Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements

Round 1

Reviewer 1 Report

The paper presents an extensive review on the subject of nonlocal elasticity for nanostructures, starting from the seminal works of Eringen to the most recent publications, among which several of the Authors themselves who presented an innovative approach able to solve the paradox well-known in the literature.

The paper is very well written, the structure of the work is clear and the results and comments are very valuable.

The references are adequate.

In my opinion, the paper deserves publication.

Author Response

Thanks to the Reviewer for positive and constructive comments. 

Reviewer 2 Report

The paper presents a review of the recent achievements in the field of nano-structures as far as nonlocal elasticity theories are concerned. In particular, nonlocal methodologies used to model structural assemblages are discussed along with benchmark applicative examples. In general, the paper represents a complete review of the aforementioned topic. Overall, the main achievements and developments in such an interesting field have been touched. Nonlocal models are accurately described by providing the adequate references to the reader. The theoretical frameworks are illustrated in a clear manner. In addition, the applications are an extremely useful tool to understand better the theories presented in the paper. Therefore, the judgement about the paper is positive. The present research should be taken as a reference for the future development of nonlocal models and theories since it represents a remarkable review. The paper should be accepted for publication in the current form.

Author Response

Thanks to the Reviewer for positive and constructive comments. 

Reviewer 3 Report

Re; Non-local elasticity for nano-structures: A review of recent achievements

              R. Barretta, F.M. de Sciarra, M. S. Vaccaro

 The present extensive review provides information on recent developments in modeling and analysis of nano-sized structural elements, concentrated mostly on beams and plates. The topic is of actual research interest. The starting point of the analysis is Eringen formulation (1972)  relating stress to distributed strain through kernel function integral form, next generalized as the combined parallel connection of local and non local models. The more  recent modification relating strain to distributed stress in the integral kernel form  and  series connection of local and non-local formulations proved to provide different and probably more reasonable results. The text concentrates on formal manipulations with no physical background.

1.Referring to constitutive relation (1), it is not clear whether it represents elastic strain properties , that is strain reversibility and path independence, also existence of stress and strain energy functions. The transition from continuum to beam theory can be questioned since the constraint of Navier hypothesis corresponds more to the Cosserat continuum model.

2.There are numerous papers modeling nano- beams  within local strain theory by accounting for the effect of surface layers imposing tractions on the beam body. Then the relations similar to Eqs. (13) and (32) are generated, that is distributed actions on beam surfaces. In the paper there is no reference to these works.

3.The analysis and results presented are not related to any empirical data. This makes impression that the modeling and analysis  are suspended in the vacuum. The paper should clearly discuss some available data. For instance, the well known book by I. Elishakoff et al. : Carbon nanotubes and nanosensors, Wiley, 2012 is not mentioned. It contains  both mathematical analysis and material data.

The paper can deserve publication, but it requires some improvements in order to increase its consistency.

Author Response

Thanks to the Reviewer for reading the manuscript whose contents provide critical discussion of classical and recent innovative contributions on nonlocal continuum mechanics. Changes in the revised manuscript are typed in red.

1. As underlined in the revised manuscript on page 3, Equation (1) represents the elastic constitutive law adopted by Eringen [6,7] to model nonlocal 3D Cauchy continua. The nonlocal elastic law for Bernoulli-Euler beams has been consistently formulated in Equation (3) of the revised manuscript following the Eringen’s approach as made in [42] and subsequently in literature.
2. By following the Reviewer’s suggestion, some contributions accounting for surface effects have been added and commented upon on page 2, although it is not the specific topic of the manuscript.
3. Outcomes presented and discussed in the paper represent innovative advancements in nonlocal continuum mechanics. The illustrated nonlocal methodologies are currently exploited by the scientific community to effectively assess size effects in nano-continua avoiding computationally expensive atomistic approaches. The presented nonlocal formulations are able to reproduce complex scale phenomena by tuning the length-scale parameters, in agreement with experimental data collected in provided references (e.g. [63,64]) as discussed in Sections 4, 6 and 7 of the revised manuscript.

Reviewer 4 Report

Contents

Recent achievements and developments in the field of nonlocal elasticity are presented in the manuscript with reference to nano-mechanical problems of current interest. Classical concepts of nonlocal continuum mechanics are first discussed with peculiar attention to the Eringen’s (strain-driven) model and to issues emerged from its application to structural problems. Secondly, mixture theories combining local and Eringen’s nonlocal responses are shown as possible alternative approaches to recover consistency of strain-driven theories.  Integral formulations based on stress-driven techniques are then illustrated and their effectiveness is shown with reference to a wide variety of applications concerning static and dynamic behaviors of nano-structures. Nonlinear problems of nonlocal continua are also examined and critically discussed. Nonlocal methodologies to model size-dependent behaviours of structural assemblages at small scales are inspected by making recourse to recent achievements in literature. Analysis of nanobeams on nano-foundations are finally investigated to address challenging nano-mechanical problems.

General Comments

The manuscript provides a comprehensive overview of recent advances in nonlocal continuum mechanics and deals with a thematic of major interest in current literature. The submitted review, suitable for Encyclopedia of Engineering, is critical, constructive and supplemented with good-quality figures. Applicative difficulties of classical strain-driven formulations are discussed and compared with modern stress-driven nonlocal strategies recently conceived by the research team of the authors to overcome paradoxes and issues raised in Mechanics of Nano-Structures and Nano-Engineering. 

The bibliography collects essential innovative highly-cited contributions of literature. 

The review is well-written with mathematical analyses clearly provided and discussed. 

Minor improvements 

·      More details about time domain analyses of nanobeams in random vibrations could be usefully provided.     

·      Constitutive boundary conditions in Eq. (58) - on page 19 - could be explicitly formulated in terms of eigenfunctions ψ.

·      Definition of the symbol k* in Section 9 could be explicitly provided.

Author Response

Thanks to the Reviewer for positive and constructive comments. Changes in the revised manuscript are typed in red.

1. To comply with the Reviewer’s request, more details about time domain analyses are provided on page 20 of the revised manuscript.
2. By following the Reviewer’s suggestion, constitutive boundary conditions in Eq. (56) (on page 19 of the revised manuscript) have been explicitly formulated in terms of eigenfunctions ψ.
3. By following the Reviewer’s observation, symbol k* has been defined on page 26 of the revised manuscript.

Reviewer 5 Report

This review paper illustrates the recent developments in modeling and analysis of nano-structures. Static and dynamic problems as well as geometrically nonlinear problems of nanorods, nanobeams, and nanoplates are investigated based on various nonlocal elasticity theories as stress- and strain-driven models. Nonlocal methodologies to model structural assemblages are shown along with benchmark applicative examples.

From my point of view, this topic is interesting and timely. It is also matched with the journal scope. Hence, I recommend its publication after a revision based on the following comments:

1.     The article is overall well-written and free from typesetting and grammatical errors. However, there is a mixture of American and British usage in this study such as “behavior-behaviour”. One of them should be selected.

2.     Determination of size effect for small-sized structures can be quoting in introduction.

3.     The authors are encouraged to provide a new table showing the various nonlocal elasticity theories in chronological order. Also references section can be extended so this is a review manuscript.

Author Response

Thanks to the Reviewer for positive and constructive comments. Changes in the revised manuscript are typed in red.

1. By following the Reviewer’s suggestion, English style has been uniformed throughout the manuscript.
2. To comply with the Reviewer’s advice, assessment of size effects in small-scale structures has been quoted in the introductory section on page 1 of the revised manuscript.
3. Nonlocal elasticity theories developed in last decades have been organically illustrated following the chronological order. Moreover, to comply with the Reviewer’s advice, Bibliography has been improved and extended by adding new references. 

Round 2

Reviewer 3 Report

The paper in the present form can be accepted for publication. However, provides review of one formal way of simulating nano sized structure response,  so far not empirically validated.

Reviewer 4 Report

This revised manuscript can be accepted for publication in Encyclopedia. 

Reviewer 5 Report

Required corrections have been made.