Stress-Driven Two-Phase Nonlocal Model for Nanobars Embedded in Elastic Substrates with Surface Effects

Authors

• Suchart Limkatanyu, Department of Civil and Environmental Engineering, Faculty of Engineering, Prince of Songkla University, Songkhla, 90110, Thailand
• Worathep Sae-Long, Civil Engineering Program, School of Engineering, University of Phayao, Phayao, 56000, ThailandFollow
• Piti Sukontasukkul, Construction and Building Materials Research Center, Department of Civil Engineering, King Mongkut's University of Technology North Bangkok, Bangkok, 10800, Thailand
• Amorn Pimanmas, Department of Civil Engineering, Kasetsart University, Bangkok, 10900, Thailand
• Preeda Chaimahawan, Civil Engineering Program, School of Engineering, University of Phayao, Phayao, 56000, Thailand

Corresponding Author

Worathep Sae-Long

Authors ORCID

Suchart Limkatanyu: https://orcid.org/0000-0002-4343-7654

Worathep Sae-Long: https://orcid.org/0000-0001-8149-4409

Piti Sukontasukkul: https://orcid.org/0000-0002-9580-7063

Amorn Pimanmas: https://orcid.org/0000-0001-9543-2700

Preeda Chaimahawan: https://orcid.org/0000-0002-3515-0506

Document Type

Article

Keywords

Local/nonlocal mixture, Surface-energy effect, Small-scale effect, Stress-driven two-phase mixture of elasticity, Winkler model

Abstract

This paper develops a nonlocal bar–substrate medium model to study the axial (longitudinal) response of nanobars embedded in an elastic foundation. Small-scale behavior is represented through a two-phase mixture, stress-driven, nonlocal integral formulation. To reflect the surrounding medium and surface-related size effects, the Winkler foundation and the Gurtin–Murdoch surface elasticity framework are incorporated, respectively. The local bar–substrate governing differential equilibrium equation (GDEE) and associated boundary conditions (BCs) are first obtained via the virtual displacement principle, and the resulting local formulation is then generalized to the nonlocal setting using the adopted two-phase mixture model. The proposed framework is assessed through three numerical studies: (i) resolving inconsistent predictions previously reported for related nonlocal bar models, (ii) quantifying the role of bar–substrate coupling on axial displacement, and (iii) investigating how key parameters influence the effective Young's modulus (EYM). The results show that the proposed formulation resolves inconsistencies in classical nonlocal models observed in the Eringen differential nonlocal bar model under end loading. Increasing the material length-scale parameter and decreasing the mixture parameter produce a stiffer axial response, while a stiffer substrate further amplifies the size-dependent behavior. The EYM increases as the nanobar diameter decreases and is influenced by the nonlocal parameters, surface elasticity, and substrate restraint.

How to Cite This Article

Limkatanyu, Suchart; Sae-Long, Worathep; Sukontasukkul, Piti; Pimanmas, Amorn; and Chaimahawan, Preeda (2026) "Stress-Driven Two-Phase Nonlocal Model for Nanobars Embedded in Elastic Substrates with Surface Effects," Babylonian Journal of Mechanical Engineering: Vol. 4: Iss. 1, Article 3.