Friction Effects in Shape Dynamics

• Friction-like effects in shape dynamics are defined by the interplay between morphological changes and friction, influencing energy dissipation and motion.
• Analytical and computational methods, including variational inequalities and convex analysis, elucidate the impact of boundary conditions on friction behavior.
• Controlling system shapes can optimize frictional responses in practical applications such as nanoscale devices and soft robotic locomotion.

Friction-like effects in shape dynamics encompass the interplay between geometric or morphological changes of a physical system and the emergence, scaling, or control of frictional forces. In diverse contexts—from variational shape optimization with boundary friction, to nanoscale interfacial sliding, soft-matter contacts, and geometric locomotion—the shape, boundary conditions, and evolution of a system critically determine frictional dissipation and the transformation of actuation into motion or resistance. This article details the analytical frameworks, computational formalisms, and representative phenomena underlying friction-like effects as mediated by shape dynamics, synthesizing developments in convex variational inequalities, geometric mechanics, nanoscale tribology, and soft matter physics.

1. Shape-Dependent Friction in Variational Inequalities

The shape of an elastic domain directly modulates boundary frictional behavior in PDE-constrained variational problems. Scalar and vectorial Tresca friction laws impose nonlinear constraints on the admissible flux through the boundary, resulting in nonsmooth variational inequalities for the field variable $u$:

$$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$

where $g(x)$ is a spatially varying friction threshold. The associated energy functional,

$$\mathcal{J}(\Omega) = \frac12\int_\Omega(|\nabla u|^2 + u^2)\,dx + \int_\Gamma g(x)|u|\,ds - \int_\Omega f u\,dx,$$

includes an explicit boundary friction term enforcing the Tresca threshold (Adly et al., 2024).

The sensitivity of $\mathcal{J}$ to domain deformations relies on tools from convex and variational analysis—specifically, the computation of shape derivatives via proximal operators and twice epi-differentiability. Under regularity conditions, the material derivative $u'_0$ satisfies a Signorini-type boundary value problem characterized by unilateral conditions (complementarity between displacement and stress), and the energy shape gradient admits an explicit boundary integral formula reflecting the coupling of curvature, boundary stress, and friction:

$$\mathcal{J}'(\Omega_0)(V) = \int_{\Gamma_0} (V \cdot n) \left[\frac12 (|\nabla u_0|^2 + |u_0|^2) - f u_0 + H g |u_0| - \partial_n(u_0 \partial_n u_0) + g u_0 \nabla (\partial_n u_0/g) \cdot n \right] ds.$$

In 2D linear elasticity models, this approach extends to tangential Tresca friction with the directional shape derivative yielding a variational inequality with tangential Signorini-type boundary conditions (Bourdin et al., 2024). As the friction parameter is varied, optimized shapes interpolate between Dirichlet-type (clamped) and Neumann-type (free) boundaries (Adly et al., 2024, Bourdin et al., 2024).

2. Geometry-Induced Friction in Soft Interface Dynamics

Shape-driven stresses—arising purely from interfacial geometric incompatibility—govern friction even in the absence of dissipative surface chemistry. When a planar elastic disc is forced to conform to a spherical substrate with mismatched Gaussian curvature, geometric confinement induces in-plane tensile stresses, which generate a normal pressure proportional to the substrate mean curvature. For a thin polymer sheet of radius $W$ on a sphere of radius $R$, the induced mean tensile strain and ensuing frictional force scale as:

$$\epsilon_\text{geom} \sim \begin{cases} (W/R)^2 & W/R \leq \zeta \ (W/R)^{2/3} & W/R > \zeta \end{cases}$$

$$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$0

where $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$1 marks the wrinkling transition, and $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$2 are the Young's modulus and thickness, respectively (Chawla et al., 2024). Planar and cylindrical substrates (zero Gaussian curvature mismatch) yield negligible geometry-induced friction, demonstrating the central role of global shape incompatibility in friction emergence at soft interfaces.

3. Nanoscale Friction Scaling and Moiré Superstructures

At atomically thin interfaces, the finite size and shape of flakes in incommensurate contacts (e.g., twisted bilayer graphene or graphene/h-BN) result in nontrivial scaling laws for static friction that are dictated by the interplay between the flake shape and moiré superstructural periodicity:

• Circular flakes: Only short-period oscillations of friction with size, with envelope $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$3.
• Polygonal flakes (square, hexagon): Superposition of two geometric periods $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$4 (short) and $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$5 (long), linked to the moiré pattern and edge orientation. The envelope of friction exhibits both oscillatory nulls (frictionless points) and a scaling $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$6 or nearly constant with size, in contrast to monotonic area scaling.
• Edge orientation and alignment: Flakes with edges parallel to moiré directions may suppress long-period compensation, recovering area-like scaling $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$7 (Yan et al., 2023).

Proper tailoring of twist angle, shape, and edge orientation enables large-scale superlubricity by exploiting these geometric compensation mechanisms.

4. Shape Control of Friction in Soft Robotic Locomotion

Locomotion in soft piezoelectric robotic systems exemplifies active exploitation of shape-induced frictional asymmetry. Coordinated actuation bends a segmented, beam-like body, leading to time-dependent redistribution of contact-normal force. This modulates local Coulomb frictional anchoring at distinct ends of the robot, enabling inchworm-like displacement:

• Analytical modeling: Euler–Bernoulli beam equations with piezo-induced pre-curvatures determine beam deflection $$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$8.
• Contact force and friction: Redistribution of normal forces and consequent frictional stick/slip transitions are computed as functions of actuator voltages and gravity.
• Motion cycle: At each actuation stage, anchorage alternates between ends via controlled shape change; net displacement per cycle analytically matches geometric contraction/extension (Zheng et al., 2021).

Experimental and modeling results confirm quantitative agreement within 20–30%, and highlight the sensitivity of motion and frictional asymmetry to mass and shape parameters.

5. Geometric Mechanics, Motility Maps, and Finsler Metrics in Locomotion

The translation of shape change into system displacement—in viscous or friction-dominated regimes—admits a geometric mechanics formalism where dissipative power defines a Riemannian (symmetric) or more generally a Finsler (asymmetric) metric on generalized velocities:

$$\begin{align*} & -\Delta u + u = f \quad \text{in } \Omega \ & |\partial_n u| \leq g(x) \text{ and } u\,\partial_n u + g(x)\,|u| = 0 \quad \text{on } \Gamma, \end{align*}$$9

$g(x)$0

for asymmetric drag/friction coefficients $g(x)$1 (Hatton et al., 27 Dec 2025).

Subject to sub-Riemannian or sub-Finslerian constraints, the resulting motility map—encoding the mapping from internal shape velocities $g(x)$2 to net body velocities $g(x)$3 via least dissipation—exhibits pronounced friction-induced effects:

• Riemannian (symmetric drag): Net motion to leading order depends on area traversed in the shape space, and vanishes for one-DOF systems.
• Finslerian (asymmetric drag): Linear, orientation-independent advance emerges, admitting net locomotion for even one-DOF non-area-enclosing gaits. The motility map and constraint curvature now feature piecewise-linear, sign-dependent contributions, enabling new forms of controllability and enhanced “geometric efficiency.”
• Controllability and efficiency: Asymmetric friction expands reachable directions and introduces a first-order contribution to net displacement and energetic efficiency (Hatton et al., 27 Dec 2025).

Biological appendages with intrinsic frictional asymmetry, such as scales or claws, and robotic appendages engineered for anisotropic drag, are effectively described within this framework.

6. Friction-Like Effects in Dynamic and Wetting Systems

Frictional resistance in dynamic systems can also bear explicit signatures of shape and contact history. In sessile droplet motion, the depinning and receding of the contact line is governed by an overdamped dynamical equation in which the frictional drag (and possibly static friction threshold) depends on both the current withdrawal rate and the time since initial contact (“state” variable), paralleling the solid-on-solid rate-and-state friction laws (Lindeman et al., 2022):

$g(x)$4

$g(x)$5

This construction captures time- and rate-history dependent frictional hysteresis and stick–slip depinning in a manner directly analogous to solid systems.

7. Broader Implications and Outlook

The effects described above collectively establish that frictional phenomena are inextricably linked to the global and local shape of materials, robots, and interfaces. From mathematical shape optimization in elastostatics (Adly et al., 2024, Bourdin et al., 2024), through geometric friction enhancement in soft matter (Chawla et al., 2024), to nanoscale superlubricity in 2D materials (Yan et al., 2023), and asymmetric-drag optimization in locomotion (Hatton et al., 27 Dec 2025, Zheng et al., 2021), shape dynamics provide both a framework for controlling and predicting friction and a target for engineering novel behaviors.

In higher dimensions and complex geometries, significant challenges remain, such as the tractability of twice epi-differentiability for non-scalar functionals in 3D Tresca-type problems (Bourdin et al., 2024). The ongoing integration of convex analysis, computational geometry, and geometric mechanics is anticipated to further illuminate frictional regularization, exotic scalability, and optimization in shape-driven systems.