Abstract
In a quasi-static regime with cyclic loading, the force–displacement curve of flexible polyurethane exhibits complicated behavior: nonlinearity, visco-elasticity, hysteresis, residual force, etc. Beside nonlinearity and visco-elasticity, this material displays high dependence on the displacement rate and past loading history. Its dependence on compression rate helps to appropriately identify the force–displacement curve. Based on the new curve identification, the overall foam response is assumed to be a composite of a nonlinear elastic component and a visco-elastic component. The elastic component is expressed as a polynomial function in displacement, while the visco-elastic one is formulated according to the hereditary approach to represent the foam visco-elastic damping force during the loading phase and according to the fractional derivative approach during unloading to represent the visco-elastic residual force in the material. The focus of this study was to develop mathematical formulations and identification parameters to faithfully characterize the visco-elastic behavior of flexible polyurethane foam under multi-cycle compressive tests. A parameter calibration methodology based on the separation of the measurement data of each component force was established. This optimization process helps to avoid the parameter values admixture problem during the phase of numeric calculations of the same component force. The validity of the model results is checked according to the simulation accuracy, the physical significance of results and their agreement with the obtained force–displacement curve identification.
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Abbreviations
- \(x\) :
-
Displacement [mm]
- \(\dot{x}\) :
-
Displacement rate [\(\mbox{mm}\,\mbox{s}^{-1}\)]
- \(t\) :
-
Time [s]
- \(T\) :
-
Total testing duration [s]
- \(A\) :
-
Maximum displacement level [mm]
- \(F_{L}\) :
-
Total foam response during loading half-cycle [N]
- \(F_{U}\) :
-
Total foam response during unloading half-cycle [N]
- \(E_{L}\) :
-
Nonlinear elastic force during loading half-cycle [N]
- \(E_{U}\) :
-
Nonlinear elastic force during unloading half-cycle [N]
- \(V_{D}\) :
-
Visco-elastic damping component [N]
- \({V}_{{R}}\) :
-
Visco-elastic residual force component [N]
- \({Q}\) :
-
Objective function [\(\mbox{N}^{2}\)]
- \({a}_{{i}}\) :
-
Visco-elastic damping coefficient of order \(i\) [\(\mbox{N}\,\mbox{mm}^{-1}\,\mbox{s}^{-1}\)]
- \({\alpha}_{{i}}\) :
-
Visco-elastic damping parameter [Hz]
- \({k}_{{i}}\) :
-
Elastic stiffness of order \(i\) [\(\mbox{N}\,\mbox{mm}^{-i}\)]
- \({r}_{{i}}\) :
-
Visco-elastic residual force coefficient of order \(i\) [\(\mbox{N}\,\mbox{s}^{\beta}\,\mbox{m}^{-1}\)]
- \({\delta x}\) :
-
Displacement sampling [mm]
- \({\beta}_{{i}}\) :
-
Fractional order
- \({\varGamma}\) :
-
Gamma function
- \({M}\) :
-
Order of fractional derivative terms
- \({N}\) :
-
Order of the visco-elastic kernel
- \({P}\) :
-
Order of elastic polynomial function
- \({S}\) :
-
Sampling number
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Elfarhani, M., Jarraya, A., Abid, S. et al. Fractional derivative and hereditary combined model for memory effects on flexible polyurethane foam. Mech Time-Depend Mater 20, 197–217 (2016). https://doi.org/10.1007/s11043-016-9291-2
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DOI: https://doi.org/10.1007/s11043-016-9291-2