Space-fractional Euler-Bernoulli beam model - Theory and identification for silver nanobeam bending
Introduction
It is well recognised in the existing literature that the majority, if not all, natural or human made materials at small scales exhibit behaviours which can be significantly different from those observed at the levels of their bulks [1], [2]. This was experimentally proved for, among others, micro-sized copper [3], [4], gradient nanostructures [5], [6], [7], honeycombs [8], composites [9], lattice structures [10], [11], bones [12], [13], nanocrystalline MoS2 [14], and silver [15]. In consequence, engineering applications of those materials result in bodies which manifest the so-called size effect. Examples of such bodies are oscillators [16], [17], plates [18], rotating discs [19], pipes conveying fluid [20], nanorings [21], or piezoelectric nanoplates [22].
An important class of engineering structures, exhibiting a strong size effect are beams identified in the seminal works of Leonhard Euler and Daniel Bernoulli around 1750 beams. In beams, two spatial dimensions are considerably smaller than the third one. Since the 1750s many theories have been proposed to model such structural elements, including those which are based on different generalisations of classical (local) mechanics e.g. [23], [24], [25], [26], [27], [28] (for review of selected theories see [29]). Importantly, since seminal works of [30], [31], some commonly used formulations (e.g. pure nonlocal strain-driven elasticity) were found ill-posed [32] - in contrast to well-posed ones like stress-driven nonlocal model [31], the two-phase local/nonlocal strain driven elasticity [33], or the two-phase local/nonlocal stress-driven elasticity [34]. Having in mind the constant needs of miniaturisation in industry 4.0, constant development of such theories, often utilising new mathematical concepts [35], [36], [37], is necessary.
This paper is a development of the previous concept of space-Fractional Euler-Bernoulli beam theory (s-FEBB) [38], where generalisation of classical Euler-Bernoulli beam theory was introduced utilising fractional calculus [39], [40], [41]. The current formulation, in contrast to other fractional beam models [42], [43], [44], [45], [46], [47], introduces:
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New kinematics with a fractional definition of cross-section rotations,
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A variable length scale on beam length,
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Arbitrary boundary conditions,
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Arbitrary transverse load conditions,
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A new numerical algorithm.
Furthermore, the considerations include an extensive parametric study of the influence of the boundary conditions, the order of fractional continua, the size and the distribution of the length scale on the beam deflection. The model is identified and validated based on experimental data for the silver nanobeam bending test.
The paper is structured as follows. Section 2 deals with the s-FEBB definition. Section 3 is devoted to the numerical scheme and parametric study. Section 4 provides experimental validation and Section 5 concludes the paper.
Section snippets
Fractional elasticity
The presented study considers static bending behaviour of the beam based on small strain fractional elasticity theory proposed in [48], [49]. This concept of elasticity is governed by the equationswhere σ is the Cauchy stress tensor, b is the body force, is the small fractional strain tensor (for physical interpretation see [50]), is the stiffness tensor, n is the outward unit
Discretization
Due to the non-existence of the analytical solution of Eq. (20), the equation was solved numerically, using the finite difference method and trapezoidal rule [56], [57], [58] to approximate Caputo fractional derivatives (for a detailed discussion of the role of the applied numerical procedure together with the applied discretisation level see [58]). For this reason, a homogeneous grid of points with fictitious nodes and was assigned to the beam (Fig. 1).
The
Experimental validation
The proposed model was validated based on the silver nanobeam bending experiment presented in [15]. The nanobeams were geometrically determined by length L, circular cross-section with diameter d, and were loaded at or near the midpoint by a point load P. The static scheme was the fixed beam for cases a) and b) and the simply supported beam for cases c) and d) (see Fig. 6). It was found that GPa, and . All data of the considered nanowires are listed in Table 3.
Fig. 6 shows
Conclusions
This paper develops the space-Fractional Euler-Bernoulli beam theory by introducing new kinematics and a variable length scale. Moreover it provides a numerical algorithm to solve the bending problem. Static bending behaviour of different beam types has been analyzed, and the influence of the order of fractional derivative and the size and distribution of the length scale on beam deflection has been discussed. Finally, the model has been experimentally validated. The obtained results lead to
CRediT authorship contribution statement
P. Stempin: Formal analysis, Investigation, Software, Validation, Writing - original draft, Writing - review & editing, Visualization. W. Sumelka: Conceptualization, Funding acquisition, Methodology, Formal analysis, Investigation, Supervision, Writing - original draft, Writing - review & editing.
Acknowledgement
This work is supported by the National Science Centre, Poland under Grant No. 2017/27/B/ST8/00351.
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