. (1) by a form defined by the additive orthogonal split of the strain tensor e into spherical sph e and deviatoric dev e parts. The split allows to define di ff erent material degradation related to volumetric or shear strain, and separation of tensile ( + ) and compressive (-) parts of the spherical part simulates the idea that there is no degradation (no crack propagation) in compression. Additionally, the changing of the phase field parameter α causes degradation of the material via the function Φ (analogous to the interface function ϕ ). The function has the properties Φ (1) = 1, Φ (0) = δ , Φ ′ ( x ) > 0 for all x ∈ [0; 1] (for computational purposes also Φ ′ (0) = 0, Φ ′′ ( x ) > 0 for all x ), δ being a small positive number to guarantee positiveness of the domain energy even in the case of a crack, which would pertain to α = 0. If the crack propagates in other than opening mode (Mode I, cf. using superscript I in fracture energies), there may appear another nonlinear process which dissipates the energy. The process can be imitated by introducing the mode dependent fracture energy as e.g. done by Hutchinson and Suo (1991) at interfaces. Crack propagation is a unidirectional process, in terms of internal parameters it corresponds to the constraints ˙ ζ ≤ 0 on Γ i , and ˙ α ≤ 0 in Ω , where ’dot’ means the time derivative. These assumptions can be indicated by a dissipation pseudo-potential 3 8 G η I c 1 ϵ (1 − α η ) + ϵ ( ∇ α η ) 2 d Ω + Γ i 1 2 κϕ ( ζ ) u · u + 1 2 κ G u − n 2 c (1 − ζ ) + ϵ 2 i ( ∇ s ζ ) 2 d Γ , valid in an admissible state expressed by the constraints u η | Γ η D = g η , u = u A − u B , u n = u · n B , 0 ≤ α η ≤ 1, 0 ≤ ζ ≤ 1, (2) with n referring to the outward normal vector n η . The term introducing the parameter κ