Spherical Lame-Type Problem in Second Strain Gradient Theory

Document Type : Research Article

Author

Engineering Department, Alzahra University, Tehran, Iran

Abstract

Second strain gradient theory is employed to examine the spherical single/double-phase Lame-type problem. Due to the capability of strain gradient theory to capture the effects of the surface, size, and discrete nature of materials, the pertinent relaxed configuration is sought. The theory is written in the spherical coordinate system and the equilibrium equations, stress/strain components, constitutive relations, and tractions are derived. The relaxed configuration is obtained for both the diamond carbon and carbon-coated crystalline silicon shell. Afterwards, the external symmetric loading is applied to the relaxed configuration to analyze the mechanical response. The elastic material parameters are calculated via the quantum computations, lattice dynamics, and material continuum description. The analysis shows that the mechanical response in the augmented theory is significantly different from that in the classical elasticity. For example, in the single-phase problem with an inner and outer radius equal to two and ten lattice parameter, respectively, under a normalized external pressure of about 0.0001, the classic elasticity predicts an approximately constant normalized radial stress of about -0.0001 in the nanoshell. However, in the framework of strain gradient theory, the normalized radial stress is varying from about -0.001 and -0.0002 in the vicinity of the inner and outer boundaries, respectively, to about 0.0003 in the middle of the hollow nanoshell. With increasing the inner radius, the difference between the two results in the middle points decreases.

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References

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Amirkabir Journal of Mechanical Engineering
Volume 53, Issue 2
May 2021
Pages 709-728

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