PSI - Issue 42

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

928

2

s A n A

Γ B N

n B Γ i c

g B

Γ B D

Ω A

Γ c

s B

Γ i

Ω B

2 ε

x 2

Γ B D

n B

s B

x 1

g B = 0 0 Fig. 1. Description of the deformable body, cracks, boundary conditions and constraints.

Many of present quasi-brittle fracture computational approaches, implemented within the finite-element world, are possibly indirectly founded on the work Francfort and Marigo (1998), which formulated the problem variationally in terms of total energy which includes strain energy in domains and surface energy of arisen cracks. The main issue in general is that the crack location is not known. To eliminate the problem in calculations, a rearrangement of the fracture was proposed which is now called phase-field model (PFM) of fracture, Bourdin et al. (2008); Miehe et al. (2010). The crack energy was reformulated by introducing a damage parameter whose extreme value pertains to an actual crack. For such a mathematical regularisation, the model needs a length parameter which may be used to describe crack formation process as presented in Tanne´ et al. (2018); Wu (2017), identified by a characteristic material length. Assumption that the fracture energies in shearing and opening modes are di ff erent was addressed in some newer approaches of PFM, e.g Wang et al. (2020); Feng and Li (2022). Additionally, in multi-material structures flaws can appear also along material interfaces. Linking them in com putational way to PFM provokes an idea to think of these faults within the theory of interface damage or adhesive contact as developed in Raous et al. (1999); Del Piero and Raous (2010). Similar computational approach was also developed by the author as shown in Vodicˇka (2016); Vodicˇka and Manticˇ (2017); Vodicˇka (2021). Also for interface cracks, their formation in a shearing mode needs more energy than in opening one, see Hutchinson and Suo (1991); Manticˇ and Par´ıs (2004), expressed by introducing so called mode mixity parameter. The computational models define an internal parameter of damage, too. Its extreme value belonging to a crack as well. In the following text, a computational model for described phenomena is proposed and its demonstration in sim plified material elements is provided.

Nomenclature

u α

[m] displacement

[-] phase-field damage parameter [-] interface damage parameter [Pa] plain strain bulk elastic modulus

ζ

K p

µ [Pa] shear elastic modulus κ [Pa m − 1 ] elastic sti ff ness of the adhesive κ G [Pa m −

1 ] compressive sti ff ness (non-degradable)

2 ] bulk fracture energy 2 ] interface fracture energy

G c [J m − c [J m − G i

[m] phase-field length-scale parameter

ϵ

[m] non-locality length parameter for interface damage

ϵ i

2. Description of the computational model

Let a deformation state is searched in a bounded deformable body Ω containing at least one internal curve splitting it into parts, Fig. 1 (includes two parts: Ω A and Ω B ). The respective boundaries of the separate parts are denoted Γ A and Γ B . The internal curve, interface, is denoted Γ i (here, an inclusion). In computational models, the cumulation of micro-faults is interpreted by decreasing of the material sti ff ness, represented by internal parameters of damage. The state of the structure then necessary involves these parameters,