On nth order linear strain gradient viscoelasticity

Technical Summary: On nth Order Linear Strain Gradient Viscoelasticity

Problem Statement
While strain gradient elasticity theories (including first-order and general $n$-th order models) have been established for elastic media to capture size effects in nanomaterials and composites, their extension to viscoelasticity remains incomplete. Existing viscoelastic gradient theories [43–45] are typically restricted to a single gradient order (often termed "first-order" in those contexts) and lack a consistent formulation of kinetic energy involving higher-order microinertia tensors. Furthermore, there is no established criterion to determine which specific gradient orders are physically admissible when relating continuum models to discrete microstructures. This paper addresses the gap by formulating a general $n$-th order linear strain-gradient viscoelasticity theory that includes consistent kinetic energy and establishes a rigorous correspondence with the dispersion properties of discrete viscoelastic lattices.

Methodology
The authors develop a continuum theory based on small deformations, introducing a displacement field $u(x,t)$ and its gradients up to order $n$. The methodology proceeds through three main stages:

1. Constitutive Formulation: The stress $T^{(1)}$ and hyperstresses $T^{(k)}$ (for $k=2, \dots, n$) are defined as functionals of the history of strain $\varepsilon$ and strain gradients $K_j = \nabla^j u$. The constitutive equations utilize relaxation tensors $C^{(k)}$ and $C_j^{(k)}$ in a Stieltjes convolution form. A simplified diagonal form is also presented where each hyperstress depends only on its corresponding gradient history.
2. Kinetic Energy and Equations of Motion: A consistent kinetic energy density $K$ is introduced, incorporating mass density $\rho$ and microinertia tensors $J_{jj}$ associated with strain rates up to order $n-1$. This leads to equations of motion of order $2n$, balancing inertial terms (including higher-order spatial derivatives of acceleration) with the divergence of stress and hyperstress tensors.
3. Discrete-to-Continuum Correspondence: To determine admissible gradient orders, the authors analyze a 1D infinite viscoelastic chain (masses connected by Kelvin-Voigt springs). They derive the exact dispersion relation for this discrete system and approximate the non-local sine-squared operator using Padé approximants $P[p, q]$. By comparing the phase and group velocities of the continuum approximations against the discrete model, they identify which Padé orders yield physically stable and causal behavior (finite, non-zero velocities at high wavenumbers).

Key Contributions
The paper introduces three genuinely new elements to the field:

• Arbitrary Order Extension: An extension of strain-gradient viscoelasticity to an arbitrary order $n$, allowing hyperstresses to depend on strain gradients up to that order.
• Consistent Kinetic Energy: The introduction of a kinetic energy formulation consistent with the $n$-th order gradient, including associated microinertia tensors, which were absent in previous viscoelastic gradient works.
• Admissibility Criterion: An explicit criterion for the admissibility of gradient orders derived from lattice dynamics. The analysis reveals that only Padé approximants of the specific order $[4m+2, 4m]$ (where $m$ is an integer) satisfy causality and stability requirements. Specifically, even-order approximations (e.g., $[4,2]$) lead to dynamically unstable modes with exponentially growing solutions, whereas odd-order continuum models (corresponding to $[6,4]$, $[10,8]$, etc.) yield finite, positive phase and group velocities.

Results

• Dispersion Analysis: The study demonstrates that standard Taylor series expansions (Padé order $[p, 0]$) result in infinite phase and group velocities as wavenumber increases, violating causality. In contrast, the specific Padé orders $[4m+2, 4m]$ ensure that velocities remain finite and non-zero.
• Odd-Order Correspondence: The admissible Padé orders correspond exactly to odd-order continuum models. For instance, the minimal admissible order $[6, 4]$ corresponds to a third-order gradient elasticity model.
• Vibration Analysis: As a case study, the authors analyze the forced vibrations of a third-gradient viscoelastic bar (corresponding to the $[6,4]$ approximation). They derive the eigenfrequencies for the elastic limit and solve the forced vibration problem for harmonic excitation. The results show that viscosity effects become more pronounced at higher frequencies and that the third-gradient model produces smoother displacement profiles near clamped boundaries compared to classical models.

Significance and Claims
The authors claim that this work provides the first general $n$-th order formulation of strain-gradient viscoelasticity that is consistently coupled with kinetic energy and constrained by a discrete-to-continuum correspondence. The primary significance lies in the identification of a physical admissibility criterion: only specific odd-order gradient models (derived from specific Padé approximants) are physically realizable when modeling dispersive viscoelastic media derived from nearest-neighbor interactions.

The paper acknowledges limitations, noting that the correspondence is established for a specific nearest-neighbor Kelvin-Voigt chain and that the causality argument was explicitly carried out in the elastic limit ($\eta = 0$), with the extension to the full viscoelastic dispersion relation left for future work. It further states that the constitutive and microinertia tensors introduced are not yet tied to measurable quantities, requiring future experimental identification procedures analogous to those in elastic strain-gradient theories. The work suggests potential applications in modeling surface phenomena in solids and the behavior of lattice-like metamaterials (e.g., origami-kirigami structures) where internal friction and higher-order gradients are relevant.