Compliant rolling-contact architected materials for shape reconfigurability

Architected materials can achieve impressive shape-changing capabilities according to how their microarchitecture is engineered. Here we introduce an approach for dramatically advancing such capabilities by utilizing wrapped flexure straps to guide the rolling motions of tightly packed micro-cams that constitute the material’s microarchitecture. This approach enables high shape-morphing versatility and extreme ranges of deformation without accruing appreciable increases in strain energy or internal stress. Two-dimensional and three-dimensional macroscale prototypes are demonstrated, and the analytical theory necessary to design the proposed materials is provided and packaged as a software tool. An approach that combines two-photon stereolithography and scanning holographic optical tweezers is demonstrated to enable the fabrication of the proposed materials at their intended microscale.


Consider a CRJ that consists of alternating layers like the layer shown in
. Suppose that the CRJ's straps are fabricated to be the exact length, Lexact, required to wrap around their corresponding cams without becoming sloppy or needing to be stretched when the joint is assembled. This length is defined by where Rb1 is the base-circle radius of Cam 1, t is the thickness of the strap, β1 is the fixed angle that defines where the strap attaches to Cam 1 as labeled in Supplementary Fig. 1, Rb2 is the base-circle radius of Cam 2, and β2 is the fixed angle that defines where the strap attaches to Cam 2. As long as the CRJ's straps are fabricated with this length (i.e., Lexact), the total strain energy, Ujoint(Φ2), stored in the joint as a function of how much Cam 2 has rotated, Φ2, from its originally assembled angular position relative to Cam 1 is given by ( ) where O is the number of layers that constitute the joint and alternate according to the pattern shown in Fig where E is the Young's modulus of the layer's constituent material, W is the layer's out-ofplane thickness, ψ is the angle between the x-axis and the line that connects the centers of both cams, Rpi(ϕi) is the pitch radius of Cam i defined as Rpi(ϕi)=Rbi(ϕi)+(t/2), and Rbi(ϕi) is the base-circle radius of Cam i as a function of the angle ϕi, labeled in Supplementary Fig. 1, where i is either 1 or 2 corresponding to either Cam 1 or Cam 2. Note that supplementary equation (2) and (3) are general for CRJs with cams of any shape-not just circles.
Supplementary equation (3) was derived 1 under the assumption that the layer's strap is initially fabricated straight with no stored strain energy and possesses no variation in its thickness, width, or material properties along its length. For CRJs consisting of perfectly circular cams like the one shown in Supplementary Fig. 1, supplementary equation (3) simplifies to 3 1 2 1 layer 2 p1 p2 π π ( ) 24 because Rbi(ϕi) would remain constant as ϕi varies. If the straps within each layer perfectly enforce rolling-contact kinematics so that the cams are not permitted to slip as they rotate according to supplementary equation (4) reduces to ( ) ( ) If supplementary equation (6) is used in conjunction with supplementary equation (2), the total strain energy, Ujoint(Φ2), stored in the CRJ as a function of Φ2 is found to be Note that this strain energy remains constant as Cam 2 rotates to various angular positions (i.e., as Φ2 changes) and thus no moment is required to rotate the joint. We have, therefore, proven that CRJs will achieve zero rotational stiffness if their straps satisfy the ideal conditions specified at the beginning of this section.
. The fabricated lengths of the CRJ's straight straps are Lo=L-Δ, where L is the length of the straps once the CRJ is assembled and Δ is the amount that the straps are stretched during assembly. The assembled length, L, is defined according to where Rpi are the pitch radii of the CRJ's cams defined by Rpi=Rbi+(t/2) for i=1 or 2. The width of each alternating layer within the CRJ is W.
We will now derive the equations that relate the relative locations (i.e., x2 and y2 labeled in Fig. 5(f, g)) and angular orientation (i.e., Φ2) of Cam 2 with respect to Cam 1 to the loads that would need to be imparted on Cam 2 to move it there. To this end, we will consider two scenarios-tension and compression.
The tension scenario, shown in Fig. 5f, occurs when D ≤ 0, where For this scenario, the angle over which the strap is bent around Cam 1, θ1, is where the angle ψ, labeled in Fig. 5a, is ( ) Note that the x2 variable is shown as a negative value in Fig. 5a according to the coordinate system defined at the center of Cam 1. The angle λ1 from supplementary equation (10), The angle over which the strap is bent around Cam 2, θ2, is where the angle λ2, labeled in Fig. 5a, is defined in supplementary equation (12), and Φ2 is how much Cam 2 has rotated from its originally assembled position relative to Cam 1.
The tension, T, in the strap at any location within one of the CRJ's layers can be derived by applying equilibrium to an infinitely small portion of the strap curved around either cam as shown in Supplementary Fig. 2a (15) and using more small-angle approximations, supplementary equation (15) simplifies to (dT/dθ)=Tμ. By solving the resulting differential equation, it can be shown that the tension in the straight portion of the strap between the two cams, Ts, (i.e., the tension at point P labeled in Fig. 5f) is where Tαi are the tensions in the straight portions of the strap between the angles αi on either cam for i=1 or 2, and θ1 and θ2 are defined in supplementary equation (10) and (13) respectively. Thus, according to supplementary equation (16) The total amount that the strap is stretched, S, for a given position of Cam 2 (i.e., for certain values of x2, y2, and Φ2) is where dαi are the distances the strap is stretched in the regions between the angles αi on either cam for i=1 or 2 according to The Et parameter in supplementary equation (19) is the tensile Young's modulus of the strap.
The values dθi in supplementary equation (18) are the distances the strap is stretched in the regions bent over the cams between the angles θi for i=1 or 2 according to The distance that the straight portion of the strap is stretched between the two cams, ds, in where Lin is the initial length of that portion of the strap before the joint is assembled or loaded, and is defined according to The total amount that the strap is stretched, S, from supplementary equation (18) can also be defined as s in where Ls is defined as Thus, as long as S>0, the tension in the straight portion of the strap on Cam 2, Tα2, between the angle α2 can be derived by combining supplementary equation (17)- (21) and (23) according to where Ls is given in supplementary equation (24) and Lin is given in supplementary equation The tension, Tα2, is the magnitude of three forces, shown blue in Fig. 5f, that the strap imparts on Cam 2 at its position defined by x2, y2, and Φ2. One of these forces acts at point OA The component of the same force along the y-axis, FOBy, is given by The moment, MOBz, resulting from this force acting at point OB on Cam 2 about the coordinate system shown in Fig. 5f is The sum of the normal force increments, dN, shown green in Fig. 5f, from the strap on where ζ is defined in Fig. 5f. The component of the force along the y-axis, FNy, that results from summing the same increments together can be derived similarly according to The moment, MNz, resulting from this force on Cam 2 about the coordinate system shown in Fig. 5f is The sum of the force increments, μdN, shown purple in Fig. 5a, from the friction between the strap and Cam 2 on Cam 2 can be similarly calculated. The component of force along the x-axis, Ffx, that results from summing these friction increments together is The component of the force along the y-axis, Ffy, that results from summing the same friction increments together can be derived similarly according to The moment, Mfz, resulting from this force on Cam 2 about the coordinate system shown in Fig. 5f is There is also a pure moment, which acts on Cam 2, that is produced by the strap because it is bent. This moment can be determined by calculating the strain energy in the strap due to bending. This energy, Ubend, is where Uθ is the bending energy stored in the strap as a result of it being bent around both cams over the angles θi. By using a similar approach to that introduced in Supplementary Note 1, this energy is determined to be Since the straps within each layer will enforce rolling-contact kinematics about the point where the alternating straps crisscross (i.e., the point P labeled in Fig. 5a), we can apply to supplementary equation (10) and (13) where Rppi from supplementary equation (38) are the distances from the centers of each Cam i to the point P, labeled in Fig. 5f. These distances are The energies UOA and UOB from supplementary equation (36) are the bending energies that result from the tensioned strap being sharply bent at the points OA and OB, labeled in Fig. 5f.
These energies can be calculated according to where Khinge can be approximated as the stiffness of a compliant living hinge according to 3 ( ) 3 12 To determine the total load that must be imparted on a full CRJ consisting of O alternating layers in the tension scenario depicted in Fig. 5f such that the entire joint will be in static equilibrium at the location, x2 and y2, and the orientation Φ2, the loads determined using are greater than zero for all the layers within the CRJ, the moments from each alternating layer due to strap bending, Mm2, will always cancel so that the only loads imparted on Cam 2 that have any effect on the joint's stiffness are those caused by the strap being stretched.
We will now consider the compression scenario shown in Fig. 5g that occurs when D>0 (see supplementary equation (9)). For a given x2 and y2 that satisfy D>0, the magnitude of the compression force, Fc, imparted on Cam 2 from Cam 1 shown in Fig. 5g where Ec is the compressive Young's modulus of the CRJ material. Once these values are known, the same geometric-compatibility equations used for the tension scenario (i.e., supplementary equation (10) (54) The component of the same force along the y-axis, Fcompy, is given by Note that no moment is applied to Cam 2 by this compressive force about the coordinate system shown in Fig. 5g.
There is also a pure moment, which acts on Cam 2, that is produced by the strap because it is bent. This moment can be determined by calculating the strain energy in the strap due to bending. This energy, Ubend, is similar to that given in supplementary equation (36) Note that the derivatives of all of the energies in supplementary equation (56) become zero (even the two new terms) except for dUθ/dΦ2. Additionally, note from supplementary equation (58) that as long as the straps are bent around both cams such that θ1 and θ2 are greater than zero for all the layers within the CRJ, the moments from each alternating layer due to strap bending, Mm2, will always cancel for the compression scenario as well as for the tension scenario.
If the friction force, shown orange and labeled by its magnitude, μFc, in Fig. 5g The moment, Mfricz, resulting from the same friction force on Cam 2 about the coordinate system shown in Fig. 5g is where t* is the compressed thickness of the strap labeled in Supplementary Fig. 2b. This thickness is given by