DMMCTE - DFEE - Artículos de revistas

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Hard-magnetic phenomena enable autonomous self-healing elastomers
(Elsevier, 2023-01-01) García González, Daniel; Ter-Yesayants, Tigran; Moreno-Mateos, Miguel Ángel; López-Donaire, María Luisa; Comunidad de Madrid; European Commission; Ministerio de Ciencia e Innovación (España); Agencia Estatal de Investigación (España)
We propose a new concept of self-healing soft material that does not require external actuation to heal after rupture. This consists of a sticky and soft elastomeric matrix filled with hard-magnetic particles. When the material breaks into two or more parts and these are approached to each other, the magnetic particles with residual magnetisation interact closing the crack. Then, if new mechanical loading is applied, the composite acts as a continuum structure transmitting internal forces homogeneously until reaching a new failure strain threshold. We demonstrate this novel concept with experiments under tensile loading and with a magneto-mechanical constitutive model that explains the healing mechanisms. The results indicate that the proposed soft materials are able to, after several fracture cycles, sustain mechanical loading beyond 20 % strains without compromising the structural integrity. The healed material experiences no mechanical degradation during cyclic loading and healing cycles. Moreover, on its healed state, it can sustain higher stresses than the original material without residual magnetisation. We demonstrate that the strain at failure is determined by the combined effect of magnetic and adhesive contributions, which are modulated by the applied deformation rate. Furthermore, we reveal that the combination of soft and hard magnetic particles in a hybrid magnetorheological elastomer enables superior healing performance.
Dynamic analysis and non-standard continualization of a Timoshenko beam lattice
(Elsevier, 2022-01-15) Gómez Silva, Francisco; Zaera, Ramón; Ministerio de Ciencia e Innovación (España); Agencia Estatal de Investigación (España)
In this paper, a Timoshenko beam lattice, made up of a chain of masses and straight segments, is proposed, considering bending and shear deformation by means of linear rotational and transverse springs, respectively. Different standard and non-standard continualization methods are applied to it, highlighting here for the first time the suitability of taking the coupled discrete governing equations as a starting point for deriving new continuum models. Several novel low order non-classical continuum models are obtained, with the aim of reliably capturing size-effects and reflecting the dispersive behaviour of the discrete system. Low order governing equations prevents the need for extra boundary conditions when finite (bounded) solids are treated. An extensive analysis of the transition frequency, which initiates the shear propagation spectrum, has been carried out, examining its influence for the discrete and non-standard continuum models. The natural frequencies of a finite solid with two different boundary conditions are obtained through an edge treatment applied here for the first time to this kind of lattices, thus making it possible to solve the clamped-free edges configuration. The reliability of these approaches is evaluated by comparing their dynamic behaviours with that of the discrete system (taken as a reference), through both dispersion and vibration analyses, some of the new proposed continuum models successfully capturing the behaviour of the discrete one, even for high wavenumbers. Moreover, the appearance of physical inconsistencies is examined.
Homogenization of magnetoelastic heterogeneous solid bodies based on micropolar magnetoelasticity
(Springer, 2022-11) Reda, Hilal; Mawassy, Nagham; Aranda Ruiz, Josué; García González, Daniel; Ganghoffer, J. F.; Comunidad de Madrid; European Commission; Ministerio de Ciencia e Innovación (España); Agencia Estatal de Investigación (España)
A variational-based homogenization method for magnetoelastic composite materials is established in a small strains framework. The existence of a non-symmetrical stress tensor motivates the elaboration of a homogenized Cosserat type magnetoelastic effective medium at the macroscale. Generic expressions of the effective magnetic and elastic properties are derived, showing the existence of couplings between the elastic and magnetic behaviors at the macrolevel. Applications of the developed homogenization methodology are done for periodic heterogeneous media prone to local bending at the scale of a few unit cells. The validation of the homogenized medium is performed by comparing its predictions versus those of fully resolved computations. The influence of the magnetic field intensity and orientation on the strength of micropolar effects is assessed. The proposed formulation opens new possibilities for the efficient design of multifunctional metamaterials via computational modelling.
Modeling and design of SHPB to characterize brittle materials under compression for high strain rates
(MDPI, 2020-01-01) Jankowiak, Tomasz; Rusinek, Alexis; Voyiadjis, George Z.
This paper presents an analytical prediction coupled with numerical simulations of a split Hopkinson pressure bar (SHPB) that could be used during further experiments to measure the dynamic compression strength of concrete. The current study combines experimental, modeling and numerical results, permitting an inverse method by which to validate measurements. An analytical prediction is conducted to determine the waves propagation present in SHPB using a one-dimensional theory and assuming a strain rate dependence of the material strength. This method can be used by designers of new SPHB experimental setups to predict compressive strength or strain rates reached during tests, or to check the consistencies of predicted results. Numerical simulation results obtained using LS-DYNA finite element software are also presented in this paper, and are used to compare the predictions with the analytical results. This work focuses on an SPHB setup that can accurately identify the strain rate sensitivities of concrete or brittle materials.
Constitutive models for dynamic strain aging in metals: Strain rate and temperature dependences on the flow stress
(MDPI, 2020-01-01) Song, Yooseob; García González, Daniel; Rusinek, Alexis; Comunidad de Madrid
A new constitutive model for Q235B structural steel is proposed, incorporating the effect of dynamic strain aging. Dynamic strain aging hugely affects the microstructural behavior of metallic compounds, in turn leading to significant alterations in their macroscopic mechanical response. Therefore, a constitutive model must incorporate the effect of dynamic strain aging to accurately predict thermo-mechanical deformation processes. The proposed model assumes the overall response of the material as a combination of three contributions: athermal, thermally activated, and dynamic strain aging stress components. The dynamic strain aging is approached by two alternative mathematical expressions: (i) model I: rate-independent model; (ii) model II: rate-dependent model. The proposed model is finally used to study the mechanical response of Q235B steel for a wide range of loading conditions, from quasi-static loading (𝜀˙=0.001 𝑠−1 and 𝜀˙=0.02 𝑠−1) to dynamic loading (𝜀˙=800 𝑠−1 and 𝜀˙=7000 𝑠−1), and across a broad range of temperatures (93 𝐾−1173 𝐾). The results from this work highlight the importance of considering strain-rate dependences (model II) to provide reliable predictions under dynamic loading scenarios. In this regard, rate-independent approaches (model I) are rather limited to quasi-static loading.
Low-order continualization of an anisotropic membrane lattice with next-nearest interactions. Enhanced prediction of its dynamic behaviour
(Elsevier, 2023-01) Gómez Silva, Francisco; Zaera, Ramón; Agencia Estatal de Investigación (España)
In this paper, a novel anisotropic membrane lattice with nearest and next-nearest interactions (long-range forces) has been continualized through different standard and non-standard continualization procedures, which enables the development of new non-classical continuum models capable of accurately capturing the scale effects, present in the matter due to its discrete nature. The performance of these continuum models is assessed by means of both dispersion and natural frequencies analyses, where the discrete model is considered as a reference. In addition, the appearance of certain physical inconsistencies in some of the developed models is analysed, concluding that these only appear for those developed with continualizations based on Taylor expansion. Interestingly, the non-standard models suitably capture the dispersive behaviour of the discrete one, without both physical inconsistencies and higher-order spatial derivatives, thus avoiding the need for extra boundary conditions when finite solids are involved.
Mass matrices for elastic continua with micro-inertia
(Elsevier, 2023-01-15) Gómez Silva, Francisco; Askes, H.
In this paper, the finite element discretization of non-classical continuum models with micro-inertia is analysed. The focus is on micro-inertia extensions of the one-dimensional rod model, the beam bending theories of Euler-Bernoulli and Rayleigh, and the two-dimensional membrane model. The performance of a variety of mass matrices is assessed by comparing the natural frequencies and their modes with those of the associated discrete systems, and it is demonstrated that the use of higher-order mass matrices reduces errors and improves convergence rates. Furthermore, finite element sizes larger than the corresponding physical length scale are shown to be sufficient to capture the natural frequencies, thus facilitating numerical models that are not only reliable but also computationally efficient.
Novel Enriched Kinetic Energy continuum model for the enhanced prediction of a 1D lattice with next-nearest interactions
(Elsevier, 2022-02-01) Gómez Silva, Francisco; Zaera, Ramón; Ministerio de Ciencia e Innovación (España)
In this paper, a novel Enriched Kinetic Energy model is proposed for an enhanced prediction of the dynamic behaviour of a one-dimensional lattice with next-nearest interactions playing an important role. The lattice system here considered is made up of a chain formed by particles equally spaced and connected with nearest and next-nearest neighbours, through linear springs with different stiffness. The ability of the novel model proposed in this work in capturing the dynamic behaviour of the lattice system is compared with that of others presented in the literature, concluding that it is the one that shows the best performance around the limit of the Irreducible Brillouin Zone (small wavelengths) when next-nearest interactions are relevant. For this purpose, natural frequencies provided by the continuum models for the finite solid are compared with those provided by the discrete system, considered as a reference. Moreover, the novel Enriched Kinetic Energy model does not present physical inconsistencies, nor higher-order spatial derivatives in its governing equation, so it does not need non-classical boundary conditions to be solved when finite solids are treated.
Low-order non-classical continuum models for the improved prediction of an anisotropic membrane lattice's dynamics
(Elsevier, 2022-10) Gómez Silva, Francisco; Zaera, Ramón; Ministerio de Ciencia e Innovación (España)
In this work, different standard and non-standard continualization methods are applied to an anisotropic membrane lattice to obtain non-classical continuum models, capable of accurately capturing its dynamic behaviour. Their performance is evaluated by comparing the continuum and discrete models through their dispersion relations, as well as their natural frequencies for the fixed-edge configuration. Moreover, the occurrence of certain physical inconsistencies in some of these models is examined. Interestingly, the novel Enriched Kinetic Energy model shows the best performance, presenting neither physical inconsistencies nor higher-order derivatives in its governing equation, thus not needing extra boundary conditions when dealing with finite solids.
New low-order continuum models for the dynamics of a Timoshenko beam lattice with next-nearest interactions
(Elsevier, 2022-11) Gómez Silva, Francisco; Zaera, Ramón; Ministerio de Ciencia e Innovación (España)
In this paper, the dynamic behaviour of a novel Timoshenko beam lattice with long-range interactions, accounting for both bending and shear deformations, is investigated. Several new non-classical continuum models are developed with the aim of capturing its dispersive behaviour with a lower computational cost. For this, innovative continualization procedures are used, comparing them with techniques commonly used in lattices continualization, as well as with advanced ones. Moreover, low-order continuum governing equations are pursued, thus avoiding the need for extra boundary conditions, whose physical meaning is unclear. A comprehensive analysis of the transition frequency, which initiates the shear propagation spectrum, has been performed here for the first time for this lattice and the corresponding continuum models. The capability of these continuum models in capturing the behaviour of the lattice is assessed by conducting both dispersion and natural frequency analyses, for the latter providing an original method to treat the edges for the three possible boundary conditions in Timoshenko beam lattices. The influence of long-range interactions is analysed, and the way shear effect affects the shape vibration modes of the discrete model is interestingly illustrated, finally concluding that some of the new developed continuum models accurately capture the behaviour of the lattice.
All-optical nanosensor for displacement detection in mechanical applications
(MDPI, 2022-11-02) Escandell Varela, Lorena; Rodríguez Álvarez, Carlos; Barreda, Ángela; Zaera, Ramón; García Cámara, Braulio; Comunidad de Madrid; Ministerio de Ciencia e Innovación (España); Universidad Carlos III de Madrid
In this paper, we propose the design of an optical system based on two parallel suspended silicon nanowires that support a range of optical resonances that efficiently confine and scatter light in the infrared range as the base of an all-optical displacement sensor. The effects of the variation of the distance between the nanowires are analyzed. The simulation models are designed by COMSOL Multiphysics software, which is based on the finite element method. The diameter of the nanocylinders (d = 140 nm) was previously optimized to achieve resonances at the operating wavelengths (lambda = 1064 nm and 1310 nm). The results pointed out that a detectable change in their resonant behavior and optical interaction was achieved. The proposed design aims to use a simple light source using a commercial diode laser and simplify the readout systems with a high sensitivity of 1.1 × 106 V/m2 and 1.14 × 106 V/m2 at 1064 nm and 1310 nm, respectively. The results may provide an opportunity to investigate alternative designs of displacement sensors from an all-optical approach and explore their potential use.
Recent advances in hard-magnetic soft composites: Synthesis, characterisation, computational modelling, and applications
(Elsevier, 2022-01-01) Lucarini, Sergio; Hossain, Mokarram; García González, Daniel; Comunidad de Madrid; European Commission; Ministerio de Ciencia e Innovación (España)
Hard-magnetic soft composites consist of magneto-active polymers (MAPs) where the fillers are composed of hard-magnetic (magnetically polarised) particles. These novel multifunctional materials are experiencing a great advance from the last few years. This rise has been motivated by the possibility of controlling ferromagnetic patterns during the manufacturing process. Thus, structures with programmable functionalities can be conceptualised and implemented, opening new routes into the design of smart components with great opportunities in the biomedical engineering and soft robotics fields. In this work, we provide an overview of the state of the art of such MAPs, providing the key fundamentals and reference works. To this end, we present the current synthesis and experimental characterisation methods, the different computational modelling approaches across scales, and a detailed presentation of their current potential applications. Finally, we provide an overall discussion on future perspectives.
Computationally guided DIW technology to enable robust printing of inks with evolving rheological properties
(Wiley, 2023-02-10) Lopez Donaire, Maria Luisa; Aranda-Izuzquiza, Gonzalo de; Garzón Hernández, Sara; Crespo Miguel, Javier; Fernandez-de la Torre, Miguel; Velasco Bayón, Diego; García González, Daniel; Comunidad de Madrid; European Commission
Soft multifunctional materials allow for mechanical sensing or actuation as a response to multiple physical stimuli, while providing material stiffness that mimic soft biological tissues (≈1–10 kPa). One of the main bottlenecks in the state of the art relates to the difficulty for manufacturing complex shapes when using inks whose properties significantly change over the printing time. To overcome this issue, the implementation of a hybrid (theoretical-experimental) framework that allows optimal printability of time-dependent viscosity inks by using the direct ink writing technology. Although the rheological properties of the ink vary during printing time, a combination of theoretical and experimental methods provides evolving printing conditions that ensure efficient and robust printability over the process. The method removes the need of introducing additives to the ink. To enable this technology, an in-house printer that provides flexibility to modulate the extrusion pressure over printing time is developed. The method is validated by manufacturing magnetorheological elastomers and conductive soft materials for specific bioengineering and soft electronics applications.
Insights into the viscohyperelastic response of soft magnetorheological elastomers: Competition of macrostructural versus microstructural players
(Elsevier, 2022-12-01) Lucarini San José, Sergio; Moreno Mateos, Miguel Ángel; Danas, Konstantinos; García González, Daniel; Comunidad de Madrid; European Commission; Ministerio de Ciencia e Innovación (España)
Magnetorheological elastomers (MREs) are multifunctional composites that consist of an elastomeric matrix filled with magnetic particles. These materials respond to an external magnetic field by mechanically deforming and/or changing their magnetorheological properties. Such a multi-physical response has made them extraordinary candidates for a wide variety of applications in soft robotics and bioengineering. However, there are still some gaps of knowledge that prevent the optimal design and application of these MREs. In this regard, the effect of viscoelastic mechanisms remains elusive from a microstructural perspective. To the best of the authors' knowledge, this work provides for the first time a numerical homogenization analysis for various magneto-active microstructures accounting for viscous deformation mechanisms. To this end, we propose an incremental variational formulation that incorporates viscoelasticity via internal variables, which is properly modified to deal with the continuity of Maxwell stresses. The proposed framework is applied to study the magneto-mechanical couplings in extremely soft MREs (stiffness 10 kPa). Such a soft matrix promotes microstructural rearrangements while transmitting internal forces leading to macrostructural synergistic responses. The constitutive parameters are calibrated with experimental tests. The numerical results are accompanied with original magnetostriction tests considering different sample geometries and confined magneto-mechanical tests, reporting the macroscopic response. The results obtained in this work suggest that the effective magneto-mechanical response of the MRE is the outcome of a competition between macrostructural and local microstructural responses, where viscous mechanisms play a relevant role.
Beam formulation and FE framework for architected structures under finite deformations
(Elsevier, 2022-11) Perez Garcia, Carlos; Aranda Ruiz, Josué; Zaera, Ramón; García González, Daniel; Comunidad de Madrid; European Commission; Ministerio de Ciencia e Innovación (España)
The breakthrough in additive manufacturing (AM) techniques is opening new routes into the conceptualisation of novel architected materials. However, there are still important roadblocks impeding the full implementation of these technologies in different application fields such as soft robotics or bioengineering. One of the main bottlenecks is the difficulty to perform topological optimisation of the structures and their functional design. To help this endeavour, computational models are essential. Although 3D formulations provide the most reliable tools, these usually present very high computational costs. Beam models based on 1D formulations may overcome this limitation but they need to incorporate all the relevant mechanical features of the 3D problem. Here, we propose a mixed formulation for Timoshenko-type beams to consistently account for axial, shear and bending contributions under finite deformation theory. The framework is formulated on general bases and is suitable for most types of materials, allowing for the straightforward particularisation of the constitutive description. To prove validity of the model, we provide original experimental data on a 3D printed elastomeric material. We first validate the computational framework using a benchmark problem and compare the beam formulation predictions with numerical results from an equivalent 3D model. Then, we further validate the framework and illustrate its flexibility to predict the mechanical response of beam-based structures. To this end, we perform original experiments and numerical simulations on two types of relevant structures: a rhomboid lattice and a bi-stable beam structure. In both cases, the numerical results provide a very good agreement with the experiments by means of both quantitative and qualitative results. Overall, the proposed formulation provides a useful tool to help at designing new architected materials and metamaterial structures based on beam components. The framework presented may open new opportunities to guide AM techniques by feeding machine learning optimisation algorithms.
Evaluation of different methodologies for primary human dermal fibroblast spheroid formation: automation through 3D bioprinting technology
(IOP Science, 2022-09) Quílez López, Cristina; Cerdeira Valtierra, Enrique; Gonzalez-Rico Iriarte, Jorge; Aranda Izuzquiza, Gonzalo De; López-Donaire, María Luisa; Jorcano Noval, José Luis; Velasco Bayón, Diego; Comunidad de Madrid; Ministerio de Ciencia e Innovación (España)
Cell spheroids have recently emerged as an effective tool to recapitulate native microenvironments of living organisms in an in vitro scenario, increasing the reliability of the results obtained and broadening their applications in regenerative medicine, cancer research, disease modeling and drug screening. In this study the generation of spheroids containing primary human dermal fibroblasts was approached using the two-widely employed methods: hanging-drop and U-shape low adhesion plate (LA-plate). Moreover, extrusion-based three-dimensional (3D) bioprinting was introduced to achieve a standardized and scalable production of cell spheroids, decreasing considerably the possibilities of human error. This was ensured when U-shape LA-plates were used, showing an 85% formation efficiency, increasing up to a 98% when it was automatized using the 3D bioprinting technologies. However, sedimentation effect within the cartridge led to a reduction of 20% in size of the spheroid during the printing process. Hyaluronic acid (HA) was chosen as viscosity enhancer to supplement the bioink and overcome cell sedimentation within the cartridge due to the high viability values exhibited by the cells -around 80%- at the used conditions. Finally, (ANCOVA) of spheroid size over time for different printing conditions stand out HA 0.4% (w/v) 60 kDa as the viscosity-improved bioink that exhibit the highest cell viability and spheroid formation percentages. Besides, not only did it ensure cell spheroid homogeneity over time, reducing cell sedimentation effects, but also wider spheroid diameters over time with less variability, outperforming significantly manual loading.
Magneto-mechanical system to reproduce and quantify complex strain patterns in biological materials
(Elsevier, 2022-06) Moreno Mateos, Miguel Ángel; González-Rico, Jorge; Nunez-Sardinha, Emanuel; Gómez Cruz, Clara; López-Donaire, María Luisa; Lucarini San José, Sergio; Arias Hernández, Ángel; Muñoz Barrutia, María Arrate; Velasco Bayón, Diego; García González, Daniel; Comunidad de Madrid; European Commission; Ministerio de Ciencia, Innovación y Universidades (España); Universidad Carlos III de Madrid
Biological cells and tissues are continuously subjected to mechanical stress and strain cues from their surrounding substrate. How these forces modulate cell and tissue behavior is a major question in mechanobiology. To conduct studies under controlled varying physiological strain scenarios, a new virtually-assisted experimental system is proposed allowing for non-invasive and real-time control of complex deformation modes within the substrates. This approach is based on the use of extremely soft magneto-active polymers, which mimic the stiffness of biological materials. Thus, the system enables the untethered control of biological substrates providing reversible mechanical changes and controlling heterogeneous patterns. Motivated on a deep magneto-mechanical characterization across scales, a multi-physics and multi-scale in silico framework was developed to guide the experimental stimulation setup. The versatility and viability of the system have been demonstrated through its ability to reproduce complex mechanical scenarios simulating local strain patterns in brain tissue during a head impact, and its capability to transmit physiologically relevant mechanical forces to dermal fibroblasts. The proposed framework opens the way to understanding the mechanobiological processes that occur during complex and dynamic deformation states, e.g., in traumatic brain injury, pathological skin scarring or fibrotic heart remodeling during myocardial infarction.
One-dimensional dispersion phenomena in terms of fractional media
(Springer, 2016-09-19) Sumelka, Wojciech; Zaera, Ramón; Fernández-Sáez, José; Ministerio de Economía y Competitividad (España)
It is well know that structured solids present dispersive behaviour which cannot be captured by the classical continuum mechanics theories. A canonical problem in which this can be seen is the wave propagation in the Born-Von Karman lattice. In this paper the dispersive effects in a 1D structured solid is analysed using the Fractional Continuum Mechanics (FCM) approach previously proposed by Sumelka (2013). The formulation uses the Riesz-Caputo (RC) fractional derivative and introduces two phenomenological/material parameters: 1) the size of non-local surrounding l(f), which plays the role of the lattice spacing; and 2) the order of fractional continua a, which can be devised as a fitting parameter. The results obtained with this approach have been compared with the reference dispersion curve of Born-Von Karman lattice, and the capability of the fractional model to capture the size effects present in the dynamic behaviour of discrete systems has been proved.
Propagation of solitons in a two-dimensional nonlinear square lattice
(Elsevier, 2018-11-01) Zaera, Ramón; Vila Morán, Javier; Fernández-Sáez, José; Ruzzene, Massimo; Ministerio de Ciencia e Innovación (España)
We investigate the existence of solitary waves in a nonlinear square spring-mass lattice. In the lattice, the masses interact with their neighbors through linear springs, and are connected to the ground by a nonlinear spring whose force is expressed as a polynomial function of the masses out-of-plane displacement. The low-order Taylor series expansions of the discrete equations lead to a continuum representation that holds in the long wavelength limit. Under this assumption, solitary wave solutions are sought within the long wavelength approximation, and the subsequent application of multiple scales to the resulting nonlinear continuum equations. The study focuses on weak nonlinearities of the ground stiffness and reveals the existence of 3 types of solitons, namely a bright, a dark, and a vortex soliton. These solitons result from the balance of dispersive and nonlinear effects in the lattice, setting aside other relevant phenomena in 2D waves such as diffraction that may lead to a field that does not change during propagation in nonlinear media. For equal constants of the in-plane springs, the governing equation reduces to the Klein-Gordon type, for which bright and dark solitons replicate solutions for one-dimensional lattices. However, unequal constants of the in-plane springs aligned with the two principal lattice directions lead to conditions in which the soliton propagation direction, defined by the group velocity, differs from the wave vector direction, which is unique to two-dimensional assemblies. Furthermore, vortex solitons are obtained for isotropic lattices, which shows similarities with results previously found in optics, thermal media and quantum plasmas. The paper describes the main parameters defining the existence of these solitary waves, and verifies the analytical predictions through numerical simulations. Results show the validity of obtained solutions and illustrate the main characteristics of the solitary waves found in the considered n
Non-standard and constitutive boundary conditions in nonlocal strain gradient elasticity
(Springer, 2020-03-01) Zaera, Ramón; Serrano González, Óscar; Fernández-Sáez, José; Ministerio de Ciencia e Innovación (España)
Zaera et al. (Int J Eng Sci 138:65-81, 2019) recently showed that the nonlocal strain gradient theory (NSGT) is not consistent when it is applied to finite solids, since all boundary conditions associated to the corresponding problems cannot be simultaneously satisfied. Given the large number of works using the NSGT being currently published in the field of generalized continuum mechanics, it is pertinent to evince the shortcomings of the application of this theory. Some authors solved the problem omitting the constitutive boundary conditions. In the current paper we show that, in this case, the equilibrium fields are not compatible with the constitutive equation of the material. Other authors solved it omitting the non-standard boundary conditions. Here we show that, in this case, the solution does not fulfil conservation of energy. In conclusion, the inconsistency of the NSGT is corroborated, and its application must be prevented in the analysis of the mechanical behaviour of nanostructures.

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