Modular energy absorbing capabilities achieved with compounded deformation mechanisms in composite AA6061-T6/PVC foam structures

Abstract

There is a significantly growing need for energy absorbing systems comprised of lightweight, low-cost and recyclable materials which can achieve high-capacity energy absorption with an ideal (near-constant) force response—these objectives are often only partially accomplished due to the many inherent engineering challenges. This study characterized a novel compounded energy dissipation system which could simultaneously exploit axial cutting, radial clamping, foam compression and extrusion/foam interactions or any desired combination of these mechanisms, for high-capacity energy absorption. Experiments were conducted for AA6061-T6 extrusions, with H-series PVC foam cores with relative densities between 0.09 and 0.18, subjected to 6-bladed cutting and 10-bladed cutting/clamping. The extrusion/foam interaction effects were newly observed, characterized by localized in-plane expansion of the foam cores and exhibited a transition from near-constant reaction forces to progressive quasi-linear increases. Introducing maximum density, H250 PVC foam consequently increased the average mean force and effectiveness (EAEF) by 33.8% with a marginal 12.3% CFE reduction. Numerical modeling was completed with a combined Lagrangian/Eulerian approach capable of predicting the experimental findings with an average validation metric of 0.92. An extended analytical modeling procedure was also developed with the ability to predict the entire force/displacement response with an average validation metric of 0.95. This modeling procedure was utilized in a parametric investigation which revealed a broad 10–200 kN steady-state load bearing capacity (86.0% average CFE) for the proposed, component-scale system since each deformation mechanism could theoretically be included or omitted as desired.

Appendix

The following geometric expressions represent necessary terms for the expressions presented and derived in Sect. 4.3 to obtain the predictions contained in Sect. 5. Only the most essential details were replicated for brevity. A simplified schematic of the axial cutting and radial petal clamping deformation modes is also included in Fig. 

Fig. 32

Simplified illustrations of the principal deformation mechanisms and associated geometric parameters for the axial cutting and radial petal clamping energy dissipation modes, recall that combining the former with the latter yields the hybrid cutting/clamping mode

32 to provide a convenient visualization of these secondary variables. Readers are encouraged to seek out the following sources [74, 75] for comprehensive discussions on the underlying theory, assumptions and step-by-step derivations.

First and foremost, the plastic deformation force which develops at the cutting interface, F_p, and the corresponding rolling radius, R_r, which minimizes the cutting resistance force in this strain field are defined in Eqs. (41) and (42), respectively.

$${F}_{p}=\frac{{2\sigma }_{o}t}{\sqrt{3}}\left[\frac{t\left(B+{R}_{r}\right)}{4\sqrt{3}}\cdot \left(\frac{1}{{R}_{r}\mathrm{cos}\theta }+ \frac{1}{{r}_{0}+{r}_{m}}- \frac{1}{{r}_{m}+{r}_{i}}\right)+B\theta +T+\frac{{K}_{\theta }{R}_{r}}{2}+\frac{\pi t{r}_{m}}{2{nR}_{a}}\right]$$

(41)

$$R_{r} = \left[ { \left( {\frac{Bt}{{4\sqrt 3 K_{\theta } \cos \theta }} - \frac{{T^{2} }}{2}} \right) + \sqrt {\left( {\frac{{T^{2} }}{2} - \frac{Bt}{{4\sqrt 3 K_{\theta } \cos \theta }}} \right)^{2} + \frac{{T^{2} }}{{K_{\theta } }}\left( {\frac{Bt}{{2\sqrt 3 \cos \theta }} + \pi tr_{m} \tan^{2} \theta } \right)} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$$

(42)

The following terms represent critical dimensions which develop and are maintained in the steady-state axial cutting field (i.e., at the extrusion/blade interfaces) and are named as follows: the steady-state radial increment, Δr_ss, axial bend radius, R_a, membrane stretching coefficient, K_θ, and deformed shoulder length, D_ss, respectively. These parameters are defined in Eqs. (43) through (46), respectively.

$$\Delta r_{{{\text{ss}}}} = \frac{{n_{{\text{b}}} \left( {2B - T} \right)}}{2\pi } + \frac{t}{2}$$

(43)

$$R_{{\text{a}}} = \frac{{\left( {D_{{{\text{ss}}}} + l_{{\text{b}}} } \right)^{2} + {\Delta }r_{t}^{2} }}{{2\Delta r_{t} }}$$

(44)

$$D_{{{\text{ss}}}} = R_{{\text{r}}} \left( {\csc \theta - \sin \theta } \right) - l_{{\text{b}}}$$

(45)

$$K_{\theta } = 0.366\left( {1 + 0.55\theta^{2} } \right) \cdot \cos^{2} \theta$$

(46)

The remaining terms are arise for radial petal clamping and the hybrid cutting/clamping mode, which generally occurs for cutting tools with 8 or more evenly spaced blades and a solid exterior ring [74, 75]. The onset displacement of petal clamping, δ_P, unconstrained (free) horizontal, h_f, and radial (lateral), l_f, positions of the petalled sidewall, blade tip offset distance, l_T, axial deflection of the petal tip, ${\Delta }_{v}$, second moment of area for a petalled sidewall, ${{I}_{G}}_{P}$, and bending moment arm, X_P, are provided in Eqs. (47) through (53), respectively.

$${\delta }_{P}=\begin{array}{c}2{R}_{\mathrm{a}}\left[{\mathrm{sin}}^{-1}\left(\sqrt{\frac{{\Delta r}_{\mathrm{ss}}\left({r}_{\mathrm{ring}}-{r}_{o}-{l}_{b}\right)}{2{R}_{a}{l}_{b}}}\right)-{\mathrm{sin}}^{-1}\left(\sqrt{\frac{{l}_{T}}{2{R}_{a}}}\right)\right]+\left[\left({r}_{\mathrm{ring}}-{r}_{o}\right){l}_{b}-{\Delta r}_{ss}\left({r}_{\mathrm{ring}}-{r}_{o}-{l}_{b}\right)\right]\\ \cdot \sqrt{1+{\left(\frac{{\Delta r}_{ss}\left({r}_{\mathrm{ring}}-{r}_{o}-{l}_{b}\right)-{{l}_{b}R}_{a}}{\sqrt{{\Delta r}_{ss}\left({r}_{\mathrm{ring}}-{r}_{o}-{l}_{b}\right)\left(2{R}_{a}{l}_{b}-{\Delta r}_{ss}\left({r}_{ring}-{r}_{o}-{l}_{b}\right)\right)}}\right)}^{2}}\end{array}$$

(47)

$$h_{{\text{f}}} = D_{{{\text{ss}}}} - \sqrt {l_{{\text{f}}} \left( {2R_{{\text{a}}} - l_{{\text{f}}} } \right)}$$

(48)

$$l_{{\text{f}}} = 2R_{{\text{a}}} \sin^{2} \left( {\frac{{L_{{\text{P}}} }}{{2R_{{\text{a}}} }} + \sin^{ - 1} \left( {\sqrt {\frac{1}{2}\left( {1 - \frac{1}{{R_{{\text{a}}} }}\sqrt {R_{{\text{a}}}^{2} - D_{{{\text{ss}}}}^{2} } } \right)} } \right)} \right)$$

(49)

$$l_{{\text{T}}} = R_{{\text{a}}} - \sqrt {R_{{\text{a}}}^{2} - D_{{{\text{ss}}}}^{2} }$$

(50)

$$\Delta_{v} \approx - 0.0127e^{{0.418n_{b} }} \left[ {{\text{mm}}} \right]$$

(51)

$$I_{{{\text{GP}}}} = \frac{1}{4}\left( {r_{o}^{4} - r_{i}^{4} } \right)\left[ {\cos \left( {\frac{\pi }{{n_{b} }}} \right)\sin \left( {\frac{\pi }{{n_{b} }}} \right) + \frac{\pi }{{n_{b} }}} \right] - \frac{{4n\left( {r_{o}^{3} - r_{i}^{3} } \right)^{2} }}{{9\pi \left( {r_{o}^{2} - r_{i}^{2} } \right)}}\sin^{2} \left( {\frac{\pi }{{n_{b} }}} \right)$$

(52)

$${X}_{P}=\frac{1}{{\mu }^{2}}\left[\begin{array}{c}\frac{2\mu }{3}\left({\left({h}_{f}+{D}_{ss}\right)}^{3}-{D}_{ss}^{3}\right)+\frac{\left({\mu }^{2}-1\right)}{3}\cdot \left({l}_{f}^{3}-{l}_{T}^{3}\right)\\ +{h}_{f}\left({l}_{f}\left(\mu \left({l}_{f}-{R}_{a}\right)-{h}_{f}-{D}_{ss}\right)+\mu {R}_{a}\left({R}_{a}-{l}_{f}\right)+{R}_{a}\left({h}_{f}+{D}_{ss}\right)\right)\\ +\left({R}_{a}-{\mu }^{2}{l}_{f}+\mu \left({h}_{f}+{D}_{ss}\right)\right)\cdot \left({l}_{f}^{2}-{l}_{T}^{2}\right)-{D}_{ss}\left(\left({l}_{f}-{l}_{T}\right)\left(\mu \left({l}_{f}-{R}_{a}\right)-{h}_{f}-{D}_{ss}\right)\right)\\ +\left({\mu }^{2}{l}_{f}^{2}-2\mu {l}_{f}\left({h}_{f}+{D}_{ss}\right)+2{l}_{f}{R}_{a}-{l}_{f}^{2}\right)\cdot \left({l}_{f}-{l}_{T}\right)\\ +{R}_{a}^{2}\left(\mu \left({R}_{a}-{l}_{f}\right)+\left({h}_{f}+{D}_{ss}\right)\right)\cdot \left({\mathrm{sin}}^{-1}\left(\frac{{R}_{a}-{l}_{f}}{{R}_{a}}\right)-{\mathrm{sin}}^{-1}\left(\frac{{R}_{a}-{l}_{T}}{{R}_{a}}\right)\right)\end{array}\right]$$

(53)