Strain gradient viscoelasticity - a possible spatiotemporal framework

A introduction of the strain gradient viscoelasticity and its application

Bridging time and scale: strain gradient viscoelasticity

In the world of advanced materials, from nanostructured alloys to living biological tissues, mechanical behavior is often a complex dance between where things are and how long they have been there. My research focuses on a phenomenon where microstructural size effects and time-dependent dissipative processes are inextricably linked.

The problem: why classical theories fall short

Classical continuum mechanics has been remarkably successful for macroscopic engineering, but it relies on a “scale-invariant” assumption. As we move to the micro- and nano-scales, materials often exhibit a “smaller is stiffer” effect that classical theories cannot explain because they lack intrinsic length scales. Furthermore, advanced materials frequently show viscous behavior, such as creep or stress relaxation, where their properties evolve over time.

The solution: strain gradient viscoelasticity (SGV)

To bridge this gap, we proposed a Strain Gradient Viscoelasticity (SGV) framework (Lin & Wei, 2020)(Lin et al., 2026). This higher-order continuum theory provides a unified “spatiotemporal” link between spatial gradients and temporal evolution. The core innovation of this theory lies in how it treats the internal length scale. In traditional gradient theories, this is a fixed constant. In our work, the gradient parameter c(t) is time-dependent, allowing the model to capture how the “size effect” itself evolves during deformation. This was first established through a rigorous correspondence principle in the Laplace phase space and later expanded into a full thermoviscoelastic framework that ensures energy conservation and physical consistency.

Spatiotemporal effect captured by the strain gradient viscoelasticity (Lin et al., 2026)

Numerical mastery: from finite elements to AI

Solving the higher-order equations that govern SGV is a significant mathematical challenge. To address this, our research developed two distinct numerical engines (Lin et al., 2026):

• Mixed-Field Finite Element Method (Mixed-FEM): This approach overcomes the stringent “C1 continuity” requirements of high-order derivatives by introducing additional degrees of freedom, creating a robust path for complex structural analysis.
• Physics-Informed Neural Networks (PINNs): We pioneered a variational PINN framework that uses the power of deep learning to handle higher-order derivatives naturally. Interestingly, we found that the inherent oscillations during the AI’s training process act as a “natural perturbation” to capture structural instabilities.

Schematic of the physics-informed neural network for strain gradient viscoelasticity (Lin et al., 2026)

Possible application: from Engineering to life

The power of SGV is its universality. We have demonstrated its ability to resolve the multiscale viscoelasticity of biological tissues, such as predicting the distinct relaxation times of cytoplasm versus cell-to-cell junctions in embryo development. Beyond biology, this framework offers a way to predict the long-term performance of thermal barrier coatings and other engineered nanomaterials operating under extreme conditions.

Spatiotemporal effect of biological tissue and the characterization of strain gradient viscoelasticity (Lin et al., 2026)

By constructing spatiotemporal phase portraits, we can now visualize how microstructural features and viscosity fundamentally modulate mechanical properties over time. This research provides the theoretical and computational foundation for a new generation of “active” materials that remember their history and adapt to their scale

References

• [lin2026spatiotemporal] A spatiotemporal framework for strain gradient viscoelasticity: Thermodynamic derivation and numerical implementation. International Journal of Solids and Structures (2026). DOI: 10.1016/j.ijsolstr.2026.113989
• [lin2020strain] A strain gradient linear viscoelasticity theory. International Journal of Solids and Structures (2020). DOI: 10.1016/j.ijsolstr.2020.08.008