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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material strip. Implementing this approach utilising the present computational power provided enormous progress on the simulations with cracks considered in the phase-field sense Sargado et al. (2018); Wu (2017); Feng and Li (2022); Vicentini et al. (2024). The analysis therein considered structures exposed to various loading conditions, in particular trying to validate the applicability in cases where compression prevails in the analysed structural element. Moroever, in a kind of problem like this, the speed at which the processes occur is also important, thus inertia should be a natural part of fracture models. Considering phase-field modelling, the inertial approaches were intensively studied in contri butions as e. g. Li et al. (2023); Zhang et al. (2023). Additionally, structural parts are usually multimaterial and cracks inside them nucleate due to an interface or directly along it. The computational treatment may be based on the models of delamination or of adhesive contact, like in Del Piero and Raous (2010); Roub´ıcˇek et al. (2015) to gain a similar approach which represents damage and fracture by an internal variable. The author’s previous works Vodicˇka (2023, 2024) included those considerations into simulations. In what follows, it is intended to describe how the computa tional model for dynamic fracture may be modified so that when a structure is exposed to compression, it provides practically observable results. The computational analysis of fracture is based on a concept of energy release which is supposed to reach a critical value, fracture energy G c , required for a crack to grow. The modern techniques implement a variational formulation as introduced by Bourdin et al. (2008), which replaced the crack itself by material degradation in narrow bands and defined a regularised crack. The body where the solution is found is given by a bounded domain Ω , split into several sub-domains Ω i (including layers, inclusions, etc.). The domain is exposed to a time dependent loading so that a kind of degradation arises. The prescribed loading introduces boundary conditions and also constraints for the displacement field u and traction vector p . The load itself is represented by a time dependent function u ( t ) = g ( t ) on a part of the domain boundary Γ D , or a time dependent function p ( t ) = f ( t ) on a part of the domain boundary Γ N (including zero traction vector for a free boundary). The boundary condition may also include spring support supposing p ( t ) + cu ( t ) = 0 on Γ R . If necessary, a supersript is used to refer to a particular subdomain, as e. g. for an interface between Ω i and Ω j , Γ i j is used. The degradation causes damage of material (inside a sub-domain) or of a material joint (an interface of sub-domains) and ultimately leads to a crack. The crack inside Ω i as a new surface could be distinguished as Γ i c , while if it appears along an interface between Ω i and Ω j , it is denoted Γ i j c . Instead of considering Γ i c , the arising crack is regularised so that a new internal variable α is introduced (or with the superscript if referring to Ω i ). The phase-field damage variable α ∈ [0; 1] is defined so that α = 0 pertains to the intact material and α = 1 reflects the actual crack. Similarly considering Γ i j another new internal variable ζ is introduced to identify the extent of the interface crack in Γ i j . This interface damage variable ζ ∈ [0; 1] is defined analogously: ζ = 0 pertains to a pristine interface and ζ = 1 belongs to the crack. The energy functional containing dependence on deformation and damage is finally represented in the following form E ( t ; u ,α ) = i Ω i Φ ( α ) K p sph + e ( u ) 2 + µ | dev e ( u ) | 2 + K p sph − e ( u ) 2 + 3 8 G I i c 1 α + ( ∇ α ) 2 d Ω (1) 2. An interface and phase-field fracture model c is the crack regularising term, where a length-scale parameter determines the width of the band of degraded material representing the regularisation of Γ i c is energy pertinent to the interface crack. The elastic energy is expressed in terms of orthogonal split of small strain e ( u ) into spherical and deviatoric parts, and also introduces degradation function Φ , controlling the process of damage Wu (2017); Sargado et al. (2018). Along the interface represented be a vanishing layer of elastic adhesive, the energy is given in terms of displacement jump [[ u ]] jk = u j − u k and its normal n and tangential components s. The contact condition is penalised by introducing the (high) compressive sti ff ness κ G . Also here, the degradation function φ controls the process of damage Vodicˇka (2016). c . Similarly, the term containing G I jk + Γ i R 1 2 cu · u d Γ + jk Γ jk 1 2 φ ( ζ ) κ n u + n 2 + κ s u s 2 + 1 2 κ G u − n 2 + G I jk c ζ + ( ε i ∇ s ζ ) 2 d Γ . The term containing G I i