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Background <p>Classical elasticity theory becomes inadequate when applied to micro‑ and nanoscale structures, where small‑scale effects significantly influence the mechanical response. In this context, refined continuum theories are required to accurately describe the vibrational behavior of nanostructures. The strain gradient framework provides a suitable approach to incorporate these size‑dependent effects.</p> Methods <p>This study develops a new analytical solution for the radial vibration of an anisotropic nanosphere immersed in an inviscid fluid, using strain gradient theory. The analysis focuses on nanospheres exhibiting a cubic crystalline structure. A numerical example is constructed to evaluate the influence of small‑scale effects on the radial vibrational response and to assess the stability of the strain gradient formulation.</p> Results <p>Numerical simulations reveal that small‑scale effects substantially modify the radial vibration characteristics of the nanosphere. The key result highlights an instability inherent to the strain gradient formulation. An instability criterion is established to clarify the vibrational behavior predicted by the model. To address this issue, a straightforward mathematical reformulation of the constitutive equation is proposed. Additionally, the study shows that acoustic quasi‑impedance plays a significant role in shaping the vibrational response, bridging the limiting cases of a nanosphere in vacuum and in an inviscid incompressible fluid.</p> Conclusions <p>The findings demonstrate that strain gradient model is essential for accurately modeling the radial vibration of anisotropic nanospheres, as classical elasticity fails to capture key small‑scale effects. The identified instability and the proposed criterion provide valuable insight into the limitations and applicability of the strain gradient model. The influence of acoustic quasi‑impedance further emphasizes the need to account for fluid-structure interaction when analyzing nanoscale vibrational phenomena.</p>

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An Alternative Based on Nonlocal Elasticity to Avoid the Instability of Gradient Elasticity : Application to the Vibration of an Immersed Anisotropic Nanosphere

  • 期刊论文

  • 发表日期：

    2026年6月24日

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  • Adil El Baroudi

摘要

Background

Classical elasticity theory becomes inadequate when applied to micro‑ and nanoscale structures, where small‑scale effects significantly influence the mechanical response. In this context, refined continuum theories are required to accurately describe the vibrational behavior of nanostructures. The strain gradient framework provides a suitable approach to incorporate these size‑dependent effects.

Methods

This study develops a new analytical solution for the radial vibration of an anisotropic nanosphere immersed in an inviscid fluid, using strain gradient theory. The analysis focuses on nanospheres exhibiting a cubic crystalline structure. A numerical example is constructed to evaluate the influence of small‑scale effects on the radial vibrational response and to assess the stability of the strain gradient formulation.

Results

Numerical simulations reveal that small‑scale effects substantially modify the radial vibration characteristics of the nanosphere. The key result highlights an instability inherent to the strain gradient formulation. An instability criterion is established to clarify the vibrational behavior predicted by the model. To address this issue, a straightforward mathematical reformulation of the constitutive equation is proposed. Additionally, the study shows that acoustic quasi‑impedance plays a significant role in shaping the vibrational response, bridging the limiting cases of a nanosphere in vacuum and in an inviscid incompressible fluid.

Conclusions

The findings demonstrate that strain gradient model is essential for accurately modeling the radial vibration of anisotropic nanospheres, as classical elasticity fails to capture key small‑scale effects. The identified instability and the proposed criterion provide valuable insight into the limitations and applicability of the strain gradient model. The influence of acoustic quasi‑impedance further emphasizes the need to account for fluid-structure interaction when analyzing nanoscale vibrational phenomena.