Sami Mehennaoui

This paper examines a one-dimensional integral constitutive model for uncoupled nonlocal shear elasticity, constructed by superposing a symmetric exponential kernel with a bounded skew-symmetric correction. The integral law is reduced exactly to a second-order ordinary differential equation supplemented by constitutive boundary conditions derived from the kernel itself. Uniqueness of the resulting strain-field boundary-value problem is proved. The symmetric part of the compliance operator is shown to define a strictly positive quadratic form, while the skew-symmetric part contributes zero to the quadratic form evaluated at a fixed stress field. Thermodynamic consistency is established for proportional loading paths, a class that includes the constant-torque benchmark considered here. For general non-proportional loading, the skew-symmetric term contributes a non-vanishing and sign-indefinite component to the internal power, and it is proved that no stored-energy functional of the stress field can reproduce the full constitutive response when the skew coefficient is non-zero. This limitation is stated explicitly, and the scope of the thermodynamic claim is restricted accordingly. The corrected constant-torque solution yields the end-twist ratio F = (1+η)[1 − (L_c/L)(1 − e^(−L/L_c))], where η denotes a symmetric compliance enhancement. In the uncalibrated case η = 0, the model predicts a stiffening of approximately 19.9% for L = 30 nm and L_c = 6 nm. With a symmetric enhancement scaling as η = η₀ L_c/L (η₀ = 2.5), the model yields a softening of about 20.2% while preserving the local limit F → 1 as L_c/L → 0. The symmetric enhancement is presented as a calibration, not as a prediction.

The mathematical modeling of singular shear-strip boundary layers in continuous media represents a profound challenge due to the breakdown of smooth differential bounds and the emergence of localized, non-smooth shear displacements. In this paper, we establish a rigorous, closed-form analytical framework to stabilize coupled transport gradients using a nonlocal Friedrichs commutator mapped within an anisotropic Sobolev functional space. By mapping the coupled transport gradients through a gauge-covariant intrinsic tensor Friedrichs mollifier, we demonstrate how high-frequency ultraviolet interactions can be structurally neutralized. We rigorously prove the exact integral formulation of the commutator and demonstrate that its L^p norm vanishes uniformly in the local limit (ε → 0), provided the velocity field maintains W^{1,∞} regularity. To justify the necessity of this regularity condition, we construct an explicit counterexample using a discontinuous shear flow, proving that when velocity gradients contain singular jumps, the commutator does not vanish but instead generates concentrated singular layers (distributions). To control these layers, we introduce a fourth-order biharmonic hydrodynamic dissipative spectral cage (κΔ²) on a bounded domain with Navier-slip boundary conditions. We rigorously establish the energy identity, prove the exponential decay of the energy, and utilize Galerkin approximations and the Aubin-Lions compactness lemma to prove the global existence and uniqueness of weak solutions. This framework provides a mathematically robust foundation for understanding boundary layer singularities, analytically isolating the anomalous dissipation profile without relying on ad-hoc empirical truncations.