Mechanical and material heterogeneity, strain localization and deformation rate effects in crushable expanded polystyrene foams

Low-density expanded polystyrene (EPS) foams are widely used in lightweight energy absorption systems such as helmets due to their ability to readily mold into complex geometries. However, varying material flow and cooling rates during manufacturing produce exterior skin layers with substantially higher density and aspect ratio from the core, and the resultant mechanical properties have not been quantified. Previous studies assumed EPS foams were homogeneous, overlooking or intentionally removing the skin from test specimens and constrain their scopes to out-of-plane compression. In this study, closed-cell EPS foam pucks of 30, 50, 80, and 100 g/L were tested under in-and out-of-plane compression at loading rates spanning 0.001–10/s. Specimens were prepared with as molded and core (skin removed) configurations to quantify anisotropy from heterogeneity. Measurements revealed a 98% ± 8% higher density in the skin layers relative to nominal material density and cells skewed 41% ± 6% in the in-plane direction. As-molded specimens exhibited a 38% ± 4% higher plateau stress for in-plane loading compared to out-of-plane, highlighting foam cell elongation as a key strengthening mechanism. Quasi-orthotropic behavior was observed for the core foam material, which possessed more evenly sized cells. Digital image correlation quantified rate-dependent strain localization, providing novel evidence of internal pressure redistribution from viscous gas dynamics within the EPS beads, with 39% lower peak true strains, on average, measured at 10/s compared to 0.001/s. Unloading data also revealed progressive increases in post-crushing strain recovery, increasing an order of magnitude from 0.04 mm/mm to 0.42 mm/mm between 0.001-10/s for the 30 g/L group, confirming more even load distribution and cell fracture mitigation at elevated rates.

1 Introduction

Closed-cell foams such as expanded polystyrene (EPS) and polypropylene (EPP) are among the most widely used crushable cellular materials for impact energy absorption. They are especially favorable in load-limiting applications such as personal safety (Fernandes and Alves De Sousa, 2013; Farajzadeh Khosroshahi et al., 2018; Bhudolia et al., 2021b), protective infrastructure (Ichino et al., 2022; Yuan et al., 2022), and high-value product packaging (Razza et al., 2015; Li et al., 2025). Closed-cell foams possess hierarchical structures, comprised of microscopic thin-walled cells that form within macroscopic beads due to gas expansion during foam injection molding processes. Rapid cooling at the mold surface and internal pressure promotes the formation of an inherently dense exterior (skin) layer during manufacture (Lee et al., 2017; Jiang et al., 2021). The densified exterior layer is generally 1.5 to 2.5 times denser than the core (interior) foam material and occupies 5–10 mm of the rise direction height (Bouix et al., 2009; Malekinejad et al., 2024) before a quasi-uniform density distribution is achieved. This effect can be mitigated when thin rectangular sheets are extracted from large molded blocks (Horvath, 1994; Rahimidehgolan et al., 2023). However, there are many applications (helmets, bumpers, etc.) where a complex geometry is needed and thus mechanical removal of the molded skin layer is unfeasible (Morton et al., 2020; Bhudolia et al., 2021; Kumar et al., 2024), underscoring the need for detailed characterization of this effect.

Most experimental studies considering 10–150 g/L density crushable foams prioritize uniaxial crushing tests in an out-of-plane orientation, parallel to the material rise direction (Rahimidehgolan and Altenhof, 2023; Rieth et al., 2025). The post-yield stress/strain behavior of these materials (EPS, EPP, etc.) is characterized by a prolonged phase with minimal (i.e., plateau) hardening (Fernandes et al., 2015; Xuan et al., 2024), followed by a sharp increase in stress beyond 55%–65% global strain referred to as densification (Tan et al., 2002; Çıbıkçı and Yaman, 2024). EPS foams generally exhibit moderately increased strain hardening in the plateau region with increasing foam density (Andena et al., 2016; Rodríguez-Sánchez and Plascencia-Mora, 2022). Quasi-static testing regularly considers the global compressive strain during collapse, but observations reveal localized bands of strain within the foam specimen with localized shear-induced foam bead rotations/kinking and material fracture (Rahimidehgolan et al., 2023; Zhu et al., 2024). Impact testing of polymeric foams is commonly performed using drop tower, pendula, and Split-Hopkinson Pressure Bar apparatuses (Ouellet et al., 2006; Liu et al., 2020; Zhao et al., 2021). Loading rate and foam density are the most significant parameters influencing energy absorption in EPS foams (Avalle et al., 2007; Caserta et al., 2010; Krundaeva et al., 2016; Bhudolia et al., 2021a), with a significant upturn in plateau stress observed above 1,000/s (Ouellet et al.,2006; Chen et al., 2015). The elevated rate mechanical response of EPS foams is heavily influenced by viscous effects due to gas dynamics associated in the closed-cell structure (Saha et al., 2005; Castiglioni et al., 2017; Sun and Li, 2018). Crushing tests performed on EPP foams immersed in water (Bouix et al., 2009) confirmed that air remains trapped in the foam beads at elevated rates (i.e., above 1 s^-1), mitigating cell wall fracture and contributing towards energy absorption.

Anisotropy was quantified for polyurethane (PU) foams subjected to uniaxial crushing by varying the direction of sample extraction (Li et al., 2019), and for divinyl cell foams using an Arcan fixture (Gdoutos et al., 2002) and orthogonal crushing tests (Liu et al., 2020) – a common outcome was distinctly higher plateau stresses for the out-of-plane direction compared to in-plane. Extraction of cubic samples from different locations within a geometrically complex EPP bumper also produced distinct, heterogeneous stress/strain responses (Morton et al., 2020). An experimental study of anisotropy in PVC foams with quasi-uniform density distributions revealed that out-of-plane loading, along the cell elongation direction, exhibited an increased elastic modulus, compressive stress and a more pronounced strain rate sensitivity (Liu et al., 2020). Increased stress in a principal material direction (e.g., out-of-plane) for cellular materials, when it occurs, is often attributed to bead/cell elongation (i.e., structural heterogeneity) parallel to the stronger direction (Ridha and Shim, 2008; Zu and Yao, 2012; Hu et al., 2021).

Widely used characterization tests (e.g., ASTM D 1621) and constitutive modeling techniques acknowledge the heterogeneity of structural foams but ultimately simplify their analyses to consider a quasi-uniform (average) global compressive strain during collapse (Zhang et al., 1998; Ashby, 2006; Avalle et al., 2007; Jeong et al., 2012). Digital image correlation (DIC) techniques have emerged as a thorough method to measure in-plane strain field heterogeneity by iteratively comparing images of evolving speckle patterns in foam samples during progressive deformation (Wang and Cuitiño, 2002). This technique has been used to examine strain distribution in varying densities of polymeric foams undergoing compression (Liu et al., 2014; Ling et al., 2018b; Bhagavathula et al., 2022), 3-point bending and tension (Tang et al., 2019). Significant rate sensitivity and strain rate localization has been observed in crushable foams, accompanied by pronounced compressive-shear stress anisotropy (Ling et al., 2018a). However, studies in this area have not considered the context of macroscale heterogeneity arising from the mentioned densified exterior (skin) layer effect, and its implications on direction-dependent strain localization.

Isotropic crush properties are expected in the (uniform) core region of closed-cell foams which could explain how some studies concluded EPS as isotropic (Horvath, 1994; Hazarika, 2006; Ossa and Romo, 2011), however, material anisotropy arising from the skin layer (Ahn et al., 2002) changes the strain pattern depending on the direction of loading, especially in thin sections. Out-of-plane loading causes more localized deformation whereas in-plane loading results in a more homogenous strain field (Liu et al., 2020). An examination into the effect of the densified skin layer is not well documented with most sources omitting mention this material phenomenon (Rahimidehgolan and Altenhof, 2023). Another major gap in existing literature is quantifying the effect of the foam’s rate sensitive unloading response, which is currently omitted from impact modelling. When evaluating the accuracy of existing constitutive EPS models against experimental data, it was determined that none of them effectively captured the unloading response of the foam (Arnesen et al., 2024).

In this paper, we present the mechanical characterization of four EPS foams considering the influence of density, load orientation, and a comprehensive range of strain rates encompassing 5 orders of magnitude, tested under all conditions for as molded (with skin) and core (no skin) samples. To our knowledge, this is the first published study to establish and evaluate the influence of manufacturing-induced material heterogeneity on the anisotropic energy absorbing capabilities of a crushable polymeric foam. Out-of-plane and in-plane compression was considered for both sample configurations to evaluate the relationship between mechanical and material heterogeneity, newly quantifying anisotropic plateau strength and hardening behavior arising from heterogeneous cell size and shape in the as molded samples. Unloading data, which is scarce in the literature for crushable foams, was also measured and revealed a distinct hysteretic rate effect with prolonged post-crushing material recovery for increasing strain rates. Planar DIC was applied to the outer surfaces of the specimens to evaluate the spatiotemporal evolution of true compressive strain within the EPS foam materials, contrasted against the bulk compressive strain, and analyzed to elucidate the underlying mechanisms responsible for rate-sensitive unloading. Together, the findings outlined in this study provide valuable mechanics and materials-based insights to advance the understanding of energy absorbing mechanics in crushable foams, most notably: the link between material heterogeneity and anisotropic energy absorption, rate sensitive compressive strain localization, and post-crushing recovery.

2 Materials and methods

2.1 EPS foam specimen preparation

The closed-cell EPS foam materials used in this study were molded as cylindrical pucks (Figure 1a) with a nominal diameter and 128 mm and height of 31 mm. A densified skin layer was observed within the EPS foam cross-sections, visibly evident from the darker region of expanded plastic at the upper and lower surfaces of the puck as shown in Figure 1b. This layer is an inherent product of the manufacturing process due to a combination of boundary pressure from expanding interior foam beads and rapid cooling at the mold surfaces and was present throughout the puck exterior. Therefore, all samples were extracted at least 10 mm from the puck edges to ensure that densified skin layers were only present, when applicable, within the upper and lower layers of each sample. Unlike prior studies that remove densified skin layers or assumed uniform density (Rahimidehgolan and Altenhof, 2023), we preserved the molded skin-to-core gradient inherent to helmet-grade pucks to quantify how the resultant higher density and bead influence the direction-dependent energy absorption mechanics, outlined in Section 3.

Figure 1

Figure 1. (a) Widescale view of an EPS foam puck, 128 mm diameter, as molded (50 g/L sample shown) with (b) an accompanying section view of the interior bead structure, and (c) schematic for the as molded (31 mm) and core (20 mm) cubic sample extraction. Average densities obtained from the as molded and core foam materials are provided in (d).

Cubic specimens with nominal 31 mm and 21 mm side lengths were extracted from the EPS foam pucks for the as molded and core samples, respectively, schematized in Figure 1c. The through-thickness density variation is quantified further in Section 3.1, confirming that the 21 mm side length was sufficient for the core samples used in this study. Preparation was completed using an ultra-fine finish miter saw (Diablo Tools, High Point, NC, USA) to prevent foam bead pullout and minimize cell damage. Average measurements of the EPS foam material densities are summarized in Figure 1d for the as molded and core sample configurations, respectively, revealing an average difference of 26.2% between their nominal densities. These apparent density measurements were obtained from the average of 6 samples per configuration, determined using the masses and volumes of cubic samples visualized in Figure 1c; an in-depth slice analysis is provided in Section 3.1.1.

2.2 Orthotropic, rate sensitive compression testing

The experimental scope considered various loading conditions for the four nominal EPS foam densities as identified in Section 2.1. Uniaxial compression tests were completed over 5 orders of magnitude, at nominal rates from 0.001/s to 10/s (Table 1). The as-molded samples were compressed in out-of-plane (OOP) and in-plane (IP) orientations, consistent with material directions normal and parallel to the foam rise direction, respectively (Figure 1).

Table 1

Table 1. Parametric scope for the present study on crushable EPS foams.

The compression tests were completed using a hydraulically driven test frame, shown in Figure 2a, (KJJ2HPS23A, Parker-Hannifin, Owen Sound, ON, CA) operated at steady crosshead speeds up to 250 mm/s to achieve the mentioned strain rates. The frame was instrumented with a linear variable differential transformer and 55.7 kN capacity strain gauge-based load cell (1020 AF, Interface Solutions, Scottsdale, AZ, USA) to measure the crosshead displacement and crushing force, respectively. Operation of this apparatus and the outlined sensors, testing environment conditions, and the mechanical property (elastic modulus, yield strength, etc.) measurement procedures were all completed in accordance with the ASTM D 1621 test standard. Note that cubic specimens (per Section 2.1), consistent with previous studies (Cronin and Ouellet, 2016; Krundaeva et al., 2016), were necessary to complete the direction-dependent testing outlined in Table 1. The specimen unloading phase immediately followed the foam compression by reversing the crosshead displacement direction, at the same rate as the loading phase, once a nominal volumetric compression of 80% was achieved (i.e., using displacement constraints) and continuing the force and displacement/time measurements up to a return to the zero position.

Figure 2

Figure 2. (a) Schematic of the low-to elevated-rate compressive test frame, with (b) a focused view of an as molded EPS foam sample speckled with paint for planar/2D DIC analysis. Schematics of (c) out-of-plane, and (d) in-plane crushing are shown for an as molded EPS foam sample.

2.2.1 Digital image correlation on foam surface

A high-speed camera (GRAS-5055M, Point Grey Pictures, Vancouver, BC, CA) was used to photograph the crushing tests performed up to 0.1/s, substituted with a FASTCAM SA4 (Photron, Imagica Group Inc., Tokyo, JP) for the elevated rates. Each specimen was painted with a random black-on-white speckle pattern (Figure 2b) for subsequent planar/2D DIC analysis, completed using the VIC2D® software package (Correlated Solutions, Irmo, SC, USA). The orientation of the as molded samples (from Figure 1) for the out-of-plane and in-plane tests are shown in Figures 2c,d, respectively. At least 3 repeats were completed per test configuration. At least 3 repeats were completed per test configuration (1 test per sample), with both the loading and unloading phases recorded for each test. As such, error bars on all subsequent results in Sections 3, 4 represent the standard deviations, quantifying repeatability.

2.2.2 Through-thickness compressive strain distribution

The true compressive strain was tracked on the exterior surface of the EPS foam specimens using 2D DIC to quantify the spatiotemporal strain distribution during crushing and unloading, at least 500 images were collected per test. The area of each strain contour was calculated using a MATLAB script employing the Image Processing Toolbox (Mathworks, Natick, MA, USA) to assign a true strain magnitude to each color band and determine the through-thickness strain distribution. The overall area reduction during crushing was synchronized with the global (volumetric) compression to observe the differences between global engineering strain and true strain localization. This analysis provided novel quantification of the orientation- and rate-dependent strain localization patterns observed in the EPS foam materials, analyzed further in Sections 4.2, 4.3.

2.3 Foam mechanical properties

The energy absorbing efficiency (Tan et al., 2002), $\eta \left(\epsilon \right)$, of a crushable foam can be obtained using the compressive stress/strain response, $\sigma \left(\epsilon \right)$, as defined in Equation 1. The densification strain, ${\epsilon }_d$, coincides with point of maximum efficiency, ${\eta }_{\mathit{max}}$, (Equation 2).

$\eta \left(\epsilon \right)=\frac{1}{\sigma \left(\epsilon \right)}{\int }_0^{\epsilon }\sigma \left(\epsilon \right)d\epsilon (1)$

${\left.\frac{d\eta \left(\epsilon \right)}{d\epsilon }\right|}_{\epsilon ={\epsilon }_d}=0(2)$

The plateau stress, ${\sigma }_p$, which quantifies the energy absorbing potential for a crushable foam, can therefore be evaluated as the average value between the strains at the onset of yielding, ${\epsilon }_y$, and densification, ${\epsilon }_d$, from the stress/strain response, as defined in Equation 3:

${\sigma }_p=\frac{1}{{\epsilon }_d-{\epsilon }_y}{\int }_{{\epsilon }_d}^{{\epsilon }_y}\sigma \left(\epsilon \right)d\epsilon (3)$

A representative stress/strain curve for crushable foam is provided in Figure 3 with annotations for the outlined mechanical properties and performance metrics.

Figure 3

Figure 3. Schematic of a compressive engineering stress/strain curve (σ,ε) and the corresponding energy absorbing efficiency curve, η, for a closed-cell EPS foam, annotated with key mechanical performance metrics.

3 Results

3.1 EPS foam properties

3.1.1 Macrostructure and through-thickness density measurements

The through-thickness density variation was quantified using strip analysis, a total of 5 EPS foam strips were extracted parallel to the foam rise direction, per the Section 2.1 sample preparation procedure, with the graph ticks nominally coinciding the sample height shown in Figure 4. Each strip had a nominal thickness of 5.5 mm and was extracted perpendicular to the foam rise direction from the cubic samples, weighed, and exterior dimensions were measured to quantify the density variation associated with the densified skin layer. The upper and lower strips corresponded to the densified exterior layers (skins), distinctly visible as darker regions of plastic in the molded EPS, with the remaining 3 interior strips containing the core EPS material. The measurements represent an average of 6 samples per strip location and are summarized in Figure 4a with a complementary sectioned view (Figure 4b) from a representative foam sample. A quasi-symmetric density gradient was observed for all 4 densities with the skin layers observed to be between 65% and 154% denser than the core material, there was no correlation between the extent of density variation and average density.

Figure 4

Figure 4. (a) Average density measurements taken at increments along the foam rise direction of an EPS foam puck with a corresponding (b) sectioned view (30 g/L sample shown) revealing distinct interior core and exterior skin layers, with. Each vertical tick mark on the graph corresponds to the nominal location where foam strips were extracted for density measurements.

3.1.2 EPS foam bead size and roundness distributions

The cross-sectional area of the EPS foam beads was analyzed on the outer surfaces of the cubic samples, histograms summarizing the bead cross-sectional area, A_bead, distributions for the 30, 50, 80, and 100 g/L density groups are provided in Figures 5a–d, respectively. The distributions are provided for the exterior/skin and interior core regions, recall Figure 4b, for each foam group. Similar trends were observed for each density group, with a symmetric area distribution observed about the mean area for the core material and left-skewed distributions for the exterior layers, quantifying the heterogeneous macrostructure of the EPS foams. The average areas for each density and material region are inversely proportionate to the apparent density, summarized in Table 2.

Figure 5

Figure 5. In-plane EPS foam bead area, A_bead, distributions in the core and exterior (skin) regions of the (a) 30 g/L, (b) 50 g/L, (c) 80 g/L, and (d) 100 g/L density materials. See Table 2 for corresponding average values and standard deviations.

Table 2

Table 2. Average foam bead cross-sectional area, diameter and roundness in the core and exterior (skin) regions for the EPS foam materials with apparent densities of 30, 50, 80, and100 g/L.

The EPS foam bead shape was also evaluated by considering the in-plane roundness (circularity), as defined in Equation 4:

$R_{bead}=\frac{4\pi A_{bead}}{{\left(P_{bead}\right)}^2}(4)$

where P_bead represents the foam bead perimeter corresponding to a given planar area. The EPS foam bead expansion (i.e., roundness) distributions were highly consistent between density groups (Table 2), therefore, the histogram provided in Figure 6a represents the average roundness distribution within the exterior and core foam regions. The error bars represent the standard deviation between 30, 50, 80, and 100 g/L foam distributions, highlighting the consistency between density groups for core and exterior (skin) regions. Most (>60%) of the EPS foam beads in the core region had a roundness of 0.75 or greater while the exterior layers exhibited a wider distribution centered about a roundness of approximately 0.55.

Figure 6

Figure 6. (a) Average in-plane EPS foam bead roundness, R_bead, distribution in the core and exterior (skin) regions EPS foam materials. Representative regions of the 50 g/L density material are provided for the (b) exterior skin, and (c) interior core regions to visualize the bead elongation within the former and quasi-isotropic bead shape in the latter region.

The foam beads within the exterior (skin) layer were elongated perpendicular to the foam rise direction, shown for a representative material section in Figure 6b. This tendency was attributed to rapid cooling and solidification of the EPS foam at the mold surface that caused adjacent material to smear during the (pressurized) foam manufacturing process. The bead structure within the core region (Figure 6c) was significantly more uniform, with no primary elongated direction observed. The flat edges were caused by internal pressure and interactions between adjacent beads. Outliers (i.e., significantly smaller foam beads) within the core region were a byproduct of some beads forming later in the manufacturing process and being suppressed by larger neighboring beads, contributing to the inherent batch variation of the material, noted further in Section 3.2.

3.2 Orthotropic, rate sensitive compression testing

Representative stress/strain responses from for the 30 g/L 100 g/L and EPS foams are provided in Figures 7a,b and Figures 7c,d, respectively. Each configuration exhibited a distinct plateau region with the onset of densification occurring at global strains between 0.6-0.7, however, orthotropy is visibly evident. The yield and plateau stresses were distinctly higher under in-plane crushing compared to the (more familiar) out-of-plane configuration, quantified further in Section 3.2. Furthermore, the plateau region was near-constant for the in-plane loading condition, exhibiting significantly less strain hardening up to densification compared to out-of-plane. This tendency was attributed to the crushing direction being aligned with the elongated direction for the beads within the densified exterior layers (recall Figure 6) under in-plane compression, promoting simultaneous and even compaction of the exterior and core material. In contrast, the out-of-plane direction exhibited up to 65% higher post-densification compressive stress due to progressive consolidation of the lower density interior (core) EPS beads between the dense upper and lower exterior (skin) layers, visualized through DIC analysis in Section 4.2. The orthotropic nature of the EPS foam is most evident within the post-densification region of the stress/strain response, and more pronounced with increased foam density.

Figure 7

Figure 7. Engineering stress/strain curves obtained at compressive rates of 0.001/s and 10/s obtained from uniaxial crushing of (a,b) 30 g/L, and (c,d) 100 g/L EPS foams. The as molded (out-of-plane and in-plane) and core configurations are plotted together for each density/strain rate pair to highlight the orthotropic nature of the material.

Crushing tests were performed in both configurations for the core material and the differences between the in- and out-of-plane responses were within the magnitude of individual test variations, thus the core material was concluded to be quasi-isotropic. This finding confirms that cell elongation is a strengthening mechanism in cellular materials (e.g., foams) since the bead structure in the core region was similarly isotropic (Figure 6). The unloading regime of the stress/strain curve was significantly longer at 10/s compared to 0.001/s in all cases, demonstrating that interior damage and fracture within the EPS foam material is mitigated at elevated loading rates. A likely mechanism for the extended unloading (all configurations) with drastically reduced hysteretic slopes is the viscous interaction between pressurized entrapped air within the foam and the bead/cell walls promoting even internal load distribution within the foamed material (Bouix et al., 2009; Liu et al., 2020), analyzed further in Section 4.3.

The remaining stress/strain curves for the 50 g/L and 80 g/L EPS foam materials are shown in Figures 8a–c and Figures 8d–f for the as molded out-of-plane, as molded in-plane and core material configurations, respectively. Consistent trends for rate sensitive hardening, onset of densification, and progressively increasing unloading regimes were observed. The in-plane stress of the EPS material is distinctly stronger (32%, on average) than the complementary out-of-plane direction for a given foam density and exhibits less hardening within the plateau region. The core material compressive stress was also distinctly lower for a given level of strain compared to the as molded material, which is reasonable since the densified material at the exterior surfaces was not present in these samples, decreasing the overall density and material stress. Each loading configuration in Figure 8 exhibited positive rate sensitivity with generally consistent stress/strain curve profiles for increasing loading rates, with notable variation (anisotropy) in the plateau regions and vastly different post-densification histories between as molded and core configurations.

Figure 8

Figure 8. Engineering stress/strain curves obtained from 0.001-10/s out-of-plane and in-plane uniaxial crushing of (a–c) 50 g/L, and (d–f) 80 g/L EPS foam using as molded and core samples.

The rate sensitive orthotropic plateau stresses for the as molded EPS foam materials are summarized with the complementary, quasi-isotropic plateau stress the core materials in Figures 9a–d for the 30 g/L, 50 g/L, 80 g/L, and 100 g/L configurations, respectively. The (log-linear) stress/strain rate slopes were nominally consistent between core and as molded materials and increased with increasing material density, with the compressive in-plane stress distinctly higher than the remaining conditions within each group. The relative strain rate sensitivity was consistent between density groups in most cases, with an average measured increase in the compressive plateau stress of 38% ± 4% from 0.001/s to 10/s. The strongest-to-weakest orientations were as molded in-plane, as molded out-of-plane, and core (both orientations), respectively, for all test groups. Minor differences were observed in the spacing of these groups and could be attributed to corresponding variability in the density gradients for each foam material (recall Figure 4).

Figure 9

Figure 9. Average plateau stress with respect to compressive strain rate for the EPS foam materials with, clockwise from top left, (a) 30 g/L, (b) 50 g/L, (c) 80 g/L, and (d) 100 g/L nominal densities obtained from the as molded and core samples. Note that the error bars for the core data represent deviations from in- and out-of-plane loading, highlighting the quasi-isotropic nature of this configuration.

3.3 Rate- and direction-dependent constitutive modeling

The Gibson-Ashby constitutive relationship (The Royal Society, 1982) can be used to relate the elastic modulus, E_f, of a closed-cell foam to the corresponding solid material properties, E_s and σ_ys, respectively:

$\frac{E_f}{E_s}\approx {\phi }^2{\left(\frac{{\rho }_f}{{\rho }_s}\right)}^2+\left(1-\phi \right)\frac{{\rho }_f}{{\rho }_s}(5)$

where ρ_s and ρ_f represent the structural density for the solid polymeric material and foam, respectively, and $\phi$ is a dimensionless parameter that is related to the amount of (solid) material present within the foam cells/struts, determined from experimental data. The rate dependent stress-strain response, $\sigma \left(\epsilon ,\dot{\epsilon }\right)$, can be predicted based on a quasi-static baseline response, ${\sigma }_0\left(\epsilon \right)$, and the global compressive strain, $\dot{\epsilon }$, using the Nagy constitutive model (Nagy et al., 1974):

$\frac{\sigma \left(\epsilon ,\dot{\epsilon }\right)}{{\sigma }_0\left(\epsilon \right)}={\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)}^{\alpha +\beta \epsilon }(6)$

where α and β are direction- and density-dependent material constants with the reference strain rate, ${\dot{\epsilon }}_0$, taken as 0.001/s, recall Section 2.2. The compressive stress/strain data summarized in Section 3.2 was used to calibrate these constants via least squares regression with a 95% confidence interval.

The orthotropic nature of the as molded EPS foam is evident in the elastic modulus of the material, summarized for each density in the as molded configuration in Figure 10a. The in-plane elastic modulus was 42% higher, on average, compared to the corresponding out-of-plane stiffness. In contrast, less than 15% variation was observed between loading configurations for the core material (Figure 10b). This extent of variation was within the range of uncertainty quantified by the error bars, confirming the quasi-isotropic nature of the material as mentioned previously in Section 3.2, and observed within the EPS foam bead structure in Section 3.1. The core elastic moduli were 23% lower, on average, compared to the corresponding as molded out-of-plane values. Rate sensitivity on the elastic modulus was negligible for the range considered within this study, hence the values in Figure 10 (with error bars) represent the averages for the entire compressive testing scope. The Gibson-Ashby model from Equation 5 was calibrated to experimental measurements assuming an elastic modulus and density of 2.4 GPa and 1,055 kg/m³, respectively, for the solid polystyrene material.

Figure 10

Figure 10. Average in- and out-of-plane elastic moduli, E_f, for the (a) as molded, and (b) core EPS foams with 30–100 g/L apparent densities. Discrete points represent the experimental data, and continuous lines the Gibson-Ashby model (Equation 5).

Representative comparisons between experimental results and the calibrated Nagy model are provided in Figures 11a,b for the as molded material compressed at 0.01/s in the out-of-plane and in-plane loading orientations, respectively, with the corresponding core material results presented in Figure 11c. Complementary comparisons at 1/s are provided in Figures 11d–f to highlight the accuracy of the model. The remaining direction-dependent Nagy model parameters are provided in Table 3. Overall, the Nagy model could replicate the rate sensitivity of the crushable EPS foam material with a high degree of accuracy, achieving less than 10% absolute error for the entire scope. These outcomes provide a valuable reference to predict the direction-dependent dynamic response of the as molded and core EPS foams under impact in future investigations.

Figure 11

Figure 11. Comparisons between the calibrated Nagy constitutive model and representative experimental results for the 80 g/L EPS foam material compressed at (a–c) 0.01/s and (d–f) 1/s in the as molded (out-of-plane and in-plane) and core sample configurations, respectively.

Table 3

Table 3. Summary of the material- and direction-dependent exponential constants for the Nagy constitutive model, Equation 6.

4 Discussion

4.1 Collapse behavior with digital image correlation

The through-thickness collapse profiles are shown for representative samples of as molded EPS foam with an 80 g/L apparent density are provided at volumetric compressions (i.e., global engineering strains) between 10%–75% in Figure 12 with accompanying DIC. The quasi-static, 0.001/s plateau crushing (Figures 12a–c) exhibited strain localization nominally across the midplane, propagating through the specimen height as the deformation progressed. The exterior skin layers remained relatively unstrained compared to interior core, owing to their significantly higher relative densities (recall Section 3.1.1) and progressively consolidated this lower-density region of material. The irregular strain contours highlight the presence of locally denser EPS foam beads within the core material, their interactions during progressive crushing lead to shear rotation through the midplane and lateral shifting of the cell walls (Krundaeva et al., 2016). More uniform strain distributions were measured post-densification (Figure 12d) with some dense EPS beads still visible, albeit more elongated, once the core material was sufficiently consolidated (i.e., possessed increased density) to resist further compaction by the exterior skin layers.

Figure 12

Figure 12. Deformation profiles with accompanying through-thickness DIC for as molded (AM) 80 g/L density EPS foam samples compressed in the out-of-plane (OOP) direction by 10%–75% of their original heights at nominal rates of (a–d) 0.001/s, (e–h) 0.1/s, and (i–l) 10/s, respectively.

Progressive crushing of the same 80 g/L material at an elevated 0.1/s rate (Figures 12e–g) revealed a similar tendency for a localized compressive strain band to form at the midplane, but with considerably more diffused borders compared to the 0.001/s testing. The post-densification strain distribution (Figure 12h) was also more uniform and symmetric about the specimen midplane, with near-constant width horizontal bands of strain formed throughout the specimen height and significantly less lateral kinking. The strain distribution consistency further increased for 10/s compressive loading (Figures 12i–l), with no discernible strain localization within individual EPS beads observed. Since EPS is a closed-cell foam, viscous gas dynamics due to trapped air (Bouix et al., 2009) and subsequent interactions between adjacent EPS beads is most likely the dominant mechanism. The influence of the uniformly entrapped air overtaking the foam bead/cell microbuckling is evident when comparing a given level of global strain at elevated strain rates, consider Figures 12c,g,k which represent 50% specimen compression from 0.001-10/s. The magnitude of peak true strain also tended to decrease (note the scale bars in Figure 12) for a given level of global compression with increasing strain rate, supporting the notion that viscous load redistribution occurs within the EPS foam.

Complementary through-thickness collapse profiles from 10%–75% compression with planar DIC for the representative 80 g/L apparent density foam are provided in Figure 13. Unlike out-of-plane loading, the densified exterior layers (and the exterior-to-core density gradient originally shown in Figure 4) for in-plane loading were parallel to the crushing direction. The interior (core) material was therefore not consolidated by the exterior skin layers, instead, both layers were compressed concurrently and with minimal interaction within the plateau regime. Consequently, strain localization initiated within multiple planes during compression, in contrast to the single localized plane observed for out-of-plane loading (Figure 12), since the densified layers were not directly engaging the (lighter) interior core.

Figure 13

Figure 13. Deformation profiles with accompanying through-thickness DIC for as molded (AM) 80 g/L density EPS foam samples compressed in the in-plane (IP) direction by 10%–75% of their original heights at nominal rates of (a–d) 0.001/s, (e–h) 0.1/s, and (i–l) 10/s, respectively. Note the onset and progression of separation between the vertically oriented densified (skin) and core material layers in (g,h).

The through-thickness strain distribution was significantly more uniform with increasing strain rate, especially post-densification (compare Figure 13d at 0.001/s, to Figure 13l at 10/s), compared to the complementary out-of-plane testing. There was also a tendency for the exterior layers to separate from the core material and progressively buckle under crushing, visible in Figures 13g,h,l, and further highlighting the anisotropic nature of the as molded material. The EPS foam beads within the exterior layer were smaller, approximately twice as dense, and more elongated than the core material, experience widescale buckling and separation from the core material when shear localization and rotation of the interior beads progressed (Ling et al., 2018a), typically above 20% global compression. The quasi-uniform material compression observed via DIC is consistent with the near-constant plateau regions of the stress/strain curves originally shown in Section 3.2, thus confirming the link between the EPS foam’s direction-dependent heterogeneity and the orthotropic mechanical behavior.

The core specimens exhibited highly consistent behavior between in- and out-of-plane loading orientations, owed to the more heterogenous expanded cell structure observed in Section 3.1, and is presented for the representative 10/s loading rate in Figure 14 for the representative 80 g/L EPS foam material. Similar mechanical behavior was observed for each foam density considered in the present study, hence presentation of a single (apparent density) configuration for brevity. The strain distribution for a given level of compression (e.g., comparing Figures 14b,f for out-of-plane and in-plane loading, respectively) reveals highly similar strain localization trends. Strain localization initiated within multiple planes simultaneously for both loading orientations with nominally (<15% difference) true strain magnitudes. The similarity between through-thickness collapse profiles for in- and out-of-plane loading also correlates with the quasi-isotropic cell size and plateau stresses noted in Section 3.2.

Figure 14

Figure 14. Deformation profiles with accompanying through-thickness DIC for core 80 g/L density EPS foam samples compressed by 10%–75% of their original heights at 10/s in (a–d) out-of-plane, and (e–h) in-plane orientations, exhibited highly consistent collapse behavior between configurations.

4.2 Through-thickness compressive strain distribution

The strain localization phenomena described in previous sections was further quantified using the cumulative true strain distribution (recall Section 2.2.2), shown for representative EPS foam samples with 50 g/L apparent density in the current section. Cumulative strain distributions at 25%, 50% and 75% global compression obtained under quasi-static, 0.001/s out-of-plane crushing for the as molded foam are summarized in Figure 15a, vertical bars are overlaid to provide a reference between the engineering (global) compressive strains and the true strain distributions. These plots quantify the tendency for a minority (i.e., ≤30%) of the EPS foam material at the less dense midplane (Figure 15b) to experience significant strain hardening, often 2 to 4 times greater than the global strain. The densified skin layers comprise most of the under-strained material and are visibly evident using the planar DIC technique.

Figure 15

Figure 15. Through-thickness true strain distribution at various levels of global compression and an accompanying DIC image at 50% compression for the as molded (AM) 50 g/L EPS foam material crushed in the out-of-plane direction at (a,b) 0.001/s, and (c,d) 10/s, respectively. Error bars represent the averages from 3 repeat tests.

The complementary cumulative strain plot for 10/s out-of-plane crushing, shown in Figure 15c, displayed more uniform strain distribution with more diffuse transitions between the localized and uniformly strained regions (Figure 15d). The average peak strain was 35% lower under 10/s compression compared to the baseline (0.001/s) condition. Strain uniformity improved with increasing global compression in all cases, especially post-densification (i.e., ≥65% compression in most cases) as the lighter core material became sufficiently consolidated to match the density of the exterior layers, thus promoting more uniform through-thickness compression. The cumulative strain distribution for 0.001/s in-plane crushing (Figure 16a) presented greater uniformity compared to out-of-plane crushing, although a localized strain gradient was still evident. The localized strain band formation, shown in Figure 16b at 50% compression, was attributed to progressive interactions between adjacent EPS foam beads of inherently varying size and local density (Section 3.1).

Figure 16

Figure 16. Through-thickness true strain distribution at various levels of global compression and an accompanying DIC image at 50% compression for the as molded (AM) 50 g/L EPS foam material crushed in the in-plane direction at (a,b) 0.001/s, and (c,d) 10/s, respectively. Error bars represent the averages from 3 repeat tests.

While the in-plane true strain uniformity was initially greater than complementary out-of-plane loading conditions, the variation tended to increase with increasing global compression. Similar tendencies were observed for elevated rate loading, with the cumulative true strain distributions under 10/s in-plane compression (Figure 16c) exhibiting greater uniformity and 37% lower peak strains compared to the 0.001/s condition, on average. In contrast to the thin strain bands observed at 0.001/s, thick layers of uniformly compressed beads were observed for 10/s loading (Figure 16d) with localized regions of densified beads that resisted compression. The smooth borders between strain and larger regions of uniformity provide a novel visualization of viscous effects within the EPS foam material and resultant stress (strain) redistribution between adjacent closed-cell beads.

The complementary cumulative true strain distributions for 25%, 50% and 75% compression of the core material at 0.001/s are summarized in Figure 17a. The strain distribution and localization patterns shown in Figure 17b more closely resembled the in-plane loading results for the as molded material (recall Figure 16), owed to the homogeneous material structure parallel to the crushing direction. The true strain distributions became more uniform at 10/s, as shown in Figure 17c, however, the effect was milder compared to the as molded material, with a 19% reduction in the peak strains from 0.001/s to 10/s. Consistent with Section 3.2, no significant differences were observed in the strain localization between out-of-plane and in-plane loading. For reference, the cumulative strain distributions shown in Figures 17a,c provide the averages from both orientations. The contours shown in Figures 17b,d were taken from representative out-of-plane crushing tests.

Figure 17

Figure 17. Through-thickness true strain distribution at various levels of global compression and an accompanying DIC image at 50% compression for the core 50 g/L EPS foam material crushed in the out-of-plane direction at (a,b) 0.001/s, and (c,d) 10/s, respectively. Error bars represent the averages from 6 repeat experiments; 3 out-of-plane and 3 in-plane tests.

4.3 Dynamic unloading effects

The rate sensitive unloading effects revealed in the stress/strain curves (Section 3.2) can be quantified by examining the magnitude of strain release during unloading, $\Delta {\epsilon }_{unload}$, per Equation 7:

$\Delta {\epsilon }_{unload}={\epsilon }_{\mathit{max}}-{\epsilon }_0(7)$

where ${\epsilon }_{\mathit{max}}$ and ${\epsilon }_0$ represent the peak engineering (global) strain and strain at zero load (unloaded), respectively. The average $\Delta {\epsilon }_{unload}$ values with respect to strain rate are summarized in Figures 18a–d for the EPS foams with 30–100 g/L apparent densities, respectively. The 30 g/L (Figure 18a) and 50 g/L (Figure 18b) density groups exhibited a sigmoidal relationship between unloaded strain increment and global strain rate. The inflection point in the unloading trend was observed at strain rates between 0.1-1/s for the 30 g/L group compared to 1-10/s for the 50 g/L group. Unloading strain increments were generally consistent between the as molded and core EPS foam specimens up to 0.1/s for most material groups, beyond which the increments were distinctly lower for the core groups, nominally coinciding with the onset of viscous effects noted in Section 4.

Figure 18

Figure 18. Average magnitude of strain released during unloading, $\Delta {\epsilon }_{unload}$, with respect to strain rate for EPS foams with, clockwise from top left, (a) 30 g/L, (b) 50 g/L, (c) 80 g/L, and (d) 100 g/L apparent densities, respectively, in both as molded and core configurations.

The strain unloading increment was observed to exponentially increase with increasing strain rate up to 10/s for the 80 g/L (Figure 18c) and 100 g/L (Figure 18d) groups. Lower limits in the strain unloading plots (at 0.001/s) in Figure 18 were nominally consistent with the strains at yielding identified in Figures 7, 8. Furthermore, the average unloaded strain was observed to decrease for a given strain rate with increasing foam density, revealing a competing effect exists between pressure redistribution from entrapped gases and increasing wall thickness (i.e., resistance to flexure) in the denser closed-cell foam beads (Bouix et al., 2009; Bhagavathula et al., 2022). The tendency for the slope to decrease and level off for the lower density groups (≤50 g/L) suggests a maximum level of post-unloading strain recovery exists, beyond which viscous interior loading provides no further mitigation of fracture within the foam. The data presented in Figure 18 supports the notion that the strain recovery limit occurs at higher strain rates with increasing foam density to generate enough internal pressure within the thicker EPS cell walls.

A representative true strain distribution for the as molded 50 g/L EPS foam material crushed at 0.001/s in the out-of-plane direction is provided in Figure 19a to visualize the strain heterogeneity at peak compression and after unloading. The through-thickness true strain distribution at the post-densification peak strain (Figure 19b, 83.3% compression) exhibited a linear variation with a maximum compression of approximately 94.0%. The unloaded true strain distribution (Figure 19c, 79.4% compression) was near-identical to the peak strain profile, coinciding with just 4% of compressive strain recovery, with similar magnitudes of strain release measured at each increment. Uniformity of the through-thickness foam strain released at each level of true strain is consistent with elastic unloading of the base EPS material (i.e., Hooke’s law) with damage due to microfractures within the interior beads remaining.

Figure 19

Figure 19. (a) Plot of the true compressive strain distribution at (b) peak strain, and (c) remaining after unloading for as molded EPS foam with a 50 g/L density compressed out-of-plane at 0.001/s, corresponding with approximately 4% strain release. The strain release was significantly more pronounced for 10/s compression, as observed in the (d) true strain distribution and observed deformation profiles at the (e) peak strain, and (f) fully unloaded states.

In contrast, the peak and unloaded strain distributions at 10/s shown in Figure 19d reveal a 61% reduction in the peak true strain (35% reduction in the engineering strain) post-unloading. Moreover, the post-densification strain distribution (Figure 19e) varied by a factor of 5 compared to a factor of 2 post-unloading. Large portions of the specimen returned to a nearly unstrained state after unloading at 10/s (Figure 19f), with entire cross-sections of EPS beads resembling the unloaded distributions from Section 3.1 visible. These observations highlight that gas-mediated strain recovery in closed-cell foams mitigates localized strain accumulation and the onset of bead fracture by redistributing stresses more uniformly through the closed-cell foam structure. Future research will investigate higher strain rates and integration of the heterogeneity into material representations to support higher fidelity computational modeling.

5 Conclusion

The purpose of this study was to characterize the rate sensitive, anisotropic behavior of crushable EPS foams and to correlate the observations with macroscale material heterogeneity. Quasi-static to elevated rate crushing tests were performed with accompanying though-thickness DIC for novel quantification of the rate dependent strain localization and unloading phenomena as influenced by the EPS macrostructure. The findings from this study were obtained from EPS foam pucks with densified exterior skin layers and a resultant through-thickness density gradient inherent to molded helmets. Therefore, the experimental results and accompanying modeling represent one of the most complete datasets assembled for this material to date, providing a host of detailed measurements directly relevant to the design of personal safety products (e.g., helmets) for impact energy absorption (falls, low speed crashes, etc.).

The key findings of this investigation are as follows:

1. A (nominally) 5 mm deep exterior skin layer was identified in the outer volume of 30–100 g/L EPS foam pucks, the exterior skin material exhibited a 98% lower density than the core, on average. Foam beads in the exterior region possessed 35% lower equivalent diameters and were elongated by 38% perpendicular to the material rise (out-of-plane) direction compared to the core material.

2. Rate dependent compression testing revealed distinctly higher plateau stresses and elastic moduli, 38% on average, for the in-plane loading condition compared to out-of-plane loading with the as molded material, owing to the alignment of the elongated exterior/skin beads with the compression direction. In contrast, the core material samples which possessed uniform EPS beads exhibited quasi-isotropic compressive stress/strain behavior with reduced stiffness and (plateau and densification) stress compared to the as molded condition.

3. Positive rate sensitivity was observed for all test configurations over 0.001/s to 10/s compression and tended to increase with increasing foam density. The average core and as molded plateau stresses increased by 45% over the mentioned range of strain rates, with nominally equal sensitivity observed between the as molded and core material configurations. The data was also used to calibrate a rate sensitive Nagy constitutive model, which could predict the rate- and direction-dependent stress/strain response of the as molded and core EPS foam materials within 10% absolute error for the entire scope.

4. The through-thickness DIC measurements revealed intense true strain localization at the sample midplane for the as molded EPS foam compressed out of plane, due to compaction of the lighter core material between the denser exterior skin layers. Above 0.1/s, the boundaries between strain contours became more diffuse and the strains were distributed over larger regions of the specimens at lower overall magnitudes (21% reduction at 10/s compared to 0.001/s). Increased strain distribution was similarly observed when loading in-plane. This phenomenon was attributed to internal pressure redistribution arising from viscous gas dynamics within the closed-cell EPS foam.

5. The post-crushing EPS foam unloading was also shown to be rate sensitive, quantified via DIC. Specimens crushed at 0.001/s (all groups) experienced an average volumetric unload of 0.04 mm/mm, compared to 0.18 mm/mm and 0.42 mm/mm for the as molded 100 g/L and 30 g/L foams, respectively, unloaded at 10/s. Increased post-test unloading at higher rates suggests that the entrapped air mitigates fracture within the foam beads, and lower strain recovery in the higher density material groups is an outcome of the increased thickness (i.e., resistance to flexure) of the cell walls.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

FY: Data curation, Formal Analysis, Investigation, Methodology, Visualization, Writing – original draft, Writing – review and editing. MJ: Formal Analysis, Investigation, Methodology, Software, Writing – original draft, Writing – review and editing. MB-R: Investigation, Methodology, Resources, Writing – review and editing. PC: Investigation, Methodology, Writing – review and editing. DC: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Writing – review and editing. JM: Conceptualization, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Visualization, Writing – original draft, Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. The authors would like to acknowledge the financial support of this research from the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Canada Research Chair (CRC, CRC-2024-00020) Program, and the University of Waterloo Startup Grant Program.

Acknowledgements

The authors also sincerely thank Trek® Bicycle Corporation for providing the closed-cell EPS foam materials used in this study.

Conflict of interest

Author MB-R was employed by Trek Bicycle Corporation.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

Generative AI statement

The authors declare that no Generative AI was used in the creation of this manuscript.

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