Modelling of the cyclic behavior of open-cell polymeric foams: non linear elasticity, viscosity and damage
Résumé
The behavior of an open-cell polymer foam under cyclic uniaxial compression is studied in
both experimental and modeling aspects. For this purpose, a phenomenological model is used here,
while most of the existing models are based on microstructural considerations and address mainly the
first compression. The choice of a model including non linear elasticity, viscosity and damage is based
on the analysis of a large panel of experiments which are presented.
In the cyclic compression experiments conducted on a polyurethane foam various relevant
phenomena were observed:
- the deformations were localized in bands orthogonal to the loading direction,
- the response curve showed a large hysteresis loop,
- a softening phenomenon consisting of a gradual decrease of strength during the subsequent
cyclic loading,
- a rate dependence of the strength resistance (see Fig.1).
In [1], a description of strain localization and hysteresis was obtained in the context of nonlinear
elasticity by assuming a non convex strain energy density. The non-convexity of the strain
energy induces different types of equilibrium, which can be considered as different phases of the
material. Both strain localization and hysteresis are the result of phase transitions.
Extensions of these models including the rate-dependent properties of foams are presented in
[2, 3, 4]. In particular, the foam is represented as a chain, each element of which consists of a nonlinear
elastic spring connected in parallel to a linear visco-elastic element. The rheological model is
completed by adding a further visco-elastic element set in parallel with the chain.
Figure 1: the model: chain of elements coupling non-linear elasticity and viscosity
The model describes many of the observed inelastic aspects of the response, but the stress
softening phenomenon is greatly underestimated. This phenomenon is reminiscent of the Mullins
effect occurring in filler-reinforced rubbers. Despite the large literature, no consensus has been
reached as to how this effect should be interpreted. Many authors proposed alternative models
describing the stress softening effect as a form of damage. This interpretation is adopted here, and we
refine the model in accounting for the damage of the non linear elastic springs [3,4].
According the representation of the behavior of the foam by a chain of non-linear springs and
dashpots, a differential system has to be solved, for which bifurcation may occur because of the non
convexity of the deformation energy. An incremental problem is introduced and a Newton Raphson
method is used. It results a system of linear algebraic equations, of which the analysis of the
determinant associated to the matrix of the coefficients characterizes the localization phenomena, i.e.
the change of phase of one element of the chain. The numerical scheme has been implemented in a
Matlab environment.
Due to the complexity of the behavior of foams, the complete model requires a large number
of constitutive constants. A special care is devoted to the identification of these constitutive
parameters (see [3,4]). An identification procedure is proposed: it combines either a Prony method or
an analytical qualitative analysis and optimization methods. The identification is conducted on
relaxation experiments and on a loading cycle chosen as reference. The good agreement between the
experiment and the simulation is shown on Fig. 2.
Figure 2: Identification of the constitutive parameters: experiment (dash line) and simulation (full line)
Validation of the model is then established by the good agreement between the numerical
simulation results and the experimental data for a set of more complex loading cycles as it can be
noted in Fig. 3.
Simulation Experiment
Figure 3: Validation of the model: simulation of complex loading cycles
References
[1] Pampolini G., Del Piero G., Strain localization in open-cell polyurethane foams: experiments and theoretical
model. J. Mech. Mater. Struct. 3: 969-981, 2008.
[2] Del Piero G.; Pampolini G., The inelastic properties of open cell polymeric foams: experiments and
theoretical model, submitted
[3] Pampolini G., Raous M., Simulation numérique du comportement des mousses polymériques sous
compression cyclique, Proceeding of X Colloque National en Calcul des Structures, Giens, France, May
2011, 8 pages.
[4] Pampolini G., Raous M., Non linear elasticity, viscosity and damage in open-cell polymeric foams, in
preparation.