Simplified Strain Gradient Elasticity Theory Research Articles

Size effects on a one-dimensional defective phononic crystal sensor

The influence of size effects on one-dimensional defective phononic crystal (PnC) sensors based on simplified strain gradient elasticity theory (SSGET) is studied in this paper. PnCs have been widely used in high-sensitivity gas and liquid sensors by introducing defects to disrupt the perfect PnC modes. In comparison with classical elasticity theory, the SSGET includes two microstructure-related material parameters that can accurately reflect the size effects of the structure. In this paper, the stiffness matrix method was used to calculate the transmission coefficients of the proposed model, avoiding the numerical instability of the transfer matrix method. The results show that the size effects at the microscale affect the perfect PnC bandgap’s frequency range, and the microstructure constants impress the resonant frequency while detecting liquids. Consequently, the accuracy of the sensor is reduced. These findings provide a theoretical basis for designing microscale PnC sensors.

When functionally graded materials meet higher order simplified strain gradient elasticity theory

We investigate the constitutive equations when the functionally graded materials (FGMs) are considered under the strain gradient elasticity theory (SGET). The SGET considered includes both the Helmholtz and bi-Helmholtz types. The constitutive equations become more complicated and contain more terms due to the interaction between the material gradation and the nonlocal effect of the strain gradient. We also derive the governing partial differential equations (PDEs) by the balance law of linear momentum, and the PDEs also appear to be highly complicated. Under the assumption that material gradation is of the exponential form, a simpler set of governing PDEs can be reached. Solutions to those PDEs under the formulation of different modes of crack problems are discussed.

Semi-analytical solution for the Lamb’s problem in second gradient elastodynamics

In the present paper, we consider the Lamb-type problem of the dynamic response of isotropic half-space subjected to the time-harmonic vertical point load within the simplified strain gradient elasticity theory (SGET). The classical elastodynamics solution of this problem predicts singular displacements and stresses at the point of the load application. In the present study, we use the technique of Lamé potentials and Hankel transform to solve the high-order motion equations of SGET. An explicit solution accounting for traction and double traction boundary conditions is obtained in the transformed domain. The inverse transformation is performed numerically with accurate capturing of the solution behavior around the Rayleigh pole, which position is defined by the modified scale-dependent Rayleigh equation. Based on obtained results, it is shown that SGET solution is bounded at the epicenter of the load, and that it takes into account the spatial dispersion effects for the propagated waves. These effects are important for the analysis of the high-frequency processes in the structured materials. The intensity of non-classical effects in the solution is controlled by the length scale parameters of gradient model. Based on numerical calculations, we derive the universal non-dimensional relations between the amplitudes of the applied load and maximum displacements on the surface, which can be used to identify the length scale parameters of SGET based on the atomistic calculations or experimental data.

Exact solution of Eshelby's inhomogeneity problem in strain gradient theory of elasticity and its applications in composite materials

A new micromechanical approach to deal with the problem of Eshelby's inhomogeneity is developed for the prediction of the effective properties of composite materials according to the strain gradient elasticity theory. The method is based on the Green's function technique leading to an integral equation of the heterogeneous elastic problem. Within the simplified strain gradient elasticity theory, the integral equation for an infinite heterogeneous medium subjected to non-homogeneous boundary conditions is acquired. Thanks to this integral equation, the exact solution of Eshelby's inhomogeneity problem is detailed for spherical inhomogeneity and isotropic elastic behavior. From the expression of strain localization relations, the effective elastic properties of a two-phase composite material are then predicted through Mori Tanaka's homogenization scheme. To test the relevance of the suggested approach, its predictions are compared with results issued from some reference models and experimental data.

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Analytical method for the second-order homogenization of two-phase composites within Mindlin–Toupin strain gradient elasticity theory is proposed. Direct approach and self-consistent approximation are used to reduce the homogenization problem to the problem of determination of averaged Cauchy stresses, double stresses and static moments of Cauchy stresses inside the inclusions under prescribed quadratic boundary conditions. The ellipsoidal shape of inclusions and orthotropic properties of phases are assumed. Extended equivalent inclusion method with linear eigenstrain is proposed to derive the explicit relations between the Eshelby-like tensors and corresponding concentration tensors that are used to define the averaged field variables inside the inclusions. Obtained analytical solutions allow to evaluate the effective classical and gradient elastic moduli of composite materials accounting for the phases properties, volume fraction, shape and size of inclusions. Presented solution for the effective gradient moduli covers the full range of volume fraction and correctly predicts the absence of gradient effects for the homogeneous classical Cauchy medium. Examples of calculations for the composites with spherical inclusions are given. Micro-scale definition of the well-known simplified strain gradient elasticity theory is provided. Namely, it is shown that this theory is the phenomenological continuum model for the composites with isotropic matrix and with the small volume fraction of stiff isotropic spherical inclusions, which have the same Poisson’s ratio to those one of matrix phase.

Modeling the flexoelectric effect of an anisotropic dielectric nanoplate

The static bending and the flexoelectric effect induced in an anisotropic dielectric nano-plate are studied in the frame of the simplified strain gradient elasticity theory (SSGT). The Kirchhoff plate theory is considered for a simply supported nano-plate loaded by a uniformly distributed constant forces at its upper surface. The displacement and the induced spontaneous polarization are formulated such that the dimensions of the problem reduce to two dimensions only. The variational concept of the internal energy and the virtual work exerted by the external forces are used to obtain the governing equations as well as the boundary conditions. The choose of the cubic materials avoid the ambiguity between piezoelectricity and flexoelectricity, where all odd tensors are vanished. The finite Fourier transform is employed to obtain the analytical solution. The numerical calculations of the problem are presented with the help of the commercial software Maple. Some selected significant results are introduced through sets of figures with physical explanations. The results reveal that the flexoelectric effect at the surface of the clamped nanoplate should be considered and cannot be neglected. Moreover, the nanoscale dimension enhances the flexoelectric effect more than macroscale.

Bending flexoelectric effect induced in anisotropic beams with cubic symmetry

The simplified strain gradient elasticity theory (SSGT) is used to study the induced flexoelectric effect in an anisotropic dielectric beam due to bending. The beam cross-sections are uniform with internal cubic structure symmetry. Three different cases are studied in the context of the Timoshenko beam theory. In the first two cases, the simply supported beam subjected to uniformly distributed or graded forces at the lateral surface of the beam are considered, while the third case is devoted to study the clumped-free ends beam problem subjected to graded force.With appropriate boundary conditions, a closed-form solution is obtained for the problem filed equations. The study shows that, when the applied force is concentrated at the center region of the beam, the induced spontaneous polarization is directed towards the beams two ends. While, when the beam is clamped from one end, the induced spontaneous polarization moves towards the applied force.

On the strain gradient elasticity theory for isotropic materials

Classical elasticity mechanics fails to explain size-dependent phenomenon, the strain gradient elasticity theories including the couple stress theory, the simplified strain gradient elasticity theory and the general strain gradient elasticity theory are used to describe size effects. In this paper, the relations among the couple stress theory, the simplified strain gradient elasticity theory and the general strain gradient elasticity theory are derived. The general strain gradient elasticity theory contains the effects of all strain gradients while the couple stress theory and the simplified strain gradient elasticity theory include effects of only part of the strain gradients, thus, the general strain gradient elasticity theory can reduce to the couple stress theory, the simplified strain gradient elasticity theory when some strain gradients are neglected. Subsequently, a reasoning comparison with examples is performed to discuss the differences between the general strain gradient elasticity theory and the reduced theories in describing the size effects: one is the static and dynamic responses of a microbeam and the other is the tensile of a microbar. In the beam bending and vibration example, the material length-scale parameters and the deflections predicted by the reduced theories are larger while the natural frequency is smaller than that of the general strain gradient elasticity theory. The reduced theories underestimate the size-dependent response of a bending beam. In the bar tensile example, the couple stress theory fails to predict the size-dependent tensile behaviour. This illustrates clearly that compared with the reduced theories, the general strain gradient elasticity theory is capable to describe the size effects more appropriately.

Thermally induced nonlinear dynamic analysis of temperature-dependent functionally graded flexoelectric nanobeams based on nonlocal simplified strain gradient elasticity theory

In this work, thermally induced dynamic behaviors of functionally graded flexoelectric nanobeams (FGFNs) are analyzed theoretically while considering the neutral surface concept and the von Karman nonlinearity induced by thermal environment. The temperature field is assumed to vary only in the thickness direction by solving a simple steady state heat transfer equation and to be constant in the plane of the beam. The temperature-dependent material properties of FGFNs are assumed to vary continuously throughout the thickness according to a power-law form. For such FGFNs, the nonlocal simplified strain gradient elasticity theory to capture the effect of size-dependent and, the higher order shear deformation beam theory to account for rotary inertia and transverse shear strains are adopted to formulate the governing equations and associated boundary conditions, which are solved by using a two-step perturbation method. In the numerical part, comparison study is also performed to verify the present theoretical model and the parametric analysis is systematically studied. It is also found that the thermally induced bending amplitude, nonlinear frequency and frequency ratio depend enormously on the material distribution profile, the flexoelectricity, the size-dependent effect and the imposed temperature field.

Flexoelectric effect induced in an anisotropic bar with cubic symmetry under torsion

In this article, we study the flexoelectricity induced in a prismatic anisotropic bar due to torsion. The simplified strain gradient elasticity theory is considered in this study. The bar is uniform, that is, any cross-section of the bar has a rectangular shape with cubic internal structure symmetry. The traction and higher traction forces effect on the deflection and spontaneous polarization of the bar with different boundary conditions are also discussed. The induced wedge forces are also considered during this study. The magnesium oxide (MgO) physical quantities values are chosen to present a numerical example as one of the practical applications of the problem. The results are discussed and introduced graphically. The most interesting finding in this study is the wedge force directions. When the displacements inside the cross-section of the bar are uniformly distributed, the resultant wedge forces have the same inclination with the cross-section boundary. Meanwhile, if the displacement is not uniformly distributed, the wedge force inclinations with the cross-section boundary are not equal.

The influence of surface stress and surface-induced internal residual stresses on the size-dependent behaviors of Kirchhoff microplate

The present paper develops a size-dependent Kirchhoff microplate model with surface effects by using simplified strain gradient elasticity theory and surface elasticity theory. This new model is able to capture size-dependent behaviors and surface effects. The most noticeable difference of the proposed model from the existing plate models about microplates is that not only strain gradient and surface stress are taken into account, but also the surface-induced internal residual stresses are considered. Based on whether the plates having surface-induced internal residual stress or not, their governing equations have distinct differences. An extended Kantorovich method is employed to provide approximate closed-form solution for the rectangular microplate with simply supported boundary conditions. For the microplate with biaxial initial residual surface stress, the numerical results reflect that when the simply supported microplates do not have surface-induced internal residual stresses, internal length scale and biaxial surface residual stress have significant influence on the static bending of the microplates. However, when the simply supported microplates have nonzero surface-induced internal residual stresses, the effects of internal length scale and biaxial surface residual stress become very weak. It indicates that the effect of surface-induced internal residual stresses can counteract most of the effects of internal length scale and surface residual stress. This work provides a more general model for the analysis of microplate problems.

Comparison between the Mori-Tanaka and generalized self-consistent methods in the framework of anti-plane strain inclusion problem in strain gradient elasticity

In the present paper, two different averaging schemes are compared in the framework of the anti-plane inclusion problem of the strain gradient elasticity. Effective longitudinal shear modulus of fiber-reinforced composites was evaluated based on the Mori-Tanaka method (MTM) within the direct approach and based on the generalized self-consistent method (GSCM) within an energy approach. It is known that, in gradient theories, the non-uniform fields are realized inside the inhomogeneity even in the presence of a uniform strain or stress field prescribed at the infinity. Therefore, in MTM, an averaged concentration tensor is utilized. GSCM is used together with Eshelby's formula, which evaluates the total strain energy of the composite representative fragment by a particular surface integral. Thus, in this energy-based approach, one does not need to use averaged field quantities. The two mentioned approaches were compared and validated by finite element modeling in the framework of the simplified strain gradient elasticity theory. It was shown that both methods capture the inclusion size effects, however the quantitative predictions of these methods are significantly different for high volume fraction of small inclusions. It was shown that MTM underestimates the effective modulus compared to numerical simulations, while GSCM provides a sufficiently good solution. Thus, it seems that energy-based averaging schemes are preferable in the framework of gradient elasticity. Nevertheless, the MTM can be useful for the effective properties approximate evaluation of composites with small volume fraction of inclusions.

Revisiting bending theories of elastic gradient beams

In this paper, we discuss a problem of Bernoulli–Euler beam models development in the frame of the strain gradient and distortion gradient elasticity theories. Contradiction between some known size-dependent gradient beam models and analytical and numerical three-dimensional solutions found for the beam bending problems is shown. In particular, it is shown that in semi-inverse analytical solutions for a beam pure bending problem, the inverse squared dependence of normalized bending stiffness on the beam thickness could arise only due to wrong definition of the resultant bending moment and improper formulation of the boundary conditions on the top and bottom surfaces of the beam. In the correct semi-inverse solutions, finite values of normalized bending stiffness for ultra-thin gradient beams arise. The obtained results are also confirmed on the basis of 3D FE simulations realized for a pure and cantilever beam bending problems in the frame of simplified strain gradient elasticity theory. As a result, it is shown that the only correct approach for Bernoulli–Euler beam models development in gradient theories should ensures the fulfillment of boundary conditions on the top and bottom surfaces of the beam and corresponds to a model that assumed a uniaxial stress state of the beam. The consistent variational algorithm for the correct gradient beam models derivation is proposed.

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.

A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory and its applications

A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory is established in the current investigation. The generalized Euler–Lagrange equations and corresponding boundary conditions are naturally derived from the Hamilton’s principle. Then axial wave propagation of small scale bars, static bending of cantilever beams, buckling and free vibration of simply supported beams are analytically solved by using the simplified strain gradient beam theory. The influences of the Poisson’s effect as well as the weak non-local strain gradient elastic effect are discussed. The Poisson’s effect is found to increase with the increase of the beam thickness in the buckling analysis, while the higher-order bending moment induced by stretch strain gradient has an insignificant influence on the critical buckling load in our numerical analysis. However, the effect of the higher-order bending moment is very significant on axial wave propagation and static bending of micro-scale beams. The current work is very helpful in understanding the microstructure-related size dependent phenomenon.

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.

Solution of the Eshelby-type anti-plane strain polygonal inclusion problem based on a simplified strain gradient elasticity theory

The Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion of arbitrary-shape polygonal cross-section is analytically solved using a simplified strain gradient elasticity theory that incorporates one material length scale parameter. The Eshelby tensor (with four nonzero components) is obtained in a general form in terms of two scalar-valued potential functions. These potential functions, as area integrals over the polygonal cross-section, are first converted to two line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former does not contain any elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle (inclusion) size effect. For homogenization applications, the area average of the new position-dependent Eshelby tensor over the polygonal cross-section is also provided in a general form. To illustrate the newly obtained Eshelby tensor, five types of regular polygonal inclusions (i.e., triangular, quadrate, hexagonal, octagonal, and tetrakaidecagonal) are quantitatively studied by directly using the general formulas derived. The components of the induced strain and the averaged Eshelby tensor inside the inclusion are evaluated. Numerical results reveal that the induced strain varies with both the position and the inclusion size. The values of the induced strain components in a polygonal inclusion approach from below those in a corresponding circular inclusion when the inclusion size or the number of sides of the polygonal inclusion increases. The results for the averaged Eshelby tensor components show that the size effect is significant when the inclusion size is small but may be neglected for large inclusions.

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.

Strain gradient solutions of half-space and half-plane contact problems

General solutions for the problems of an elastic half-space and an elastic half-plane, respectively, subjected to a symmetrically distributed normal force of arbitrary profile are analytically derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. Mindlin’s potential function method and Fourier transforms are employed in the formulation, and the half-space and half-plane contact problems are solved in a unified manner. The specific solutions for the problems of a half-space/plane subjected to a concentrated normal force or a uniformly distributed normal force are obtained by directly applying the general solutions, which recover the existing classical elasticity-based solutions of the Flamant and Boussinesq problems as special cases. In addition, the indentation problems of an elastic half-space indented by a flat-ended cylindrical punch, a spherical punch, and a conical punch, respectively, are solved using the general solutions, leading to hardness formulas that are indentation size- and material microstructure-dependent. Numerical results reveal that the displacement and stress fields in a half-space/plane given by the current SSGET-based solutions are smoother than those predicted by the classical elasticity-based solutions and do not exhibit the discontinuity and/or singularity displayed by the latter. Also, the indentation hardness values based on the newly obtained half-space solution are found to increase with decreasing indentation radius and increasing material length scale parameter, thereby explaining the microstructure-dependent indentation size effect.

Strain gradient solution for the Eshelby-type polygonal inclusion problem

The Eshelby-type problem of an arbitrary-shape polygonal inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. The Eshelby tensor for a plane strain inclusion with an arbitrary polygonal cross section is derived in a general form in terms of three potential functions, two of which have the same definitions as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as area integrals over the polygonal cross section, are first converted to three line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson’s ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. For homogenization applications, the area average of the new Eshelby tensor over the polygonal cross section is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its area average, three types of polygonal inclusions are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for all inclusion shapes considered vary with both the position and the inclusion size. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size. By applying the newly derived solution, a homogenization analysis of a composite reinforced by polygon-shaped fibers is performed. The induced strain in the composite is evaluated, and the effective stiffness of the composite is predicted. It is found that the values of the induced strain components approach from below those in a corresponding composite reinforced by circular fibers when the fiber size or the number of sides of the polygonal fiber increases. Also, the analysis reveals that the components of the effective stiffness tensor vary slightly with the fiber shape, and thus the simpler solution for the circular inclusion problem may be used to estimate the effective stiffness of a composite reinforced by regular polygon-shaped fibers.