54 H. A. Bruck et al. To achieve equilibrium, we relate the corresponding internal stress distribution to the applied bending moment, M, and the internal balance of force from the transverse location of each interface, hi, as follows: M(x)=Eb NX i=1 − k(x) 3 hi+1 3 −hi 3 + (k(x)hi +εo,i) 2 hi+1 2 −hi 2 0= NX i=1 − k(x) 2 hi+1 2 −hi 2 +(k(x)hi +εo,i)(hi+1 −hi) From the Euler-Bernoulli model, the maximum shear stress occurs in the center of the cross section of the beam and is given by: τmax=1.5 V A where V is the shear force and A is the cross-sectional area. Interlayer shear stress, τinter is given by: τinter ,i= 3V 2A" 1− 2yi h −1 2# Where his the total thickness of the layer jamming specimen. The maximum interlayer shear stress that can be sustained by jamming layers with surface friction coefficient, under a vacuum pressure is given by: τinter max=µ∆P Therefore, the maximum load, Vmax, before slip occurs at an interface located at yi: Vmax= 2Aµ∆P 3h1− 2yi h −1 2i For our experiments, one of the specimens was biased by a mechanism to induces a constraint on the lower surface that prevents it from stretching, and imposes a bending moment opposing the loading so that the lower surface of the structure has zero strain. The resulting biased bistable composite plate has an impact on the resulting stiffness reduction for the layered jamming bistable composite structure when slip occurs at the interface between the bistable composite plates. We again can look at this from a 1-D perspective using the previous Euler-Bernoulli beam theory, where for the two layer jammed bistable composite structure with plate thickness h1 and h2, respectively, the bending stiffness, (EI)c,jam, is related to the bending stiffnes of each layer, (EI)1 and (EI)1, as follows: (EI)c,jam=(EI)1+4Eh1 3/b+(E1) 2+E(h1+0.5h2) 2 h 2b When there is slip at the interface, their stiffness becomes: (EI)c,jam=(EI)1+4Eh1 3/b+(E1) 2 Since the constraint is applied on the first plate to create the biased response. Thus, the resulting stiffness reduction ratio, (EI)c,ratio, can be determined as follows: (EI)c,ratio=[(EI)1+0.25Eh1 3/b+(E1) 2]/[(EI)1+Eh1 3/(4b)+(E1) 2+E(h1+0.5h2) 2 h 2b] For our specimens, h1 =2h2, therefore, the stiffness reduction ratio should be 3.3. Experimental Results Three-point bending of layered jamming bistable composites For the initial set of experiments, we used three-point bending to characterize the behavior of the two bistable composite plates stacked on top of each other in the stable state with maximum curvature with the directions of curvature collinear. The
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