Researchers Formulate Eight Three-Dimensional Non-Local Stress-Gradient Models for Materials
Physicists have formulated eight three-dimensional stress-gradient non-local models by generalizing the classical one-dimensional Eringen model using vector and operator methods. The models are classified into scalar-type and tensor-type categories, with compatibility conditions, non-locality kernels, and isotropy properties derived for each. The work expands the mathematical toolkit for describing materials whose mechanical response at a point depends on stress states across a surrounding region, with implications for nanoscale and advanced material modeling.
A new preprint submitted to arXiv extends Eringen's well-known one-dimensional stress-gradient non-local elasticity model into three dimensions by replacing the scalar non-locality parameter and second-order derivative with a non-locality vector and the nabla operator. This generalization yields eight distinct three-dimensional models, grouped into three scalar-type and five tensor-type variants depending on how the non-locality operator is constructed from vector products. For the tensor-type models, compatibility conditions ensuring the symmetry of the Cauchy stress tensor are explicitly derived. Using Fourier integral transforms with respect to spatial coordinates, the authors obtain non-locality kernels — effectively Green's functions — for each model, characterizing how non-local interactions decay or propagate through the material. Isotropy analysis reveals that all scalar-type models are isotropic in the classical sense, though two exhibit non-local anisotropy, while all tensor-type models are classically anisotropic, with two being non-locally isotropic and three displaying directional non-locality. All models except one scalar-type variant incorporate both local and non-local contributions to the Cauchy stress tensor. The framework provides a systematic classification that could guide future applications in continuum mechanics of micro- and nanoscale structures where long-range interatomic interactions are significant.
What's missing
As a preprint, this work has not yet undergone formal peer review. The paper does not appear to include numerical examples, benchmark comparisons with experimental data, or validation against known non-local material behavior, leaving the physical predictive accuracy of the eight models untested. It is also unclear whether the proposed models satisfy thermodynamic consistency conditions or reduce correctly to established limiting cases beyond the one-dimensional Eringen model.
What different sources said
- arXiv physicsCenter
Formulation of stress-gradient models describing three-dimensional non-local medium
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