Nonlocal Strain Research Articles

Nonlinear Free Vibration of Postbuckled PFG Micro/nanotubes in Presence of Internal Resonances via Nonlocal Strain and Velocity Gradient Theory

Nonlinear Free Vibration of Postbuckled PFG Micro/nanotubes in Presence of Internal Resonances via Nonlocal Strain and Velocity Gradient Theory

Nonlinear free vibration of pre-buckled PFG micro/nanotubes via nonlocal strain and velocity gradient theory

Nonlinear free vibration of pre-buckled PFG micro/nanotubes via nonlocal strain and velocity gradient theory

Dynamic stability of porous functionally graded nanotubes via nonlocal strain and velocity gradient theory

Dynamic stability of porous functionally graded nanotubes via nonlocal strain and velocity gradient theory

Differential geometry-based thermodynamics derivation of isotropic and anisotropic eikonal non-local gradient (ENLG) damage models using a micromorphic media framework

The Eikonal non-local damage (ENL) approach models damage as a space-deforming phenomenon that affects the interaction distance between material points. As damage increases, interactions between points decrease, ultimately resulting in no interaction. In the integral version of such an approach, non-local interaction distances between material points are computed by solving a stationary Hamilton–Jacobi equation with a damage-dependent Riemannian metric. In the implicit gradient version of ENL models, the Riemannian metric figures in the Helmholtz equation to be solved for computing the non-local field controlling damage evolution. However, one of the main criticisms of such formulation is the lack of thermodynamics basis in its derivation. This paper presents a thermodynamics derivation of the Eikonal implicit gradient formulation based on differential geometry concepts to overcome this issue. A free-energy potential is defined considering the non-local strain as a morphological descriptor belonging to the abstract differentiable manifold (where the Riemannian metric is defined). Following a micromorphic media framework, the balance equations of the model are obtained. It is shown that the stress tensor and thermodynamic force associated with damage are the sum of standard and additional non-local contributions. It is also shown that the resulting energy dissipation is always positive, thus verifying the Clausius–Duhem inequality. After presenting all development considering second-order anisotropic continuum damage, the isotropic formulation is obtained as a particular case. Finally, a two-dimensional numerical implementation of an isotropic implicit Eikonal non-local gradient damage model is illustrated. Test cases are simulated to show the relocalization features of the considered formulation and its natural capability of naturally representing damage-to-fracture transition for high damage levels.

Nonlocal strain and stress gradient elasticity of Timoshenko nano-beams with loading discontinuities

A unified approach is applied for determining both strain- and stress-driven differential formulations of Timoshenko nano-beams in presence of loading discontinuities. The consequent models can simulate small scale effects with different types of constitutive laws (such as pure nonlocal, mixture of local and nonlocal phases, and nonlocal gradient). A specific novel feature of the proposed models is the ability to consider loading discontinuities, i.e. points of discontinuities for generalized internal forces occurring in presence of external supports, forces, or couples concentrated at internal points of the nano-beam. To this end, novel constitutive continuity conditions (CCCs) are imposed at the beam interior points of loading discontinuities. CCCs contain integral convolutions of generalized forces or displacements over suitable parts of the nano-beam; they represent a valid alternative to Dirac delta function and are different from the well-known constitutive boundary conditions (CBCs) imposed at the end-points of the nano-beam. Finally, the proposed models are applied for finding closed-form solutions to cases of practical interest.

微視的な不均質構造によって生じる巨視的な不均一変形のモデル化に関する研究

Engineering materials typically have a heterogeneous microstructure, and it induces microscopic non-uniform deformation even under the macroscopically uniform stress state. With the decrease in the dimension of engineering structural components, such a microscopic non-uniform deformation affects the macroscopic deformation state. In this study, the development of the microscopic heterogeneity-induced non-uniform deformation is modeled using the rate-form nonlocal constitutive equation. First, the inelastic strain is decomposed into local and nonlocal inelastic strains. Then, the nonlocal strain is assumed to be developed by the distribution of plastic compliance, which is characterized by an intrinsic size of the material microstructure. Finally, the multiaxial rate-form constitutive equation, in which the stress on the material points is calculated by both the local and nonlocal strains, is formulated. The finite element simulations of the tensile tests of specimens with curved and parallel gauge sections are performed using the proposed model. The obtained results clarified that the mechanical response and the non-uniform deformation are characterized by the microscopic heterogeneity of the material.

Fractional Derivatives and Dynamical Systems in Material Instability

Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model.

Variationally consistent dynamics of nonlocal gradient elastic beams

The variational static formulation contributed in [International Journal of Engineering Science 143, 73–91 (2019)] is generalized in the present paper to model axial and flexural dynamic behaviors of elastic nano-beams by nonlocal strain and stress gradient approaches. Appropriate forms of non-standard boundary conditions are detected and properly prescribed. Equivalence between differential laws and integral convolutions is elucidated and invoked to analytically evaluate size-dependent axial and flexural fundamental frequencies of cantilever and fully-clamped beams which significantly characterize new-generation nano-actuators. The proposed methodology and ensuing results are tested by pertinent outcomes in literature. Advantageously, in comparison with available nonlocal gradient models, the developed formulation of elasticity leads to well-posed dynamic structural problems of nano-mechanics. Outcomes obtained by the well-established strain-driven and stress-driven nonlocal and gradient theories of Engineering Science are recovered under special ad hoc assumptions. The present study offers a simple and effective strategy to predict peculiar stiffening and softening dynamic responses of nano-scaled components of advanced technological devices, such as Nano-Electro-Mechanical-Systems (NEMS), and modern composite nano-structures.

Multiscale analysis of head-on quenching premixed turbulent flames

Multiscale analysis of wall-bounded turbulent premixed flames is performed using three-dimensional direct numerical simulation data of flame-wall interaction (FWI). The chosen configuration represents head-on quenching of a turbulent statistically planar stoichiometric methane-air flame by an isothermal inert wall. Different turbulence intensities and chemical mechanisms have been analyzed. A bandpass filtering technique is utilised to analyze the influence of turbulent eddies of varying size and the statistics of vorticity and strain rate fields associated with them. It is found that the presence of the flame does not alter the mechanism of vortex stretching in turbulent flows when the flame is away from the wall, but in the case of FWI, the mechanism of vortex stretching is altered due to a reduction in the contribution from non-local strain, and the small scales of turbulence start to contribute to the flame straining process. The results indicate that small scale eddies do not contribute to the tangential strain rate when the flames are away from the walls, whereas the contribution from the small scales to the tangential strain rate increases when the flame is in the vicinity of the wall. It is also found that the choice of chemical mechanism does not influence the underlying fluid mechanical processes involved in FWI.

Static and dynamic response of CNT nanobeam using nonlocal strain and velocity gradient theory

This paper examines the length-scale effect on the nonlinear response of an electrically actuated Carbon Nanotube (CNT) based nano-actuator using a nonlocal strain and velocity gradient (NSVG) theory. The nano-actuator is modeled within the framework of a doubly-clamped Euler–Bernoulli beam which accounts for the nonlinear von-Karman strain and the electric actuating forcing. The NSVG theory includes three length-scale parameters which describe two completely different size-dependent phenomena, namely, the inter-atomic long-range force and the nano-structure deformation mechanisms. Hamilton’s principle is employed to obtain the equation of motion of the nonlinear nanobeam in addition to its respective classical and non-classical boundary conditions. The differential quadrature method (DQM) is used to discretize the governing equations. The key aim of this research is to numerically investigate the influence of the nonlocal parameter and the strain and velocity gradient parameters on the nonlinear structural behavior of the carbon nanotube based nanobeam. It is found that these three length-scale parameters can largely impact the performance of the CNT based nano-actuator and qualitatively alter its resultant response. The main goal of this investigation is to understand the highly nonlinear response of these miniature structures to improve their overall performance.

The scaling of straining motions in homogeneous isotropic turbulence

The scaling of turbulent motions is investigated by considering the flow in the eigenframe of the local strain-rate tensor. The flow patterns in this frame of reference are evaluated using existing direct numerical simulations of homogeneous isotropic turbulence over a Reynolds number range from $Re_{\unicode[STIX]{x1D706}}=34.6$ up to 1131, and also with reference to data for inhomogeneous, anisotropic wall turbulence. The average flow in the eigenframe reveals a shear layer structure containing tube-like vortices and a dissipation sheet, whose dimensions scale with the Kolmogorov length scale, $\unicode[STIX]{x1D702}$. The vorticity stretching motions scale with the Taylor length scale, $\unicode[STIX]{x1D706}_{T}$, while the flow outside the shear layer scales with the integral length scale, $L$. Furthermore, the spatial organization of the vortices and the dissipation sheet defines a characteristic small-scale structure. The overall size of this characteristic small-scale structure is $120\unicode[STIX]{x1D702}$ in all directions based on the coherence length of the vorticity. This is considerably larger than the typical size of individual vortices, and reflects the importance of spatial organization at the small scales. Comparing the overall size of the characteristic small-scale structure with the largest flow scales and the vorticity stretching motions on the scale of $4\unicode[STIX]{x1D706}_{T}$ shows that transitions in flow structure occur where $Re_{\unicode[STIX]{x1D706}}\approx 45$ and 250. Below these respective transitional Reynolds numbers, the small-scale motions and the vorticity stretching motions are progressively less well developed. Scale interactions are examined by decomposing the average shear layer into a local flow, which is induced by the shear layer vorticity, and a non-local flow, which represents the environment of the characteristic small-scale structure. The non-local strain is $4\unicode[STIX]{x1D706}_{T}$ in width and height, which is consistent with observations in high Reynolds number flow of a $4\unicode[STIX]{x1D706}_{T}$ wide instantaneous shear layer with many $\unicode[STIX]{x1D702}$-scale vortical structures inside (Ishihara et al., Flow Turbul. Combust., vol. 91, 2013, pp. 895–929). In the average shear layer, vorticity aligns with the intermediate principal strain at small scales, while it aligns with the most stretching principal strain at larger scales, consistent with instantaneous turbulence. The length scale at which the alignment changes depends on the Reynolds number. When conditioning the flow in the eigenframe on extreme dissipation, the velocity is strongly affected over large distances. Moreover, the associated peak velocity remains Reynolds number dependent when normalized by the Kolmogorov velocity scale. It signifies that extreme dissipation is not simply a small-scale property, but is associated with large scales at the same time.

Pull-In Instability of Electrostatically Actuated Nano-Beams Using the Nonlocal-Gradient Elasticity Theory

A nonlocal-gradient elasticity beam model with two independent gradient coefficients based on the classical nonlocal elasticity theory and strain gradient theory is used to study the pull-in instability of electrostatically actuated nanobeams. The numerical results show that the pull-in voltage of nanobeams are affected by the small scale. And the two independent gradient coefficients play the different roles in it. This paper broadens the way of studying scale effect.

Scale Effect on Dynamic Analysis of Electrostatically Actuated Nano Beams Using the Nonlocal-Gradient Elasticity Theory

A nonlocal-gradient elasticity beam model with two independent gradient coefficients based on the classical nonlocal elasticity theory and strain gradient theory is used to study dynamic analysis of electrostatically actuated nanobeams. The numerical results show that the deflection response and frequency response of nanobeams are all affected by the small scale. And the two independent gradient coefficients play the different roles in it. This paper broadens the way of studying scale effect.

Multi-scale failure analysis of plain-woven composites

A numerical model capable of dealing with progressive degradation of plain woven composites in a computationally efficient manner is presented in this article. A semi-analytical homogenization method is used to derive effective properties of the composite from the material properties of the constituents. The progressive failure is described using nonlocal continuum damage mechanics where the driving internal variable for the damage is the nonlocal strain. The model was implemented into Abaqus/Explicit, where the failure of a longitudinal tension and an open hole tension specimens were simulated in a multi-scale manner and verified experimentally.

Modeling the mechanisms of the photomechanical response of a nematic liquid crystal elastomer

Recent studies of azo-dye doped liquid crystal elastomers show a strong photomechanical response. We report on models that predict experimental results that suggest photothermal heating is the dominant mechanism in a planar constrained geometry. We compare our models with experiments to determine key material parameters, which are used to predict the dynamical response as a function of intensity. We show that a local strain from photothermal heating and a nonlocal strain from thermal diffusion is responsible for the observed length changes over time. This work both elucidates the fundamental mechanisms and provides input for the design of photomechanical optical devices, which have been shown to have the appropriate properties for making smart materials.

Experimental studies of the mechanisms of photomechanical effects in a nematic liquid crystal elastomer

Azo-dye-doped liquid crystal elastomers (LCEs) are known to show a strong photomechanical response. We report on experiments that suggest that photothermal heating is the underlying mechanism in surface-constrained geometry. In particular, we use optical interferometry to probe the length change of the material and direct temperature measurements to determine heating. LCEs with various dopants and optical density were used to study the individual mechanisms. In the high dye-doped limit, most of the light is absorbed near the entry surface, which causes a local strain from photothermal heating and a nonlocal strain from thermal diffusion. The results of our research on the microscopic mechanisms of the photomechanical response can be applied to designing photomechanical materials for actuating/sensing devices, the potential basis of smart structures.

Rhizobial strain involvement in symbiosis efficiency of chickpea–rhizobia under drought stress: plant growth, nitrogen fixation and antioxidant enzyme activities

Chickpea plants were inoculated with two strains of Mesorhizobium ciceri: local strain (C-15) and non-local strain (CP-36) in order to evaluate plant growth parameters, activities of nitrogenase and antioxidant enzymes under drought stress as well as control condition within 15 days of imposition of drought stress. Biomass production, nodulation, nitrogen fixation and antioxidant enzyme activities under drought condition were compared. Under control condition, symbiotic efficiency in symbiosis formed by C-15 was higher than that in symbiosis derived by CP-36. Although drought stress decreased shoot dry weight, root dry weight, nodule dry weight and nitrogen fixation in both symbioses, the rate of decline in plants inoculated with CP-36 was higher than that in symbiosis chickpea with C-15. Therefore, symbioses showed different tolerance level under drought condition which was essentially correlated with symbiotic performance at non-stressful conditions. Under drought stress, nodular peroxidase (POX) activity increased in both symbioses but was higher in nodules produced by C-15. Ascorbate peroxidase (APX) increased significantly in nodules of symbiosis of chickpea with C-15. Catalase (CAT) and glutation reductase (GR) declined in both symbioses which decline extent in symbiosis with C-15 was lower than that in the nodules of CP-36. These results suggested contribution of rhizobial partner in enhancing the tolerance of symbioses to drought stress, which was related with the increase of antioxidant enzyme activities (APX and POX) under drought conditions.

Dispersion of dilatation wave propagation in single-wall carbon nanotubes using nonlocal scale effects

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This paper investigates a model of wave propagation in single-wall carbon nanotubes (SWCNTs) with small scale effects are studied. The equation of motion of the dilatation wave is obtained using the nonlocal elastic theory. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. The SWCNTs structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The SWCNT was the (40,0) zigzag tube with an effective diameter of 3.13 nm. Nonlinear frequency equations of wave propagation in SWCNTs are described through the effect of small scale. The phase velocity and the group velocity are derived, respectively. The nonlinear dispersion relation is analyzed with different wave numbers versus scale coefficient. It can be observed from the results that the dispersion properties of the dilatation wave are induced by the small scale effects, which will disappear in local continuous models. The dispersion degree can be strengthened by increasing the scale coefficient and the wave number. Furthermore, the characteristics for the group velocity of the dilatation wave in carbon nanotubes can also be tuned by these factors.

Micromechanics Study of Fatigue Damage Incubation Following an Initial Overstrain

Understanding and quantifying the effects of overloads/overstrains on the cyclic damage accumulation at a microscale discontinuity is essential for the development of a multistage fatigue model under variable amplitude loading. Micromechanical simulations are conducted on a 7075-T651 Al alloy to quantify the cyclic microplasticity in the matrix adjacent to intact or cracked, life-limiting intermetallic particles. An initial overstrain followed by constant amplitude cyclic straining is simulated considering minimum to maximum strain ratios of 0 and −1. The nonlocal equivalent plastic strain at the cracked intermetallic particles reveals overload effects manifested in two forms: (1) the cyclic plastic shear strain range is greater in the cycles following an initial tensile overstrain than without the overstrain and (2) the initial overstrain causes the nonlocal cumulative equivalent plastic strain to double in subsequent tensile-going half cycles and triple in subsequent compressive-going half cycles, as compared with cases without an initial tensile overstrain. The cyclic plastic zone at the microdiscontinuity corresponds to that of the maximum strain during the initial overstrain and the nonlocal cyclic plastic shear strain range in the matrix near the intact or cracked inclusion is substantially increased for the same remote strain amplitude relative to the case without initial overstrain. These results differ completely from the effects of initial tensile overload on the response at a macroscopic notch root or at the tip of a long fatigue crack in which the driving forces for crack formation or growth, respectively, are reduced. The micromechanical simulation results support the incorporation of enhanced cyclic microplasticity and driving force to form fatigue cracks at cracked inclusions following an initial tensile overstrain in a fatigue incubation model.

Length Scales Interaction in Nonlocal Plastic Strain Localization of Bars of Varying Section

In computational mechanics, strain softening generates ill-posed boundary value problems, which cannot be solved without being regularized, e.g., through the introduction of an internal length scale. This paper investigates how the internal length scale that is introduced by regularization may interact with the external length scale arising from boundary conditions in the particular case of a strain-softening bar of varying cross section and a nonlocal averaging regularization. The interaction of internal and external length scales is examined using an analytical closed-form solution for overnonlocal softening plasticity that derives from a Fredholm equation of the second kind. In the absence of external length (bars of constant section), the analysis shows that the overnonlocal averaging confines and smoothly distributes plastic strain into a localized band. The localization width, plastic strain distribution inside the band, and load-displacement response are controlled by the internal length of the averaging function and the overnonlocal weighting factor. In the presence of external length (bar of varying section), the analytical solution shows that the localization width is controlled by the interaction of external and internal length scales. This interaction is significant when the external and internal lengths are of comparable magnitude, and decreases when the external length becomes large compared to the internal length. The bandwidth is found to depend on the internal and external lengths and stress level while strain localizes, and to relate only to the internal length when the bar collapses.