Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory

TLDR: In this paper, the in-plane free vibrations (axial and bending) of a Bernoulli-Euler nanobeam using the mixed local/non-local Eringen elasticity theory is studied.

About: This article is published in International Journal of Engineering Science.The article was published on 2017-10-01 and is currently open access. It has received 149 citations till now. The article focuses on the topics: Dynamic problem & Elasticity (physics).

Figures

Figure 6: Bending vibration. First four natural frequencies of the free beam as a function of the mixture parameter ξ1, for four different values of the nonlocal parameter h. The frequency ωwn has been normalized by the frequency ωlocalwn corresponding to the local case (ξ1 = 1).

Table 2: Nondimensional eigenfrequencies (n = 1, ..., 4) corresponding to bending vibration for different boundary conditions. Fully local case (ξ1 = 1).

Figure 1: Axial vibration. First four natural frequencies of the supported beam as a function of the mixture parameter ξ1, for four different values of the nonlocal parameter h. The frequency ωun has been normalized by the frequency ωlocalun corresponding to the local case (ξ1 = 1).

Figure 4: Bending vibration. First four natural frequencies of the supported beam, as a function of the mixture parameter ξ1, for four different values of the nonlocal parameter h. The frequency ωwn has been normalized by the frequency ωlocalwn corresponding to the local case (ξ1 = 1).

Figure 2: Axial vibration. First four natural frequencies of the cantilever beam as a function of the mixture parameter ξ1, for four different values of the nonlocal parameter h. The frequency ωun has been normalized by the frequency ωlocalun corresponding to the local case (ξ1 = 1).

Table 1: Nondimensional eigenfrequencies (n = 1, ..., 4) corresponding to axial vibration for different boundary conditions. Fully local case (ξ1 = 1).

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Vibrations of bernoulli-euler beams using the two-phase nonlocal elasticity theory" ?

In this work the problem of the in-plane free vibrations ( axial and bending ) of a Bernoulli-Euler nanobeam using the mixed local/nonlocal Eringen elasticity theory is studied. To the best knowledge of the authors, this is the first time an exact solution is presented for a dynamic problem involving structures with constitutive equations corresponding to nonlocal integral Eringen ’ s elasticity. 

Q2. What future works have the authors mentioned in the paper "Vibrations of bernoulli-euler beams using the two-phase nonlocal elasticity theory" ?

In this paper the authors presented an exact solution for the free in-plane vibrations ( axial and bending ) of a Bernoulli-Euler beam using the mixed local/nonlocal constitutive equations related to the Eringen elasticity theory. The method has been applied to study the in-plane free vibration of a Bernoulli-Euler beam with different classical boundary conditions: supported, cantilever and free. 

Q3. What is the eigenvalue problem for axial and bending vibrations?

For the case of axial vibrations the integro-differential eigenvalue problem was transformed to a fourth-order differential equation with four boundary conditions: two of them correspond to the classical ones (one for each end), while the other5 two come from the transformation process. 

Q4. What is the reason why the study of nonlocal elasticity theories has soared?

The explosive growth of the nanotechnology and of the applications in the field of nanostructures has soared the studies related to nonlocal elasticity theories, among other generalized continuum mechanics approaches. 

Q5. What is the case of the nonlocal elasticity theory?

The case ξ2 = 0 corresponds to the pure local elasticity approach, while ξ1 = 0 deals with the fully nonlocal integral elasticity formulation. 

Q6. what is the kinematics of the Bernoulli-Euler beam?

Using the kinematics of the Bernoulli-Euler beam, the authors have:Ux(x, y, z, t) = u(x, t)− z∂xw(x, t); Uy(x, y, z, t) = 0; Uz(x, y, z, t) = w(x, t) (2)where u and w represent, respectively, the axial and transverse displacements of the crosssection’s centroid. 

Q7. What is the main purpose of this paper?

In this paper the authors formulate and analytically solve the problem of the free in-plane (axial and bending) vibrations of a beam using the mixed local/nonlocal Eringen elasticity theory. 

Q8. What are the parameters of the nonlocal elasticity theory?

The parameters ξ1 and ξ2 represent the volumen fraction of material complying with local and nonlocal integral elasticity, respectively. 

Q9. Why are nonlocal elasticity theories so attractive?

The reasons are: (i) the clasical elasticity is a scale-free theory which cannot adequately address the size effect commonly present in nanotecnology applications, and (ii) they are an attractive alternative to the huge computational cost of the Molecular Dynamic techniques. 

Q10. What are the eigenfrequencies for the axial and bending vibrations?

Figs. 1 to 3 show, for each boundary condition, the first four eigenfrequencies ωun , n = 1, . . . , 4, as a function of the mixture parameter ξ1, and of the nonlocal parameter h. 

Q11. What is the eigenvalue of the Euler-Bernoulli beams?

The vibrational behaviour of Euler-Bernoulli beams involving the two-phase Eringen nonlocality has been studied by Eptaimeros et al. (2016), using a FEM approach to obtain the eigenfrequencies for different boundary conditions. 

Q12. What is the influence of the material parameters in the vibrational behavior?

The influence of the material parameters in the vibrational behavior has been analysed by considering four values of the nonlocal parameter h = {0.010, 0.025, 0.050, 0.075}, and ranging the mixture parameter ξ1 from 0.1 to 1.0. 

Q13. What is the main reason why Eringen and Zhou proposed the nonlocal approach?

The nonlocal approach enabled different authors (Eringen, 1977; Eringen et al., 1977; Zhou et al., 1999) to address problems related with stress singularities which arise in classical fracture mechanics formulations, showing that these disappear using the nonlocal treatment.