# Modeling the Nonlinear Deformation of Highly Porous Cellular Plastics Filled with Clay Nanoplatelets

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## Abstract

**:**

## 1. Introduction

## 2. Structural Model

#### 2.1. Stiffness of the Composite Material of the Load-Carrying Element

#### 2.2. Stresses in the Structural Element

#### 2.3. Evaluation of Foam Stresses and Foam Stiffness

## 3. Nonlinearity

#### 3.1. Parabolic Functions

#### 3.2. Elliptic Functions

## 4. Comparison of Theoretical Calculations with Experimental Data

_{f}as ${V}_{f}={W}_{f}{\rho}_{m}/\left[{W}_{f}{\rho}_{m}+\left(1-{W}_{f}\right){\rho}_{f}\right]$, using the densities of monolithic PUR ${\rho}_{m}$ reported in [29] and montmorillonite clay ${\rho}_{f}$ [32]. For filled foams, it was assumed that Cloisite

^{®}30B clay filler had fully exfoliated during the rigid PUR foam production; geometrical dimensions and mechanical characteristics of the clay nanoplatelets used in calculation were taken from [31]. Introduction of clay filler led to a slight reduction of cell size, while the geometrical anisotropy of the foam cells remained the same within the experimental scatter [33] (see also SEM images of neat [30] and filled [33] foams), therefore the average value of cell shape anisotropy ratio (i.e., the degree of cell extension) k = 1.5 [30,33] was used in modeling.

^{®}30B clay-filled foams with the same composition of the PUR matrix as the monolithic polymer [30]. The formulation and manufacturing procedure of foams, as well as preparation and testing of foam specimens, are described in [34]. For completeness and ease of reference, we briefly recapitulate the relevant information. Foam specimens of dog-bone shape, with a rectangular test section of 85 mm length, 22 mm width and 20 mm thickness, where cut from slices of the free-rise foam blocks so that the mechanical response in the direction normal to foam rise could be characterized. Metallic plates with hooks were glued to the ends of the specimens, and short chains were attached to the hooks to enable gripping. Such a gripping system [29] provided alignment of the specimen with the line of action of the applied load, eliminating bending and ensuring pure tension during the test. A clip-on extensometer with 50 mm base length was used for strain measurement in the loading direction. Tensile tests were performed at a displacement rate of 8 mm/min.

^{3}density, with 0, 1, 2, and 5 wt.% loading of the clay filler [34], are shown in Figure 3 together with the model prediction obtained using the elliptic nonlinear function (19) with ${\stackrel{\pm}{k}}_{a}={\stackrel{\pm}{k}}_{b}=0$. Three foam specimens were tested for each filler loading level; the respective experimental stress–strain diagrams are plotted by dashed lines in Figure 3.

^{3}, density and 0, 1, 2, and 3 wt.% loading of Cloisite

^{®}30B [34] with theoretical prediction performed as described above.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Highly porous cellular plastic: (

**a**) Micrography showing the reticulated morphology of foams and global (i.e., foam) and local (strut) coordinate systems; (

**b**) Schematic of the structural element comprising the load-carrying element (gray); (

**c**) Schematic of the load-carrying element showing the loads applied to it; (

**d**) Calculation element for estimation of stiffness tensor $\mathit{C}$ of the nanoplatelet-filled material of load-carrying element.

**Figure 3.**Stress–strain diagrams in tension transverse to the rise direction of ca. 40 kg/m

^{3}density foams with (

**a**) 0, (

**b**) 1, (

**c**) 2, and (

**d**) 5 wt.% loading of Cloisite

^{®}30B clay. Foam test results are plotted by dashed lines, and model predictions—by solid lines for clay platelets aligned with strut axis and by dash-dot lines for a random orientation of clay platelets in foam struts.

**Figure 4.**Stress–strain diagrams in tension transverse to the rise direction of ca. 50 kg/m

^{3}density foams with (

**a**) 0, (

**b**) 1, (

**c**) 2, and (

**d**) 3 wt.% loading of Cloisite

^{®}30B clay. Foam test results are plotted by dashed lines, and model predictions—by solid lines for clay platelets aligned with strut axis and by dash-dot lines for a random orientation of clay platelets in foam struts.

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**MDPI and ACS Style**

Lagzdiņš, A.; Zilaucs, A.; Beverte, I.; Andersons, J.
Modeling the Nonlinear Deformation of Highly Porous Cellular Plastics Filled with Clay Nanoplatelets. *Materials* **2022**, *15*, 1033.
https://doi.org/10.3390/ma15031033

**AMA Style**

Lagzdiņš A, Zilaucs A, Beverte I, Andersons J.
Modeling the Nonlinear Deformation of Highly Porous Cellular Plastics Filled with Clay Nanoplatelets. *Materials*. 2022; 15(3):1033.
https://doi.org/10.3390/ma15031033

**Chicago/Turabian Style**

Lagzdiņš, Aivars, Alberts Zilaucs, Ilze Beverte, and Jānis Andersons.
2022. "Modeling the Nonlinear Deformation of Highly Porous Cellular Plastics Filled with Clay Nanoplatelets" *Materials* 15, no. 3: 1033.
https://doi.org/10.3390/ma15031033