PSI - Issue 61

Bekir Kaçmaz et al. / Procedia Structural Integrity 61 (2024) 130–137 Author name / Structural Integrity Procedia 00 (2024) 000–000

132

3

the dominance of some other, it is plausible to claim that there exists interaction between micro-cracks. Therefore, representing strain tensor by ¯ ϵ and elasticity tensor by C , the free energy density of the following form,

1 2

1 2

1 2 (1 − D ) ¯ ϵ : C : ¯ ϵ +

h ( ϵ − ˜ ϵ ) 2

gh l 2 ∇ ˜ ϵ ·∇ ˜ ϵ

(1)

ψ =

+

is considered to be well motivated where the second term captures the microscopic heterogeneity and the third term reflects the interaction between micro-cracks through the gradient of ˜ ϵ , ∇ ˜ ϵ . In equation 1, h , g , D and l are the micro-macro coupling modulus, interaction function, damage variable and characteristic length scale, respectively. Furthermore, it is expected that the interaction between micro-cracks weakens as the deformation localizes to a specific region / band. Extending internal and external power with ˜ ϵ, ∇ ˜ ϵ and the associated stresses and tractions (please see Poh and Sun (2017) for details), and employing the Coleman-Noll procedure, the following relations σ = (1 − D ) C : ¯ ϵ + h ( ϵ − ˜ ϵ ) N (2) ˜ ϵ − ϵ = ∇· ( g l 2 ∇ ˜ ϵ ) (3) are obtained, where N = ∂ϵ ∂ ¯ ϵ . The interaction function g ( D ) decays with damage (i.e. ∂ g /∂ D < 0) and takes values between 1 and residual interaction R ≊ 0 depending on the degree of damage. Due to its proven e ff ectiveness, the interaction function proposed and used in Poh and Sun (2017); Sarkar et al. (2019); Shedbale et al. (2021), is used in this work as well. It is worthy to note that by setting g = 1 and h ≪ E , the conventional gradient damage can be recovered. As far as the influence of h is concerned, the comparative analysis presented in Sarkar et al. (2019) reveals that h is not a significant parameter. Almost identical results are obtained by setting h = 0 as compared to the case where h > 0 and h ≪ E. Therefore h is set to zero in this work as well. Referring back to 2, with a (non-constant) decaying interaction function, the non-local part of ˜ ϵ diminishes and eventually disappears. However, in case of conventional implicit gradient damage formulation, non-locality persists even for highly damaged regions which in turn causes non-physical coupling of ‘cracked’ and intact regions. This is the source of non-physical damage di ff usion / growth and wrong failure mode initiation. By setting h = 0 and rearranging Equation 3, governing equations at a material point within the domain V can be written as, ∇· σ + ρ⃗ b =⃗ 0 (5) ˜ ϵ −∇· ( g l 2 ∇ ˜ ϵ ) − ϵ = 0 (6) where the first equation set results from the conservation of linear momentum in the absence of inertia e ff ects. ∇· is the divergence operator, ρ is the density of the material point and⃗ b is the body force vector. These di ff erential equations are complemented by the following boundary conditions, σ ·⃗ n =⃗ ¯ t on Γ t and⃗ u =⃗ ¯ u on Γ u (7) ∇ ˜ ϵ ·⃗ n = 0on Γ (8) where Γ t and Γ u are the non-overlapping sub-boundaries i.e., Γ t ∩ Γ u = ∅ and Γ t ∪ Γ u = Γ , over which tractions and displacements are prescribed, respectively. Introducing test functions⃗ ϕ u & ϕ ˜ ϵ and following a standard Galerkin 3. Element Formulation and Implementation g = (1 − R ) exp( − η D ) + R − exp( − η ) 1 − exp( − η ) (4)