Strain Gradient Terms Research Articles

Second-order Willis metamaterials: Gradient elasto-momentum coupling in flexoelectric composites

Willis materials are composites whose the overall constitutive relations exhibit coupling between momentum and strain. Recently, piezoelectric Willis materials have been studied, allowing the macroscopic momentum to be additionally coupled to the non-mechanical stimulus. Such metamaterials classified as first-order Willis materials generate cross-couplings due to their asymmetric microstructures in order to realize novel phenomena in wave propagation. In this work, we study Willis materials that are flexoelectric and offer an electric field induced by a strain gradient. We show that in the case of flexoelectric Willis materials, the momentum also gets coupled to the strain gradient term under an effective description. Hereby, an ensemble averaging-based dynamic homogenization theory is developed for flexoelectric composites to compute constitutive relations of the macroscopic fields. This second-order Willis metamaterial offers a novel coupling termed gradient elasto-momentum coupling. The presence of non-uniform strain that can break the inversion symmetry of a unit cell is thus significant in generating the imaginary portion of all cross-couplings in the absence of asymmetric microstructures.

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

We study inextensional vibrations of the spherical shell using strain gradient elasticity theory under Kirchhoff-Love hypotheses. For admissibility of the inextensional deformations the shell is punctured by making a tiny hole. The shell is assumed to be made of linearly elastic, homogeneous, and isotropic material. The closed form expression for the frequencies of inextensional modes of vibration of spherical caps of various polar angles and that of nearly complete spherical shell are derived using Rayleigh's method. Towards this, the displacement and velocity fields are obtained by setting the membrane strains to zero. Further, to facilitate the existence of inextensional vibrations, boundaries are assumed to be traction free. The frequencies are computed for the positive and negative signs in the strain gradient term of the constitutive law. The computed frequencies, employing negative sign are found to be higher than those computed using classical elasticity theory. The opposite effect is seen when the frequencies are computed with the positive sign. Further, the frequencies are found to saturate when the positive sign is used in the constitutive law. Parametric studies viz. variation in frequencies with the circumferential wavenumbers for different values of length scale parameter, variation in frequencies with circumferential wavenumbers for different values of polar angles and a given length scale parameter, variation of frequency with increasing polar angle for a given circumferential wavenumber, variation of saturation wavenumber with increasing polar angle, and increasing length scale parameter are carried out. The frequencies of nearly a spherical shell with increasing circumferential wavenumber are also computed.

A weak form quadrature element formulation of geometrically exact strain gradient shells

A geometrically exact strain gradient shell model and its weak form quadrature element formulation are proposed. The governing equations, boundary conditions and corresponding variational form of the shell model are presented. The generalized differential quadrature analogue is employed for derivative approximations of displacements and rotations with C1 continuity conditions brought about by the second-order strain gradient terms. A novel scheme based on auxiliary vectors is developed to represent and update the derivatives of constrained rotations on element edges for the numerical approach. The present quadrature shell element is feasible to incorporate strain gradient terms and free from shear and membrane locking problems due to its adjustable high-order approximation property in nonlinear shell analysis. To demonstrate the potential of this formulation for nonlinear analysis of shells with size effects, five benchmark numerical examples are presented. The results verify the feasibility of this formulation and illustrate the strain gradient effects on elastic shells undergoing large displacements and rotations.

Study of the interplay between lower-order and higher-order energetic strain-gradient effects in polycrystal plasticity

Strain-gradient (SG) plasticity refers to a class of non-local theories in which gradients of plastic slip determine the storage of geometrically necessary dislocations, introducing a length-scale dependence in the mechanical behavior of crystalline materials, which is otherwise lacking in local theories. In this work, we incorporate lower-order (LO) and higher-order energetic (HOE) strain-gradient effects into a crystal plasticity fast Fourier transform (FFT)-based formulation to investigate the interplay of the length scale that each strain-gradient term introduces at the microscale, and the mechanical properties that result at the macroscale. For an applicable range of length scales, we consider two systems: a 1-D two-phase face centered cubic (FCC) laminate and a 3-D FCC polycrystal, and two uniaxial deformation modes: monotonic tension and cyclic tension–compression. We show that increases in the individual LO and HOE length scales increase the hardening rate and strength of the material, respectively. When combined, the strong LO hardening is less pronounced than the effect alone due to the lowering of the gradients due to the HOE microstress. We demonstrate that the LO and HOE hardening manifest as “isotropic” (yield surface expansion) and “kinematic” (yield surface shift) effects, respectively, consistent with their theoretical origins. We show that in cyclic loading, the Bauschinger effect emerges in both local and non-local calculations and link its origins and severity to the behavior in the strain field, slip-system rates, and the HOE microforce.

On the dissipation at a shock wave in an elastic bar

This paper aims to relate the energy dissipated at a shock wave in a nonlinearly elastic bar to the energy in the oscillations in two related dissipationless, dispersive systems. Three, one-dimensional, dynamic impact problems are studied: Problem 1 concerns a nonlinearly elastic bar, Problem 2 a discrete chain of particles, and Problem 3 a continuum with a strain gradient term in the constitutive relation. In the impact problem considered, the free boundary of each initially quiescent body is subjected to a sudden velocity, that is then held constant for all subsequent time. There is energy dissipation at the shock in Problem 1, but Problems 2 and 3 are conservative. Problem 1 is solved analytically, Problem 2 numerically, and an approximate solution to Problem 3 is constructed analytically. The rate of increase of the oscillatory energy in Problems 2 and 3 are calculated and compared with the dissipation rate at the shock in Problem 1. The results indicate that the former is a good qualitative measure of the latter. The quantitative agreement is satisfactory at larger impact speeds but less so at smaller speeds, some possible reasons for which are discussed.

Bending analysis of nanoscopic beams based upon the strain-driven and stress-driven integral nonlocal strain gradient theories

Strain gradient and nonlocal influences have important roles in the mechanical characteristics of structures as they are scaled down to nanoscopic dimensions. In this research, an attempt is made to capture both effects on the static bending response of nanoscopic Bernoulli–Euler beams by means of a comprehensive model. For this purpose, the strain gradient theory of Mindlin is combined with the integral (original) form of Eringen’s nonlocal theory. Also, the integral nonlocal formulation is written on the basis of both strain-driven and stress-driven versions of nonlocal theory. This mixed nonlocal strain gradient formulation is capable of reducing to the size-independent and combination of integral nonlocal strain gradient family theories by employing simple substitutions. Through constructing finite difference-based differential and integral matrix operators, numerical solution approaches are developed to obtain the deflection of nanobeams under various end conditions. In addition, for the solution of governing equation of strain-driven nonlocal strain gradient theory, an efficient technique is presented that is applied to the variational statement of the problem in a direct approach. Selected numerical results are provided to explore the simultaneous influences of strain gradient terms and nonlocality based on different theories on the static bending response of beams. It is revealed that the developed size-dependent model has the ability of considering strain gradient and nonlocal influences in the most general way.

Hygro–thermo–magnetically induced vibration of nanobeams with simultaneous axial and spinning motions based on nonlocal strain gradient theory

In this paper, based on the nonlocal strain gradient theory (NSGT), the coupled vibrations of nanobeams with axial and spinning motions under complex environmental changes are modeled. A detailed parametric investigation is also performed to determine the effect of size-dependent parameters, boundary conditions, hygro–thermo–magnetic loads, axial and spin velocities on the dynamical behavior and stability regions of the system. Adopting the Galerkin discretization technique, the eigenvalue problem is solved, and natural frequencies, divergence and flutter instability thresholds of the system are extracted accordingly. The acquired outcomes are compared with the reported results in the literature. Besides, the accuracy of the numerical method is compared with analytical approaches and a good agreement is observed. The obtained results demonstrate that considering the nonlocal and hygro–thermal effects in modeling has destabilizing impacts on the dynamical response of the system. While imposing the strain gradient term and magnetic field leads to the enhancement of vibrational frequencies and enlargement of stability areas. In addition, it is concluded that in hygro–thermal environments, by ascending the spin velocity, instead of the occurrence of divergence instability, the system experiences the flutter conditions. Meanwhile, the attained outcomes indicated that the variation of spin velocity does not affect the flutter instability threshold of the system.

A second strain gradient damage model with a numerical implementation for quasi-brittle materials with micro-architectures

In this paper, a quasi-brittle damage model for micro-architectural materials is presented within the framework of isogeometric analysis to exploit the high-order continuity of the non-uniform B-spline basis functions. The constitutive relation depends not only on the strain field, but also on their first and second strain gradient terms. The simplified second-gradient elasticity formulation from Mindlin’s theory is employed with corresponding micro-architecture-related length scales to capture the material nonlocality and size effects. The strain-based damage is modelled by a nonlocal independent field coupled to the displacement field. Influences of the two types of nonlocalities (manufactured micro-architectures and damage-induced micro-defects) on the response of structures, as well as the damage initiation and propagation, are analysed through numerical experiments. A formula to determine the micro-defect-related length scale from macroscopic measurements is proposed, boosting the accuracy and applicability of the model. In addition, relevant open problems and further developments of this damage model are discussed.

Three‐dimensional elastic analysis of cracks with the g2 constant displacement discontinuity method

SummaryA new triangular element was created that could be used for the improvement of the accuracy of the constant displacement discontinuity method (CDDM). This element is characterized by three degrees of freedom in the three‐dimensional space as in the classical CDDM approach. The element is based on strain gradient elasticity theory that accounts for the difference of the average value of stress with the local stress at surfaces with large curvature (eg, crack borders, corners, and notches) in elastic bodies. The new element is characterized by a strain gradient term in addition to the two Lamè constants that gives a more representative value of the stresses at the centroid of crack edge elements compared with the classical elasticity solution and thus an accurate stress intensity factor. In this approach, special crack border elements with square‐root radius dependent displacements and numerical integrations are avoided. The extra strain gradient term is calibrated once only on the analytical solution for the penny‐shaped crack. In a verification stage, the accuracy of the computational algorithm for the elliptic and rectangular crack problems is demonstrated. Then, the algorithm, which also accounts for crack closure in compression, is applied for the modeling of crack propagation and crack interaction in uniaxial tension and compression loading. It is illustrated that the numerical predictions are in accordance with experimental evidence pertaining to uniaxial compression of transparent precracked specimens in the lab.

Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms

We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars. It has been shown that such structures can be modeled as discrete systems of nodes linked by extensional, flexural/torsional interactions corresponding to frame lattices and that the corresponding homogenized models can be strain-gradient models, i.e., models whose effective elastic energy involves components of the first and the second gradients of the displacement field. However, in the existing models, there is no coupling between the classical strain and the strain-gradient terms in the expression of the effective energy. In the present article, under some assumptions on the positions of the nodes of the unit cell, we show that classical strain and strain-gradient strain terms can be coupled. In order to illustrate this coupling we compute the homogenized energy of a particular structure that we call asymmetrical pantographic structure.

Characterizing Flexoelectricity in Composite Material Using the Element-Free Galerkin Method

This study employs the Element-Free Galerkin method (EFG) to characterize flexoelectricity in a composite material. The presence of the strain gradient term in the Partial Differential Equations (PDEs) requires C 1 continuity to describe the electromechanical coupling. The use of quartic weight functions in the developed model fulfills this prerequisite. We report the generation of electric polarization in a non-piezoelectric composite material through the inclusion-induced strain gradient field. The level set technique associated with the model supervises the weak discontinuity between the inclusion and matrix. The increased area ratio between the inclusion and matrix is found to improve the conversion of mechanical energy to electrical energy. The electromechanical coupling is enhanced when using softer materials for the embedding inclusions.

A Brief Note on the Nix–Gao Strain Gradient Plasticity Theory

The mathematical nature of the flow rule for the strain gradient plasticity theory proposed by Nix and Gao (W.D. Nix and H. Gao, J Mech Phys Solids 46(3), 411(1998)) is discussed based on the paradigm developed by Gurtin and Anand (M.E. Gurtin and L. Anand, J Mech Phys Solids 57 (3), 405 (2009)). It is shown that, when investigated on the basis of Gurtin–Anand theory, the Nix–Gao flow rule is a combination of constitutive equations for microstresses, balance law, and a constraint. As an accessory, we demonstrate that the strain gradient term introduced in the model is energetic. The results are obtained by combining a virtual-power principle of Fleck and Hutchinson, and the free-energy imbalance under isothermal conditions.

Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading

Levy's analytical solution approach is extended for analysis of rectangular strain gradient elastic plates under static loading for the first time with different boundary conditions at the edges using the method of superposition. The governing equation of equilibrium and the corresponding classical/non-classical boundary conditions for strain gradient flexural Kirchhoff plate under static loading are considered. Numerical examples on static bending of Kirchhoff nanoplates involving five different combinations of simply supported, clamped and free edge boundary conditions are presented. The effect of negative strain gradient terms is of hardening nature thus resulting in decrease in the deflection. Plates with geometry comparable to the microstructural length scale show significant size effect and this size dependency diminishes with the increase in the plate size.

Exact solutions for elastic response in micro- and nano-beams considering strain gradient elasticity

This paper presents the exact solutions derived for the static bending response of simply supported isotropic micro- and nano-beams. The governing differential equations of equilibrium and the associated boundary conditions for the beam are derived by applying the variational principle over the internal energy functional involving strain and strain gradient terms. The Mindlin’s Form II model for higher order metrics of energy involving strain gradient terms and three additional material length constants has been used for the current formulation. Constitutive relations for stress and the higher order stress in an isotropic solid, which are the conjugates of strain and strain gradient, respectively, are provided in this paper. The exact solutions for the displacement vector are obtained by solving the governing differential equations. The significance of the gradients and the corresponding conjugates are studied using numerical examples of micro- and nano-beams. These results are compared with those obtained from the classical model of elasticity for structures which does not involve strain gradients. Due to the consideration of strain gradients, the structures exhibit a significant increase in stiffness as the beam dimensions are reduced. The profiles for stress developed across the thickness are presented to study the change in the behavior of the structures at a micro- and nano-level. The importance of considering strain gradients for the design of low-dimensional structures in sensitive applications is noted. The exact solutions developed in this paper can be used as benchmark solutions for validating further research on the behavior of low-dimensional bodies using computational models and experiments. The present study calls for the development of new failure theory of isotropic materials possessing strain gradients.

Bending, buckling and vibration of small-scale tapered beams

In this paper, a comprehensive study on mechanical behavior of non-uniform small scale beams in the framework of nonlocal strain gradient theory is presented. The governing equations and boundary conditions have been developed using Hamilton's principle and solved with the aid of finite element method for introducing the bending, buckling and free vibration response of nano-beams. The current formulation could be used for all types of non-uniformities by varying the width and thickness of nano-beams along the length. The accuracy of the current model and formulation is verified by comparing the results with previous literature and those obtained by analytically solving simplified problems. In order to understand the influence of having non-uniform cross section on static and dynamic behavior of small scale beams, parametric study is presented and the effects of different parameters such as non-uniformity, non-local and strain gradient terms on natural frequency, buckling load and deformation are observed and discussed. It is seen that having non-uniform cross section in nonlocal strain gradient beams could lead to significant changes in mechanical behavior of such structures.

Material length scale of strain gradient plasticity: A physical interpretation

The physical basis and the magnitude of the material length scale in theories of strain gradient plasticity are crucial for accounting for size effects in the plastic behavior of metals at small scales. However, the underlying physics of the length scale is ambiguous. The length scales in strain gradient plasticity theories in which the plastic work density can be expressed as a function of the gradient-enhanced plastic strain are here derived from known physical quantities via critical thickness theory. A connection between the length scale and the fundamental physical quantities is elucidated. The combination of the strain and strain-gradient terms within the deformation theory of strain gradient plasticity is addressed. It is shown that, compared with the harmonic sum of the strain and strain-gradient terms in Fleck-Hutchinson theory, the linear combination gives a more reasonable value of length scale, several micrometers, which is close to that in the gradient theory of Aifantis. In contrast, the value of length scale in Nix-Gao theory is much larger, in the millimeter range. The length scales determined by critical thickness theory are in good agreement with those obtained by fitting to experimental data of wire torsion.

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.

Wave propagation in strain gradient poroelastic medium with microinertia: closed-form and finite element solutions

This article is about ultrasonic wave propagation in microstructured porous media. The classic Biot’s model is enriched using a strain gradient approach to be able to capture high-order effects when the wavelength approaches the characteristic size of the microstructure. In order to reproduce actual transmission/reflection experiments performed on poroelastic samples, and to validate the choice of the model, the computation of the time domain response is necessary, as it allows for a direct comparison with experimental results. For obtaining the time response, we use two strategies: on the one hand we compute the closed form solution by using the Laplace and Fourier transforms techniques; on the other hand we used a finite element method. The results are presented for a transmission/reflection test performed on a poroelastic sample immersed in water. The effects introduced by the strain gradient terms are visible in the time response and in agreement with experimental observations. The results can be exploited in characterization of mechanical properties of poroelastic media by enhancing the reliability of quantitative ultrasound techniques.

Thermodynamic coupling between gradient elasticity and a Cahn–Hilliard type of diffusion: size-dependent spinodal gaps

In the electrode materials of lithium ion batteries, the large variations of Li concentration during the charge and discharge processes are often accompanied by phase separations to lithium-rich and lithium-poor states. In particular, when the composition of the material moves into the spinodal region (linearly unstable uniform compositions) or into the miscibility gap (metastable uniform compositions), it tends to decompose spontaneously under composition fluctuations. If the lattice mismatch of the two phases is not negligible, coherency strains arise affecting the decomposition process. Furthermore, when the dimensions of a specimen or a grain reduce down to the nanometer level, the phase transition mechanisms are also substantially influenced by the domain size. This size effect is interpreted in the present article by developing a thermodynamically consistent model of gradient elastodiffusion. The proposed formulation is based on the coupling of the standard Cahn–Hilliard type of diffusion and a simple gradient elasticity model that includes the gradient of volumetric strain in the expression of the Helmholtz free energy density. An initial boundary value problem is derived in terms of concentration and displacement fields, and linear stability analysis is employed to determine the contribution of concentration and strain gradient terms on the instability leading to spinodal decomposition. It is shown that the theoretical predictions are in accordance with the experimental trends, i.e., the spinodal concentration range shrinks (i.e., the tendency for phase separation is reduced) as the crystal size decreases. Moreover, depending on the interplay between the strain and the concentration gradient coefficients, the spinodal region can be completely suppressed below a critical crystal size. Spinodal characteristic length and time are also evaluated by considering the dominant instability mode during the primary stages of the decomposition process, and it is found that they are increasing functions of the strain gradient coefficient.

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.