54 V. Ilari et al. InEq.(4), gi are the relative moduli, so that g ∞+Pgi =1; G0 represents the short term, or instantaneous, stiffness that is obtained as the sum of all branches’ stiffness. The long term and short term stiffnesses are related as G ∞=g∞· G0 . The stress along time in the i-th layer of the Generalized Maxwell model is given by: (S2d i )n+1 =exp − ∆t τGi (S2d i )n +α Gi exp − ∆t 2τGi dΦ dCn+1 − dΦ dCn (5) where S2d i is the second Piola-Kirchhoff stress tensor and Cis the right Cauchy-Green deformation tensor. The true stress can be obtained from: σ = 2FS2dFT J (6) where Fis the deformation gradient [13]. The total stress can be obtained from the sum of the hyperleastic stress with the NGviscoelastic stresses. In order to correctly model permanent energy dissipation effects and stress softening in agglomerate cork, Mullins effect is implemented, providing an extension to the elastomeric foam model [14]. Therefore, this damage model is used to include the damage present in elastomeric foams, modeling the energy absorption in foam components subjected to dynamic loading, with high strain rates compared to the characteristic relaxation time of the foam. In this model, energy dissipation effects are considered by introducing an augmented strain energy density function of the shape: W(λi,η)=ηW(λi)+ϕ(η) (7) The function W(λi,η) is a continuous function of the damage variable, η, and is related to the damage function ϕ(η). The damage variable, η, varies continuously during the course of deformation and always satisfies 0 <η<1, withη =1at the points of the primary curve (described by the Hyperfoam model). Taking into account the Mullins effect, the stresses are calculated by: σ(λi,η)=ησ(λi) (8) where σ(λi) is the stress corresponding to the primary behavior of the foam at the current strain level λi. Then, the stress is obtained by simply scaling the stress of the primary behavior of the foam by the damage variable η. Eq.8 is used when the material is at an energy potential Wthat is lower than the maximum energy potential Wmexperienced by the material itself at the end of the loading phase. The damage variable is generally assumed to be represented by the so called “error function”: η =1− 1 r erf Wm−W m+βWm (9) where r,mandβ are material parameters that govern the shape of the unloading curve. Note that r>1, m≥0, β>0[15]. Validation In order to obtain the coefficients of the models used to describe the cork behavior, an optimization technique was implemented. Specially, a cost function, which represents the difference between the experimentally obtained stresses and those predicted analytically, is minimized. To describe the compressible hyperelastic behavior, a fourth-order Hyperfoam model was adopted, which requires the identification of 8 parameters, namely the coefficients µi and αi withi =1. . . 4. The viscoelastic behavior was instead modeled by a 10-term Prony series, where the Maxwell branches have relaxation times uniformly distributed on a logarithmic scale. Consequently, the optimization problem focuses on the determination of the two extreme relaxation times τmin and τmax and the 10 related moduli gi. Lastly, Mullins effect requires the calibration of three parameters r,mandβ. The optimization process was implemented in Matlab. All the optimized parameters are listed in Table 1, 2 and 3 for the Ogden hyperfoam model, Prony series model and Mullins damage, respectively. Table 1 4th order Ogden hyperfoam parameters µ1 α1 µ2 α2 µ3 α3 µ4 α4 β1···4 4.236 7.532 -5.579 5.580 2.543 4.146 0.007 -3.137 0 After material model calibration, the impact tests were simulated by means of the commercial Finite Element software. In particular, the Abaqus/Explicit code was adopted because it is well suited for implementing user-defined constitutive models by means of user-defined Fortran subroutines (VUMAT).
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