6. Material model assembling of selected PU foam

Figure 20. FEM model of sample dynamically compressed without the initial deformation.

PAM CRASH (right).

100 Aspects of Polyurethanes

Model Element type

Rigid plates Element size [mm]

Table 3. FEM model dynamically compressed sample.

Figure 21. Modification of the input parameters in the FEM model dataset (left); visualization of the simulation model in

Friction in contact

2D shell 4 <sup>3600</sup> 0.1 0.5 5.297 <sup>10</sup><sup>1</sup>

PU foam 3D solid 4 <sup>6250</sup> 0.1 0.5 0.1471 <sup>10</sup><sup>4</sup>

Distance between contacts [mm]

Time step Δt [s]

Element number

Material models in the PAM CRASH chosen to describe the nonlinear behavior of selected samples are prepared from material library, namely:

• Material model describing the mechanical behavior of PU foam has been defined through a nonlinear material model 45—General Nonlinear Strain Rate Dependent Foam with Optional Energy Absorption. This material model has already been used and published, for example, in Refs. [3, 9, 12]. Its advantage is to especially allow to assess the influence of rigidity of polyurethane foams depending on the strain rate. The model is based on the rheological behavior of the modified Kelvin model that allows to express Cauchy stress σ<sup>i</sup> in the loading axis in accordance with Eq. (41).

$$
\sigma\_i = E \cdot \varepsilon\_i(t) + \eta\_t \cdot \dot{\varepsilon}\_i(t), \tag{41}
$$

where E is modulus of elasticity, εiðtÞ and ε\_iðtÞexpress the strain and strain rate in a single direction, and η<sup>t</sup> is a damping of the material.

• Another suitable material can be material model 37—Viscoelastic Ogden Rubber for Solid Elements—which allows to describe not only viscoelastic but also hyperelastic material properties (suitable for the study of rubber, polymers, fibers, foams, etc.). This is based on the description of the functional dependence of the strain energy density Eðλ1, λ2, λ3Þ defined by Eq. (42), expressing the energy that is required for the structure deformation. Practically, it is analogy to Eqs. (25) and (30).

$$E(\lambda\_1, \lambda\_2, \lambda\_3) = \sum\_{i=1}^3 \frac{\mu\_p}{\alpha\_p} \cdot \left(\sum\_{i=1}^3 \lambda\_i^{\infty\_o} - 3\right) \tag{42}$$

where i ¼ 1, …, 3, Eðλ1, λ2, λ3Þis the strain energy density, μ<sup>p</sup> and α<sup>p</sup> are material constants, and let <sup>X</sup><sup>n</sup> p¼1 μ<sup>p</sup> � α<sup>p</sup> <sup>2</sup> <sup>¼</sup> <sup>G</sup>, where <sup>G</sup> is the shear modulus defined by Eq. (43), and <sup>λ</sup><sup>∝</sup><sup>σ</sup> <sup>i</sup> vectors are elongation in principal directions.

$$G = \frac{F}{2 \cdot (1 + v)} \,\prime \tag{43}$$

where E is elastic modulus and ν is Poisson's ratio.

Using strain energy Eðλ1, λ2, λ3Þ, the Cauchy stress in principal directions σ<sup>i</sup> can be expressed in Eq. (44).

$$\sigma\_i = p\_k + \lambda\_i^{\infty\_\sigma} \cdot \frac{\partial E(\lambda\_1, \lambda\_2, \lambda\_3)}{\partial \lambda\_i^{\infty\_\sigma}} \tag{44}$$

where pk is compression stress.

The input material parameters of the simulation model are given in Table 4.


Table 4. Material properties of FEM model dynamically loaded sample.
