Strain Gradient Elasticity Model Research Articles

A microsphere-homogenized strain gradient elasticity model for polymers

A microsphere-homogenized strain gradient elasticity model for polymers

A non-second-gradient model for nonlinear elastic bodies with fibre stiffness

In the past, to model fibre stiffness of finite-radius fibres, previous finite-strain (nonlinear) models were mainly based on the theory of non-linear strain-gradient (second-gradient) theory or Kirchhoff rod theory. We note that these models characterize the mechanical behaviour of polar transversely isotropic solids with infinitely many purely flexible fibres with zero radius. To introduce the effect of fibre bending stiffness on purely flexible fibres with zero radius, these models assumed the existence of couple stresses (contact torques) and non-symmetric Cauchy stresses. However, these stresses are not present on deformations of actual non-polar elastic solids reinforced by finite-radius fibres. In addition to this, the implementation of boundary conditions for second gradient models is not straightforward and discussion on the effectiveness of strain gradient elasticity models to mechanically describe continuum solids is still ongoing. In this paper, we develop a constitutive equation for a non-linear non-polar elastic solid, reinforced by embedded fibers, in which elastic resistance of the fibers to bending is modelled via the classical branches of continuum mechanics, where the development of the theory of stresses is based on non-polar materials; that is, without using the second gradient theory, which is associated with couple stresses and non-symmetric Cauchy stresses. In view of this, the proposed model is simple and somewhat more realistic compared to previous second gradient models.

Causality in strain gradient elasticity: An internal variables approach

The absence of higher order time derivatives in equations of motion causes the infinite velocity of wave propagation in strain gradient elasticity models. This issue needs to be avoided from both a theoretical and practical viewpoint. It is shown that dual internal variables approach solves the causality problem. Furthermore, the primary internal variable can be interpreted as the Laplacian of the strain gradient resulting in an Aifantis-type strain gradient model.

Reduced strain gradient elasticity model with two characteristic lengths: fundamentals and application to straight dislocations

In this paper, the reduced strain gradient elasticity model with two characteristic lengths is proposed and presented. The reduced strain gradient elasticity model is a particular case of Mindlin’s first strain gradient elasticity theory with a reduced number of material parameters and is a generalization of the simplified first strain gradient elasticity model to include two different characteristic length scale parameters. The two characteristic lengths have the physical meaning of longitudinal and transverse length scales. The reduced strain gradient elasticity model is used to study screw and edge dislocations and to derive analytical solutions of the dislocation fields. 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 two characteristic lengths. For the numerical analysis of the dislocation fields, the material parameters including the characteristic lengths have been used, computed from a second nearest neighbor modified embedded-atom method (2NN MEAM) potential for aluminum.

On the elastic wedge problem within simplified and incomplete strain gradient elasticity theories

In this paper, we investigate the behavior of the strain gradient elasticity solutions around the sharp edges of the body loaded by the concentrated loads. We consider the general Mindlin–Toupin strain gradient elasticity (SGET) and several simplified models, including the dilatation gradient elasticity theory (DGET) and the couple stress theory (CST). In the framework of a plane strain problem for an elastic wedge with prescribed tip displacement we obtain analytical assessments for the displacement field and the stress state around the wedge apex. Involving the Papkovich–Neuber solution we show that the nonsingular solutions for this problem can be obtained within the general SGET and within simplified gradient models that provide the regularization of the dilatational and the rotational parts of the displacement field both. In the incomplete theories such as DGET or CST the singularity in the displacements and stresses at the wedge apex cannot be avoided. This result is validated also based on the full-field numerical solutions for the wedge-type domains. It is shown that the stress field around the wedge apex is bounded in the complete gradient theories, while the mesh dependent solutions are realized in DGET and CST. Generally, obtained results means that the boundary value problems with the edge-type loading should be abandoned within the incomplete gradient theories like DGET and CST or it can be considered only under some additional assumptions (similarly to classical elasticity).

A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory

This paper is concerned with the analysis of deflection, stresses, buckling and vibration analysis for a porous functionally graded (FG) curved nanobeam with different boundary conditions using a nonlocal strain gradient theory. The curved nanobeam is made of porous FG material. This property varies according to a power-law function over the thickness. The stresses can be calculated on the basis of the nonlocal strain gradient elasticity model which contains both the nonlocal stress and strain gradients stress field. The Hamilton’s principle is adopted in order to develop differential equation and boundary condition. Numerical results with various cases of boundary conditions are carried out with a view to discuss the influences of porosity factor, nonlocal, length-scale parameters and gradient index on the deflection, stresses, buckling and vibration of porous FG curved nanobeam.

Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam

The fundamental motivation of this research is to investigate the effect of flexoelectricity on a piezoelectric nanobeam for the first time involving internal viscoelasticity. To date, the effect of flexoelectricity on the mechanical behavior of nanobeams has been investigated extensively under various physical and environmental conditions. However, this effect as an internal property of materials has not been studied when the nanobeams include an internal damping feature. To this end, a closed-circuit condition is considered taking converse piezo–flexoelectric behavior. The kinematic displacement of the classical beam using Lagrangian strains, also applying Hamilton’s principle, creates the needed frequency equation. The natural frequencies are measured in nanoscale by the available nonlocal strain gradient elasticity model. The linear Kelvin–Voigt viscoelastic model here defines the inner viscoelastic coupling. An analytical solution technique determines the values of the numerical frequencies. The best findings show that the viscoelastic coupling can directly affect the flexoelectricity property of the material.

Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior

A computational homogenization method to determine the effective parameters of Mindlin’s Strain Gradient Elasticity (SGE) model from a local heterogeneous Cauchy linear material is developed. The devised method, which is an extension of the classical one based on the use of Quadratic Boundary Conditions, intents to correct the well-known non-physical problem of persistent gradient effects when the Representative Volume Element (RVE) is homogeneous. Those spurious effects are eliminated by introducing a microstructure-dependent body force field in the homogenization scheme together with alternative definitions of the localization tensors. With these modifications, and by a simple application of the superposition principle, the higher-order stiffness tensors of SGE are computed from elementary numerical calculations on RVE. Within this new framework, the convergence of SGE effective properties is investigated with respect to the size of the RVE. Finally, a C1-FEM procedure for simulating the behavior of the effective material at the macro scale is developed. We show that the proposed model is consistent with the solutions arising from asymptotic analysis and that the computed effective tensors verify the expected invariance properties for several classes of anisotropy. We also point out an issue that the present model shares with asymptotic-based solutions in the case of soft inclusions. Applications to anisotropic effective strain-gradient materials are provided, as well as comparisons between fully meshed structures and equivalent homogeneous models.

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. https://doi.org/10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

An MD simulation study to the indentation size effect of polystyrene and polyethylene with various indenter shapes and loading rates

Abstract Several nanoindentation MD simulations were conducted to investigate the indentation size effect (ISE) of Polystyrene (PS) and Polyethylene (PE) at nano-scale using different indenters and loading rates. A strain gradient elasticity model in which the ISE is related to the Frank energy caused by finite bending stiffness and neighboring interactions of chains was imported to describe the ISE in this work. Two large-scale molecular substrates for PS and PE were constructed to compare the ISE performance during indentation simulations. The spherical indenter was firstly considered, and an uptrend of the calculated hardness and modulus with increasing indenter radius was found for both PS and PE. A conical indenter consisting of 25 virtual rigid spheres used in LAMMPS was designed. Considering different tip sizes and loading rates, indentation simulations using these conical indenters were conducted. The uptrend of hardness curves at the initial indentation stage was also found in conical cases. Through fitting the hardness curves, the strengthening effects of tip size and loading rate were certified, especially for PS. At nano-scale, the dominant effect of benzene ring component on ISE performance was confirmed. The downtrend of ISE was found to exist at nano-scale for PE although it was not obviously found in experimental cases and the contributing factors for this kind of ISE were discussed.

Analytic solutions of a reducible strain gradient elasticity model for solid cylinder with a cavity and its application in zonal failure

• Reducibility condition for the fourth-order equilibrium equation . • Analytic solutions of the fourth-order equilibrium equation in the strain gradient elasticity model for a solid cylinder with a cavity. • Classification of analytic solutions in the strain gradient elasticity model for a solid cylinder with cavity. • Quasi periodic solution explaining the occurrence of zonal failure. In this work, the reducibility condition of the fourth-order equilibrium equation in the strain gradient elasticity (SGE) model for solid cylinder with a cavity is obtained. When the reducibility condition is satisfied, the analytic displacement, generalized radial stress , and generalized angular stress can be solved out, and according to the higher-order coefficients, internal length scale, and Lamé constants , the displacement, generalized radial stress, and generalized angular stress are classified into four types: (1) conventional elasticity solution, (2) quasiperiodic SGE solution, (3) monotonous SGE solution, and (4) non-real-number solution. Quasiperiodic generalized radial stress and generalized angular stress are used to explain the occurrence of zonal failure of surrounding rock of a circular roadway. Numerical analysis with MATLAB is applied to study the influence of loading on zonal failure of surrounding rock of a circular raodway.

Size effects of small-scale beams in bending addressed with a strain-difference based nonlocal elasticity theory

A strain-difference based nonlocal elasticity model devised by the authors elsewhere (Polizzotto et al., Int. J. Solids Struct. 25 (2006) 308–333) is applied to small-scale homogeneous beam models in bending under static loads in the purpose to describe the inherent size effects. With this theory —belonging to the strain-integral nonlocal model family, but exempt from anomalies typical of the Eringen nonlocal theory— the relevant beam problem is reduced to a set of three mutually independent Fredholm integral equations of the second kind (each independent of the beam’s ordinary boundary conditions, only one depends on the given load), which can be routinely solved numerically. Applications to five cases of beam samples (usually addressed in the literature) are performed, the obtained results are graphically illustrated and compared with analogous results from the literature. Size effects of stiffeningtype are found for all beam samples, in agreement with the analogous results obtained with the well-known and widely accepted strain gradient elasticity model. Analogous size effects are expected to be predicted for other multi-dimensional structures, all of which seems to confirm the smaller-is-stifferphenomenon.

Kirchhoff–Love shells within strain gradient elasticity: Weak and strong formulations and an [formula omitted]-conforming isogeometric implementation

A strain gradient elasticity model for shells of arbitrary geometry is derived for the first time. The Kirchhoff–Love shell kinematics is employed in the context of a one-parameter modification of Mindlin’s strain gradient elasticity theory. The weak form of the static boundary value problem of the generalized shell model is formulated within an H3 Sobolev space setting incorporating first-, second- and third-order derivatives of the displacement variables. The strong form governing equations with a complete set of boundary conditions are derived via the principle of virtual work. A detailed description focusing on the non-standard features of the implementation of the corresponding Galerkin discretizations is provided. The numerical computations are accomplished with a conforming isogeometric method by adopting Cp−1-continuous NURBS basis functions of order p≥3. Convergence studies and comparisons to the corresponding three-dimensional solid element simulation verify the shell element implementation. Numerical results demonstrate the crucial capabilities of the non-standard shell model: capturing size effects and smoothening stress singularities.

Free vibrations of elastic beams by modified nonlocal strain gradient theory

Size-dependent axial and flexural free vibrations of Bernoulli–Euler nano-beams are investigated by the modified nonlocal strain gradient elasticity model presented in (Barretta & Marotti de Sciarra, 2018). The corresponding elastodynamic problem, with the natural constitutive boundary conditions, is solved by an effective analytical methodology. Axial and flexural fundamental frequencies are determined for cantilever and fully-clamped nano-beams. Effects of nonlocal and gradient scale parameters on fundamental frequencies are examined and compared with those obtained by Eringen's local/nonlocal mixture model. New benchmarks are found for vibrations of beams. The adopted nonlocal strain gradient model, with the appropriate constitutive boundary conditions, is capable of capturing both softening and stiffening dynamical responses. Accordingly, it provides an advantageous approach for design and optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS).

Nonlocal second-order strain gradient elasticity model and its application in wave propagating in carbon nanotubes

Recent experimental studies indicate that Young’s modulus of carbon nanotubes increases steeply with tube diameter decreasing. The consideration of this effect is of great importance for the fabrication and exploitation of nano-electromechanical devices. Nevertheless, the rapid stiffness enhancement effect noticed from experimental observation maybe unable to be predicted by using size-dependent elasticity models available in literatures. It is strongly necessary to further shed light on the size-dependent mechanical mechanism and characterize the rapid strengthening effect of stiffness for nano-sized materials. To achieve this goal, the nonlocal second-order strain gradient elasticity model is established by introducing the second-order strain gradient field with nonlocal effect into the stored energy function of nonlocal first-order strain gradient elasticity theory. With the aids of the laws of thermodynamics, the constitutive relations are obtained. The Hamilton principle is used to derive the governing equations of equilibrium and boundary conditions. The proposed model is applied to investigate the problem of wave propagating in carbon nanotubes. The new dispersion relations derived are presented for evaluating the influences of size-dependent parameters on the characteristics of wave propagation. The results show that present model can predict the rapid increasing effect of carbon nanotubes with the decrease of tube size.

Exact solution for size-dependent elastic response in laminated beams considering generalized first strain gradient elasticity

In this paper, exact solutions for the static bending response of laminated composite beams have been derived considering the effect of strain gradient elasticity. Separate solutions for the laminate comprising of isotropic and orthotropic laminae have been presented here. The sixth-order tensor for the higher order elastic coefficients has been evaluated using the generalized first strain gradient elasticity model with three independent material length constants. The significance of the gradient elasticity in low-dimensional structures has been established employing numerical examples of micro and nano-beams, and comparing the current results with the classical elasticity results. A parametric study over the variation of the axial, transverse Cauchy and physical stresses across the thickness of the beam has been conducted. Also, the discontinuity of the inter-laminar transverse Cauchy stresses and higher order stresses across the thickness has been studied in the current framework. The exact solutions developed in this paper may be used as benchmark results for validating further research involving the strain gradient elastic response of low-dimensional laminated composite beams.

Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses*

Here we discuss the similarities and differences in anti-plane surface wave propagation in an elastic half-space within the framework of the theories of Gurtin–Murdoch surface elasticity and Toupin–Mindlin strain-gradient elasticity. The qualitative behaviour of the dispersion curves and the decay of the obtained solutions are quite similar. On the other hand, we show that the solutions relating to the surface elasticity model are more localised near the free surface. For the strain-gradient elasticity model there is a range of wavenumbers where the amplitude of displacements decays very slowly.

Temperature and porosity effects on wave propagation in nanobeams using bi-Helmholtz nonlocal strain-gradient elasticity

In this paper, applying a general nonlocal strain-gradient elasticity model with two nonlocal and one strain-gradient parameters, wave dispersion behavior of thermally affected and elastically bonded nanobeams is investigated. The two nanobeams are considered to have material imperfections or porosities evenly dispersed across the thickness. Each nanobeam has uniform thickness and is modeled by refined shear deformation beam theory with sinusoidal transverse shear strains. The governing equations of the system are derived by Hamilton’s rule and are analytically solved to obtain wave frequencies and the velocity of wave propagation. In the presented graphs, one can see that porosities, temperature, nonlocal, strain gradient and bonding springs have great influences on the wave characteristics of the system.

Element-free Galerkin model of nano-beams considering strain gradient elasticity

This paper deals with the investigation of the static bending response of nano-beams upon application of a mechanical load, considering the effect of gradient elasticity. The gradient elasticity is modelled using the strain gradient elasticity model with three independent material length constants. For the numerical modelling of the structural response, the higher-order displacement theory is used. The element-free Galerkin method is employed to interpolate the displacement field in the continuum in terms of the nodal displacements. Thereafter, using the variational principle, the governing equations of motion in terms of the nodal displacements are derived in the algebraic form. The model developed in this paper is validated by comparison with the exact solutions available in the literature, for the specific case of a simply supported beam under a sinusoidally distributed load. Following this, the numerical model is solved for various other boundary conditions and loading functions. The effect of the strain gradient elasticity is studied for different aspect ratios, and the results are compared. The reduction of the bending response due to the strain gradient effect is compared for each case. A comparison of the effects of each of the material length scales, corresponding to different forms of the higher-order gradients, over the structural response is performed. The present study reveals that the strain gradient elasticity must be considered for an accurate analysis of nanostructures.