Gradient Elasticity Theory Research Articles

Vibration analysis of a strain gradient plate model via a mesh-free moving Kriging Interpolation Method

The strain gradient elasticity theory is adopted to capture the small-scale effect of micro/nanostructures to derive the governing equation by using Hamilton's principle. A mesh-free method on the basis of moving Kriging interpolation, of which shape function possesses C2 continuum, is developed to investigate the vibration of the strain gradient plate. Owing to the inherent Kronecker delta function property of the moving Kriging interpolation, essential boundary conditions can be directly implemented. One superiority of the moving Kriging interpolation over the traditional moving least square approximation is that it can ensure the continuity and stability at the third order derivatives, which is confirmed by comparison with analytic solutions. Parametric studies in terms of small-scale parameters, boundary conditions and side length on the frequencies and mode shapes are carried out. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of classical plate model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the mesh-free moving Kriging interpolation method has high precision and good stability in the construction of the C2 continuum shape functions.

3D strain gradient elasticity: Variational formulations, isogeometric analysis and model peculiarities

This article investigates the theoretical and numerical analysis as well as applications of the three-dimensional theory of first strain gradient elasticity. The corresponding continuous and discrete variational formulations are established with error estimates stemming from continuity and coercivity within a Sobolev space framework. An implementation of the corresponding isogeometric Ritz–Galerkin method is provided within the open-source software package GeoPDEs. A thorough numerical convergence analysis is accomplished for confirming the theoretical error estimates and for verifying the software implementation. Lastly, a set of model comparisons is presented for revealing and demonstrating some essential model peculiarities: (1) the 1D Timoshenko beam model is essentially closer to the 3D model than the corresponding Euler–Bernoulli beam model; (2) the 3D model and the 1D beam models agree on the strong size effect typical for microstructural and microarchitectural beam structures; (3) stress singularities of reentrant corners disappear in strain gradient elasticity. The computational homogenization methodologies applied in the examples for microarchitectural beams are shown to possess disadvantages that future research should focus on.

Static and dynamic modeling of functionally graded Euler–Bernoulli microbeams based on reformulated strain gradient elasticity theory using isogeometric analysis

Strain Gradient Theories (SGTs) are often considered to capture the intrinsic microstructural behaviors of the microbeams. In the present work, the recent Reformulated Strain Gradient Theory (RSGT), which accounts the effect of the strain gradient, couple stress (rotation gradients) and velocity gradients collectively, is extended for the static and dynamic modeling of Functionally Graded (FG) Euler–Bernoulli microbeams to account for the effect of the strain gradient, couple stress (rotation gradients) and velocity gradients collectively. The present formulation also considers the shift of neutral axis due to material inhomogeneity in the FG microbeam. The problems of static deflection and free vibrations with different boundary conditions are addressed. Numerical solutions are obtained using isogeometric analysis. Analytical solutions of the present formulation for simply supported FG microbeam for static deflection and free vibration problems are also obtained and employed to validate the numerical results. The influence of the material index, different length scale parameters on the static deflection and free vibration responses of the FG microbeams are discussed. The present FG RSGT microbeam shows the prominent size scale effect in thin beams and predicts stiffer response than the FG microbeam based on modified couple stress theory.

Strain gradient elasto-plasticity model: 3D isogeometric implementation and applications to cellular structures

In the present work, we combine Mindlin’s strain gradient elasticity theory and Gudmundson–Gurtin–Anand strain gradient plasticity theory to form a unified framework. The gradient plasticity model is enriched by including the gradient of elastic strains into the expression of the internal virtual work and free energy. This augments the modelling capabilities by incorporating elasticity-related length scales along with plasticity-related energetic and dissipative ones. The strong form governing equations are derived via the principle of virtual work addressing a complete set of boundary conditions. The fourth-order boundary value problem of the gradient elasto-plasticity model is then formulated in a variational form within an H2 Sobolev space setting. Conforming Galerkin discretizations for numerical results are obtained utilizing an isogeometric approach with NURBS basis functions of degree p≥2 providing Cp−1-continuity. The implementation follows a viscoplastic constitutive framework and adopts the backward Euler time integration scheme. A set of benchmark examples is considered to illustrate convergence properties and to accomplish parameter studies. It is shown that the elastic length scale parameter controls the slope of the elastic part and causes an additional hardening in the plastic part of the material response curves. Finally, an illustrative example is considered in order to demonstrate the applicability of both the continuum model and the numerical method in capturing the size-dependent torsion response of cellular structures.

Fracture behaviour of nanobeams through Two-Phase Local/Nonlocal Stress-Driven model

The aim of the present work is to employ the two-phase local/nonlocal Stress-Driven integral Model (SDM) to analyse the size-dependent Mode I fracture behaviour of Bernoulli-Euler cracked nanobeams, in terms of energy release rate, stress intensity factor and nonlocal stress field near the crack tip. Both edge- and centrally-cracked nanobeams, subjected to concentrated forces, are examined. The edge-cracked nanobeam is modelled as a pair of cantilever nanobeams, whereas the centrally-cracked nanobeam as a pair of double-clamped nanobeams with internal discontinuity due to concentrated loads. Moreover, a comparison with the results obtained from a gradient elasticity theory based model, available in the literature, is performed. From the present study, it is observed that the energy release rate decreases by increasing the nonlocality, showing the superior fracture performance of nanobeams with respect to large-scale beams.

New approach to failure of pre-cracked brittle materials based on regularized solutions of strain gradient elasticity

In this paper, we use the feature of the strain gradient elasticity theory (SGET) related to the regularization of classical singularity problems. We suggest that the structural analysis of pre-cracked materials can be reduced to failure analysis within SGET by using the appropriate failure criteria formulated in terms of the Cauchy stresses. These stresses are work-conjugated to strains, and they have non-singular values in SGET solutions for the problems with cracks and sharp notches. We confirm our suggestion based on the full-field numerical simulations and provide examples of identification of the length scale parameters of SGET for the known experimental data with pre-cracked brittle and quasi-brittle materials. We show that the identified length scale parameters allow us to predict the failure loads for the experimental samples with different type of cracks by using the maximum principal Cauchy stress criterion. In the experiments with a mixed-mode fracture of the quasi-brittle material (PMMA with inclined cracks), we found that there may arise a transition between the failure determined by the maximum principal stress criterion and the criterion related to the second invariant of the Cauchy stress tensor that can be explained by the change of the stress state triaxiality.

On the wave dispersion in functionally graded porous Timoshenko-Ehrenfest nanobeams based on the higher-order nonlocal gradient elasticity

Dispersion characteristics of flexural waves in a functionally graded (FG) porous nanobeam are analytically examined. Kinematics of the nano-sized beam is assumed to be consistent with the Timoshenko-Ehrenfest beam model. Material behavior of the FG porous beam is considered to vary symmetrically along the beam thickness while appropriate symmetric porosity models are applied to account for two diverse distributions of porosity with variable volume of voids. For the first time, size-dependent response of the symmetric FG porous nanobeam is realized within the framework of the higher-order nonlocal gradient elasticity theory. The integro-differential constitutive laws of the stress resultant fields are established and reinstated with the equivalent differential relations equipped with non-standard boundary conditions. The closed-form solution of the phase velocity of flexural waves is analytically determined. The ensuing numerical results of the flexural wave dispersion detect new benchmarks for numerical analysis and can be effectively exploited in design and optimization of composite nano-structural elements of advanced nano-electro-mechanical systems.

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.

Nonlinear forced vibrations of a slightly curved nanotube conveying fluid based on the nonlocal strain gradient elasticity theory

Nonlinear forced vibrations of a slightly curved nanotube conveying fluid based on the nonlocal strain gradient elasticity theory

Колебания слоя с расслоением в рамках градиентной теории упругости

The direct problem of antiplane oscillations of a layer with delamination in the context of the gradient theory of elasticity is considered. The gradient model proposed by Aifantis is taken as a mathematical model. The main attention has been paid to the analysis of mechanical fields at the crack bank and at its tips - stress concentrators. The study is carried out using the method of boundary integral equations (BIE). The BIE for the gradient of displacement field on the crack is constructed. The analysis of the constructed BIE is carried out and the cubic singularity is explicitly revealed. The solution of singular BIE is constructed using approximating Chebyshev polynomials. A study for cracks of small relative length - asymptotic approach is carried out, simple semi-analytical expressions for constructing the crack swap function are obtained, the range of efficiency of the asymptotic approach is obtained. The stress fields in the area of the crack tips are constructed. Numerical results of computational experiments are presented.

Vibrations of a Layer with Delamination in the Framework of the Gradient Elasticity Theory

Vibrations of a Layer with Delamination in the Framework of the Gradient Elasticity Theory

A novel spatial–temporal nonlocal strain gradient theorem for wave dispersion characteristics of FGM nanoplates

It is clear that there is a relationship between time and space in nanostructures that are always attacked by waves with wavelengths within the intrinsic characteristic length of the nanostructure. Also, the viscoelastic behavior of nanostructures is affected by this relationship between nonlocal time and nonlocal space. It is worth noting that the conventional tempo-spatially decoupled nonlocal viscoelasticity does not give accurate results regarding the viscoelastic response of these structures. Hence, herein and for the first time, a novel fractional nonlocal time–space strain gradient viscoelasticity is implemented to be able to give a correct answer in addressing the dispersion of elastic waves inside functionally graded material (FGM) viscoelastic nanoplates. The constructive equations of this theory are based on the theory of nonlocal strain gradient elasticity. The properties of the FGM studied in this research are obtained by the power-law model, and the kinematic relations are extracted based on the refined higher-order plate theory. Subsequently, motion equations are written using Hamilton’s principle and solved analytically. The results of this work reveal that an increase in the nonlocal parameter can reduce the loss factor in a remarkable way due to the coupling between the nonlocalities in the spatial and temporal domains.

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin\u2013Mindlin anisotropic first strain gradient elasticity

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, 3+11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1+6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

A phase field model for fracture based on the strain gradient elasticity theory with hybrid formulation

In this paper, a novel phase field (PF) model for fracture is developed in the framework of strain gradient elasticity. The strain energy decomposition methods initially proposed for linear elastic fracture problems are extended to the gradient elasticity situation. The PF model is numerically implemented through Abaqus subroutine UEL (user defined element) with nine-node quatrilateral C1-continous element. The finite element implementation is validated by studying the stress field near the static crack tip. When the length-scale parameter of the gradient elasticity is zero, the stress field is consistent with the analytical solution predicted by linear elastic fracture mechanics (LEFM). For non-zero length-scale parameters, the Cauchy stress at the crack tip is found to be non-singular and the numerical results exhibit less mesh sensitivity in the crack tip region compared with the linear elastic case. After that, four different strain energy decomposition methods (Method I-IV) are compared and discussed by studying Mode I fracture behavior under tensile load. It is found that Method I can properly characterize the toughening effect of gradient elasticity. Based on Method I, the influences of the fracture length-scale parameter lc and the strain gradient length-scale parameter l on the characteristics of the smeared crack model and the load–displacement curves are systematically discussed. The specimen size effect of Mode I crack propagation is also investigated. It is found that the length-scale parameter l has a larger influence on the load–displacement curve as the specimen size decreases. Finally, crack propagation behavior in pure shear test, three point bending test and tensile test of a notched plate with hole is investigated. It is demonstrated that the proposed PF model can well predict the curvilinear crack propagation path and properly characterize the effect of gradient elasticity. Thus, the PF model developed in this paper can be applied to study complex fracture behaviors in the framework of gradient elasticity, where the effect of internal structure on the mechanical responses become significant.

Strain Gradient Theory Based Dynamic Mindlin-Reissner and Kirchhoff Micro-Plates with Microstructural and Micro-Inertial Effects

In this study, a dynamic Mindlin–Reissner-type plate is developed based on a simplified version of Mindlin’s form-II first-strain gradient elasticity theory. The governing equations of motion and the corresponding boundary conditions are derived using the general virtual work variational principle. The presented model contains, apart from the two classical Lame constants, one additional microstructure material parameter g for the static case and one micro-inertia parameter h for the dynamic case. The formal reduction of this model to a Kirchhoff-type plate model is also presented. Upon diminishing the microstructure parameters g and h, the classical Mindlin–Reissner and Kirchhoff plate theories are derived. Three points distinguish the present work from other similar published in the literature. First, the plane stress assumption, fundamental for the development of plate theories, is expressed by the vanishing of the z-component of the generalized true traction vector and not merely by the zz-component of the Cauchy stress tensor. Second, micro-inertia terms are included in the expression of the kinetic energy of the model. Finally, the detailed structure of classical and non-classical boundary conditions is presented for both Mindlin–Reissner and Kirchhoff micro-plates. An example of a simply supported rectangular plate is used to illustrate the proposed model and to compare it with results from the literature. The numerical results reveal the significance of the strain gradient effect on the bending and free vibration response of the micro-plate, when the plate thickness is at the micron-scale; in comparison to the classical theories for Mindlin–Reissner and Kirchhoff plates, the deflections, the rotations, and the shear-thickness frequencies are smaller, while the fundamental flexural frequency is higher. It is also observed that the micro-inertia effect should not be ignored in estimating the fundamental frequencies of micro-plates, primarily for thick plates, when plate thickness is at the micron scale (strain gradient effect).

Incompatible strain gradient elasticity of Mindlin type: screw and edge dislocations

The fundamental problem of dislocations in incompatible isotropic strain gradient elasticity theory of Mindlin type, unsolved for more than half a century, is solved in this work. Incompatible strain gradient elasticity of Mindlin type is the generalization of Mindlin’s compatible strain gradient elasticity including plastic fields providing in this way a proper eigenstrain framework for the study of defects like dislocations. Exact analytical solutions for the displacement fields, elastic distortions, Cauchy stresses, plastic distortions and dislocation densities of screw and edge dislocations are derived. For the numerical analysis of the dislocation fields, elastic constants and gradient elastic constants have been used taken from ab initio DFT calculations. The displacement, elastic distortion, plastic distortion and Cauchy stress fields of screw and edge dislocations are non-singular, finite, and smooth. The dislocation fields of a screw dislocation depend on one characteristic length, whereas the dislocation fields of an edge dislocation depend on up to three characteristic lengths. For a screw dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type agree with the corresponding ones in simplified incompatible strain gradient elasticity. In the case of an edge dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type are depicted more realistic than the corresponding ones in simplified incompatible strain gradient elasticity. Among others, the Cauchy stress of an edge dislocation obtained in incompatible isotropic strain gradient elasticity of Mindlin type looks more physical in the dislocation core region than the Cauchy stress obtained in simplified incompatible strain gradient elasticity and is in good agreement with the stress fields of an edge dislocation computed in atomistic simulations. Moreover, it is shown that the shape of the dislocation core of an edge dislocation has a more realistic asymmetric form due to its inherent asymmetry in incompatible isotropic strain gradient elasticity of Mindlin type than the dislocation core possessing a cylindrical symmetry in simplified incompatible strain gradient elasticity. It is revealed that the considered theory with the incorporation of three characteristic lengths offers a more realistic description of an edge dislocation than the simplified incompatible strain gradient elasticity with only one characteristic length.

A nonlocal operator method for finite deformation higher-order gradient elasticity

We present a general finite deformation higher-order gradient elasticity theory. The governing equations of the higher-order gradient solid along with boundary conditions of various orders are derived from a variational principle using integration by parts on the surface. The objectivity of the energy functional is achieved by carefully selecting the invariants under rigid-body transformation. The third-order gradient solid theory includes more than 10.000 material parameters. However, under certain simplifications, the material parameters can be greatly reduced; down to 3. With this simplified formulation, we develop a nonlocal operator method and apply it to several numerical examples. The numerical analysis shows that the high gradient solid theory exhibits a stiffer response compared to a ’conventional’ hyperelastic solid. The numerical tests also demonstrate the capability of the nonlocal operator method in solving higher-order physical problems.

Analytical and meshless numerical approaches to unified gradient elasticity theory

The unified gradient elasticity theory with applications to nano-mechanics of torsion is examined. The Reissner stationary variational principle is invoked to detect the differential and boundary conditions of equilibrium along with the consistent form of the constitutive laws. An efficient meshless numerical approach is established by making recourse to the Reissner variational functional wherein independent series solution of the kinematic and kinetic field variables are proposed. Suitable forms of the coordinate functions, in terms of the Chebyshev polynomials, are introduced to fulfill a set of kinematic and higher-order boundary conditions in the elastic torsion of nano-bars with practical kinematic constraints. Torsional behavior of the unified gradient elastic bar is studied for structural schemes of applicative interest. An excellent agreement between the torsional responses of the nano-bar detected based on the established meshless method and obtained exact analytical solution is realized. The proposed meshless numerical approach is confirmed to have a fast convergence rate and an admissible convergence region in determination of the torsional rotation field with high accuracy. The introduced meshless method is demonstrated to be highly efficacious in characterizing both the softening and stiffening structural behaviors at nano-scale. The presented numerical approach therefore paves the way ahead in mechanics of nano-structures.

Isogeometric analysis of size-dependent Bernoulli–Euler beam based on a reformulated strain gradient elasticity theory

In this paper, we present an efficient and accurate numerical approach for static bending and free vibration analyses of microstructure-dependent Bernoulli-Euler beams. The current approach includes strain gradient, couple stress (rotation gradient) and velocity gradient effects simultaneously through a reformulated strain gradient elasticity, which applies only one material parameter for each gradient effect. Based on the Hamilton’s principle and an Isogeometric Analysis (IGA) approach, the governing equations are derived and solved, respectively, which effectively fulfills the higher continuity requirements in the present microstructure-dependent Bernoulli-Euler beam formulation. The static bending and free vibration analyses of simply supported microstructure-dependent beams are studied by directly applying the current approach and compared with the corresponding analytical solutions. The physical mesh convergence and numerical results prove the high performance and accuracy of the present numerical approach. In addition, the cantilever and clamped microbeams are also carried out to show the applicability of the present approach.

Strain gradient enhanced chemo-mechanical modeling of fracture in cathode materials for lithium-ion batteries

Next generation lithium-ion batteries (LIBs) cathodes are composed of polycrystalline microstructures where spatial heterogeneity such as grain size is often comparable to the geometric dimension of the cathodes. Incorporating the influence of the aforementioned heterogeneity in a chemo-mechanical continuum description is necessary to address the diffusion induced stress field during electrochemical cycling. Therefore, in the present study, the micro-structural size-dependent characteristic length of the cathode is embedded through a non-classical continuum mechanics approach known as the strain gradient elasticity (SGE) theory. Accordingly, a thermodynamically consistent multi-physics framework is developed where nonlinear diffusion kinetics is included along with the energetic interaction between the lithium-ions and host lattice of the electrode. The coupled governing equations are derived for the spherical cathode, and the generalized differential quadrature (GDQ) method is utilized to build the numerical procedure. The role of material heterogeneity length scale is addressed for strongly coupled chemo-mechanical process. Subsequently, the growth of potential surface flaws is examined under multiple length scales, particle size, and charge rate conditions. Ultimately, a comprehensive fracture map is established to illustrate the regime of flaw insensitivity, partial breakage and complete disintegration. Thus, the present findings can provide a suitable design perspective for fail-safe particulate type composite cathodes.