Stiff and strong, lightweight bi-material sandwich plate-lattices with enhanced energy absorption

Abstract

Plate-based lattices are predicted to reach theoretical Hashin–Shtrikman and Suquet upper bounds on stiffness and strength. However, simultaneously attaining high energy absorption in these plate-lattices still remains elusive, which is critical for many structural applications such as shock wave absorber and protective devices. In this work, we present bi-material isotropic cubic + octet sandwich plate-lattices composed of carbon fiber-reinforced polymer (stiff) skins and elastomeric (soft) core. This bi-material configuration enhances their energy absorption capability while retaining stretching-dominated behavior. We investigate their mechanical properties through an analytical model and finite element simulations. Our results show that they achieve enhanced energy absorption approximately 2–2.8 times higher than their homogeneous counterparts while marginally compromising their stiffness and strength. When compared to previously reported materials, these materials achieve superior strength-energy absorption characteristics, making them an excellent candidate for stiff and strong, lightweight energy absorbing applications.

Graphic Abstract

Appendix

A: Fabrication of the bi-material isotropic cubic + octet lattices

A hollow CAD model of an isotropic cubic + octet plate-lattice with $\overline{\rho }_{{{\text{cubic}} + {\text{octet}}}} = 0.3$ (shown in Fig. 7) was printed in CFRP via the projection micro-stereolithography (PμSL) system developed in the previous studies [37, 50]. After printing, the samples were cleaned in ethanol using an ultrasonic cleaner. This process was repeated several times until the trapped resin was removed entirely. The sample was then left to dry and post-cured under UV light. One of the as-fabricated samples was cut into pieces to verify that the inner hollowed channels are interconnected (Fig. 7a). To realize bi-material plate-lattices, the soft phase, comprising methacrylate monomers and oligomers and a thermal initiator (2,2′-Azobis(2-methylpropionitrile), was injected into the structure via a small hole at the top of each sample (Fig. 7a). This process was followed by thermal post-curing at 150°F for 24 h. We ground off the extra materials (over 25 × 25 × 25 mm) on six faces of the samples. One sample after grinding is displayed in Fig. 7b, clearly showing the boundary of the two material phases. Note that CFRP and soft phases were strongly bonded at their interface allowing the transfer tensile/compression loads between the two phases; this was verified through experimental observations in our previous work [37].

Figure 7

Fabrication of bi-material isotropic cubic + octet plate-lattices. (a) Schematic of the injection thermal curing method. Soft resin is injected into the 3D-printed octet-cubic shell made by CFRP then cured via heating. (b) Photograph showing the printed sample after injection and sanding.

B: Mechanical properties of the constituent materials

Development of CFRP and Flexible

Consistent with the methods used in our previous study [37], an ultraviolet (UV) curable CFRP composite was made with a UV-sensitive resin (Formlabs Rigid, Formlabs Inc) reinforced with 5 vol% short carbon fibers (PC100, E&L Enterprises, Inc). A high-energy ball mill was used to mix the monomer and carbon fiber thoroughly. The resulting CFRP composites are stiffer than the monomer, benefiting from the high stiffness of the carbon fibers and the interfacial friction [51,52,53] between fibers and monomer. On the other hand, the soft material was composed of methacrylate monomers and oligomers (Formlabs Flexible, Formlabs Inc) and a thermal initiator 2,2′-Azobis(2-methylpropionitrile) (Sigma-Aldrich).

Mechanical testing

To quantify the mechanical properties of CFRP and soft materials, we built ASTM standard (D3039) bulk samples to test along the same built direction via projection micro-stereolithography (PμSL). Two mechanical testing methods were performed: uniaxial tension and dynamic mechanical analysis (DMA). The uniaxial tension tests were performed using an Instron 5944 equipped with Bluehill data acquisition software and a 2000 N load cell to evaluate the stress–strain curve of the base material. A strain rate of 10⁻³/s (quasi-static strain rate) was conducted on each sample until fracture. The dynamic mechanical properties (storage and loss modulus) of the constituent materials were measured via a DMA apparatus (TA Instruments DMA 850) at 0.1 Hz (equivalent frequency for quasi-static condition [29]). The measured material properties are listed in Table 1; the measured stress–strain curves under uniaxial tension are compared with those obtained via constituent material modeling (see ‘Constituent material modeling’) in Fig. 8.

Figure 8

The comparison of the tensile stress–strain curves between the simulation and experiment for CFRP and Formlabs flexible constituent materials under uniaxial tension.

TABLE 1 Bulk material properties.

C: Yield and failure strains of the bi-material isotropic cubic + octet plate-lattices

Figure 9 shows yield and failure strains of the bi-material isotropic cubic + octet plate-lattices that were obtained from simulated stress–strain curves in Fig. 2.

Figure 9

Yield and failure strains measured from simulated stress–strain curves of bi-material isotropic cubic + octet plate-lattices.

D: Quasi-static response of the isotropic bi-material cubic + octet sandwich plate-lattice

Consider a cubic + octet unit cell oriented in a global Cartesian coordinate system as shown in Fig. 6b. We apply two different strain fields separately as follows:

Uniaxial strain (${\upvarepsilon }_{{{\text{xx}}}} = {\upvarepsilon }_{{{\text{yy}}}} = {\upvarepsilon }_{{{\text{xy}}}} = {\upvarepsilon }_{{{\text{xz}}}} = {\upvarepsilon }_{{{\text{yz}}}} = 0$ and ${\upvarepsilon }_{{{\text{zz}}}} = {\upvarepsilon }$)

First, we transform the uniaxial strain tensor from global $xyz$ coordinate to each plate’s local $x^{\prime}y^{\prime}z^{\prime}$ coordinate. Second, we enforce the plane-stress condition and obtain the principal stress ($\sigma_{{\text{I}}}$, $\sigma_{{{\text{II}}}}$, and $\sigma_{{{\text{III}}}}$) and principal strain ($\varepsilon_{{\text{I}}}$, $\varepsilon_{{{\text{II}}}}$, and $\varepsilon_{{{\text{III}}}}$) components. Strain energy density of each plate then can be calculated via $U_{{{\text{el}}}} = \frac{1}{2}\left( {\sigma_{{\text{I}}} \varepsilon_{{\text{I}}} + \sigma_{{{\text{II}}}} \varepsilon_{{{\text{II}}}} + \sigma_{{{\text{III}}}} \varepsilon_{{{\text{III}}}} } \right)$. This leads to the effective strain energy density of the unit cell under uniaxial strain as

$$U_{{{\text{el}},{\text{cubic}} + {\text{octet}},{\text{uni}}}} = \frac{{E_{{\text{p}}} \left( {2\overline{\rho }_{{{\text{octet}}}} + 3\overline{\rho }_{{{\text{cubic}}}} } \right)\varepsilon^{2} }}{{9\left( {1 - v_{{\text{p}}}^{2} } \right)}}$$

(11)

Hydrostatic strain ($\varepsilon_{{{\text{xy}}}} = \varepsilon_{{{\text{xz}}}} = \varepsilon_{{{\text{yz}}}} = 0$ and $\varepsilon_{{{\text{xx}}}} = \varepsilon_{{{\text{yy}}}} = \varepsilon_{{{\text{zz}}}} = \varepsilon$)

Similarly, we obtain the effective strain energy density of the unit cell under hydrostatic strain as

$$U_{{{\text{el}},{\text{cubic}} + {\text{octet}},{\text{hydro}}}} = \frac{{E_{{\text{p}}} \left( {\overline{\rho }_{{{\text{octet}}}} + \overline{\rho }_{{{\text{cubic}}}} } \right)\varepsilon^{2} }}{{1 - v_{{\text{p}}} }}$$

(12)

Once we obtain the effective strain energy density of the unit cell, we enforce the isotropy with $\overline{\rho }_{{{\text{cubic}}}} = \frac{2}{3}\overline{\rho }_{{{\text{octet}}}}$ [19, 21]. Equations (11)–(12) are then reduced to

$$U_{{{\text{el}},{\text{cubic}} + {\text{octet}},{\text{uni}}}} = \frac{{4E_{{\text{p}}} \overline{\rho }_{{{\text{cubic}} + {\text{octet}}}} \varepsilon^{2} }}{{15\left( {1 - v_{{\text{p}}}^{2} } \right)}}$$

(13)

$$U_{{{\text{el}},{\text{cubic}} + {\text{octet}},{\text{hydro}}}} = \frac{{E_{{\text{p}}} \overline{\rho }_{{{\text{cubic}} + {\text{octet}}}} \varepsilon^{2} }}{{\left( {1 - v_{{\text{p}}} } \right)}}$$

(14)

Since the unit cell is isotropic, we can write the effective constitutive relation as follows:

$$\left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\sigma_{{{\text{xx}}}} } \\ {\sigma_{{{\text{yy}}}} } \\ {\sigma_{{{\text{zz}}}} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\tau_{{{\text{yz}}}} } \\ {\tau_{{{\text{xz}}}} } \\ {\tau_{{{\text{xy}}}} } \\ \end{array} } \\ \end{array} } \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {C_{11} } & {C_{12} } & {C_{12} } \\ {} & {C_{11} } & {C_{12} } \\ {} & {} & {C_{11} } \\ \end{array} } & {{\text{Symm}}} \\ {{\text{Symm}}} & {\begin{array}{*{20}c} {\frac{{C_{11} - C_{12} }}{2}} & {} & {} \\ {} & {\frac{{C_{11} - C_{12} }}{2}} & {} \\ {} & {} & {\frac{{C_{11} - C_{12} }}{2}} \\ \end{array} } \\ \end{array} } \right]\left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varepsilon_{{{\text{xx}}}} } \\ {\varepsilon_{{{\text{yy}}}} } \\ {\varepsilon_{{{\text{zz}}}} } \\ \end{array} } \\ {\begin{array}{*{20}c} {2\varepsilon_{{{\text{yz}}}} } \\ {2\varepsilon_{{{\text{xz}}}} } \\ {2\varepsilon_{{{\text{xy}}}} } \\ \end{array} } \\ \end{array} } \right),$$

(15)

where $C_{11} = 2U_{{{\text{el}},{\text{cubic}} + {\text{octet}},{\text{uni}}}} /\varepsilon^{2}$ and $C_{11} + 2C_{12} = 2U_{{{\text{el}},{\text{cubic}} + {\text{octet}},{\text{hydro}}}} /\left( {3\varepsilon^{2} } \right)$. Using Eqs. (13) and (14), we can obtain elastic constants C₁₁ and C₁₂ as

$$C_{11} = \frac{{8E_{{\text{p}}} \overline{\rho }_{{{\text{cubic}} + {\text{octet}}}} }}{{15\left( {1 - v_{{\text{p}}}^{2} } \right)}}$$

(16)

$$C_{12} = \frac{{E_{{\text{p}}} \overline{\rho }_{{{\text{cubic}} + {\text{octet}}}} \left( {1 + 5v_{{\text{p}}} } \right)}}{{15\left( {1 - v_{{\text{p}}}^{2} } \right)}}$$

(17)

Finally, using $E = \left( {C_{11} - C_{12} } \right) \cdot \left( {C_{11} + 2C_{12} } \right)/\left( {C_{11} + C_{12} } \right)$, we can obtain the linearly elastic effective modulus as

$$E = \frac{{2\left( {7 - 5\nu_{{\text{p}}} } \right)E_{{\text{p}}} \overline{\rho }_{{{\text{cubic}} + {\text{octet}}}} }}{{\left( {1 - v_{{\text{p}}} } \right)\left( {27 + 15v_{{\text{p}}} } \right)}},$$

(18)

where E_p and ν_p were defined in Eqs. (2) and (3), respectively.

E: Convergence study

Since both buckling and fractures can occur in the simulations (two situations that can break the material symmetry required for periodic boundary conditions³), we first performed both the mesh and unit cell convergence studies and determined that a 3 × 3 × 3 lattice configuration and an average mesh size ratio $\tilde{e}_{{{\text{avg}}}}$ (the ratio of the average element size to the unit cell size) of 0.04 are close to the converged values (see Figs. 10 and 11). The determined number of unit cells and mesh size ratio were then used in all simulations of the isotropic cubic + octet lattices with $\overline{\rho }_{{{\text{cubic}} + {\text{octet}}}}$ = 0.1, 0.2, and 0.3 and several V_soft from 0 to 40% under compression.

Quasi-periodic boundary conditions (QPBCs) discussed in “Finite element simulations” were used in Abaqus to perform both the mesh and the unit cell convergence study on the isotropic cubic + octet lattice shell models with $\overline{\rho }_{{{\text{cubic}} + {\text{octet}}}}$ = 0.3. For each convergence study, two constituent materials (100% volume fraction of CFRP, V_CFRP = 1 or 100% volume fraction of soft phase, V_soft = 1) are considered, hence representing two extreme ends of material behaviors. In the case of CFRP, a displacement δ, corresponding to 2% effective strain, is applied in the QPBCs; in the case of soft phase, a δ, corresponding to 10% effective strain is applied instead. The details are discussed below:

Mesh convergence study

Only a single unit cell was used. The average mesh size ratio $\tilde{e}_{{{\text{avg}}}}$ was then refined from 0.06 to 0.03 to study the convergence of Young’s modulus, peak strength, and strain at the peak strength. $\tilde{e}_{{{\text{avg}}}} = 0.04$ was deemed appropriate for all cases (Fig. 10a and b) and then used in the unit cell convergence study.

Figure 10

The Young’s modulus, peak strength, and strain at peak strength of the isotropic cubic + octet plate-lattice with $\overline{\rho }_{{{\text{cubic}} + {\text{octet}}}}$ = 0.3, made of (a) 100% volume fraction of CFRP, V_CFRP = 1 or (b) 100% volume fraction of the soft phase (V_soft = 1) is plotted against the decreasing average mesh ratio for the mesh convergence study.

Unit cell convergence study

The chosen average mesh size ratio $\tilde{e}_{{{\text{avg}}}}$ = 0.04 was used to mesh the models and the number of unit cell was increased cubically from 1 × 1 × 1 to 4 × 4 × 4 to study the convergence of Young’s modulus, peak strength, and strain at the peak strength. The optimal number of unit cells is determined to be 3 $\times$ 3 $\times$ 3 from Fig. 11.

Figure 11

The Young’s modulus, peak strength, and strain at peak strength of the isotropic cubic + octet plate-lattice with $\overline{\rho }_{{{\text{cubic}} + {\text{octet}}}}$ = 0.3, made of (a) 100% volume fraction of CFRP, V_CFRP = 1 or (b) 100% volume fraction of the soft phase (V_soft = 1) is plotted against the increasing number of unit cells per side for the unit cell convergence study.