1,721,079 research outputs found
Internal Constraints and Gauge Relations in the Theory of Uniaxial Nematic Elastomers
No full textWe apply the formalism of analytical mechanics for constrained systems to reformulate the equilibrium theory of uniaxial nematic elastomers, allowing for constitutive dependence on the gradient
of the director
. In this setting, inextensibility is enforced by requiring that
and that
. Starting from these constraints, and using the principle of virtual work within a thermomechanically consistent framework, we derive boundary-value problems for determining equilibrium configurations. We show that the original formulation yields an underdetermined system for the Lagrange multiplier fields unless ancillary gauge conditions are imposed. To resolve this indeterminacy, we introduce two effective Lagrange multiplier fields: one defined in the interior of the referential region and the other on that portion of the boundary where the director traction is prescribed
Kinematic and static characterization of everting Möbius kaleidocycles with slightly incongruent links
No full textA Möbius kaleidocycle is a closed kinematic chain of n≥7 identical links connected by revolute joints, forming a linkage with the nonorientable topology of a Möbius band. If its joints are set at a critical, n-dependent twist angle — the smallest that allows closure without forcing — then, despite formally having n−6 internal degrees of freedom, the linkage admits only a single one: a reversible, periodic everting motion. Focusing on the case n=7, we determine the kinematic matrix via the Denavit–Hartenberg construction, under closure and congruence constraints. A geometric mechanism arises alongside the topological one due to a matrix-rank deficiency, accompanied by a corresponding state of self-stress. The geometric mechanism is infinitesimal and stiffened by self-stress, while eversion is enabled by the finite mechanism. Using a variational argument, we confirm that the sum of squared joint rotations remains constant throughout eversion. We further categorize the states of self-stress, identifying conserved quantities — including the sum of twisting moments raised to any positive integer power λ≥1 — which enable estimates of self-stresses in moderately incongruent linkages requiring elastic forcing to close
Monotonicity formulae for smooth extremizers of integral functionals
Full text linkA general monotonicity formula for smooth constrained local extremizers of first- order integral functionals subject to non-holonomic constraints is established. The result is then applied to recover some known monotonicity formulae and to discover some new monotonicity formulae of potential value
Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra
Full text linkWe discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves.journal articl
Instability Paths in the Kirchhoff–Plateau Problem
No full textThe Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy
Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization
No full textA new slender-body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is approximated by lower-dimensional objects, and a hyperviscous term is added to the flow equation. The hyperviscosity is given by the product of the ordinary viscosity with the square of a length that is shown to play the role of effective thickness of any lower-dimensional object. Explicit solutions of simple problems illustrate how the proposed method is able to represent with good approximation both the velocity field and the drag forces generated by rigid motions of the immersed bodies, in analogy with classical slender-body theories. This approach has the potential to open up the way to more effective computational techniques, since geometrical complexities can be significantly reduced. This, however, is achieved at the expense of involving higher-order derivatives of the velocity field. Importantly, both the dimensional reduction and the hyperviscous regularization, combined with suitable numerical schemes, can be used also in situations where inertia is not negligible
Wrinkling of a Stretched Thin Sheet
No full textWhen a thin rectangular sheet is clamped along two opposing edges and stretched, its inability to accommodate the Poisson contraction near the clamps may lead to the formation of wrinkles with crests and troughs parallel to the axis of stretch. A variational model for this phenomenon is proposed. The relevant energy functional includes bending and membranal contributions, the latter depending explicitly on the applied stretch. Motivated by work of Cerda, Ravi-Chandar, and Mahadevan, the functional is minimized subject to a global kinematical constraint on the area of the mid-surface of the sheet. Analysis of a boundary-value problem for the ensuing Euler-Lagrange equation shows that wrinkled solutions exist only above a threshold of the applied stretch. A sequence of critical values of the applied stretch, each element of which corresponds to a discrete number of wrinkles, is determined. Whenever the applied stretch is sufficiently large to induce more than three wrinkles, previously proposed scaling relations for the wrinkle wavelength and, modulo a multiplicative factor that depends on the Poisson ratio of the sheet and the applied stretch and arises from the more general and weaker nature of geometric constraint under consideration, root-mean-square amplitude are confirmed. In contrast to the scaling relations for the wrinkle wavelength and amplitude, the applied stretch required to induce any number of wrinkles depends on the in-plane aspect ratio of the sheet. When the sheet is significantly longer than it is wide, the critical stretch scales with the fourth power of the length-to-width ratio but, when the sheet is significantly wider than it is long, the critical stretch scales with the square of that same ratio
Solution of the Kirchhoff-Plateau Problem
Full text linkThe Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of noninterpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments
- …
