PSI - Issue 35

S. YaŞayanlar et al. / Procedia Structural Integrity 35 (2022) 18– 24

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Yas¸ayanlar et al. / Structural Integrity Procedia 00 (2021) 000–000

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satisfied and they simply reflect the conditions that the amount of incremental plastic flow is supposed to be positive and the yield criterion φ cannot be violated. The paraboloidal yield criterion by Tschoegl (1971)

φ ( σ, σ c , σ t ) = 6 J 2 + 2 I 1 ( σ t − σ c ) − 2 σ c σ t ≤ 0

(2)

is used, where σ c and σ t are the compressive and tensile yield strengths, respectively. In case of equal compressive and tensile strengths, i.e. σ c = σ t , this yield criterion simplifies to von Mises criterion. J 2 and I 1 are the second invariant of the deviatoric stress tensor and the first invariant of the total stress tensor, respectively. A non-associated flow rule is used due to plastic compressibility of epoxy type materials and the following plastic potential, g = 3 J 2 + α p 2 (3) is used where α is a material parameter and related with the plastic Poisson’s ratio ν p through α = 9 / 2(1 − 2 ν p ) / (1 + ν p ). In case of plastic incompressibility, i.e. ν p = 0 . 5, the flow regains an isochoric character such that tr p = 0. The type of damage considered here is introduced through the yield criterion with the following form,

2 σ

6 J 2 + 2 I 1 ( σ t − σ c ) − 2 (1 − D )

t σ c ≤ 0

(4)

where D is the damage variable and it progressively reduces the yield stress σ y upon material deterioration, Geers (2004). It is assumed that pressure dependency is not a ff ected by damage. Furthermore, the type of damage considered here does not influence the elastic properties but the extension is rather straight forward. The coupling between the damage field and deformation is achieved by means of a local history variable κ . Depending on the type of material considered, di ff erent damage evolution laws in terms of κ is proposed in the literature, Geers et al. (1998), Xu et al. (2020). In this work, the following forms

κ c − κ κ c − κ i

κ i κ (1 − α + α exp( − β ( κ − κ i )))

κ i κ

) β and D = 1 −

) α (

(5)

D = (

are used where κ i is the damage initiation threshold and κ c is the critical value corresponding to full damage state, α and β are material properties. The evolution of κ is coupled to the non-local equivalent plastic strain ¯ eqv p by means of an additional set of Karush-Kuhn-Tucker conditions ˙ κ ≥ 0, ¯ eqv p − κ ≤ 0, ˙ κ (¯ eqv p − κ ) = 0, which simply enforces κ to take the largest value of ¯ eqv p reached at a particular material point. Non-local equivalent plastic strain is obtained through the solution of the following partial di ff erential equation (PDE),

¯ eqv p − ∇ ( g ( D ) l 2

eqv p ) = 0

c ∇ ¯

(6)

in which l c is the internal length scale and g ( D ) is the interaction function that controls the evolution of localization bandwidth. In fact this specific form corresponds to the microforce balance which naturally arises in Poh and Sun (2017) where a micromorphic continuum framework is used to construct a mesh objective damage formulation for quasi-brittle materials. Furthermore, when g ( D ) is set to unity, this equation boils down to the Helmholtz type PDE that is well known from implicit gradient type damage formulations, Peerlings et al. (1996). The specific form of g