PSI - Issue 35

S. YaŞayanlar et al. / Procedia Structural Integrity 35 (2022) 18– 24 Yas¸ayanlar et al. / Structural Integrity Procedia 00 (2021) 000–000

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Particularly, the implicit gradient enhancement has been extensively adapted for modelling of failure of both brittle and ductile materials. Actually, soon after its introduction, it was observed that a spurious damage growth takes place when a constant internal length scale parameter (interaction domain size) l c is used in the enhanced formulation, Geers et al. (1998). Furthermore, in some cases, constant interaction domain size may lead to erroneous propagation of damage zones particularly in shear band formation problems. The source of spurious damage growth was identified as the energy transfer from the damaged zone to the neighbouring elastically unloading region. The earliest remedy to this non-physical di ff usion of damage zone was using an evolving internal length scale. An increasing internal length scale was proposed which was later supported by micromechanical arguments. However, this formulation introduces an additional continuity equation in its implementation and retards the initiation of damage di ff usion rather than removing it. Recently, an alternative formulation that addresses the limitations of the conventional implicit gradient formu lation is proposed which uses a decreasing internal length scale and therefore diverts from most of the existing non-local damage models, Poh and Sun (2017). The model is initially proposed for the modelling of quasi-brittle failure and afterwards extended to ductile failure, Xu et al. (2020). As deformation proceeds, coalescence of mi crovoids / microcracks occur rapidly along a particular band meanwhile similar microscopic features tend to close and / or are inactivated in the neighbouring unloading regions. This corresponds to narrowing of the process zone width which finally evolves into a macroscopic crack. This is the physical motivation for the decreasing interaction domain size. In fact, the model is a multi-scale one that captures the microstructural fluctuations through a micromor phic variable (i.e. non-local equivalent plastic strain in this work). Following the Coleman-Noll procedure, structure of the constitutive model is elaborated and it is shown that the energetic interactions between the undamaged and damaged regions can be diminished (disappears at the limit l c → 0) with a decreasing length scale parameter, see Poh and Sun (2017) for further details. In this work, this formulation (called as Localizing Implicit Gradient Damage (LIGD)) is utilized to treat the mesh dependency and also to avoid artificial widening of the softening zone. Apart from mesh objective modelling of softening, another problem associated with elasto-plasticity is volumetric locking due to volume preserving nature of plastic deformations in Von Mises plasticity. Therefore it is essential to resolve this issue in order to reach a reliable model. There are a number of methods to address volumetric locking and in this work, mixed displacement / pressure (u / p) formulation is adopted. In order to be able to model 3D geometries, a 10-noded tetrahedra element with quadratic displacement, linear pressure and linear non-local equivalent plastic strain interpolation is formulated. For the compressible plasticity model introduced in the next section, a hexahedra element with linear displacement and linear non-local equivalent plastic strain interpolation is preferred and both elements are implemented in commercial finite element software Abaqus through user element (UEL) subroutine. In the next section, problem statement is presented concisely in terms of governing equations. Thereafter, the discretization scheme and the solution procedure are introduced. The following section is reserved for two di ff erent problems illustrating the capabilities of the element. The paper is finalized by the conclusions and the outlook section.

2. Problem Definition

In a geometrically linear setting, in the absence of body forces, the static equilibrium of a deformable body is governed by,

∇ · σ = 0

(1)

where ∇· is the divergence operator and σ is the stress tensor which can be decomposed into the deviatoric part ( σ ) and the volumetric part ( p I ) in terms of pressure p as σ = σ + p I . The boundary conditions are specified on non-overlapping parts of the surface such that tractions are specifed over Γ t as t = ¯ t and displacements over Γ u as u = ¯ u . The starting point for the plasticity model employed here is the additive decomposition of total strain into elastic and plastic strains as = e + p . The evolution of plastic strain is controlled by ˙ p = ˙ γ ∂ g ∂σ , where g is the plastic potential and ˙ γ represents the plastic multiplier. Karush-Kuhn-Tucker conditions, ˙ γ ≥ 0, φ ≤ 0, ˙ γ φ = 0 have to be