We study the spontaneous out-of-plane bending of a planar untwisted ribbon composed of nematic polymer networks activated by a change in temperature. Our theory accounts for both stretching and bending energies, which compete to establish equilibrium. We show that when equilibrium is attained these energy components obey a complementarity relation: one is maximum where the other is minimum. Moreover, we identify a bleaching regime: for sufficiently large values of an activation parameter (which measures the mismatch between the degrees of order in polymer organization in the reference and current configurations), the ribbon’s deformation is essentially independent of its thickness.
We present a study on planar equilibria of a terminally loaded elastic rod wrapped around a rigid circular capstan. Both frictionless and frictional contact between the rod and the capstan are considered. We identify three cases of frictionless contact – namely where the rod touches the capstan at one point, along a continuous arc, and at two points. We show that, in contrast to a fully flexible filament, an elastic rod of finite length wrapped around a capstan does not require friction to support unequal loads at its two ends. Furthermore, we classify rod equilibria corresponding to the three aforementioned cases in a limit where the length of the rod is much larger than the radius of the capstan. In the same limit, we incorporate frictional interaction between the rod and the capstan, and compute limiting equilibria of the rod. Our solution to the frictional case fully generalizes the classic capstan problem to include the effects of finite thickness and bending elasticity of a flexible filament wrapped around a circular capstan.
We present a theory of deformation of ribbons made of nematic polymer networks (NPNs). These materials exhibit properties of rubber and nematic liquid crystals, and can be activated by external stimuli of heat and light. A two-dimensional energy for a sheet of such a material has already been derived from the celebrated neo-classical energy of nematic elastomers in three space dimensions. Here, we use a dimension reduction method to obtain the appropriate energy for a ribbon from the aforementioned sheet energy. We also present an illustrative example of a rectangular NPN ribbon that undergoes in-plane serpentine deformations upon activation under an appropriate set of boundary conditions.
In this work, we revisit experimentally and theoretically the mechanics of a tape loop. Using primarily elastic materials (polydimethylsiloxane, PDMS, or polycarbonate, PC) and confocal microscopy, we monitor the shape as well as the applied forces during an entire cycle of compression and retraction of a half-loop compressed between parallel glass plates. We observe distinct differences in film shape during the cycle; points of equal applied force or equal plate separation differ in shape upon compression or retraction. To model the adhesion cycle in its entirety, we adapt the ‘Sticky Elastica’ of [T. J. W. Wagner et al., Soft Matter, 2013, 9, 1025–1030] to the tape loop geometry, which allows a complete analytical description of both the force balance and the film shape. We show that under compression the system is generally not sensitive to interfacial interactions, whereas in the limit of large separation of the confining parallel plates during retraction the system is well described by the peel model. Ultimately, we apply this understanding to the measurement of the energy release rate of a wide range of different cross-linker ratio PDMS elastomer half-loops in contact with glass. Finally, we show how the model illuminates an incredibly simple adhesion measurement technique, which only requires a ruler to perform.
P. Grandgeorge, C. Baek, H. Singh, P. Johanns, T. G. Sano, A. Flynn, J. H. Maddocks, and P. M. Reis
Networks of flexible filaments often involve regions of tight contact. Predictively understanding the equilibrium configurations of these systems is challenging due to intricate couplings between topology, geometry, large nonlinear deformations, and friction. Here, we perform an in-depth study of a simple yet canonical problem that captures the essence of contact between filaments. In the orthogonal clasp, two filaments are brought into contact, with each centerline lying in one of a pair of orthogonal planes. Our data from X-ray tomography (μCT) and mechanical testing experiments are in excellent agreement with the finite element method (FEM) simulations. Despite the apparent simplicity of the physical system, the data exhibits strikingly unintuitive behavior, even when the contact is frictionless. Specifically, we observe a curvilinear diamond-shaped ridge in the contact pressure field between the two filaments, sometimes with an inner gap. When a relative displacement is imposed between the filaments, friction is activated, and a highly asymmetric pressure field develops. These findings contrast to the classic capstan analysis of a single filament wrapped around a rigid body. Both the μCT and the FEM data indicate that the cross-sections of the filaments can deform significantly. Nonetheless, an idealized geometrical theory assuming undeformable tube cross-sections and neglecting elasticity rationalizes our observations qualitatively and highlights the central role of the small but finite tube radius of the filaments. We believe that our orthogonal clasp analysis provides a building block for future modeling efforts in frictional contact mechanics of more complex filamentary structures.
Geometry and Mechanics of Solids 