Design and Optimization of Shock Absorbers Made of Grade Density Foams 47 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 (7) where Fis the deformation gradient. 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 PVC foam, Mullins effect is implemented, providing an extension to the elastomeric foam model [1]. 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) (8) 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) (9) 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. Equation 9 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 (10) where r,mand β are material parameters that govern the shape of the unloading curve. Note that r>1, m≥0, β>0 [1]. Model validation Before proceeding with the simulation of a puncture test, the proposed model was validated by reproducing the compression tests performed experimentally at different strain rates and densities. The specimen was modeled as a 2D body, constrained at the lower edge, applying the axial symmetry of the problem. Compression at various strain rates was applied by varying the velocity of the rigid body loading the PVC specimen. The simulation results for the density of 130 at strain rates of 10−3, 101, and 103 are shown in Fig. 3a. Fig. 3 (a)Hyperelastic and viscoelastic results,(b) Mullins effect result
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