Stress Gradient Model Research Articles

Integral nonlocal stress gradient elasticity of functionally graded porous Timoshenko nanobeam with symmetrical or anti‐symmetrical condition

AbstractUtilization of symmetrical or anti‐symmetrical condition could improve the calculation efficiency. In this paper, a mathematical formulation is proposed to deal with the symmetrical or anti‐symmetrical condition in an integral nonlocal stress gradient model (INSGM), which is transformed equivalently into differential form with constitutive boundary condition as well as constitutive symmetrical or anti‐symmetrical condition. Unlike general constitutive boundary conditions, an integral item is introduced to constitutive symmetrical and anti‐symmetrical conditions, and they are opposite to each other. Based on INSGM with symmetrical or anti‐symmetrical conditions, static bending of simply‐supported (SS) and clamped‐clamped (CC) functionally graded porous Timoshenko nanobeams is investigated for symmetrical loads, including uniformly distributed load (UDL) and middle point force, as well as anti‐symmetrical loads, including anti‐symmetrical UDL and middle point moment. The exact solutions are deduced and expressed in explicit form for different boundary and loading conditions. Calculation shows that, under UDL, bending deflections of half Timoshenko nanobeams based on current model agree well with those for whole Timoshenko nanobeams based on general INSGM for both SS and CC boundary conditions. Numerical study is performed to show the effectiveness of current model.

Alongshore variability of a southern California beach, before and after nourishment

A moderate-size (344,000 m 3 ) subaerial beach nourishment was placed in 2012 in front of flood-prone homes at Imperial Beach, California. The subaerially persistent coarser-than-native nourishment sand spread alongshore over a ∼ 4 km span. By winter 2018/19 subaerial sand volumes in front of the flood-prone homes were eroded to near pre-nourishment levels, while sand accumulated near a river mouth ∼ 1.5 km to the south. The observed alongshore patterns of post-nourishment erosion/accretion are consistent with the modeled convergence/divergence of alongshore currents driven by wave radiation stress S x y gradients associated with refraction over a relic cobble ebb shoal. The same alongshore patterns of subaerial erosion/accretion were observed in the 6 years preceding the nourishment, indicating a wave-driven erosion hotspot in front of the flood-prone homes, and an accretion hotspot near the region of river mouth migration. Over the 13-yr observation period, the river mouth migrated 600 m southward and closed in 2016, possibly owing to nourishment sand and El Niño intensified local drift convergence. The computationally efficient drift (radiation stress) gradient model uses simplified wave and surfzone dynamics, assumes proportionality between alongshore sediment transport and the radiation stress component S x y , but nevertheless captures the overall patterns of alongshore nourishment evolution and provides a framework to inform sand management. • The subaerial influence of relatively coarse beach nourishment sand spans 4 km of coast 6 years after placement. • Accretion and erosion hotspots are persistent, 6 years before and after nourishment. • Patterns in the divergence of the drift, controlled by an ebb shoal, cause the hotspots.

Application of the Higher-Order Hamilton Approach to the Nonlinear Free Vibrations Analysis of Porous FG Nano-Beams in a Hygrothermal Environment Based on a Local/Nonlocal Stress Gradient Model of Elasticity.

Nonlinear transverse free vibrations of porous functionally-graded (FG) Bernoulli–Euler nanobeams in hygrothermal environments through the local/nonlocal stress gradient theory of elasticity were studied. By using the Galerkin method, the governing equations were reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency was then established using the higher-order Hamiltonian approach to nonlinear oscillators. A numerical investigation was developed to analyze the influence of different parameters both on the thermo-elastic material properties and the structural response, such as material gradient index, porosity volume fraction, nonlocal parameter, gradient length parameter, mixture parameter, and the amplitude of the nonlinear oscillator on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.

A mixed variational framework for higher-order unified gradient elasticity

The higher-order unified gradient elasticity theory is conceived in a mixed variational framework based on suitable functional space of kinetic test fields. The intrinsic form of the differential and boundary conditions of equilibrium along with the constitutive laws is consistently established. Various forms of the gradient elasticity theory, in the sense of stress or strain gradient models, can be retrieved as particular cases of the introduced generalized elasticity theory. The proposed stationary variational principle can effectively realize the nanoscopic structural effects while being exempt of restrictions typical of the nonlocal gradient elasticity model. The well-posed generalized gradient elasticity theory is invoked to study the mechanics of torsion and the torsional behavior of elastic nano-bars is analytically examined. The closed-form analytical formulae of the size-dependent shear modulus of nano-sized bar is determined and efficiently applied to reconstruct the shear modulus of SWCNTs with dissimilar chirality in comparison with the numerical simulation data. A practical approach to calibrate the characteristic lengths associated with the higher-order unified gradient elasticity theory is introduced. Numerical results associated with the torsion of higher-order unified gradient elastic bars are demonstrated and compared with the counterpart size-dependent elasticity theories. The conceived generalized gradient elasticity theory can beneficially characterize the nanoscopic response of advanced nano-materials.

Extended stress gradient elastodynamics: Wave dispersion and micro-macro identification of parameters

In its original formulation by Forest & Sab (Math. Mech. Solids, 2017), stress gradient elastodynamics incorporate two inner-lengths to account for size effects in continuum theory. Here, an extended one-dimensional stress gradient model is developed by means of Lagrangian formalism, incorporating an additional inner-length and a fourth-order space derivative in the wave equation. Dispersive properties are characterised and hyperbolicity and stability are proven. Group velocity remains bounded in both original and extended models, proving causality is satisfied for both contrary to a usually-accepted postulate. By means of two-scale asymptotic homogenization, the high-order wave equation satisfied by the stress gradient model is shown to stand for an effective description of heterogeneous materials in the low-frequency range. An upscaling method is developed to identify the stress gradient material parameters and bulk forces on the parameters of elastic micro-structures. Application of the micro–macro procedure to periodic multi-laminates demonstrates the accuracy of the stress gradient continuum to account for the dispersive features of wave propagation. Frequency and time-domain simulations illustrate these properties.

Nonlocal nonlinear analysis of functionally graded plates using natural neighbour Galerkin method

In the present work, flexural response of functionally graded plates subjected to transverse loads have been investigated using the meshless natural neighbor Galerkin method (NNGM). The plate formulation has been developed based on the Reddy’s (Mechanics of laminated composite plates and shells: theory and analysis, 2nd edition, CRC Press, Boca Raton, 2014) third-order shear deformation theory (TSDT) using the von Karman nonlinear strains. The governing equations of the TSDT have been derived accounting for the length scale/size effects considering the Eringen’s nonlocal stress-gradient model (Eringen in Microcontinuum filed theories—I: foundations and solids, Springer-Verlag, 1998). The C1 continuous shape functions have been computed using the sibson’s interpolant and generalizing a Bezier patch over the domain. The nonlocal nonlinear model of the resulting governing equations has been developed, and Newton’s iterative procedure is used for the solution of nonlinear algebraic equations. The mechanical properties of functionally graded plate are assumed to vary continuously through the thickness and obey a power-law distribution of the volume fraction of the constituents. The variation of volume fractions through the thickness have been computed using two different homogenization techniques, namely, the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the effect of side-to-thickness ratio, power-law index, and nonlocal parameter on the load-deflection characteristics of plates have been presented. The central deflections obtained using (NNGM) have been compared with the results from literature based on finite element method. The results have been compared with the two homogenization schemes and also with results computed with the first-order shear deformation theory (FSDT) to show the accuracy of nonlocal nonlinear formulation based on TSDT.

Free vibration of a nanogrid based on Eringen’s stress gradient model

This article deals with the free vibration analysis of a two-member nanogrid. First, the governing differential equations of flexural and torsional vibrations are found based on nonlocal elasticity theory. Second, the problem under study is formulated, considering the interaction of bending and torsion by direct method. It contains four partial differential equations and twelve boundary and compatibility conditions. Third, the problem is derived by using the variational method. Fourth, the problem at hand is analytically solved and natural frequencies of the structure are obtained. Finally, a finite element solution is developed for the problem at hand. A two-node six-degree-of-freedom nanogrid element is introduced. Comparison of both exact and numerical results shows an excellent agreement between findings via both methods. Findings can be served as benchmarks for future researchers. Communicated by Chandrika Prakash Vyasarayani

Theoretical analysis for static bending of Euler–Bernoulli using different nonlocal gradient models

In this work, the bending response of Euler–Bernoulli beam is analyzed with three nonlocal strain gradient model with high-order boundary conditions and modified nonlocal strain and stress gradient models, in which the integral constitutive equation is transformed to an equivalent differential equation with two extra constraint equations. The corresponding differential equations are solved uniquely by the Laplace transformation and its inverse version. Under different boundary and loading conditions, the inconsistency of a nonlocal strain gradient model is examined extendedly, and consistent softening and toughening responses can be observed while using modified nonlocal strain and stress gradient model.

Springback Analysis of AA5754 under Warm Stamping Conditions

Prediction of springback has been thoroughly investigated for cold forming processes; however with the rising prominence of lightweight materials and new forming technologies, predicting springback at elevated temperatures has become essential. In this paper, three analytical models and one empirical model were proposed to predict springback of an aluminium alloy AA5754 at warm forming conditions. The analytical models developed were based on the effect of the linear bending moment, uniform bending moment and through-thickness stress gradient respectively on springback, while the empirical model was developed using the results of L-shape bending tests. The model predictions were compared with the experiment results for various forming conditions. At room temperature, all four models had very good agreement. At elevated temperatures, the linear bending moment model was preferred for a die radius of 8mm, whereas empirical and stress gradient models were more suitable for a die radius of 4mm; in both cases, very close agreement was achieved where errors were within 5%.

Variational nonlocal gradient elasticity for nano-beams

In the paper (Zaera, Serrano, & Fernández-Sáez, 2019) it is claimed that the nonlocal strain gradient theory (NSGT) leads to ill-posed structural problems. This conclusion was motivated by observing that Constitutive Boundary Conditions (CBC) conflict with non-standard Kinematic and Static Higher-Order Boundary Conditions (KHOBC) - (SHOBC). In the present study, it is shown that no ill-posedness holds if NSGT is established by an adequate variational formulation, with appropriate test fields. KHOBC and SHOBC have nothing to do with the proper mathematical formulation and thus they have not to be prescribed, while standard kinematic and static boundary conditions and CBC have to be imposed to close the relevant nonlocal gradient problem. This conclusion follows from a well-posed abstract variational scheme conceived for nonlocal gradient inflected beams. The treatment provides as special cases most of the size-dependent models adopted in Engineering Science to assess size-effects in nanostructures, such as NSGT, strain-driven and stress-driven local-nonlocal elasticity approaches. Additionally, a well-posed Nonlocal Stress Gradient (NStressG) model is presented, coupling the stress-driven nonlocal strategy (Romano and Barretta, 2017) with the stress gradient elasticity. The presented methodology is elucidated and validated by investigating the structural behavior of a variety of inflected nano-beams of nanotechnological interest, such as sensors and actuators. NStressG is able to predict both softening and hardening nonlocal responses and, unlike the special stress gradient elasticity model, leads to well-posed structural problems in Nanomechanics.

Deriving the Oil-Water Seepage Pressure Distribution from Two-Phase Oil-Water Flow Stress Gradient

The characteristics of oil-water two-phase flow in low-permeability reservoirs have been insufficiently studied. In the present article, we estimate the oil-water seepage parameters using a flow stress gradient model with a nonuniform radial distribution of the water saturation coefficient in a low-permeability reservoir. We apply the oil-water relative permeability curves and the Buckley-Leverett equation to determine the oil-water mobility ratio and the functional relationship between the two-phase flow stress gradient, the radial distribution, and the oil saturation. In this way, we construct an oil-water radial flow model based on the oil-water two-phase flow stress to derive the pressure distribution equation. Calculation results show that there is a linear relationship between the oil-water two-phase flow stress gradient and the oil saturation and a quadratic relationship between the oil-water two-phase mobility ratio and the oil saturation. The strata/ pressure distribution for low-permeability reservoirs is highly sensitive to the oil-water two-phase flow stress gradient, and the effect is more noticeable near the wellbore.

Interface-coupling-dependent mechanical behaviors of sandwich-structured Ni/Cu/Ni composites

Maximizing the interface coupling effect is a key to improve mechanical properties of laminate/gradient structures. In this work, Ni/Cu/Ni sandwich-structured composites with the middle Cu layers cyclically deformed to different cycles were fabricated to investigate the influence of initial dislocation density on the interface-coupling-dependent mechanical behavior. Extra yield strength contributed by the interface coupling was found to decrease with decreasing the degree of mechanical incompatibility across the interface during uniaxial tensile loading. In addition, the width of geometrical necessary dislocation (GND) pile-up regions adjacent to the interfaces was observed to decrease with increasing the initial dislocation density of Cu layers, which is attributed to the change of source-obstacle configuration in the stress gradient model. These findings suggest that the GND strengthening effect caused by the interface coupling and the cyclic hardening effect in the constituent layers appears to be competitive, and the interface coupling effect is gradually weakened with increasing the initial dislocation density of Cu layers.

Lattice and continualized models for the buckling study of nonlocal rectangular thick plates including shear effects

The present study investigates the exact buckling behavior of a new lattice plate theory, called microstructured thick plate model, that accounts for the shear effect. Three different kinds of continuous nonlocal elasticity plate theories for capturing the small length scale effect of this bending-shear lattice plate have been compared: a fourth and a sixth order continualized models based on the derivation of continuous equations from discrete ones and a phenomenological nonlocal Uflyand–Mindlin plate model. The phenomenological nonlocal model is the stress gradient model of Eringen (1983) applied at the plate scale for the bending part and eventually the shear part of the constitutive law. Each nonlocal plate model is calibrated with respect to the lattice plate model. By definition, the length scale coefficients are constant for the two proposed continualized models. For the phenomenological model, these coefficients are structural dependent and vary with different parameters, such as the aspect ratio or the buckling mode. More especially, it is shown that the calibrated small length scale coefficient is influenced by the shear effect. The comparison of the nondimensional buckling loads calculated with the optimal value of the small length scale coefficient, which differ for each nonlocal model, shows that the continualized models constitute a much better, more consistent and more accurate approximation than the phenomenological nonlocal approach.

Investigation both actions of elastic foundation parameters and small scale effect on axisymmetric bending of annular single-layered graphene sheet resting on an elastic medium

Many different researches show that stiffness of nanoplates or nanosheets and other nanostructures are decreased by existing or increasing the small scale effect. But few studies that applied other models or approaches to get relationships between stiffness of nano/microstructures and the small scale effect are found results contrary to the results of the above mentioned researches. In this paper, axisymmetric bending behaviour of the annular single-layered graphene sheet resting on an elastic medium is studied by comparing results of the nonlocal elasticity and classical theories. The Pasternak-type foundation model is employed to simulate the interaction between the graphene sheet and the surrounding elastic medium. To get numerical results, the Ritz method is applied. Also, an aspect ratio is defined, and a parametric study is carried out varying the small scale parameter, elastic foundation parameters and the mentioned aspect ratio of the annular graphene sheet. The applied nonlocal elasticity model to analyze annular graphene sheet resting on an elastic medium is an exact nonlocal stress gradient model which have presented correct results in cases of with and without the elastic medium and whose results are in accord with experimental researches, studies based on other continuum approaches and some molecular dynamics simulations. Results of present paper represent that increase of the small scale effect increases the stiffness of nanostructures. For the first time, it is also shown that the mentioned result is in agreement with experimental researches, other continuum approaches, and some molecular dynamics simulations studies. Also, the simultaneous effects of all the mentioned parameters on the stiffness of nanostructures are investigated.

Application of Green's function method to bending of stress gradient nanobeams

In this article, the deflection functions of nanobeams are found by employing the well-known Green's function method. Based on Eringen's nonlocal elasticity theory, stress gradient model is employed to formulate the problem. Different boundary conditions, including, clamped–clamped, clamped-free, simply supported-simply supported and clamped-simply supported are studied. The beams are assumed to be under arbitrary distributed or concentrated loads. By presenting some new Green's integral formulations, three different ways of treating the nonlocal terms are suggested. The specificity of inhomogeneous boundary conditions for the stress gradient nonlocal model is found. A particular property is a possible loss of the symmetry of Green's functions for nonlocal boundary conditions. To validate the proposed results, findings are compared with the available ones in the literature.

Free axial vibration of nanorods with elastic medium interaction based on nonlocal elasticity and Rayleigh model

In this paper, the free axial vibration of single walled carbon nanorod embedded in an elastic medium is investigated by the use of Rayleigh model. The stress gradient model introduced by Eringen is used to formulate the governing equations. Explicit expressions are derived for eigenfrequencies of fixed-fixed and fixed-free boundary conditions.

Research on the influence of nonlocal effect on specific roll pressure by nonlocal stress gradient model in strip rolling

In this paper, a nonlocal stress gradient model based on nonlocal theory has been developed to research the influence of the nonlocal effect on the specific roll pressure in strip rolling. In order to obtain the higher-order stress terms in the nonlocal stress gradient model, the local stress in Eringen’s nonlocal integral model is expressed as the form of a series by Taylor expansion. By comparisons of the experimental data with the results predicted by the nonlocal stress gradient model and the local model, it is found that the application of the nonlocal stress gradient model can lead to a better solution. Meanwhile, the effect of the material characteristic parameter, the friction coefficient, and the reduction on the non-dimensional pressure difference (nonlocal effect) is discussed.

Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models

Germain’s general micromorphic theory of order is extended to fully non-symmetric higher-order tensor degrees of freedom. An interpretation of the microdeformation kinematic variables as relaxed higher-order gradients of the displacement field is proposed. Dynamical balance laws and hyperelastic constitutive equations are derived within the finite deformation framework. Internal constraints are enforced to recover strain gradient theories of grade . An extension to finite deformations of a recently developed stress gradient continuum theory is then presented, together with its relation to the second-order micromorphic model. The linearization of the combination of stress and strain gradient models is then shown to deliver formulations related to Eringen’s and Aifantis’s well-known gradient models involving the Laplacians of stress and strain tensors. Finally, the structures of the dynamical equations are given for strain and stress gradient media, showing fundamental differences in the dynamical behaviour of these two classes of generalized continua.

Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity

Abstract Thermoelastic analysis at micro and nano-scale is becoming important along with the miniaturization of the device and wide application of ultrafast lasers, even the novel laser burst technology, where size effect on heat conduction and elastic deformation increase and classical theory of thermoelastic coupling does not hold any more. In this work, a size-dependent thermoelastic model is established for higher order simple material by adopting both the size effect of heat conduction and elasticity with the aids of extended irreversible thermodynamics and generalized free energy. It is proven that the present model is in essence identical to the coupling of nonlocal heat conductive law (GK model) and nonlocal elastic (stress gradient) model. Also, higher order boundary conditions for stress tensor and heat flux are presented and discussed. For numerical evaluation, a bi-layered structure is considered with interfacial thermal contact resistance and elastic wave impendence incorporated. From numerical results, the effects of size-dependent characteristic lengths and material constants of each layer on the transient responses are discussed, systematically. This study is expected to be helpful for theoretical modeling of thermoelasticity at nano-scale, and may be beneficial to design of nano-sized and multi-layered devices.

Eringen’s Stress Gradient Model for Bending of Nonlocal Beams

This paper is concerned with the bending response of nonlocal elastic beams under transverse loads, where the nonlocal elastic model of Eringen, also called the stress gradient model, is used. This model is known to exhibit some paradoxical responses when applied to beams with certain types of boundary conditions. In particular, for clamped-free boundary condition, this nonlocal model is not able to predict scale effects in the presence of concentrated loads, or it leads to an apparent stiffening effect for distributed loads in contrast to other boundary conditions for which softening effect is observed. In the literature, these paradoxes have been resolved by changing the kernel of the nonlocal model or by modifying the standard boundary conditions. In this paper, the paradox is solved from the nonlocal differential model itself via some related discontinuous nonlocal kinematics. It is shown that the kinematics related to the nonlocal constitutive law lead to the use of moment or shear discontinuities. With such a nonlocal differential model coupled with the nonlocal discontinuity requirements, the beam effectively shows a softening response irrespective of the boundary conditions studied, including the clamped-free boundary conditions, and thereby resolves the paradox. The model is also compared to lattice-based solutions where an excellent agreement between the present nonlocal model and the lattice one is obtained. Finally, the stress gradient model is shown to be cast in a stress-based variational framework, which coincides with a Timoshenko-type model where the shear effect is shown to play the nonlocal role.