# Parameter estimation of a hyperelastic constitutive model for the description of polyurethane foam in large deformation

## Summary (3 min read)

### 1. Introduction

- Today, polymeric foam materials, such as polystyrene (PS), expanded polypropylene (EPP) and polyurethane (PU), are widely used in numerous industrial applications in engineering, sport, medical care.
- Polyurethane foams are cellular materials characterized by the spectrum of mechanical properties [1] such as: low stiffness, low Poisson rate, low density (less than 80kg· m -3 for flexible foam), the ability to absorb the strain energy, high compressibility and slow recovery rate.
- In the literature, there are numerous models designed to fit experimental results for hyperelastic materials.
- Conventionally, the determination of material parameters is based on the use of test samples with a standardized geometry under a simplified strain state.

### 2. Experimental

- The three types of polyurethane foams, designated by foam Type A, Type B and Type C, have characteristics similar to those of automotive seat foam.
- This device includes a basis frame and an upper block which moves vertically.
- Before starting the tests, the top plate had to move down slightly for full contact with the material because the top and bottom of each foam samples were not exactly parallel.
- All the test conditions including the strain rate, the maximum compression level, the number of cycles, the sampling period, and the test mechanical parameters exported were conducted using the BLUEHILL software configuration window.
- Each specimen had been quasi-statically loaded and then unloaded with a constant speed during the test process.

### 3.1. Constitutive Theory

- According to the empirical results, polyurethane foams show large strains, highly non-linear elastic and some inelastic properties.
- This study is restricted to the elastic properties of polyurethane foam and their stress-strain relationship can be characterized by a strain energy function which is related to the principal stretches.
- Hyperelastic constitutive models are adapted for this description [25] .
- If the principal stretches are denoted by i , then the most common quoted triad of invariants are given by: (5) A strain energy function can represent the stress-strain behaviour of hyperelastic materials and the stress tensor can be generated by the derivation of the strain energy function with regard to the strain tensor.

### 3.2. Ogden’s model

- From a phenomenological standpoint several attempts have been made to obtain a realistic mathematical explanation of the mechanical behaviour of highly elastic materials.
- In the present paper, this model is used to determine the mechanical behaviour of polyurethane foams.

### 3.3. Parameter Optimization

- To identify the parameters, the optimization methods are used as basic tools.
- The value of stressstrain experimental data can be obtained from the uniaxial compression test described in Section 2.
- Then the Ogden model (12) helps to calculate the model results which are compared with the experimental data.
- The trust region reflective, Levenberg-Marquardt and Gradient methods are three examples of the deterministic methods which are effective when the objective function (function to optimize) changes rapidly.
- In order to find the minimum results, the authors used the optimization tool in MATLAB with the solver FMINCON (Constrained nonlinear minimization).

### 4.1. Experimental results

- As can be seen in figure 2, polyurethane foam deformation in uniaxial compression presents three stages: initial elastic deformation, collapse deformation and compaction deformation.
- In the third stage, a region of densification occurs, where the cell walls crush together, resulting in a rapid increase of compressive stress.
- It can also be seen from the figure that there are large differences in the stress corresponding to the same strain level under the loading and the unloading processes.
- Figure 3 indicates the stress-strain curves of Foam A in three different strain rates (test 1, test 2 and test 3).
- This means that the model parameters for loading and unloading phases should be calculated separately.

### 4.2. Model results

- The first step was to determine the material parameters which describe the loading process for polyurethane foam using the loading experimental curves of three foams.
- The loading and unloading parameters were determined using Ogden's model in equation (12), with a three-term expression.
- The parameter results for three foams in the three tests are given in Table 4.
- It can be seen in Table 4 that the values of parameters are different between the loading and unloading phases, which corresponds to the experimental results.
- Figure 4 shows the best set of loading-unloading data for three foams in test 1.

### 4.3. Discussion

- In Figures 4 and 5, it appears that there is a good correspondence between the Ogden model and the experimental results, especially in the high compression phase.
- These two parameters are obtained from the identification results of the preliminary test with 15 test samples.
- The Student law is used for all parameters in this paper.
- So the conclusion is that the unloading phase is more sensitive than the loading phase because of the residual stress which has a great effect at the end of the test.
- After a sufficiently large period, the foams will return to the original configuration.

### 5. Conclusion

- This paper has presented numerous experiments with three different polyurethane foams in three different strain rates for loading-unloading uniaxial compression tests.
- The model consists of an incompressibility component and a compressibility component and the stress has been derived from the model in terms of principal stretches.
- The stability conditions of the Ogden model have been proposed.
- The results show that the Ogden model can predict the quasi-static mechanical behaviour of polyurethane foam under large strain compression.
- The model results agree with the experimental results and a detail of the unloading phase in the compression test has been discussed.

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "Parameter estimation of a hyperelastic constitutive model for the description of polyurethane foam in large deformation" ?

For further studies, other models will be used and compared with Ogden ’ s model for polyurethane foams and the viscoelasticity behaviour will be presented in a future paper.