Identification of mechanical properties of 1D deteriorated nonlocal bodies
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Abstract
Deterioration, understood as a change of mechanical properties of devices exhibiting a strong scale effect, is a phenomenon of great importance in modern industry. This effect is simultaneously an open question in theoretical mechanics, due to the fact that none of existing nonlocal theories (which are necessary to describe the scale effect) seem to be universal  thus constant development of new mathematical models is necessary. This paper presents the study of the above emphasised problem in the framework of fractional continuum mechanics, as an optimization (identification) task. In the identification routine, the steering parameters are: density of material, order of material, and material length scale.
Keywords
Optimization Nonlocal models Fractional calculus Deterioration1 Introduction
It was in 1926 when Werner Heisenberg said “... it is the theory which first determines what can be observed ...” (Heisenberg 1989). This fundamental statement was related to atomic physics at that time, nevertheless it also holds nowadays and even for higher scales of observation, e.g. micro, meso, macro. This is because of high maturity of computer aided decision making in many branches of human activity, which causes that ‘virtually’ obtained results give in most cases more comprehensive insight into an analysed problem than even a very sophisticated real experiment. The same applies to the scaleeffect phenomenon, which plays the central role in this paper and whose hidden aspects will be extracted hither utilising the theory of spaceFractional Continuum Mechanics (sFCM).
Fractional Continuum Mechanics (FCM) is in a broad sense a class of mechanical models that utilise the fractional calculus (FC) (Podlubny 1999; Kilbas et al. 2006; Malinowska et al. 2015). One can say that FMC is a generalisation of classical Continuum Mechanics (CCM) because the latter is a special case of the former, or more precisely, when orders of all FC in the FCM model become integers, smooth passage to the CCM case is obtained. FCM can be classified in several ways, however the most popular classification in the literature includes the name of the variable on which the fractional derivative (FD) operates in the model. Therefore we have, e.g.: (i) timefractional models (Nan et al. 2017; Suzuki et al. 2016; Liao et al. 2017; Zhilei et al. 2016; Wu et al. 2016; Ansari et al. 2016; Faraji Oskouie and Ansari 2017; Sumelka and Voyiadjis 2017); (ii) spacefractional models (Klimek 2001; Drapaca and Sivaloganathan 2012; Sumelka et al. 2015b; Tomasz 2017; Lazopoulos and Lazopoulos 2017; Peter 2017); (iii) stressfractional models (Sumelka 2014a; Sun and Shen 2017, 2017a, b). It is crucial to emphasise that all FCM models are nonlocal, although the physical interpretation of this nonlocality depends on variable on which FD acts (Sumelka and Voyiadjis 2017). Herein, as mentioned, the nonlocality in space should be pointed out as being the main constituent of sFCM.
The sFCM theory, being space nonlocal, enables us to model scaleeffect which is of extreme importance, taking into account constant miniaturisation in many areas of human activity i.e. the production of microstructured and nanostructured materials or micro or nanoelectromechanical (MEMS or NEMS) devices, nanomachines, as well as in biotechnology, biomedical fields and astronomy (Drexler 1992; Martin 1996; Han et al. 1997; Fennimore et al. 2003; Bourlon et al. 2004; Saji et al. 2010). Modelling of such bodies is a challenging task by itself; however, in this paper, we try to go further, asking (theoretically): “What are the mechanical properties of deteriorated nonlocal bodies?”. This question is of fundamental importance considering the durability of miniaturised devices.
Identification of mechanical properties frequently requires solving an inverse problem. There are plenty of examples which confirm the usefulness of this approach when CCM is applied (Constantinescu and Tardieu 2001; Uhl 2007; Garbowski et al. 2012; Bonnet and Constantinescu 2005). However, there is little to none studies in this area when it comes to nonlocal models. From the few, the Vosoughi and Darabi work can be mentioned. In Vosoughi and Darabi (2017) the authors used a conjugate gradient method hybridised with a genetic algorithm to identify a volume fraction coefficient and a small scale parameter in a functionally graded nanobeam together with the Eringen nonlocal elasticity and a firstorder shear deformation theory. The frequencies of a beam were compared for calibration. Similarly, a dynamic response was used by Kiris and Inan (2008) to estimate upper bounds of elastic modulus in a material model based on Eringen’s microstretch theory. The optimization problem was then solved by a direct search method along with a microgenetic algorithm. Another work by Diebels and Geringer (2014) dealt with calibration of the Cosserat model parameters for a foam material (an internal length and an additional stiffness parameter), based on shear stresses from a spatial finite element model. A genetic algorithm optimizer was used. None of the cited works, however, deal with a structure whose mechanical properties locally deteriorate.
A few papers devoted to the optimization problem coupled with nonlocal models, although not directly connected with the inverse problem, indicate that the optimization results can vary strongly when the size of a body approaches a length scale. Liu and Su (2010) performed topology optimization within a rectangular domain filled with material where the couplestress theory was applied (Mindlin 1964). The work generalises the conventional Solid Isotropic Material with Penalisation (SIMP) model (Bendsøe 1989; Bendsøe and Sigmund 2004) and the results clearly demonstrate that the optimal solution changes when the length scale is varied. The optimal material layout approaches the solution for classical continuum mechanics when the proportion of the length scale to the minimal size of the rectangular domain tends to zero. Similarly, Veber and Rovati (2007) solved the minimum compliance problem (Bendsøe 1989) for a micropolar body (Eringen 1966) finding the optimal material distribution in a rectangular domain (plain stress) for different values of the characteristic length for bending. The final topology diversity is considerable for all analysed boundary and load schemes when the characteristic length becomes comparable with the body size. The observation is also confirmed by Sun and Zhang (2006) who optimized the topology of lightweight structures with a cellular core. This research output indicates that the solution depends on the length scale when the macrostructure has size comparable to a microstructure. The mentioned papers prove that the optimization result is sensitive to nonlocal model parameters. Thus, one should expect the same when it comes to the identification problem especially for deteriorated bodies.
The primary goal of this paper is to capture the mechanical proprieties of 1D deteriorated nonlocal bodies, understood as a change of mechanical properties of devices exhibiting a strong scale effect, under the assumption that topology remains constant. The problem is formulated within sFCM defined in Sumelka (2014b) (together with the variable length scale concept presented in Sumelka 2017), as a sensitivity analysis of mass and stiffness distribution in reference to fractional model parameters and loading scheme. The inverse approach as an optimization procedure is proposed, where without loss of the primary goal importance, test data are prepared as a displacement field for a corroded fractional body. The obtained results show sensitivity of identified corrosion depending on fractional constitutive parameters. Furthermore the results are contrasted with the solution of CCM. In this sense, the meaningful role of nonlocal (fractional) modelling is pointed out. In other words, the considerations show the complexity of behaviour of bodies exhibiting the scale effect which can go beyond the standard engineering intuition.

modified numerical scheme compared to Sumelka (2017) for better efficiency of computations  for each single optimization task it was required the execution of few thousands of iterations;

extension of numerical approximation procedure to include the inhomogeneous linear elasticity (and furthermore the nonlinear stiffness to density relationship);

methodology of deterioration including especially the stiffness to density relationship;

penalty terms to make the inverse problem successful.
The paper is structured as follows. In Section 2 sFCM fundamentals are presented. Section 3 deals with a general idea for capturing deterioration, details on implementation of a numerical model and an algorithm as well as an inverse problem formulation. Section 4 presents results from almost 150 analyses for various load schemes and fractional body parameters, along with a discussion. Finally, Section 5 provides the conclusions.
2 Governing equations and numerical approximation
2.1 Fractional derivative
It is commonly accepted that studies on FC has been initiated in 1695 by Leibniz and L’Hospital (Leibniz 1962). Since that times the investigation on differential equations of an arbitrary (even complex) order has become an individual branch of pure mathematics with many practical applications, as mentioned above. To understand the fundamental concepts and techniques of FC one should follow the monographs (Nishimoto 1984; Podlubny 1999; Kilbas et al. 2006; Malinowska et al. 2015; Changpin and Fanhai 2015) and papers cited therein.
Derivative given by (3) play a central role in succeeding fractional elasticity definition.
2.2 1D inhomogeneous fractional elasticity
We consider 1D reduction of a general 3D formulation for small strains and a linear constitutive law formulated in Sumelka (2014b). It is important that this concept was validated with experimental result in Sumelka et al. (2015a), where the sFCM concept was used to mimic the behaviour of microbeams made of the polymer SU8 (where strong scale effect was observed), and furthermore, in Sumelka et al. (2016) it was presented that sFCM can correctly mimic the BornVon Karman (BK) lattice (discrete system). This result are crucial for correctness of physical interpretation of the results obtained below.
 for both ends of a 1D body displacements are prescribed$$ U(x_{0})=U_{L}, \quad U(x_{r})=U_{R}, $$(5)
 for the left end displacements and for the right end forces are given$$ U(x_{0})=U_{L}, \quad \overset{\Diamond}{\varepsilon}(x_{r})=\overset{\Diamond}{\varepsilon}_{R}, $$(6)
2.3 Discretization
2.3.1 Internal points
2.3.2 Boundary points
2.4 Algebraic problem
3 Identification
3.1 Structure
3.2 Capturing deterioration

topology of the structure does not change,

deterioration is related to density reduction,

longitudinal stiffness depends on density,

mass lost is known (optional).
The second assumption, in turn, determines that density is a deterioration indicator. As a consequence, identification of density is enough to assess the condition of a device. In the general case, the density distribution can be free; however, in order to parameterize it using a reasonable number of variables, a spline function is utilized. Let us assume vector \(\bar {\boldmath {\rho }} = [\bar {\rho }_{1}, ..., \bar {\rho }_{i}, ..., \bar {\rho }_{s}]\) where components represent density values in discrete points (knots). Then the intermediate values are interpolated using a uniform spline function of order ≤ 3. The general idea is presented in Fig. 3. Note that the knot points indexed by 1,...,i...,s are not the same as the ones used for the body discretization (cf. Section 2.3.1). Their number is much lower, which allows us to reduce number of parameters. In the current study eight knots were assumed as a tradeoff between computation time related to the number of variables and capability to capture gradients in the density distribution.
The final assumption refers to optimization requirements. The mass stabilises the process and leads to unique solutions. If the mass cannot be measured, it should be considered as one more parameter in optimization leading to minimisation of the original formulation (cf. Section 3.4).
Note that because of the above assumptions, in the current study the corrosion phenomenon is reduced to a density change and a corresponding stiffness modification. Further investigations of corrosion effects should be done to verify the approach for any structure.
3.3 Numerical model
The behaviour of the analysed structure is described using sFCM according to (11). In all analysed computational scenarios the body is discretized into 100 subintervals, what leads to Δx = const. = 0.01.
A dedicated procedure in Python was developed in order to build governing equations dynamically for the particular points and then assemble them into a system of linear equations and solve (cf. github.com/szajek). The main idea behind the library was to create a template of a governing equation as a combination of elements (etc. numerical approximations of a differentiate, variables, constants, ...) bound using lazy operators (Watt and Findlay 2004). This concept allows us to easily capture varying definitions of elements and changing values within a body, which is an important case in the analysed problem. The system of (15) is solved using an LU decomposition (Strang 1980) with partial pivoting and row interchanges. The final output is a series of displacements along the body U_{i}.
3.4 Optimization problem
For all parameters the boundary limits were defined. The boundary values of ρ are ρ_{lower} = 0 and ρ_{upper} = 1, while the c limits equal c_{lower} = 0, c_{upper} = ∞. To compute the penalty terms, the preliminary studies based on trials and errors approach were done, and the following coefficients were estimated: p_{1} = 10^{− 2}, p_{3} = 10^{− 4} and p_{2} = p_{4} = 2.5.
3.5 Algorithm
3.6 Test data
4 Results
The problem described by (23) in the variant with a known mass was solved using the algorithm presented in Section 3.5. The optimization was carried out for seven different length scale distributions (\(\ell _{f}^{max} \in \{0.05,\) 0.10,0.15,0.2,0.25,0.30,0.35}, see Fig. 4), six orders of fractional continua (α ∈{0.4,0.5,0.6,0.7,0.8, 0.999}) and three load scenarios depending on the field intensity function, g(x) (cf. (17)).
The last observation, based on the obtained results, is that the inverse problem in (23), where exterior static penalty function is used for both mass constraint and (what is more important here) the design parameters limits is in numerical sense wellposed. For each configuration of the fractional parameters (ℓ_{f}, α) the solution was found. The results for a classical model (when α → 1) were confirmed with the analytical solution (which is wellposed) and moreover the inverse problem solution changes continuously (smoothly) with the fractional parameters for both density distribution ρ (cf. Figs. 9, 10 and 11), as well as exponent c (cf. Figs. 16, 17 and 18). Additionally, for the proposed test data (Fig. 6) the inverse problem converges to the same results as the assumed for the forward analysis (Fig. 7) in all cases.
5 Conclusions
The paper presents the study of mechanical properties of 1D deteriorated bodies exhibiting the scale effect. The overall problem has been formulated in the framework of the spaceFractional Continuum Mechanics as an inverse problem. Steering parameters of the optimization task were: density of material, order of material, and material length scale. The obtained results can be crucial, especially from the point of view of durability of microstructured/nanostructured materials/devices whose popularity in real life applications is growing rapidly and manufacturing is becoming easier in view of constant progress of 3D printing techniques. Furthermore, the outcomes show the meaningful role of proper modelling of nonlocal bodies, specifically taking into account their maintenance which can be in the near future one of the most challenging engineering tasks.

classical continuum mechanics is not able to properly predict the deterioration of 1D nonlocal bodies  both qualitative and quantitative difference with respect to sFCM outcomes was observed;

the obtained results are sensitive with respect to the length scale to body dimension ratio, being more emphasised for ratios closer to one;

the order of fractional continua acts as a scaling factor, and makes the difference more pronounced approaching to zero;

for more complex displacement field/loading, considerable increase of outcomes sensitivity is observed.
The paper is the first step in research on optimal spaceFractional Continuum Bodies, and it is a base for further investigations including optimal distribution of the length scale and order of sFCM continua, including anisotropy (Sumelka 2016).
Notes
Acknowledgements
This work is supported by the National Science Centre, Poland under Grant No. 2017/27/B/ST8/00351.
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