Soft mechanical metamaterials with unusual swelling behavior and tunable stress-strain curves

RESULTS

Figure 1 presents a schematic illustration of the design strategy and fabricated samples. Here, the material system is constructed in a periodic two-dimensional (2D) triangular lattice configuration, in which the filamentary, horseshoe composite microstructures of hydrogel and passive materials serve as the building blocks. This type of reconfigurable, network composite material can be fabricated using the techniques of multimaterial 3D printing (see Materials and Methods for details), similar to the concepts of 4D printing (50–52). The horseshoe-shaped microstructure consists of two identical circular arcs, with an arc angle of θ and a radius of r, as shown in Fig. 1B. Each arc segment has a sandwich structure (see fig. S1A for more details) of a supporting layer (a digital polymeric material, ~65 MPa; RGD8530, Stratasys; in blue; width W₁), an active layer (a hydrogel, ~0.2 MPa; SUP705, Stratasys; in red; width W₃), and an encapsulation layer (an elastomer, ~0.5 MPa; TangoBlackPlus, Stratasys; in green; width W₂). Here, a thin encapsulation layer (90 μm in width) serves to maintain a well-bonded interface of the hydrogel layer and the supporting layer during the swelling and deswelling processes. Small holes arranged on top of the encapsulation layer allow water (or other solutions) to flow in and out, similar to the design adopted by Mao et al. (19, 20). The horseshoe-shaped sandwich microstructure can be represented mainly by four nondimensional parameters, including the arc angle θ, width ratio W₂/W₁, normalized width W₁/r, and length ratio (s/rθ) of the active layer to the supporting layer, as long as the width of the encapsulation layer is fixed in the design. By tuning these design parameters, the middle of the hydrogel layer can be well separated from the neutral mechanical plane of the sandwich structure such that the hydraulic swelling deformations of hydrogel can be converted into the bending deformations. Additional details of a specific example that illustrate this mechanism appear in fig. S2. Both an effective shrinkage and expansion of the entire network material are possible with this mechanism, depending on the magnitude of the arc angle and the placement of the active layer relative to the supporting layer (fig. S1B).

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Figure 1 (C and D) demonstrates a representative network composite material that can achieve a reversible negative swelling, with a fractional area change up to ~70% (and a relative length change up to ~45%). This design adopts a straight microstructure (θ = 0°) such that the bending deformations can induce a relatively large change of the end-to-end distance. When the network material is immersed in water, the hydration process begins to expand the hydrogel layer. Because of an offset (~420 μm) of the neutral mechanical plane with respect to the middle of the hydrogel layer, a considerable bending deformation is evident, reshaping the straight microstructure into the horseshoe geometry and resulting in shrinkage of the network material. As the hydration proceeds, this level of shrinkage increases until the hydration time reaches ~45 min, at which point the horseshoe microstructures begin to contact each other. As the saturated network material is put in a drying oven maintained at 75°C, the hydrogel deswells gradually due to the water evaporation. After dehydration for ~35 min, the network material recovers almost completely. Figure 1D presents the effective strain components (ε_x-swelling and ε_y-swelling) of the network material that quantify the shrinking deformations during the processes of hydration and dehydration. The slight deviation between the responses of ε_x-swelling and ε_y-swelling results mainly from the PolyJet 3D printing techniques that cause slightly different mechanical properties between the microstructures aligned horizontally and those aligned diagonally. After dehydration for ~40 min, the residual strain is below ~5%, indicating relatively good reversibility, consistent with the results on a single straight microstructure (fig. S2). Quantitative mechanics modeling based on the finite element analyses (FEA; see Materials and Methods for details) allows the prediction of the deformed configurations at different stages of hydration and dehydration, as shown in Fig. 1C. The detailed strain distribution and deformations of a representative unit cell of the network material are shown in fig. S3. The high strains occur mainly at the hydrogel layer and the encapsulation layer, while the maximum strains of the supporting layer are typically below ~3%, thereby ensuring the integrity of the lattice network. Quantitative agreements of the computational and experimental results can be observed (Fig. 1, C and D), suggesting the FEA-based modeling as an effective tool to guide the microstructure design for achieving desired swelling ratios.

Because of the capability of altering the microstructure geometries (Fig. 1C), the mechanical properties of the network material can be tuned continuously by controlling the time of water absorption. As an example, Fig. 1 (E and F) presents the stress-strain curves of the above design after water absorption for different times. Without any water absorption, the network material, consisting of straight microstructures, undergoes stretching-dominated deformations under a uniaxial stretching, leading to a relatively linear stress-strain response. After water absorption (for example, for 45 min), the network material with reshaped horseshoe microstructures undergoes bending-dominated deformations at the initial stage of uniaxial stretching, as shown in Fig. 1G and fig. S4. The stress thereby increases slowly with increasing strain during this stage. As the horseshoe shapes begin to reach full extension (~65% strain), the slope of the stress-strain curve increases rapidly due to the transition into a stretching-dominated deformation mode. This leads to a J-shaped, nonlinear stress-strain response that widely exists in biological materials (for example, human skins and heart valves) (53, 54). The complete extension of the horseshoe microstructures typically defines a critical strain (ε_cr) that marks the transition point of the J-shaped stress-strain curves. As shown by Fig. 1E, the J-shaped stress-strain curves can be effectively tuned by controlling the time of water absorption. As the saturated network material experiences the dehydration process, the stress-strain curve moves leftward gradually until the J shape vanishes, indicating a good recovery. FEA can capture the tunability of the stress-strain response in a quantitative manner, as evidenced by the reasonably good correspondence with experiments (Fig. 1F).

Because of the design flexibility enabled by tailoring the microstructure geometries, the above network composite design provides an ideal platform to offer desired swelling ratios in a wide range. Figure 2 summarizes a collection of theoretical and experimental results that elucidate the effects of three dominant design parameters, including the arc angle θ, width ratio W₂/W₁, and length ratio s/rθ. For relatively slender horseshoe microstructures, we develop a mechanics model that simplifies the sandwich microstructures as planar beams consisting of only the supporting layer (W₁) and the active layer (W₂), with a negligible stiffness contribution from the soft, thin encapsulation layer. Assuming planar cross-sections during deformation, the theory of composite beam (see Supplementary Methods and fig. S5 for details) shows that the curved beam (with curvature radius r) maintains an arc shape, as the active layer undergoes a swelling ratio (ε_hydrogel). Because of the strain mismatch between the active layer and the supporting layer, the arc radius changes to r_deformed and is solved asrdeformed=1+(E2E1)2(W2W1)4(1+εhydrogel)+E2E1W2W1[4(W2W1)2+6W2W1+4+εhydrogel]1+(E2E1)2(W2W1)4+2E2E1W2W1[2(W2W1)2+3W2W1+2+3(1+W2W1)rW2εhydrogel]r(1)where E₁ and E₂ are the elastic moduli of the supporting layer and the active layer, respectively (see Supplementary Methods for details). Without any self-contact of microstructures, the effective strain of the network material induced by the swelling of hydrogel layer can then be obtained asεx‐swelling=εy‐swelling={(stotal−s)cos(λs2rdeformed)+2rdeformed sin(λs2rdeformed)stotal−1for 1r=0r sin(λs2rdeformed+rθ−s2r)+(rdeformed−r)sin(λs2rdeformed)rsin(θ/2)−1for 1r≠0(2)where s_total is the half-length of the straight microstructure in the limit of θ = 0°, and λ is the stretch ratio at the middle axis of the supporting layer covered with hydrogel, as given byλ=1+(E2E1)2(W2W1)4(1+εhydrogel)+E2E1W2W1[4(W2W1)2+6W2W1+4+εhydrogel]1+(E2E1)2(W2W1)4+2E2E1W2W1[2(W2W1)2+3W2W1+2](3)The isotropic strain response arises from the homogeneous microstructure design. Figure 2A and fig. S5 show that the analytic solution (Eq. 2) corresponds reasonably closely to the results of FEA and experiment on the effective strain of the network material for a wide range of arc angle θ and length ratio (s/rθ). In Fig. 2A, the dimensionless swelling ratio ε¯hydrogel denotes the linear strain ε_hydrogel normalized by the value (0.21) at a nearly saturated state of the hydrogel. The negative arc angle represents the placement of the active layer to the inner side of the supporting layer, as opposed to the case of positive arc angle discussed above (fig. S1B). A distinct swelling behavior is evident for different arc angles, as shown in Fig. 2 (D to F). Specifically, for θ = −220°, a large positive swelling ratio (linear strain of up to ~98%) can be achieved by expanding the horseshoe microstructures with water absorption, as evidenced by the deformed configurations (Fig. 2D) from both the experiment and the FEA. For θ = −90°, the network material undergoes an expansion at the initial stage of hydration, and the associated swelling ratio (ε_x-swelling) increases to approach a peak value (10.8%) at the dimensionless swelling ratio of ε¯hydrogel=0.42. Beyond this critical point, the swelling ratio (ε_x-swelling) decreases with further water absorption and becomes negative at ε¯hydrogel=0.74. At the state of ε¯hydrogel=1, the shrinkage reaches its maximum, corresponding to a negative swelling ratio of ε_x-swelling = −4.3% based on FEA (and −5.1% based on the experiment). This represents an unusual type of swelling behavior—expanding first and then shrinking during hydration. For θ = 0°, as discussed in Fig. 1, the straight microstructures enable a large negative swelling (ε_x-swelling = −47.8 and −47.1% based on FEA and the experiment, respectively). Further increase of the arc angle (for example, to θ = 90° or 180°) leads to a reduced negative swellability, as shown in Fig. 2A, due to the self-contact of microstructures that occurs at an earlier stage of hydration (fig. S6). In the examples studied here, the network designs with θ = −220° and 0° offer the largest expansion and shrinkage, respectively, and both show a monotonous dependence on the swelling ratio (ε¯hydrogel) of hydrogel.

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Figure 2 (B and C) highlights the effect of length ratio (s/rθ) and width ratio (W₂/W₁) on the swellability of two representative network designs (θ = 0° and −220°), as characterized by the effective strain at ε¯hydrogel=1. Both figures show an increased magnitude of effective strain (that is, |ε_x-swelling|) at a larger length ratio (s/rθ). For a given length ratio (for example, s/rθ = 0.67), the magnitude of effective strain usually increases first with increasing width ratio (W₂/W₁) because the actuation force increases, as shown in Fig. 2B for the design with θ = 0°. At a critical width ratio (W₂/W₁ ≈ 3.2 in Fig. 2B), the magnitude of effective strain ε_x-swelling reaches a peak, after which |ε_x-swelling| decreases. This can be attributed to the reduced offset of the neutral mechanical plane relative to the middle of the hydrogel layer, with the further increase of the width ratio beyond W₂/W₁ = 3.2. Because of this mechanism, we observed an optimal value of width ratio (W₂/W₁) independent of the length ratio (s/rθ) (Fig. 2B), which can maximize the shrinkage of the network material. A similar dependence can be also found for the design with θ = −220° (Fig. 2C). According to these dependences, we can expect a reduced swellability with an increasing filling ratio of the cellular network material or equivalently decreasing the porosity. For the design with straight microstructures (that is, θ = 0°) and fixed length ratio (s/s_total = 0.75), the magnitude of swelling-induced strain decreases gradually from ~47.2% to 11.0% (and to 3.5%), as the filling ratio (defined on the basis of the undeformed configuration) increases from 5% to 12.5% (and to 22.5%) (fig. S7).

Excluding the isotropic expansion/shrinkage during water absorption, anisotropic swelling/deswelling is also achievable by exploiting heterogeneous horseshoe microstructures in the representative unit cell of network materials. By simply tailoring the selections of length ratio (s/rθ) and arc angle (θ) for the differently oriented horseshoe microstructures, a variety of usual swelling responses can be accessed. Figure 3 and figs. S8 to S10 demonstrate a representative collection of examples designed for this purpose. In particular, Fig. 3A presents the swelling path in the space of strain components (ε_x-swelling and ε_y-swelling) for our designed network materials, in comparison to the traditional hydrogels (31), polymers (55), and elastomers (56), as well as a buckling-guided metamaterial design (45). The traditional soft materials (for example, elastomers, polymers, and hydrogels) usually offer an isotropic positive swelling, with the linear expansion strain that can exceed 150% for specially designed hydrogels (31). In terms of the isotropic negative swelling, the materials presented here can reach a linear strain of −48%, exceeding that (−16%) reported previously (45). In terms of the anisotropic swelling, the traditional composite materials are typically constrained by the swellability of the constituent materials, which give limited levels of responses. By contrast, the current network design, because of the versatile flexibility, is able to offer a diversity of anisotropic swelling responses, including some unusual behavior (for example, expanding along one direction while shrinking along the perpendicular direction). Figure 3B shows an example that shrinks only along the horizontal direction (x) during hydration while staying undeformed along the perpendicular direction (y). This is achieved by using hydrogel layers only at the two diagonally oriented horseshoe microstructures and thickening the vertically oriented microstructures to suppress the deformations along this direction. More details of this design are in fig. S9A. Anisotropic negative swelling is also possible (Fig. 3C and fig. S9B) by exploiting different length ratios (s/rθ = 0.67 versus 0.25) for the differently oriented microstructures. In this example, the ratio (K_y/x = ε_y-swelling/ε_x-swelling) is maintained at approximately 0.27 during the entire process. Figure 3D illustrates a network material designed to respond to water absorption in the form of expansion along the y direction and shrinkage along the x direction. Here, the hydrogel layers are adopted only at the horizontally oriented microstructures, such that a lateral expansion can be achieved through a rotation of the passive components oriented diagonally (see fig. S9C for details). Figure 3E presents another type of design to offer a similar response through distinct placements of the hydrogel layer relative to the supporting layer at the different microstructures (see fig. S9D for details). In this example, the ratio (K_y/x) of swelling-induced strains varies considerably during the deformation process. Additional examples that offer anisotropic positive swelling ratios are provided in figs. S8 and S10. In all of the above cases, FEA results correspond well with the experimental measurements (figs. S9 and S10). Excluding the above experimental demonstrations as summarized in Fig. 3A, the proposed network design is also able to cover most of the blank spaces between the plotted red curves, thereby substantially expanding the accessible regime of existing soft materials in the space of swelling ratios ε_x-swelling and ε_y-swelling.

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Because of the versatile control of the microstructure geometries through water absorption, a systematic tunability of the stress-strain curve is achievable, taking into account the responses along two principal directions simultaneously. Figure 4A shows a network material that becomes softer as the hydration proceeds. The J-shaped stress-strain curves shift rightward nearly at the same pace for the uniaxial stretching along the x and y directions (Fig. 4, B and C, and fig. S11). The elastic modulus at the infinitesimal deformations decreases by nearly three orders of magnitude from the dry state to the hydrated state (fig. S12A). The critical strains (ε_cr-x and ε_cr-y) that mark the transition point of the J-shaped stress-strain curves can be tuned well by controlling the time of hydration (fig. S12B). In contrast to the case of x-directional stretching, the diagonally oriented horseshoe microstructures serve as the main load-bearing components for the y-directional stretching, as shown by the deformation sequences in Fig. 4 (B and C). Figure 4D presents a network material that expands along the x direction and shrinks along the y direction during hydration, similar to the design in Fig. 3E. As a result, the stress-strain curve along the x direction moves rightward, while the one along the y direction moves leftward, as hydration proceeds, as shown by the FEA calculations and experiments (Fig. 4D, middle and right). This indicates a decrease of elastic modulus E_x and critical strain ε_cr-y, as well as an increase of E_y and ε_cr-x (fig. S12, C and D). Figure 4E demonstrates another type of mechanical tunability by exploiting the network designs (as in Fig. 3C) with an anisotropic negative swelling. The resulting stress-strain curves move rightward for both the x and y directions, but at different rates (fig. S12, E and F).

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