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**A mixed two-phase stress/strain driven elasticity: in applications on static bending, vibration analysis and wave propagation.**
*(English)*
Zbl 1493.74012

Summary: In recent years, there are numerous papers that deal with two-phase strain-driven model, including softening effect, and two-phase stress-driven model, with stiffening effect, and also the nonlocal strain gradient model, with both the softening and stiffening effects, to seek the impacts of length scales on the mechanics of structures in small scales. The current paper is a novel well-posed mixture model to cover all previous theories through employing various proposed small scales. Concurrent accounting of stiffening and softening effects is performed through employing a combination of the two-phase strain-driven elasticity with the two-phase stress-driven elasticity, with two additional local phase fraction factors and two various type of nonlocal parameters. Accordingly, in this model the total strain or stress at a certain point is regarded as a function of the stress or strain of all neighboring points in addition to the point of interest. Compatibility between integral and differential relations without any conflict of restrictions is doable through considering constitutive boundary conditions as key point of this match. To demonstrate its application values, the proposed mixture model as an efficient theoretical tool is hired for static bending, vibration and wave propagation in a mixed two-phase stress/strain driven elasticity system and the new essential relations are developed through examples for static bending, free vibration and wave propagating in Euler-Bernoulli nanobeams. The results are obtained by proposing an efficient exact solution, and the integrity as well as reliability of the present constitutive differential relations are evaluated through some validation studies. The output results in the framework of new suggested mixture model address some points about static bending, free vibration and wave propagation that is more comprehensive than the previous results of the contemporary continuum theories. Thus, based on the observations, this mixed two-phase stress/strain driven elasticity could be cover a widespread range of dynamic response in the nano-systems that the contemporary continuum theories can be only investigated some aspects of those problems.

### MSC:

74B99 | Elastic materials |

74A20 | Theory of constitutive functions in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74H45 | Vibrations in dynamical problems in solid mechanics |

74J05 | Linear waves in solid mechanics |

74M25 | Micromechanics of solids |

### Keywords:

two-phase stress-driven local/nonlocal elasticity; two-phase strain-driven local/nonlocal elasticity; integral/differential constitutive law; Euler-Bernoulli nanobeam; harmonic wave
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\textit{S. Behdad} and \textit{M. Arefi}, Eur. J. Mech., A, Solids 94, Article ID 104558, 15 p. (2022; Zbl 1493.74012)

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### References:

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