Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach

Area of Science:

• Solid Mechanics
• Materials Science
• Nanotechnology

Background:

• Traditional models often neglect small-scale and surface-energy effects in bar-elastic substrate analysis.
• Understanding material nonlocality and surface phenomena is crucial for accurately modeling nanoscale structures.
• Existing displacement-based models may not fully capture the complex interactions in bar-substrate systems.

Purpose of the Study:

• To develop a rational bar-elastic substrate model that integrates small-scale and surface-energy effects.
• To establish a robust theoretical framework using the virtual force principle for governing equations and boundary conditions.
• To demonstrate the model's efficacy and advantages over displacement models using a nanowire-in-elastic-substrate example.

Main Methods:

• Utilized a thermodynamics-based strain gradient model to incorporate small-scale effects (material nonlocality).
• Employed Gurtin-Murdoch surface theory to account for surface-energy effects.
• Incorporated the Winkler foundation model to represent the bar-surrounding substrate interaction.
• Applied the virtual force principle to derive the governing differential compatibility equation and boundary conditions.

Main Results:

• Derived the governing differential compatibility equation and consistent end-boundary compatibility conditions.
• Identified the axial force field as the fundamental solution to the governing equation.
• The proposed model demonstrated accuracy and advantages over displacement models in the nanowire example.
• Thoroughly analyzed the influence of material nonlocality on both global and local responses.

Conclusions:

• The developed bar-elastic substrate model provides a more comprehensive approach by including small-scale and surface-energy effects.
• The virtual force principle offers a powerful framework for formulating such advanced mechanical models.
• The model's application to nanowires highlights its potential for analyzing nanoscale structures and understanding nonlocal effects.