Discrete mass-spring structure identification in nonlocal continuum space-fractional model

Abstract

This paper considers discrete mass-spring structure identification in a nonlocal continuum spacefractional model, defined as an optimization task. Dynamic (eigenvalues and eigenvectors) and static (displacement field) solutions to discrete and continuum theories are major constituents of the objective function. It is assumed that the masses in both descriptions are equal (and constant), whereas the spring stiffness distribution in a discrete system becomes a crucial unknown. The considerations include a variety of configurations of the nonlocal parameter and the order of the fractional model, which makes the study comprehensive, and for the first time provides insight into the possible properties (geometric and mechanical) of a discrete structure homogenized by a space-fractional formulation.Recently, the application of fractional calculus (FC) in mechanics for describing scale effects (in a broad sense [1]) has gained considerable attention; such concepts are called fractional mechanics (FM). FM formulations exist that describe the following materials: polymers [2], granular soils [3-7], silts [8], granites [9], biological tissues [1,10], metals [6,11], and silicon [12]. FC introduces a continuum-type description through the definition of the fractional derivative (FD) (derivative of an arbitrary order [13-16]), revealing certain hidden aspects of the discrete nature of the phenomena analysed. Furthermore, FC has an indisputable attribute, namely, that FC provides an infinite number of possible FD realizations [17,18]; thus we can identify a specific type of FD depending on the experimental evidence.An important family of FM concepts includes formulations where, in classic continuum mechanics (CCM) theory, the integer differential operators acting on a spatial variable are replaced by fractional operators; such models are called space-fractional continuum mechanics (s-FCM) [19][20][21][22][23][24][25]. The physical interpretation of s-FCM is that FD homogenizes, in a phenomenological sense, the underlying microstructure [26]; thus, the scale effect is included within FC, and the s-FCM model is nonlocal. Nevertheless, to the best of our knowledge no one has answered the following question: Is there a discrete structure that is homogenized by s-FCM?.In this paper, we provide the answer to the above question by identifying a discrete mass-spring structure, whose behaviour (static and dynamic) is mapped with high precision by an s-FCM approach. Such a concept is a kind of inverse problem to recent trends in physics/mechanics on the study of continualization techniques, such as the Taylor series-based method or the shift operator expansion method [27][28][29]. The overall problem is defined as a 1D optimization task with several variables (altogether approximately 2.5 million analyses were conducted). Furthermore, the considerations include a variety of configurations of the non-local parameter and the order of the fractional model (altogether twenty discrete spring-mass structures were identified), which makes the s...

Designing of Dynamic Spectrum Shifting in Terms of Non-Local Space-Fractional Mechanics

“…This is positive and moreover, such an answer is true in terms of the static and dynamic answer-in the last case, both in terms of eigenvalues and eigenvectors; such results have been never presented before for competitive theories. This paper extends the results presented in [32] to show the applicability of sFCM to design 1D material bodies with a specific dynamic eigenvalue spectrum. Such a formulated problem is based on the proper spatial distribution of material length scale which maps the information about the underlying microstructure (it is important that the material length scale is one of two additional material parameters of sFCM compared to the classical local continuum mechanics-the second one, the order of fractional continua-is treated herein as a scaling parameter only).…”

“…Herein, it is important, because the designing procedure to obtain the specific dynamic properties of the analysed material body is the crux of presented considerations, that the applicability of sFCM was proved based on validation with rigorous experiments [30,31]. Furthermore, as presented in [32] the answer to the question: Is there a discrete structure that is homogenized by sFCM? This is positive and moreover, such an answer is true in terms of the static and dynamic answer-in the last case, both in terms of eigenvalues and eigenvectors; such results have been never presented before for competitive theories.…”

Formulation and experimental validation of space-fractional Timoshenko beam model with functionally graded materials effects

“…The concept of variable length scale [36] f = f (x), as function decreasing at the boundaries, has been kept. These two parameters α and f are regarded as associated with microstructure [37] and responsible for SE mapping.…”

Dynamics of Space-Fractional Euler–Bernoulli and Timoshenko Beams

“…The concept of variable length scale , as function decreasing at the boundaries, has been kept [ 39 ]. These two parameters— and —are regarded as being associated with microstructure [ 40 ] and responsible for scale effect mapping.…”

“…In models defined within the framework of space-fractional mechanics, the non-locality parameters and are considered to be related to the microstructure of the material (for a discussion on the association of microstructure with the non-locality parameters see in [ 40 ]). Consequently, these parameters are considered as constant and independent of the body geometry or the analysis performed (static or dynamic).…”