PSI - Issue 52

Roman Vodička et al. / Procedia Structural Integrity 52 (2024) 242 – 251 R. Vodicˇka / Structural Integrity Procedia 00 (2023) 000–000

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be identified in early works of Bourdin et al. (2008); Miehe et al. (2010). The (surface) crack energy was reconsid ered by introducing an internal variable of damage whose extreme value pertains to an actual crack. The width of the smear crack then needs a formal length parameter which may be used to identify crack formation process as presented in Tanne´ et al. (2018); Sargado et al. (2018); Wu (2017) and it is related to some characteristic material length. Con sidering various forms or patterns of cracks, they may be initiated and propagated in various modes. An assumption that the fracture energy in shearing and opening modes may be di ff erent was addressed in some newer approaches of PFM, e.g Zhang et al. (2017); Wang et al. (2020); Feng and Li (2022). And finally, as mentioned afore, it is also important to see the attempts to implement the influence of inertia within this fracture concept. Using a similar energy formulation, the good strategy covers utilisation of the Lagrangian for dynamic fracture problems as done by Borden et al. (2012); Zhang et al. (21); Li et al. (2023). Naturally, the solution then includes the Hamilton variational principle extended to system with dissipation, here represented unidirectionality of crack propagation processes Kruzˇ´ık and Roub´ıcˇek (2019); Roub´ıcˇek and Panagiotopoulos (2017). Additionally, as a natural physical phenomenon and also as an improvement for numerical treatment of the solution, it may be useful to consider such a dynamical system with a kind of rheology pertinent to solid materials Roub´ıcˇek (2020). The main objective of the paper is to provide an extension of the author’s last works Vodicˇka (2022, 2023) of quasi static fracture propagation with the dependency on the fracture mode solved by a PFM to the cases where inertia is taken into account and where cracks are nucleated and propagated. This is demonstrated in the following text. A computational model for described phenomena is proposed and illustrated for simple material elements trying give them a context within existing referenced solutions.

Nomenclature

u e

[m] displacement

[-] small strain tensor

[Pa] stress tensor

σ α e 2

[-] phase-field damage parameter

[-] internal strain tensor in a visco-elastic material

[Pa] plain strain bulk elastic modulus

K p

[Pa] shear elastic modulus

µ

ρ [kgm − 3 ] mass density τ r

[s] time relaxation for a visco-elastic material

2 ] fracture energy

G c [Jm −

[m] phase-field length-scale parameter

ε

2. Description of the computational model

Consider a bounded deformable body Ω made of a visco-elastic material with a solid rheology, bounded by a contour Γ . This boundary is split into two non-overlapping parts Γ B D and Γ B N according to prescribed Dirichlet and Neumann boundary conditions, respectively. It means that the displacement field u is constrained by a time dependent function u ( t ) = g ( t ) on the part Γ B D , and surface tractions p are controlled be applied forces p ( t ) = f ( t ) on the part Γ B N or pertinent part of the boundary is supposed to be load free. Demonstratively, it is shown in Fig. 1. The rheological model is generally considered in a form of a 4-parametric solid schematically depicted in the same graphics. The stress strain relation corresponding to the scheme is described by the relations σ = C 1 ( e ( u ) − e 2 ) + D 1 e (˙ u ), C 1 ( e ( u ) − e 2 ) = C 2 e 2 + D 2 ˙ e 2 , where ’dot’ means the time derivative. For the material used in the model whose elastic tensor is C it is reduced to σ = C 1 + 1 γ ( e ( u ) − e 2 ) + τ r1 e (˙ u ) , 1 + 1 γ ( e ( u ) − e 2 ) = (1 + γ ) e 2 + τ r2 ˙ e 2 , where e ( u ) is the small strain tensor e ( u ) = 1 2 ∇ u + ( ∇ u ) and σ defines also aforementioned surface tractions as p = σ · n . The material parameters are now defined by the sti ff ness C (generated for the isotropic material by (plain strain) bulk modulus K p and shear modulus µ ) and three additional parameters: sti ff ness ratio γ (relating C 2 and C 1 ), and time relaxation values τ r1 and τ r2 (relating viscosity to elastic tensor: D 1 = τ r1 C , D 2 = τ r2 C ). If τ r1 = 0 the model is reduced to Poynting-