Strain Gradient Elasticity Theory Research Articles

Size Effect Impact on the Mechanical Behavior of an Electrically Actuated Polysilicon Nanobeam based NEMS Resonator

In this paper, the dynamic response of resonating nano-beams is investigated using a strain gradient elasticity theory. A nonlinear model is obtained based on the Galerkin decomposition method to find the dynamic response of the investigated beam around its statically deflected position. The mid-plane stretching, axial residual stress and nonlinear interaction due to the electrostatic force on the deflected beam are included in the proposed nonlinear beam model. Comparing the beam natural frequency using strain gradient theory with experimental data shows an excellent agreement among both approaches. The normalized natural frequency is shown to be increasing nonlinearly with the decrease of the applied DC voltage as well as beam thickness. The results also reveal that increasing the tension axial stress increases the natural frequency; however its influence decreases when decreasing the beam thickness. To investigate the effect of AC actuation voltage on the beam resonant frequency, a Lindstedt-Poincare based perturbation method is utilized and validated by comparison with experimental data. The results show that increasing the AC actuation voltage makes the beam stiffer by increasing its resonant frequency.

Analytical Treatment of the Size-Dependent Nonlinear Postbuckling of Functionally Graded Circular Cylindrical Micro-/Nano-Shells

Mindlin’s strain gradient theory (SGT) is the most popular size-dependent higher-order gradient elasticity theory capable of describing the mechanical behavior of structures at micro-/nanoscale, in which the strain gradient terms are included in the strain energy density. In this article, based on Mindlin’s SGT and classical Donnell’s shell theory, the size-dependent nonlinear postbuckling characteristics of circular cylindrical micro-/nanoscale shells under the action of axial compressive loads are studied. For some specific values of the gradient-based material parameters, the present general micro-/nano-shell formulation can be reduced to those based on simple forms of the strain gradient elasticity theory such as the modified strain gradient theory and the modified couple stress theory. The micro-/nano-shells are assumed to be made from functionally graded materials whose properties vary across the thickness direction based on a power-law distribution function. To consider the geometric nonlinearity, the von Karman relations are used. After obtaining the potential energy of the system including strain gradient effects, an analytical variational approach is utilized to solve the postbuckling problem for small-scale shells with simply supported ends. Finally, selected numerical results are presented to investigate the influence of different parameters such as volume fraction index, length scale parameter and radius-to-thickness ratio on the nonlinear postbuckling behavior of micro-/nano-shells.

A comparison of strain gradient theories with applications to the functionally graded circular micro-plate

• The strain gradient elasticity theory of Zhou et al. is re-expressed. • The formulations of present theory in cylindrical coordinates are derived. • The differences of strain gradient theories are investigated. • A FGM circular plate model is established. • The present model can degenerate to the corresponding models of other theories. The strain gradient theory of Zhou et al. is re-expressed in a more direct form and the differences with other strain gradient theories are investigated by an application on static and dynamic analyses of FGM circular micro-plate. To facilitate the modeling, the strain gradient theory of Zhou et al. is re-expressed in cylindrical coordinates, and then the governing equation, boundary conditions and initial condition for circular plate are derived with the help of the Hamilton's principle. The present model can degenerate into other models based on the strain gradient theory of Lam et al., the couple stress theory, the modified couple stress theory or even the classical theory, respectively. The static bending and free vibration problems of a simply supported circular plate are solved. The results indicate that the consideration of strain gradients results in an increase in plate stiffness, and leads to a reduction of deflection and an increase in natural frequency. Compared with the reduced models, the present model can predict a stronger size effect since the contribution from all strain gradient components is considered, and the differences of results from all these models are diminishing when the plate thickness is far greater than the material length-sale parameter.

Size-dependent axial instability of microtubules surrounded by cytoplasm of a living cell based on nonlocal strain gradient elasticity theory

Microtubules including tubulin heterodimers arranging in a parallel shape of cylindrical hollow plays an important role in the mechanical stiffness of a living cell. In the present study, the nonlocal strain gradient theory of elasticity including simultaneously the both nonlocality and strain gradient size dependency is put to use within the framework of a refined orthotropic shell theory with hyperbolic distribution of shear deformation to analyze the size-dependent buckling and postbuckling characteristics of microtubules embedded in cytoplasm under axial compressive load. The non-classical governing differential equations are deduced via boundary layer theory of shell buckling incorporating the nonlinear prebuckling deformation and microtubule-cytoplasm interaction in the living cell environment. Finally, with the aid of a two-stepped perturbation solution methodology, the explicit analytical expressions for nonlocal strain gradient stability paths of axially loaded microtubules are achieved. It is illustrated that by taking the nonlocal size effect into consideration, the critical buckling load of microtubule and its maximum deflection associated with the minimum postbuckling load decreases, while the strain gradient size dependency causes to increase them.

Geometric–material analogy for multiscale modelling of twisted plates

It is well known that the macroscopic behaviour of many engineering materials is strongly affected by the role of underlying microstructure. Currently though, mathematical expressions linking behaviour of large scale structures to the geometry of their microscopic structure are largely lacking. In this respect, establishing quantitative links across different material lengthscales may offer new pathways for engineering design. In the present work an analogy between cross sectional geometrical properties, representing macrostructure, and a material length parameter, representing microstructure, is presented. The analogy is established through the study of a thin plate subject to axial loading undergoing finite displacements from two alternative perspectives. First, we consider a thin elastic plate with a pretwist about the loading axis where a warping term is introduced accounting for the out-of-plane deformation of the cross section. The coupled governing differential equations and the corresponding coupled boundary conditions are explicitly derived employing a classical structural mechanics approach utilising an energy variational statement. Secondly, an axially loaded thin flat plate (i.e. with no pretwist) is studied with strain gradient elasticity theory incorporating only one material length parameter representing the microstructure, in addition to the two classical Lamé stiffness constants. The ensuing analogy emerges by comparison of the governing equations of the two formulations which shows a mathematical expression can be identified, which incorporates both geometric and material length variables, that formalises the link between microscale and macroscale. This mathematical expression, which constitutes the kernel of the proposed multiscale approach, admits a twofold interpretation depending on the assumed independent variable. On the one hand, the proposed multiscale modelling approach suggests that a plate with complex global geometry can be substituted by a structurally - equivalent, flat plate with constitutive relations given by a non - local, strain gradient theory. On the other hand, the material length parameter can be interpreted on a physical basis because for the first time it has been identified as a known function of geometrical features of the structure through simple algebraic relationships for various cross sectional profiles.

A nonlinear strain gradient finite element for microbeams and microframes

The aim of this paper is to develop a total Lagrangian finite element formulation for the geometrically nonlinear deformation analysis of microbeams and microframes. A general form of the first strain gradient elasticity theory of Mindlin is employed to capture size effects at micron scales. Moreover, the von Karman strain tensor and its spatial gradient are used to include the effects of geometric nonlinearity. Due to the higher-order nature of the gradient-based governing equilibrium equations, the interpolation functions of the extension and bending field variables are chosen to be the standard \(C^1\) and \(C^2\) functions, respectively. Accordingly, a novel two-node strain gradient microbeam element is introduced. The formulation is then extended to include the deformation of microbeams with arbitrary orientation in a plane, which enables us to analyze arbitrary planar microframes composed of several microbeams at different orientations. The full Newton–Raphson scheme is used to solve the resulting nonlinear algebraic equations. Three different examples are analyzed to show the accuracy and performance of the proposed element at linear as well as nonlinear regimes of deformation. It is shown that the new element can successfully capture the so-called size effect as well as geometric nonlinearity at large distortion of microbeams and microframes at micron scales.

Three-Dimensional Elastic Analysis of Rock Excavations by Using the g2 Constant Displacement Discontinuity Method

A fast computational code is presented that is dedicated for the elastic analysis of three-dimensional excavations and cracks in rocks. The problem is solved on the boundaries that are discretized with a new triangular leaf constant displacement discontinuity element with one collocation point. The creation of the new triangular element was inspired from Mindlin's special version of grade-2 or strain-gradient elasticity theory (second gradient of displacement, g2). This element is characterized by a much better measure of the average stress at the center of gravity of the triangular element compared to that of the classical elasticity element close to regions with stress or strain gradients (e.g. notches, cracks etc). In a verification stage, the accuracy of the computational algorithm for the pressurized penny-shaped and mixed-mode elliptical crack problems that have analytical solutions is demonstrated. More specifically, it is shown that the average error of the crack tip Stress Intensity Factor predicted by the gradient modified method for nine discretizations of varying density is around 3.5% with a máximum error of 5%, while the constant displacement discontinuity element displays errors varying around 14%. Moreover, the new method preserves the simplicity and hence the high speed of the constant displacement discontinuity with only one collocation point per element, but it is far more efficient compared to it, especially close to the crack tips and corners of excavations where the displacement and stress gradients are highest.

Interrelationship between flexoelectricity and strain gradient elasticity in ferroelectric nanofilms: A phase field study

With the development of the integrated circuit technology and decreasing of the device size, ferroelectric films used in nano ferroelectric devices become thinner and thinner. Along with the downscaling of the ferroelectric film, there is an increasing influence of two strain gradient related terms. One is the strain gradient elasticity and the other one is flexoelectricity. To investigate the interrelationship between flexoelectricity and strain gradient elasticity and their combined effect on the domain structure in ferroelectric nanofilms, a phase field model of flexoelectricity and strain gradient elasticity on the ferroelectric domain evolution is developed based on Mindlin's theory of strain-gradient elasticity. Weak form is derived and implemented in finite element formulations for numerically solving the model equations. The simulation results show that upper bounds for flexoelectric coefficients can be enhanced by increasing strain gradient elasticity coefficients. While a large flexoelectricity that exceeds the upper bound can induce a transition from a ferroelectric state to a modulated/incommensurate state, a large enough strain gradient elasticity may lead to a conversion from an incommensurate state to a ferroelectric state. Strain gradient elasticity and the flexoelectricity have entirely opposite effects on polarization. The observed interrelationship between the strain gradient elasticity and flexoelectricity is rationalized by an analytical solution of the proposed theoretical model. The model proposed in this paper could help us understand the mechanism of phenomena observed in ferroelectric nanofilms under complex electromechanical loads and provide some guides on the practical application of ferroelectric nanofilms.

The Strain Gradient Elasticity Theory in Orthogonal Curvilinear Coordinates and its Applications

AbstractThe strain gradient elasticity theory including only three independent material length scale parameters has been proposed by Zhou et al. to explain the size effect phenomena in micro scales. In this paper, the general formulations of strain gradient elasticity theory in orthogonal curvilinear coordinates are derived, and then are specified for the cylindrical and spherical coordinates for the convenience of applications in cases where orthogonal curvilinear coordinates are suitable. Two basic problems, one is the twist of a cylindrical bar and the other is the radial deformation of a solid sphere, are analyzed under the cylindrical and spherical coordinates, respectively. The results reveal that only the material length scale parameter l2 enters the torsion problem, while completely disappears in the problem of radial deformation of a sphere. The size effect of radial deformation of a solid sphere is controlled by the material length scale parameters l1 and l2. In addition, for the incompressible solid sphere especially, only the material length scale parameter l1 enters this radial deformation problem by neglecting the strain gradient terms associated with hydrostatic strains. Predictably, the present paper offers an alternative avenue for measuring the three independent material length scale parameters from bar twisting and sphere expansion tests.

1D HERMITE ELEMENTS FOR C<sup>1</sup>-CONTINUOUS SOLUTIONS IN SECOND GRADIENT ELASTICITY

We present a modified strain gradient theory of elasticity for linear isotropic materials in order to account for the so-called size effect. Additional material length scale parameters are introduced and the problem of static beam bending is analyzed. A numerical solution is derived by means of a finite element approach. A global C1-continuous displacement field is applied in finite element solutions because the higher-order strain energy density additionally depends on second gradients of displacements. So-called Hermite finite elements are used that allow for merging gradients between elements. The element stiffness matrix as well as the global stiffness matrix of the problem is developed. Convergence, C1-continuity and the size effect in the numerical solution is shown. Experiments on bending stiffnesses of different sized micro beams made of the polymer SU-8 are performed by using an atomic force microscope and the results are compared to the numerical solution.

Size effects on centrosymmetric anisotropic shear deformable beam structures

In this paper, the size effect on beam structures with centrosymmetric anisotropy is studied within strain gradient elasticity theory. Applying dimension reduction to the three dimensional anisotropic gradient elasticity, the third-order shear deformable (TSD) beam is analysed. A variational approach is used to determine the equilibrium equations of TSD beam together with consistent (classical and non-classical) boundary conditions. The TSD beam theory which is suitable for deep beam structures can be replaced by (less complicated) Euler-Bernoulli beam model for thin beam structures. The anisotropic Euler-Bernoulli beam model is also formulated within the framework of strain gradient theory. This anisotropic beam theory can be used to study size effects for any types of centrosymmetric anisotropy. To address the more practical cases of composite structures, the formulation is simplified for orthotropic and transversely isotropic materials. Finally, the analytical solutions are provided for bending of simply supported (TSD and Euler-Bernoulli) beams as well as clamped Euler-Bernoulli beams. The effect of the crystal orientation with respect to the beam geometry is investigated in these examples.

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.

Nonlocal strain gradient theory for damping vibration analysis of viscoelastic inhomogeneous nano-scale beams embedded in visco-Pasternak foundation

In this article, free vibration of functionally graded (FG) viscoelastic nanobeams embedded in viscoelastic foundation exposed to hygro-thermal loading is investigated based on nonlocal strain gradient elasticity theory and a higher order refined beam theory which captures shear deformation influences without the need for any shear correction factor. Also, the exact position of neutral axis is determined. The visco-Pasternak foundation is consists of parallel springs and dashpots as well as a shear layer. Temperature-dependent material properties of FGM beam are graded across the thickness based on the power-law form. Hamilton’s principle is used to obtain nonlocal governing equations of embedded strain gradient viscoelastic nanobeam which are solved analytically for various boundary conditions. The results are validated with those available in the literature. The impacts of visco-Pasternak foundation parameters, structural damping coefficient, hygro-thermal loading, nonlocal stress parameter, nonlocal gradient parameter, power-law exponent, mode number, boundary conditions and slenderness ratio on the damping frequency of nanoscale viscoelastic FG beams are evaluated.

Size effects on double cantilever beam fracture mechanics specimen based on strain gradient theory

This paper investigates large deformation of a cantilever beam which is further employed to study the fracture behavior of double cantilever beam (DCB), based on strain gradient elasticity theory. Root effect of the DCB is also included for modelling and analyses. The numerical solutions of maximum tip deflection and strain energy release rate are presented. Results demonstrate that the consideration of large deformation is crucial at small scale, especially for more slender beams, as the bending behavior of the beam in that case is different from the classical results. The strain gradient and root effects of the DCB are more prominent when thickness of the beam is less than the material length scale parameter. The strain gradient model demonstrates significant stiffening behavior at the smaller scale. In general, the root effect may not be neglected if the length to thickness ratio of the beam is smaller. Overall, the strain energy release rate of the gradient model, even with the incorporation of root part, remains less than that of non-gradient model. This conclusion is entirely different from the classical method that neglects the uncracked part of the DCB.

Gradient-elastic stress analysis near cylindrical holes in a plane under bi-axial tension fields

This paper is devoted to a gradient-elastic stress analysis of an infinite plate weakened by a cylindrical hole and subjected to two perpendicular and independent uni-axial tensions at infinity. The problem setting can be considered as an extension and generalization of the well-known Kirsch problem of the classical elasticity theory which is here extended with respect to the external loadings and generalized with respect to the continuum framework. A closed-form solution in terms of displacements is derived for the problem within the strain gradient elasticity theory on plane stress/strain assumptions. The main characters of the total and Cauchy stress fields are analyzed near the circumference of the hole for different combinations of bi-axial tensions and for different parameter values. For the original Kirsch problem concerning a uni-axially stretched plate, the analytical solution fields for stresses and strains are compared to numerical results. These results are shown to be in a full agreement with each other and, in particular, they reveal a set of new qualitative findings about the scale-dependence of the stresses and strains provided by the gradient theory, not common to the classical theory. Based on these findings, we finally consider the physicalness of the concepts total and Cauchy stress appearing in the strain gradient model.

Size-dependent thermally affected wave propagation analysis in nonlocal strain gradient functionally graded nanoplates via a quasi-3D plate theory

This article examines the application of nonlocal strain gradient elasticity theory to wave dispersion behavior of a size-dependent functionally graded nanoplate in thermal environments. The theory contains two scale parameters corresponding to nonlocal and strain gradient effects. A quasi-3D plate theory considering shear and normal deformations is employed to present the formulation. Mori–Tanaka micromechanical model is used to describe functionally graded material properties. Hamilton’s principle is employed to obtain the governing equations of nanoplate accounting for thickness stretching effect. These equations are solved analytically to find wave frequencies and phase velocities of functionally graded nanoplate. It is indicated that wave dispersion behavior of functionally graded nanoplates is significantly affected by temperature rise, nonlocality, length scale parameter, and material composition.

Size-dependent vibration of nickel cantilever microbeams: Experiment and gradient elasticity

The size-dependent elasticity of a series of nickel cantilever microbeams was investigated experimentally for the first time. The experimental results revealed that the dimensionless natural frequencies of the cantilever microbeams increase to about 2.1 times with the beam thickness decreasing from 15 to 2.1 μm. Furthermore, based on the strain gradient elasticity theory (SGT) and by using the differential quadrature method (DQM) and the least square method (LSM), the experimental results were interpreted and the material length scale parameters in the scale of micron in elastic range were obtained. This investigation will be useful and helpful for the theoretical and numerical simulation of micro-structures and important for the design of the MEMS/NEMS.

Large deformation analysis of strain-gradient elastic beams

A total Lagrangian finite element formulation for the analysis of large deformation of beams and frames, based on the strain-gradient elasticity theory and the Timoshenko beam model, is developed herein. A generalized version of the Kirchhoff-Saint Venant constitutive equation is proposed to capture geometric nonlinearities at small scales. Also, field variables are interpolated using C1 shape functions for constructing conforming elements. Accordingly, a novel 6-DOF two-node beam element is introduced. To analyze frames, the formulation is extended so that a 9-DOF two-node frame element is produced. Several examples are studied to show the accuracy of proposed elements.

Extended Hill’s lemma for non-Cauchy continua based on the simplified strain gradient elasticity theory

Hill's lemma for the Cauchy continuum has been playing an important role in micromechanics. An extended version of Hill's lemma for non-Cauchy continua is formulated using the simplified strain gradient elasticity theory (SSGET), which contains only one material length scale parameter and can account for the microstructure-dependent strain gradient effect. As a corollary of the extended Hill's lemma, the Hill–Mandel macro-homogeneity condition for non-Cauchy continua is obtained along with the general forms of kinematically and statically admissible boundary conditions that are required for constructing an energetically equivalent homogeneous comparison material. Based on these general forms, four sets of uniform boundary conditions are identified, which are implementable in material tests and can be directly used in homogenization analyses of heterogeneous materials. It is shown that when the strain gradient effect is suppressed, the extended Hill's lemma recovers the classical Hill's lemma for the Cauchy continuum and the extended Hill–Mandel condition reduces to its classical counterpart.

Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory

This paper investigates damping vibration characteristics of hygro-thermally affected functionally graded (FG) viscoelastic nanobeams embedded in viscoelastic foundation based on nonlocal strain gradient elasticity theory. The modeling of nanobeam is carried out via a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. The viscoelastic foundation is consists of Winkler-Pasternak layer together with a viscous layer of infinite parallel dashpots. Power-law model is adopted to describe continuous variation of temperature-dependent material properties of FG nanobeam. The governing equations of nonlocal strain gradient viscoelastic nanobeam in the framework of refined beam theory are obtained using Hamilton’s principle and solved implementing an analytical solution for different boundary conditions. To validate the presented model, the results are compared with those of elastic nanobeams. The effects of linear, shear and viscous layers of foundation, structural damping coefficient, hygro-thermal environment, nonlocal parameter, material characteristic parameter, power-law exponent, mode number, boundary conditions and slenderness ratio on the frequency response of viscoelastic FG nanobeams are investigated.