Higher-order Strain Gradient Theory Research Articles

Influences of External Magnetic Field on Thermo-Mechanical Vibration Analysis of Nanocomposite Beam using Higher-Order Strain Gradient Theory

Composite nanobeams with carbon nanotube (CNT) reinforcement are crucial components in various applications due to their unique mechanical and thermal properties. The present work investigates the frequency–aspect ratio relationship in composite nanobeams with CNT reinforcement, particularly under varying temperatures, magnetic field strengths, and Winkler elastic stiffnesses. The non-dimensional frequency variation with aspect ratio, influenced by temperature differences, reveals distinct behavior patterns. The introduction of a magnetic field alters frequency characteristics, with uniform distribution (UD) nanobeams displaying higher frequencies. The impact of Winkler elastic stiffness, temperature differences, and magnetic field strengths on frequency elucidates differing responses between UD, [Formula: see text]-type distribution (XD), and [Formula: see text]-type distribution (VD) nanobeams. Nonlocal strain gradient theory proves essential in capturing dynamic characteristics influenced by CNT distribution. These findings shed light on the intricate dynamics of composite nanobeams, providing valuable insights for the design of future nano-devices employing CNT-reinforced composite beams as fundamental elements.

Static bending analysis of functionally graded polymer composite curved beams reinforced with carbon nanotubes

This article deals with the static bending response of a functionally graded polymer composite (FG-PC) curved beam reinforced with carbon nanotubes (CNTs) subjected to sinusoidal and uniform loads. The effective material properties of beam are approximated according to modified rule of mixture. Four types of CNT distribution are also considered. Assuming Timoshenko beam theory and a higher-order strain gradient theory, size-dependent equilibrium equations are extracted. Using Navier solution procedure, nonlocal strain gradient governing equations are solved for simply-supported edges. Ultimately, numerical results are expanded to show the influence of weight fraction and distribution patterns of CNTs, small scale parameters, and opening angle on the static bending response of CNTs reinforced nanocomposite curved beam.

Higher order mixture nonlocal gradient theory of wave propagation

The higher order mixture nonlocal gradient theory of elasticity is conceived via consistent unification of the higher order stress‐ and strain‐driven mixture nonlocal elasticity and the higher order strain gradient theory. The integro‐differential constitutive law is established applying an abstract variational approach and appropriately replaced with the equivalent differential condition subject to nonclassical boundary conditions. The introduced higher order elasticity theory provides, as special cases, a variety of generalized elasticity theories adopted in nanomechanics to assess size effects in continua with nanostructural features. The well‐posed higher order mixture nonlocal gradient theory is elucidated and invoked to examine the flexural wave propagation. The closed‐form wave propagation relation between the phase velocity and the wave number is analytically derived. The determined wave propagation response and ensuing results are compared and calibrated with the pertinent molecular dynamic simulations. The demonstrated results of the phase velocity of the flexural wave propagation detect new benchmarks for numerical analyses. The proposed higher order size‐dependent elasticity approach can be profitably employed in rigorous analysis of pioneering nanotechnological devices.

Two‐phase local/nonlocal gradient mechanics of elastic torsion

The higher order two‐phase local/nonlocal elasticity model and the higher order strain gradient theory are unified via an abstract variational scheme. The higher order constitutive integral convolution is established in a consistent variational framework governed by ad hoc functional space of test fields. Equivalent differential constitutive law equipped with nonclassical boundary conditions of constitutive type is determined. The proposed higher order elasticity theory provides as special cases a range of well‐known size‐dependent elasticity models such as nonlocal, two‐phase local/nonlocal, strain gradient, modified nonlocal strain gradient, and nonlocal strain‐driven gradient models. Evidences of well‐posedness of the introduced higher order two‐phase local/nonlocal gradient problems are elucidated by rigorous examination of the elastostatic torsional response of structural schemes of applicative interest in nano‐mechanics. The exact analytical solution of the torsion problem of elastic nano‐beams is derived, graphically demonstrated, and compared with analogous outcomes in the literature. The conceived higher order elasticity theory can efficiently characterize advanced nano‐materials and structural elements of modern nano‐systems.

Higher–order nonlocal gradient elasticity: A consistent variational theory

A consistent variational theory of the higher–order nonlocal gradient elasticity is conceived to appropriately introduce the nonlocality to the higher-order strain gradient theory. The abstract variational approach, based on appropriate functional spaces of test fields, is applied to establish the higher–order nonlocal gradient mechanics of elastic beams in flexure. Two nonlocal and two gradient characteristic lengths are exploited to describe the size–dependent response of continua with nano–structures. Integral convolutions of the higher–order constitutive law are restored to the equivalent differential problem endowed with non–standard boundary conditions of constitutive–type. The higher–order strain gradient theory, higher–order nonlocal elasticity and modified nonlocal strain gradient theory, extensively adopted in the community of Engineering Science, are demonstrated to be particular cases of the introduced higher-order nonlocal gradient theory. The well-posedness of the developed higher-order nonlocal gradient problem is revealed by studying the flexural response of structures with wide-ranging applications in nano-engineering. Exact analytical solution for elastostatic deflections of nano-beams is derived and new benchmark examples of nano-mechanics interest are detected. The higher-order nonlocal gradient elasticity theory can effectively characterize nanoscopic phenomena in advanced nano–composites and nano–structures.

Some issues concerning the use of a single, material specific length scale parameter in theories of higher order strain gradient plasticity

Higher order strain gradient plasticity theories extend conventional ones with a view to explain the observed elevation of plastic flow stress in micron-sized structures. The essential elements of this modifications are the definition of a measure of plastic strain gradients, introduction of a conjugate higher order stress and introduction of one or more length scale parameters. We adopt a particular elasto-viscoplastic version of the higher order strain gradient theory based on a single scalar length scale parameter, formulate a large deformation based Finite Element model capable of handling strain gradients and higher order boundary conditions and simulate for three different materials, two sets of experiments where size effects manifest. The experiments differ in the level of plasticity that is induced during deformation — ranging from the order of the yield strain to approximately ten times its value. The materials, namely Cu, Al and Ni, have grain sizes that differ widely. We determine the scalar length scale parameter by demanding that the simulations match the experimental load deformation responses closely. The results indicate that single, scalar length scale parameter based higher order strain gradient theories cannot describe experimental observations across a wide range of geometries, boundary conditions, and plastic strain levels. The length scale is not intrinsic to the material but depends on the level of plastic strain induced by the deformation and also possibly on the number of grains involved in the deformation.

A numerical method for magneto-hygro-thermal postbuckling analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions

Many researchers work on the mechanical analysis of perfect graphene sheets while these structures have some defects actually. In this paper, postbuckling of the defective quadrilateral single layered graphene sheets (SLGS) is presented for the first time. Three types of defects including single vacancy (SV), double vacancy (DV) and Stone–Wales (SW) in conjunction with the influence of vacancy defect reconstruction are considered. Nonlocal higher order strain gradient theory with three parameters for size effects is used. The quadrilateral graphene sheet is subjected to temperature, moisture and in-plane magnetic loads. The elastic foundation is modelled by Pasternak medium. The equations of motion are obtained imposing higher order shear deformation theory and Hamilton’s principle. The transformed weighing (TW) and differential quadrature (DQ) method are applied for postbuckling load–deflection relation of the structure. Due to the hygrothermal load, different movable boundary conditions are used. The influences of various parameters such as defect types, defect degree, defect reconstruction, hygrothermal load, magnetic field, nonlocal parameters, 7 different quadrilateral SLGS, boundary conditions and elastic medium on the postbuckling behaviour of the quadrilateral graphene sheet are shown. The numerical results are validated with other published works in the literature. Results show that with increasing the defect degree, the buckling load decreases.

Thermal effect on the dynamic behavior of nanobeam resonator assuming size-dependent higher-order strain gradient theory

The temperature change and size scale dependent effects on the dynamical behaviors of the Carbon nanotube (CNT)-based nano-resonator are investigated in this work. The equation of motion of the CNT-based nano-resonator incorporating the higher-order strain gradient deformation, the geometric nonlinearity due to von Karman nonlinear strain, and the thermal effects is derived. A Galerkin based modal decomposition is used to investigate both the free vibration and forced vibration of the nano-resonator. The temperature variation effect is analyzed by assuming both low and high temperature environments. The results show that when assuming a high temperature in the post buckling regime, the nano-resonator natural frequencies dispersion graph shows possibilities of modes-veering and modes-crossing variation due to the initial buckling of the nano-resonator. From the forced-vibration analysis, the results demonstrate that taking into consideration the higher-order strain gradient deformation in modeling the CNT based nano-resonator completely changes the frequency response of the CNT-based nanoresonator.

Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory

This paper proposes to investigate the nonlinear size dependent behavior of electrically actuated carbon nanotube (CNT) based nano-actuator while including the higher-order strain gradient deformation, the geometric nonlinearity due to the von Karman nonlinear strain as well as the slack effect, and the temperature gradient effects. The assumed non-classical beam model adopts some internal material size scale parameters related to the material nanostructures and is capable of interpreting the size effect that the classical continuum beam model is unable to pronounce. The higher-order governing equations of motion and boundary conditions are derived using the so-called extended Hamilton principle. A Galerkin based reduced-order model (ROM) modal decomposition is developed to prescribe the non-classical nanotube mode shape as well as its static behavior under any applied DC actuation load. Results of the static analysis is compared with those obtained by both classical elasticity continuum and strain gradient theories. A Jacobian method is utilized to determine the variation of the natural frequencies of the nanobeam with the DC load as well as the slack level. A thorough parametric study is conducted to study the influences of the size scale dependent parameters, the geometric nonlinearity, the initial curvature, the gate voltage, and the temperature gradient effect on structural behavior of the CNT-based nano-actuator. It is found that the size effect based on the strain gradient deformation has significant influence on the fundamental nanotube natural frequency dispersion. Also, varying this size effect have revealed the offering of numerous possibilities of modes veering and crossing, all shown to be dependent of the strain gradient parameters as well as the CNT slack level.

Modeling dislocation absorption by surfaces within the framework of strain gradient crystal plasticity

Surfaces play important roles in plastic deformation of crystals at submicron scale by emitting/absorbing dislocations and hence may contribute to the complicated size effect. To take into account the dislocation absorption on the continuum level, an energetic surface model has been proposed by considering the change of surface energy contributed by the formation of surface steps via dislocation absorption. In combination with the conventional higher-order strain gradient theory for grain interior, a new model for single crystal plasticity has been developed. Within the model, there exists a critical threshold for the onset of dislocation absorption by surfaces, alternatively, an independent yield criterion for surfaces and the microscopic boundary conditions can be obtained naturally. As an example of application, plastic behavior of a thin film under plane constrained shear has been studied. Results demonstrate that surface yield strength strongly depends on the film thickness and the orientation of the activated slip system. Comparison with the strain gradient plasticity models under the conventional microscopic boundary conditions, i.e., the microhard and microfree conditions has also been made. It has been found that the plastic behavior is sensitive to the surface boundary conditions.

On higher-order quantum stress

In this paper, we present a novel theory of the higher-order quantum stress and its formulations in the framework of density functional theory. Specifically, we have systematically derived the third-order quantum stress expression based on the higher-order Cauchy–Born rule in combination with the higher-order strain gradient theory. The higher-order quantum stress formulation provides a theoretical foundation for first-principle calculation of stress couples at atomistic scale, which may help us understand possible quantum strain gradient effects, and related ferroelectric and flexoelectric effects at small scales.

Finite Element studies on indentation size effect using a higher order strain gradient theory

In this work, a Finite Element implementation of a higher order strain gradient theory (due to Fleck and Hutchinson, 2001) has been used within the framework of large deformation elasto-viscoplasticity to study the indentation of metals with indenters of various geometries. Of particular interest is the indentation size effect (ISE) commonly observed in experiments where the hardness of a range of materials is found to be significantly higher at small depths of indentation but reduce to a lower, constant value at larger depths. That the ISE can be explained by strain gradient plasticity is well known but this work aims to qualitatively compare a gamut of experimental observations on this effect with predictions from a higher order strain gradient theory. Results indicate that many of the experimental observations are qualitatively borne out by our simulations. However, areas exist where conflicting experimental results make assessment of numerical predictions difficult.

On some applications of gradient elasticity to composite materials

Some applications of the gradient theory of elasticity to composite materials are discussed. A brief introduction to gradient theory is given and some mathematical aspects are provided. Of particular importance is a constitutive equation of a higher-order strain gradient theory, and a particular form of gradient theory for which the Ru–Aifantis theorem holds. Applications include a fiber pullout from a matrix, stress transfer in short fiber composites, and the transverse shearing of a sandwich plate. Size effects for all three examples are provided.