Gradient Stress Field Research Articles

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

Effects of geometric configurations on the thermal-mechanical properties of parabolic trough receivers based on coupled optical-thermal-stress model

An optical-thermal-stress coupled analysis model of the Parabolic Trough Receiver (PTR) was built based on the Monte Carlo Ray Tracing method and Computational Fluid Dynamics. The temperature field and stress field of the PTR were studied by numerical simulation, and the absorber deformation and the Kovar ring failure were analyzed. Furthermore, the effects of the aperture width (W) and focal length (f) across the entire receiver temperature gradient and thermal stress field were studied. The results showed that the effect of width and focal length on the temperature field was mainly reflected in the change in the high-temperature distribution of the absorber. The maximum collector efficiency were 81.02% and 70.4% when W = 13 m and f = 1.84 m, respectively. The maximum deformation on the absorber reached 17.85 mm at W = 7 m. High deformation and high Von Mises stress were maintained with f from 3.5 m to 5.5 m. The trends of maximum temperature difference and average temperature on bellows, Kovar ring and glass tube were similar. The results also showed that width and focal length mainly affected the axial stress on Kovar rings and bellows.

Nonlinear vibration of a nonlocal functionally graded beam on fractional visco-Pasternak foundation

This paper investigates the nonlinear dynamic behavior of a nonlocal functionally graded Euler–Bernoulli beam resting on a fractional visco-Pasternak foundation and subjected to harmonic loads. The proposed model captures both, nonlocal parameter considering the elastic stress gradient field and a material length scale parameter considering the strain gradient stress field. Additionally, the von Karman strain–displacement relation is used to describe the nonlinear geometrical beam behavior. The power-law model is utilized to represent the material variations across the thickness direction of the functionally graded beam. The following steps are conducted in this research study. At first, the governing equation of motion is derived using Hamilton’s principle and then reduced to the nonlinear fractional-order differential equation through the single-mode Galerkin approximation. The methodology to determine steady-state amplitude–frequency responses via incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the perturbation multiple scales method for the weakly nonlinear case and numerical integration Newmark method in the case of strong nonlinearity. It has been shown that the application of the incremental harmonic balance method in the analysis of nonlocal strain gradient theory-based structures can lead to more reliable studies for strongly nonlinear systems. In the parametric study, it is shown that, on the one hand, parameters of the visco-Pasternak foundation and power-law index remarkable affect the amplitudes responses. On the contrary, the nonlocal and the length-scale parameters are having a small influence on the amplitude–frequency response. Finally, the effects of the fractional derivative order on the system’s damping are displayed at time response diagrams and subsequently discussed.

Effects of Spatial Scale Modification on the Responses of Surface Wind Stress to the Thermal Front in the Northern South China Sea

AbstractThe responses of surface wind stress to the mesoscale sea surface temperature (SST) anomalies associated with the SST front in the northern South China Sea (NSCS) are studied using satellite observations and reanalysis data. Both satellite and reanalysis data explicitly show the linear relationships between the spatial-high-pass filtered wind stress perturbation derivatives and the underlying SST gradient field. However, the noise in the linear relationships is much smaller in the reanalysis data than in the satellite observations. This result is rarely reported in other frontal areas. The wavelet analysis shows that the satellite scatterometer observed numerous high wavenumber perturbations within 100 km in the NSCS, but these perturbations were absent in the reanalysis data. The linear relationship between the perturbation SST gradient and derivative wind stress fields is not significant at this scale, which enhances the noise in the linear relationship. The spatial bandpass-filtered perturbation between 100 and 300 km can give reasonable estimates of the coupling coefficients between the wind stress divergence and downwind SST gradient (αd) and between the wind stress curl and crosswind SST gradient (αc) in the NSCS, with values of 1.33 × 10−2 and 0.95 × 10−2 N m−2 °C−1, respectively.

A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory

This paper is concerned with the analysis of deflection, stresses, buckling and vibration analysis for a porous functionally graded (FG) curved nanobeam with different boundary conditions using a nonlocal strain gradient theory. The curved nanobeam is made of porous FG material. This property varies according to a power-law function over the thickness. The stresses can be calculated on the basis of the nonlocal strain gradient elasticity model which contains both the nonlocal stress and strain gradients stress field. The Hamilton’s principle is adopted in order to develop differential equation and boundary condition. Numerical results with various cases of boundary conditions are carried out with a view to discuss the influences of porosity factor, nonlocal, length-scale parameters and gradient index on the deflection, stresses, buckling and vibration of porous FG curved nanobeam.

P.831Trajectories of suicidal ideation after a first-episode psychosis: a growth mixture modeling approach

Singular mode shapes around the sharp frictional contact interfaces of different materials with linear elastic behavior are derived semi-analytically using the asymptotic eigen-function expansion method. The characteristic equation which is not proposed in a short explicit expression in the literature is extracted in symbolic terms; then solved numerically and the singularity order is derived using Newton's iterative method for the first time in the literature.The extracted eigen-shapes of the high gradient stress fields and the corresponding displacement fields satisfy the compatibility of deformations and tractions on the contact interface with arbitrary frictional interface equations. These asymptotic mode shapes are then implemented into the finite element interpolation using Partition of Unity Finite Element Method (PUFEM). Since the additional enrichment degrees of freedom on the singular contact nodes have the physical meaning of generalized stress intensity factor (GSIF), the direct GSIFs obtained are more accurate and the contour integral manipulations need not be maintained in the post processing phase of the contact analysis.Finally, some engineering problems are solved which demonstrate the capacities of the approach concerning the accuracy of GSIF obtained from the direct-PUFEM proposed compared with that derived by reciprocal work contour integral method used in traditional finite element analyses.

Dynamic response and failure of rock in initial gradient stress field under stress wave loading

Once an opening is created in deep underground, the stresses surrounding the opening will be redistributed, inducing a gradient stress field. To understand how the ground rock in such a gradient stress field responses to dynamic stress loading, the gradient stress distribution at a circular opening was first analyzed and the propagation of 1D stress wave in rock mass under gradient stress field was theoretically derived. By using an implicit to explicit solution method in LS-DNA code, the dynamic mechanical behaviors of rock in gradient stress field were numerically investigated. The results indicate that the damage is mainly produced at or near the free face, partly due to the straight action of compressive stress wave and the tensile stress wave generated at the free face. The range of the induced damage zone is narrowed under the conditions of higher gradient stress rate and lower dynamic stress amplitude. However, under lower gradient stress field and higher dynamic stress, the damage becomes severer and wider with discontinuous failure regions.

Partition of unity finite element for analysis of contact interface stress singularity

Singular mode shapes around the sharp frictional contact interfaces of different materials with linear elastic behavior are derived semi-analytically using the asymptotic eigen-function expansion method. The characteristic equation which is not proposed in a short explicit expression in the literature is extracted in symbolic terms; then solved numerically and the singularity order is derived using Newton's iterative method for the first time in the literature.The extracted eigen-shapes of the high gradient stress fields and the corresponding displacement fields satisfy the compatibility of deformations and tractions on the contact interface with arbitrary frictional interface equations. These asymptotic mode shapes are then implemented into the finite element interpolation using Partition of Unity Finite Element Method (PUFEM). Since the additional enrichment degrees of freedom on the singular contact nodes have the physical meaning of generalized stress intensity factor (GSIF), the direct GSIFs obtained are more accurate and the contour integral manipulations need not be maintained in the post processing phase of the contact analysis.Finally, some engineering problems are solved which demonstrate the capacities of the approach concerning the accuracy of GSIF obtained from the direct-PUFEM proposed compared with that derived by reciprocal work contour integral method used in traditional finite element analyses.

The Strongest Size in Gradient Nanograined Metals.

Conventional polycrystalline metals become stronger with decreasing grain size, yet softening starts to take over at the nanometer regime, giving rise to the strongest size at which the predominate strengthening mechanism switches to softening. We show that this critical size for the onset of softening can be tuned by tailoring grain size gradient, and raising in the gradient shifts the size toward the smaller value. The decrease in the strongest size is prompted by mitigation of grain boundary-mediated softening processes accompanying by enhanced intragranular plastic deformations. We found that the nanograins smaller than 6 nm, mainly involving intergranular sliding in homogeneous structures, reveal anomalous plastic deformation in gradient systems, which is mediated by partial dislocation nucleation, faulting and twinning activated in a gradient stress field. The results on extended dislocation slip and gradient plasticity, stemming from the structure heterogeneity, shed light on an emerging class of heterogeneous nanostructured materials of improved strength-ductility synergy.

X-ray diffraction method application to assess the energy losses on explosive-rock contact under various blasting

The dissipative energy losses of the explosion on the explosive-rock contact are usually evaluated with comparative analysis of the particle size distribution of finely dispersed fractions (0-100 microns). The more tiny particles contained in the destruction products, the higher there is a level of energy loss during the explosion. Fine dust granulometric characteristics are determined by processing the mass measurements data of the individual smallest particles sizes when decoding microphotographs obtained by a microscope. However due to the chromatic aberrations due the wave nature of light and the optical systems imperfection, it is not possible to reliably estimate the mass and granulometric characteristics particles of micron size. X-ray diffraction method for studying ultrafine rock fracture products makes it possible to estimate the dissipative energy losses on explosive-rock contact based on the reflected X-ray beam total intensity in diffractograms. In order to establish the effectiveness of methods for reducing the level of dissipative energy losses of an explosion, X-ray diffraction patterns of finely dispersed fracture products of rock samples under various conditions of dynamic loading are analyzed (using different charge designs, attenuating the rocks by the action of a surfactant, and the force action of a different gradient stress field).

Effect of interconnect linewidth on the evolution of intragranular microcracks due to surface diffusion in a gradient stress field and an electric field

Effect of interconnect linewidth on the evolution of intragranular microcracks due to surface diffusion in a gradient stress field and an electric field

A nonlocal strain gradient theory for scale-dependent wave dispersion analysis of rotating nanobeams considering physical field effects

This paper is concerned with the wave propagation behavior of rotating functionally graded temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field. Uniform, linear and nonlinear temperature distributions across the thickness are investigated. Thermo-elastic properties of FG beam change gradually according to the Mori-Tanaka distribution model in the spatial coordinate. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The governing equations are derived by Hamilton\'s principle as a function of axial force due to centrifugal stiffening and displacement. By applying an analytical solution and solving an eigenvalue problem, the dispersion relations of rotating FG nanobeam are obtained. Numerical results illustrate that various parameters including temperature change, angular velocity, nonlocality parameter, wave number and gradient index have significant effect on the wave dispersion characteristics of the understudy nanobeam. The outcome of this study can provide beneficial information for the next generation researches and exact design of nano-machines including nanoscale molecular bearings and nanogears, etc.

Wave propagation analysis of smart strain gradient piezo-magneto-elastic nonlocal beams

This study presents the investigation of wave dispersion characteristics of a magneto-electro-elastic functionally graded (MEE-FG) nanosize beam utilizing nonlocal strain gradient theory (NSGT). In this theory, a material length scale parameter is propounded to show the influence of strain gradient stress field, and likewise, a nonlocal parameter is nominated to emphasize on the importance of elastic stress field effects. The material properties of heterogeneous nanobeam are supposed to vary smoothly through the thickness direction based on power-law form. Applying Hamilton\'s principle, the nonlocal governing equations of MEE-FG nanobeam are derived. Furthermore, to derive the wave frequency, phase velocity and escape frequency of MEE-FG nanobeam, an analytical solution is employed. The validation procedure is performed by comparing the results of present model with results exhibited by previous papers. Results are rendered in the framework of an exact parametric study by changing various parameters such as wave number, nonlocal parameter, length scale parameter, gradient index, magnetic potential and electric voltage to show their influence on the wave frequency, phase velocity and escape frequency of MEE-FG nanobeams.

Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter

A size-dependent novel hyperbolic shear deformation theory of simply supported functionally graded beams is presented in the frame work of the non-local strain gradient theory, in which the stress accounts for only the nonlocal strain gradients stress field. The thickness stretching effect (&#949z ≠ 0) is also considered here. Elastic coefficients and length scale parameter are assumed to vary in the thickness direction of functionally graded beams according to power-law form. The governing equations are derived using the Hamilton principle. The closed-form solutions for exact critical buckling loads of nonlocal strain gradient functionally graded beams are obtained using Navier's method. The derived results are compared with those of strain gradient theory.

Nonlocal strain gradient-based vibration analysis of embedded curved porous piezoelectric nano-beams in thermal environment

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory

This paper is concerned with the wave propagation behavior of rotating functionally graded (FG) temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field. Uniform, linear and nonlinear temperature distributions across the thickness are investigated. Thermo-elastic properties of FG beam change gradually according to the Mori–Tanaka distribution model in the spatial coordinate. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The governing equations are derived by Hamilton's principle as a function of axial force due to centrifugal stiffening and displacement. The solution of these equations is provided employing a Galerkin-based approach which has the potential to capture various boundary conditions. By applying an analytical solution and solving an eigenvalue problem, the dispersion relations of rotating FG nanobeam are obtained. Numerical results illustrate that various parameters including temperature change, angular velocity, nonlocality parameter, wave number and gradient index have significant effects on the wave dispersion characteristics of the nanobeam under study. The outcome of this study can provide beneficial information for the next-generation research and the exact design of nano-machines including nanoscale molecular bearings, nanogears, etc.

Cylindrical functionally graded shell model based on the first order shear deformation nonlocal strain gradient elasticity theory

In this article, a cylindrical functionally graded shell model is developed in the framework of nonlocal strain gradient theory for the first time. For this purpose, the modeled cylindrical FG nanoshell, its equations of motion and corresponding boundary conditions are derived by Hamilton’s principle and the first-order shear deformation theory. Generalized differential quadrature method is applied to discretize the equations of motion. The results of the proposed model are compared with those of the Eringen’s nonlocal, strain gradient, modified couple stress and classical theories. The conclusion of this comparison is that the nonlocal strain gradient theory combines advantages of nonlocal and strain gradient theories by applying both material length scale parameter and a nonlocal parameter in the model to consider the significance of strain gradient stress field and nonlocal elastic stress field, respectively. Furthermore, the effects of the material length scale, nonlocal parameter, FG power index, circumferential wave numbers and length of shell on vibrational behavior of the nonlocal strain gradient FG nanoshell for simply supported and clamped–clamped boundary conditions are investigated.

Filler-induced heterogeneous distribution of stretch-induced crystallization in natural rubber: An in-situ synchrotron-radiation micro-focused scanning X-ray diffraction study

Crystallinity distribution of natural rubber around filled micrometer-size glass beads under different strains are studied with in-situ micro-focused scanning X-ray diffraction (μ-SXRD), where the glass beads are taken to serve as a model of filler. The experimental results suggest, due to the gradient stress field around the glass bead, the oscillatory distribution of crystallinities around the glass bead tends to form soft-hard double network with multi-scale hierarchical structures and spontaneously responds to external strains. The double network spreads stress in larger region and avoids stress concentration, in which the hard domains with high crystallinities effectively sustain the external stress while the soft domains with low crystallinity store elastic energy. According to the equal strain and equal stress conditions, the quantitative calculation on enhancement factor indicates that the soft-hard double network structure can increase the fracture energy to about three orders of magnitude as comparing to that of structure with homogeneous distribution of crystallinity.

Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory

A size-dependent inhomogeneous beam model, which accounts for the through-length power-law variation of a two-constituent axially functionally graded (FG) material, is deduced in the framework of the nonlocal strain gradient theory and the Euler–Bernoulli beam theory. By employing the Hamilton principle, the equations of motion and boundary conditions for size-dependent axially FG beams are deduced. A material length scale parameter and a nonlocal parameter are introduced in the axially FG beam model to consider the significance of strain gradient stress field and nonlocal elastic stress field, respectively. The bending, buckling and vibration problems of axially FG beams are solved by a generalized differential quadrature method. The influences of power-law variation and size-dependent parameters on the bending, buckling and vibration behaviors of axially FG beams are investigated. The mechanical behaviors can be affected by the through-length grading of the FG material and therefore may be controlled by choosing appropriate values of the power-law index. When considering concentrated and uniformly distributed loads, the maximum deflection decreases with increasing length scale parameter. The axially FG beam may exert a stiffness-softening effect or a stiffness-hardening effect on the critical buckling force and the natural frequencies depending on the values of the two size-dependent parameters.

Vibration analysis of viscoelastic inhomogeneous nanobeams resting on a viscoelastic foundation based on nonlocal strain gradient theory incorporating surface and thermal effects

In this paper, surface and thermal effects on the vibration characteristics of viscoelastic functionally graded (FG) nanobeams embedded in a viscoelastic foundation are investigated based on nonlocal strain gradient elasticity theory and Euler–Bernoulli beam model. The present model contains two parameters to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Gurtin–Murdoch elasticity theory is incorporated into the nonlocal strain gradient theory to develop a nonclassical beam model including the surface effects. The viscoelastic foundation consists of a Winkler–Pasternak layer together with a viscous layer of infinite parallel dashpots. A power-law model is adopted to describe a continuous variation of material properties of the FG nanobeam. The governing equations of nonlocal viscoelastic FG nanobeams are obtained using Hamilton’s principle and solved implementing an analytical solution for simply supported and clamped–clamped boundary conditions. The results are validated with those available in the literature. The effects of linear, shear and viscous layers of foundation, structural damping coefficient, surface elasticity, nonlocal parameter, length scale parameter, temperature change, power-law exponent, and slenderness ratio on the frequencies of viscoelastic FG nanobeams are examined.