Open Access

REVIEW

Beyond Classical Elasticity: A Review of Strain Gradient Theories, Emphasizing Computer Modeling, Physical Interpretations, and Multifunctional Applications

Shubham Desai, Sai Sidhardh^*

SMART Lab, Department of Mechanical and Aerospace Engineering, IIT Hyderabad, Kandi, 502285, Telangana, India

* Corresponding Author: Sai Sidhardh. Email:

Computer Modeling in Engineering & Sciences 2025, 144(2), 1271-1334. https://doi.org/10.32604/cmes.2025.068141

Received 21 May 2025; Accepted 04 August 2025; Issue published 31 August 2025

Abstract

The increasing integration of small-scale structures in engineering, particularly in Micro-Electro-Mechanical Systems (MEMS), necessitates advanced modeling approaches to accurately capture their complex mechanical behavior. Classical continuum theories are inadequate at micro- and nanoscales, particularly concerning size effects, singularities, and phenomena like strain softening or phase transitions. This limitation follows from their lack of intrinsic length scale parameters, crucial for representing microstructural features. Theoretical and experimental findings emphasize the critical role of these parameters on small scales. This review thoroughly examines various strain gradient elasticity (SGE) theories commonly employed in literature to capture these size-dependent effects on the elastic response. Given the complexity arising from numerous SGE frameworks available in the literature, including first- and second-order gradient theories, we conduct a comprehensive and comparative analysis of common SGE models. This analysis highlights their unique physical interpretations and compares their effectiveness in modeling the size-dependent behavior of low-dimensional structures. A brief discussion on estimating additional material constants, such as intrinsic length scales, is also included to improve the practical relevance of SGE. Following this theoretical treatment, the review covers analytical and numerical methods for solving the associated higher-order governing differential equations. Finally, we present a detailed overview of strain gradient applications in multiscale and multiphysics response of solids. Interesting research on exploring the relevance of SGE for reduced-order modeling of complex macrostructures, a universal multiphysics coupling in low-dimensional structures without being restricted to limited material symmetries (as in the case of microstructures), is also presented here for interested readers. Finally, we briefly discuss alternative nonlocal elasticity approaches (integral and integro-differential) for incorporating size effects, and conclude with some potential areas for future research on strain gradients. This review aims to provide a clear understanding of strain gradient theories and their broad applicability beyond classical elasticity.

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