PSI - Issue 52

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

244

3

e

(a)

(b)

e 1

e 2

f

ρ

Γ N

C 1

g

C 2 D 2

Γ D

Ω

s

x 2

n

Γ c

Γ D

2 ε

D 1

x 1

g = 0

0

Fig. 1. Description of the deformable body (a), a smear crack characteristic, boundary conditions and constraints, with a scheme of the rheological model (b).

Thomson rheology , if τ r2 = 0 the rheologyl may be simplified to Kelvin-Voigt model. Anyhow, the described type of rheology is interpreted using an internal parameter which pertains to a part of strain denoted e 2 , it is also used to completely describe the state of the structure at any instant t . In many computational models, the formation of micro-faults is interpreted as a degradation of the material. This degradation is represented by another internal parameter α which has a damage-like character. The state of the struc ture then necessary involves also this parameter, whose distribution in the body simulates a crack. The actual crack is denoted Γ c inFig 1. Summarising, the current state of the body at a time instant t is described by three variables: the displacement field u and two internal parameters of partial strain e 2 and of damage α . A crack can be identified by a Gri ffi th-like energy expressed by an integral Γ c G c d Γ , which introduces the fracture energy G c as a crack characteristic. Nevertheless, the integration domain Γ c is not known a priori, as long as the crack develops according to the loading history previous to the time instant t . The energy representation for the crack may be introduced by the aforementioned internal parameter α available at each point of the material domain. Such a replacement can be obtained by a regularisation functional, Ambrosio and Tortorelli (1990), which allows to make displacements continuous across a crack, though with high gradients depending on a length parameter ε , and provides models of smeared cracks called phase-field models. There exist several regularisations, one introduced in Tanne´ et al. (2018) presents the energy equivalent to that of the cracks in the form: Ω 3 8 G I c 1 ε (1 − α ) + ε ( ∇ α ) 2 d Ω (the integral has a known integration domain Ω , though it is not a curve as the actual crack), where the parameter α ∈ [0; 1] is defined so that α = 1 pertains to the intact material and α = 0 reflects the actual crack. Along with it, the material properties ( K p and µ ) progressively decrease according to a degradation function Φ obeying the relations Φ (1) = 1, Φ (0) = δ , (0 < δ 1 to guarantee positiveness of the energy in the case of a crack), Φ ( x ) > 0 for all x ∈ [0; 1] (for computational purposes also Φ (0) = 0 and Φ ( x ) > 0). The resulting displacements generate the smeared crack (there is no discontinuity only a narrow band of degraded material) exhibiting a finite width determined by ε , see Fig. 1. As presented by Tanne´ et al. (2018); Sargado 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 , e 2 ,α ) = Ω Φ ( α ) K p sph + ( e ( u ) − e 2 ) 2 + µ | dev( e ( u ) − e 2 ) | 2 + K p sph − ( e ( u ) − e 2 ) 2 1 + 1 γ + Φ ( α ) K p sph + e 2 2 + µ | dev e 2 | 2 + K p sph − e 2 2 (1 + γ ) + 3 8 G I c 1 ε (1 − α ) + ε ( ∇ α ) 2 d Ω , (1)

valid in an admissible state at the time instant t expressed by the constraints

u | Γ D

= g ( t ), 0 ≤ α ≤ 1,

(2)

and where v ± = max(0 , ± v ). The elastic energy term is written in Eq. (1) using an additive orthogonal split of the strain tensor e into its spherical sph e and deviatoric dev e parts. The split allows to define material degradation related