these voids, only. This reduces the number of parameters to five, namely q1 , q2 , f0 , fc , κ. The parameters q1 = 1.5 and κ = 4 have the usual values known from literature, whereas q2 = 2 was assumed here for the thin panels with low triaxiality, which differs from common simulations of thick structures. The initial void volume fraction was taken to equal the volume fraction of inclusions, f0 = 0.0012. Coalescence of voids takes place if a critical void volume fraction, fc, is exceeded, and macroscopic crack growth results from f* = 0.6 ≈ 1/q1. The only adjustable parameter is the critical void volume fraction, fc. The Cohesive Zone Model (CZM) Whereas the GTN model results in constitutive equations governing inelastic deformation and evolution of damage in the continuum, the cohesive zone model intruduces a traction-separation law, Σ( δ), at the "interface" of continuum elements. Hence, the crack has to follow a presecribed path along the element boundaries where cohesive elements have been placed. A decohesion law based on a potential proposed by Rose [9] and applied by Xu and Needleman [10] has been used in the present simulations: Σn = σmax e z δn δc exp −z δn δc , (2) with maximum normal stress, σmax and separation length, δc , being model parameters; e = exp(1) and z = 16 e/9 are just numbers. This decohesion law is for pure "mode I" separation, but shear components can be added easily [10]. As the present FE model is 2D and accounts for crack growth in the ligament, only, which is assumed to be a plane of symmetry, no shear stresses can occur and the simulation is restricted to normal fracture. During the separation process, the mechanical work Γ0 = Σn d δn = 0 δc ∫ 9 16 σmax δc (3) is "released". Hence, a crack has grown by one element length if δn = δc or, equivalently, Γ = Γ0. The CZM has been implemented as "user supplied element" (UEL). MODELING OF MATERIAL AND SPECIMEN BEHAVIOR Yield Curve The yield curve of the material, σY( ε p), was determined from tensile tests and fitted by a power law, (4) σY = σ0 +C ε p ( )n with σ0 = 343 MPa, C = 670 MPa and n = 0.67. Young's modulus was taken as 65 GPa from literature. Finite Element Models The global deformation of thin specimens can be well described by plane stress models. The local triaxiality at a crack tip, however, is much higher than in plane stress, T = 0.66, as 3D analyses show. As void growth is significantly influenced by stress triaxiality, no damage evolution occurs under plane stress conditions [11] and the GTN model cannot be applied for plane stress elements. Plane strain conditions will overestimate damage evolution on the other hand. Hence, crack growth simulations using the GTN model require a 3D simulation at least in the vicinity of the crack tip and the ligament. In addition, the parameter q2, which governs the influence of stress triaxiality, see eq. (1), and is commonly set to 1., has to be increased to a value of 2. in the present situation of thin panels. A fourfold symmetry is introduced to reduce the number of elements and unknowns in the FE simulation. Due to this symmetry, damage evolution and crack growth are restricted to a plane normal to the external load, and no slant fracture can be obtained in the simulations. A 3D model as required for the GTN model is not very convenient for simulations of large amounts of crack growth. The phenomenological CZM offers the advantage of modeling in 2D, as the separation law is not dependent on the local triaxiality. The FE mesh now consists of three regions: • a process zone where separation occurs, which is modeled by a layer of cohesive elements in the symmetry line, i.e. the ligament, • a layer of elastic-plastic plane strain elements which allow for higher triaxiality and thus prevent localization of plastic deformation and necking adjacent to the cohesive elements, • elastic-plastic plane stress elements all over the rest of the specimens, which guarantee the overall plane stress deformation behavior.
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