Nonlocal stress gradient formulation for damping vibration analysis of viscoelastic microbeam in thermal environment

doi:10.1007/s10483-023-2981-7

摘要/Abstract

摘要： An integral nonlocal stress gradient viscoelastic model is proposed on the basis of the integral nonlocal stress gradient model and the standard viscoelastic model, and is utilized to investigate the free damping vibration analysis of the viscoelastic Bernoulli-Euler microbeams in thermal environment. Hamilton's principle is used to derive the differential governing equations and corresponding boundary conditions. The integral relations between the strain and the nonlocal stress are converted into a differential form with constitutive constraints. The size-dependent axial thermal stress due to the variation of the environmental temperature is derived explicitly. The Laplace transformation is utilized to obtain the explicit expression for the bending deflection and moment. Considering the boundary conditions and constitutive constraints, one can get a nonlinear equation with complex coefficients, from which the complex characteristic frequency can be determined. A two-step numerical method is proposed to solve the elastic vibration frequency and the damping ratio. The effects of length scale parameters, viscous coefficient, thermal stress, vibration order on the vibration frequencies, and critical viscous coefficient are investigated numerically for the viscoelastic Bernoulli-Euler microbeams under different boundary conditions.

关键词: damping vibration, size effect, integral nonlocal stress gradient model, standard viscoelastic model, Laplace transformation

Abstract: An integral nonlocal stress gradient viscoelastic model is proposed on the basis of the integral nonlocal stress gradient model and the standard viscoelastic model, and is utilized to investigate the free damping vibration analysis of the viscoelastic Bernoulli-Euler microbeams in thermal environment. Hamilton's principle is used to derive the differential governing equations and corresponding boundary conditions. The integral relations between the strain and the nonlocal stress are converted into a differential form with constitutive constraints. The size-dependent axial thermal stress due to the variation of the environmental temperature is derived explicitly. The Laplace transformation is utilized to obtain the explicit expression for the bending deflection and moment. Considering the boundary conditions and constitutive constraints, one can get a nonlinear equation with complex coefficients, from which the complex characteristic frequency can be determined. A two-step numerical method is proposed to solve the elastic vibration frequency and the damping ratio. The effects of length scale parameters, viscous coefficient, thermal stress, vibration order on the vibration frequencies, and critical viscous coefficient are investigated numerically for the viscoelastic Bernoulli-Euler microbeams under different boundary conditions.

Key words: damping vibration, size effect, integral nonlocal stress gradient model, standard viscoelastic model, Laplace transformation

中图分类号:

45J05

Hai QING, Huidiao SONG. Nonlocal stress gradient formulation for damping vibration analysis of viscoelastic microbeam in thermal environment[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(5): 773-786.

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