Material Characteristic Length Research Articles

Combined effects of surface energy and couple stress on the nonlinear bending of FG-CNTR nanobeams

This paper investigates the size-dependent nonlinear bending of functionally graded carbon nanotube-reinforced (FG-CNTR) nanobeams. Chen–Yao’s surface elasticity and modified couple stress theories are adopted to describe surface effects and couple stress effects, respectively. These nanobeams, in which the carbon nanotube (CNT)-reinforced phases are assumed to be distributed in a gradient along the thickness, are subjected to a uniform pressure and rest on a nonlinear elastic foundation. In accordance with the Euler–Lagrange variational principle, the governing equations and boundary conditions for the FG-CNTR nanobeams, which involve geometric nonlinearity due to the von Kármán strain relations, are obtained. Then, with the assistance of the two-step perturbation technique, the load-deflection relationship is determined for nanobeams subjected to simply supported (SS) and clamped–clamped (CC) boundary conditions. Finally, the impacts of various factors, including surface properties, characteristic material length, elastic foundation, geometric factors, layout type and volume fraction of CNTs, on the mechanical behaviors of CNT-based nanobeams are examined. The numerical results reveal that the combination of surface effects and couple stress helps to enhanceq the stiffness of the nanobeams. Furthermore, the size-dependent nonlinear bending of the FG-CNTR nanobeam is markedly affected by the content and layout type of the reinforcements.

Two types of scale effects on the nonlinear forced vibration of axially moving nanobeams

Various non-classical continuum mechanics models appearing in previous studies cannot perfectly explain the mechanical properties of micro- and nanomaterials. Establishing a reasonable continuum mechanics model that comprehensively reflects the scale effect on material deformation is of great practical significance for objectively explaining the variation law of mechanical properties of micro- and nanomaterials under the combined action of different scale effects. Based on nonlocal strain gradient theory, a new scale-dependent model is proposed for axially moving nanobeams. In this study, an asymptotic expansion is performed using the multiscale time method to obtain the amplitude-frequency response curve of the equilibrium solutions for the forced vibration problem. Afterwards, the effects of various system parameters, especially the scale parameters, on the resonance curve are examined. Finally, the effects of nonlocal parameters and material characteristic length parameters on the amplitude-frequency response curves are investigated through typical numerical examples. The numerical results show that the nonlocal parameters promote the emergence of the main resonance, whereas the material characteristic length parameters suppress the emergence of the main resonance. Moreover, these parameters also affect the response amplitude and the skewness and jumping point of the amplitude-frequency characteristic curve.

Failure criterion for highly stretchable elastomers under triaxial loading

A failure criterion is introduced for highly stretchable hyper-elastic materials under tensile-dominated triaxial loading. It is based on energy balance using Griffith approach but is presented by the critical state of load as in traditional size independent strength theories. The basic argument is made of the recent assessment about the cavity nucleation in soft materials: the volumetric energy dissipation in the growth of defect and size-insensibility of the critical load in soft materials when the defect is smaller than the material characteristic length. Thus, using the energy approach as in fracture mechanics, we are able to estimate the critical state of load using only one material parameter, which is subsequently replaced by the critical stress under one specific loading, e.g., the strength of uniaxial loading. This leads us to the failure surface in the normalized principal stress space. The predictions of the present theory agree well with the experimental results collected from the literature, including an order of magnitude difference between the strengths of the uniaxial loading and hydrostatic loading. This addresses a variety of difficulties in the previous failure analyses. For the easiness of practical application, we further analyze the data and obtain a criterion using mean stress and the first invariant of Cauchy–Green deformation tensor, I1. We are able to obtain the widely mentioned maximum I1 criterion, and also find its limitation: only applicable when one of the principal stresses is zero. For general triaxial loadings, the material can fail at any allowable I1 and the critical mean stress monotonically increases with I1; and under a constant mean stress that is above the hydrostatic strength, the material is only safe when I1 is above a mean stress-dependent lower bound but below an unknown up-bound.

Low and ultra-low-cycle fatigue behavior of X52 piping steel based on theory of critical distances

The cyclic failure observed in structural components such as pipelines subjected to extreme loading conditions highlights some limitations concerning the application of existing fatigue damage models. The evaluation and prediction of this type of failure in these steel components under large-scale plastic yielding associated with high levels of stress triaxiality are not sufficiently known nor explored. This fatigue domain is conventionally called ultra-low-cycle fatigue (ULCF) and damage features are representative of both low-cycle fatigue (LCF) and monotonic ductile fracture. Thus, in order to understand the ULCF damage mechanisms both monotonic and LCF tests are required to get representative bounding damage information to model the material damage behaviour under such extreme loading conditions. This paper aims at exploring the Theory of Critical Distances (TCD) in the LCF and ULCF fatigue regimes, including the application of the point, line and area methods. The application of the TCD theories has not been explored so far in the ULCF fatigue regimes, despite its promising results in the LCF and high-cycle fatigue. An experimental program was carried out on several specimens’ geometries made of X52 piping steel. In detail, smooth plane specimens and notched plane specimens were cyclic loaded under tension/compression loading in order to obtain fatigue lives within the range of 101–104 cycles. In addition, cyclic bending tests on notched plane specimens were also incorporated in this study. Finite element simulations of all small-scale tests were conducted allowing to derive elastoplastic stress/strain fields along the potential crack paths. The numerical data were subjected to a post-processing in order to find characteristic lengths that can be treated as a fatigue property according to the TCD. A unified strain-life relation is proposed for the X52 piping steel together with a characteristic material length, consisting of a practical relation for pipeline strain-based design under extreme cyclic loading conditions.

Couple stress-based 3D contact of elastic films

This study reports a three-dimensional (3D) size-dependent contact model of a thin elastic film indented by an elastic sphere, based on a linear couple stress-based semi-analytical method (SAM), where the thin film is perfectly bonded to a rigid substrate. The couple stress-based frequency response functions (FRFs) of the film are analytically derived, and the corresponding continuous and discrete influence coefficients (ICs) are subsequently obtained. The contact pressure is numerically determined based on the conjugate gradient method (CGM), while the elastic displacements and stresses of the thin film and indenter are calculated by means of the discrete convolution-fast Fourier transform (DC-FFT) algorithm. The combined effects of the characteristic material length and film thickness on contact behaviors are investigated for selected material elastic coefficients.

Modified LSM for size-dependent wave propagation: comparison with modified couple stress theory

A modified LSM is proposed by introducing an independent micro-rotational inertia, which may help characterize the scale-dependent effect and avoid the Poisson’s ratio limitation of regular triangle lattices in two dimensions. For this method, some factors may affect data pickup and modeling accuracy, but the ‘optimal’ inputs, like stiffness ratio, numerical damping, and micro-rotational inertia, could be obtained from parameter identification by the Dakota toolkit (Adams et al., Tech Rep SAND2010–2183, 2009), when a suitable excitation source function and lattice spacing are set up. By comparing with the modified couple stress theory, we analyze the dispersion relationship of elastic waves for the estimation of the characteristic material length. It shows that this modified LSM may provide an alternative and promising way to investigate the size-dependent wave propagation in elastic media numerically.

Computational investigation on the performance of hydrodynamic micro-textured journal bearing lubricated with micropolar fluid using mass-conserving numerical approach

The tribological characteristic of journal bearing systems can be enhanced with the integrating of textures in the contact interfaces, or using the lubricating effect of non-Newtonian fluids. In this study, the combined effects of bearing surface texturing and non-Newtonian lubricants behavior, using micropolar fluid model, on static characteristics of hydrodynamic circular journal bearings of finite length are highlighted. The modified Reynolds equation of micropolar lubrication theory is solved using finite differences scheme and Elrod's mass conservation algorithm, taking into account the presence of the cylindrical texture shape on full and optimum bearing surfaces. The optimization textured area is carried out through particle swarm optimization algorithm, in order to increase the load lifting capacity. Preliminary results are in good agreement with the reference ones, and present an enhancement in the performances of micro-textured journal bearings (load carrying capacity and friction). The results suggest that texturing the bearing convergent zone significantly increases the load carrying capacity and reduce friction coefficient, while fully texturing causes bad performances. It is also shown that the micropolar fluids exhibit better performances for smooth journal bearings than a Newtonian fluid depending on the size of material characteristic length and the coupling number. The combined effects of fully surface textured with micropolar fluids reduce the performance of journal bearing, especially at lower eccentricity ratios. Considering the optimal arrangement of textures on the contact surface, a significant improvement in terms of load capacity and friction can be achieved, particularly at high eccentricity ratios, high material characteristic lengths and high values of the coupling numbers of micropolar fluids.

Nonlocal and strain gradient effects on nonlinear forced vibration of axially moving nanobeams under internal resonance conditions

Based on the nonlocal strain gradient theory (NSGT), a model is proposed for an axially moving nanobeam with two kinds of scale effects. The internal resonance-accompanied fundamental harmonic response of the external excitation frequency in the vicinities of the first and second natural frequencies is studied by adopting the multivariate Lindstedt-Poincare (L-P) method. Based on the root discriminant of the frequency-amplitude equation under internal resonance conditions, theoretical analyses are performed to investigate the scale effects of the resonance region and the critical external excitation amplitude. Numerical results show that the region of internal resonance is related to the amplitude of the external excitation. Particularly, the internal resonance disappears after a certain critical value of the external excitation amplitude is reached. It is also shown that the scale parameters, i.e., the nonlocal parameters and the material characteristic length parameters, respectively, reduce and increase the critical amplitude, leading to a promotion or suppression of the occurrence of internal resonance. In addition, the scale parameters affect the size of the enclosed loop of the bifurcated solution curves as well by changing their intersection, divergence, or tangency.

Two-dimensional sliding friction contact analysis of a coated half-plane with coupled stress

Based on the couple stress theory, we study the two-dimensional sliding friction contact between a rigid punch and a homogeneous coated half plane. This theory describes the size effect of microstructure materials by introducing characteristic material length. Using the Fourier integral transform method, the size-dependent contact problem is reduced into a Cauchy singular integral equation of the second kind. Then, it is numerically solved to determine the distribution of the normal contact stress and in-plane stress on the contact surface. The effect of the length scale parameter, the ratio of the shear modulus and the friction coefficient on the normal contact stress and in-plane stress are discussed in detail. The results indicate that the contact stress considering the couple stress departs significantly from the prediction of the classical elastic theory, and shows a strong dependence on the characteristic material length.

An investigation of free vibrations of a strain gradient Timoshenko beams with the method of initial values

Investigated herein is the free vibrations of beams based on the strain gradient Timoshenko beam theory with the method of initial values. For the vibration of strain-gradient Timoshenko beam (SGTB), the sixth-order ordinary differential equation and three boundary conditions at each end have been obtained by using the Hamilton principle. The effect of the characteristic length on the frequencies of free vibrations is shown. The frequencies of the SGTB are compared to the frequencies of the strain gradient Euler beam (SGEB), classical Timoshenko beam (CTB) and classical Euler beam (CEB). It has been observed that the high-frequency values of conventional and strain-gradient beams are very different. This result can be used to determine the value of the material characteristic length for a nanobeam for which lengthscale effects are believed to be dominant.

Nonlinear vibrations of axially moving simply supported viscoelastic nanobeams based on nonlocal strain gradient theory

The lateral nonlinear vibration of an axially moving simply supported viscoelastic nanobeam is analysed based on nonlocal strain gradient theory. The proposed model includes the nonlocal parameters and material characteristic length parameters, investigating the two kinds of size effects of micro-nano beam structures. Firstly, the steady-state amplitude-frequency response of the subharmonic parametric resonance is analysed by a direct multiscale method, and the stability of the (non-) zero equilibrium solution determined by the Routh–Hurwitz criterion. Subsequently, the nonlinear frequencies of the nanobeams are calculated. Finally, several numerical examples are used to illustrate the influence of the scale parameters on the nonlinear vibration characteristics of nanobeams. The results show that when subharmonic parametric resonance occurs in the system, the (non-) zero equilibrium solution and the boundary of the instability region are markedly affected by the scale parameters. In addition, the nonlocal parameters soften the system, the material characteristic length parameters harden the system, and these softening and hardening effects are strengthened (or weakened) to varying degrees in the presence of nonlinearity.

Anti-plane inclusion problem in the second gradient electroelasticity theory

In this paper, we compare direct and energy based homogenization schemes in the framework of the second gradient electroelasticity theory. We consider a special type of the model with simplified constitutive equations that allow to derive an exact analytical solution for the effective properties of fiber-reinforced piezoelectric composite under anti-plane loading. To find the solution we use an approach that was proposed by Benveniste for similar problem in classical electroelasticity. Based on the transformations of the field variables, we construct a solutions for the coupled anti-plane problem of gradient electroelasticity by using known solutions for the uncoupled problem of the strain gradient elasticity and similar problem of gradient electrostatics. Based on the derived solution, we demonstrate the main effects that arise in the composite armed with thin piezoelectric fibers, which diameter is of the order of the characteristic length of material’s microstructure. We show that such composites may demonstrate an unusual behavior: With the decrease of the fiber diameter its effective properties increase and its apparent behavior becomes similar to the homogeneous material. We carry out a comparison between predictions for effective properties, which were provided by the Mori–Tanaka scheme (MTS) and generalized self-consistent scheme (GSCS). We show that these methods give significantly different predictions, which can deviate from each other by hundreds of percent, especially for the effective piezoelectric constants.

Effect of displacement loading rates on mode-I fracture toughness of fiber glass-epoxy composite laminates

Size-effect method is used in determining mode-I fracture characteristics of woven fiber glass/epoxy composite laminates for varying loading rates. Tensile testing of geometrically similar single-edge notch (SEN) specimens with three widths 30 mm, 40 mm and 50 mm is carried out for four different displacement loading rates namely 1, 10, 100, 500 mm/min. For each width D and loading rate, the crack-length a is varied as 0.125D, 0.25D, 0.375D and 0.5D. A total of 186 specimens are tested. The peak stresses σNu, the initial Youngs Modulus Exx and the shear modulus Gxy are calculated. The fracture toughness KIC, critical strain energy release rate Gf and material characteristic length Cf are calculated using linear regression analysis of non-linear fracture mechanics equations. The values of Gf and Cf first decrease and then increase monotonically as loading rate changes from 1 to 500 mm/min. This behavior was explained using Bažant size-effect equation relating nominal strength and specimen size. The study revealed change in modes of failure characterized by a jump in brittleness number as loading rate increases from 1 mm/min to 10 mm/min followed by decrease in brittleness number as loading rate increases from 10 mm/min to 500 mm/min. This conclusion was supported by highly magnified images of failed specimens using scanning electron microscope. A numerical algorithm to determine crack growth resistance curves (R-curves) from peak load is also implemented. The R-curves are geometry as well as loading rate dependent and give Gf,Cf values which agree qualitatively and to some extent quantitatively with the linear regression approach.

Energy-Based Strength Theory for Soft Elastic Membranes

In the previous studies by the authors and others, it was demonstrated that there are two possible defect growth modes and a characteristic material length for any soft material. For a pre-existing defect smaller than the material characteristic length, the energy is dissipated all around the defect as it grows and the critical load for the growth is independent of the defect size. For defects larger than the characteristic length, the growth is by cracking and the energy is dissipated along a plane. Thus, the critical load for the growth is size dependent and can be predicted by fracture mechanics. In this study, we apply the same energy-based argument to the failure of thin membranes, with the focus on the first growth mode that gives the maximum critical load. We assume that strain localization due to damage is the precursor to rupture, and hence, we model the corresponding zone as a through-thickness hole, with its size smaller than the material characteristic length. The defect grows when the elastic energy relaxed by the growth is enough to provide the energy needed for internal microstructure changes. This leads us to the size-independent failure conditions for membranes under the biaxial load. The conditions are expressed in terms of either two principal stretches or two principal stresses for two different types of materials. For verification, we test the theory using the published experimental data on natural and styrene-butadiene rubber. By using the experimental data from equal biaxial loading, we predict the critical principal stretch ratios and critical stresses for different biaxialities. The predictions agree well with the experimental results.

The Hertz contact problem in couple-stress elasticity

The classical three-dimensional Hertz contact problem is re-examined in the present work within the context of couple-stress elasticity. This theory introduces characteristic material length scales that emerge from the underlying microstructure and has proved to be very effective yet rather simple for modeling complex microstructured materials. An exact solution of the axisymmetric contact problem is obtained through Hankel integral transforms and singular integral equations. The goal is to examine to what extend this gradient theory captures the experimentally observed indentation response and the related size effects in materials with microstructure. Furthermore, the present work extends the classical Hertz contact solution in the framework of a generalized continuum theory providing thus a useful theoretical background for the interpretation of spherical indentation tests of microstructured materials. The results obtained in the present work can be used in order to model nano/micro indentation experiments of several materials, such as polymers, ceramics, composites, cellular materials, foams, masonry, and bone tissues, of which their macroscopic response is influenced by the microstructural characteristic lengths.

Three-dimensional contact analysis with couple stress elasticity

This study establishes a size-dependent three-dimensional semi-analytical model (SAM) on the contact of an elastic half-space indented by a rigid sphere within the framework of couple stress elasticity. A characteristic material length is used to describe the size-dependent contact behaviors. The frequency response functions (FRFs) of the displacements and stresses are analytically derived with incorporating the effects of the characteristic material length, which are then converted to the corresponding influence coefficients (ICs). By using the conjugate gradient method (CGM), the surface pressure distribution is obtained, and the stress field beneath the contact surface is calculated with the fast Fourier transform (FFT). The proposed model is implemented to analyze the combined effects of the characteristic material length and Poisson's ratio on the size-dependent contact behaviors. A transition value of the characteristic material length is found, above which the size-dependent contact modeling is required, and below which the size-dependent behaviors can be ignored. Finally, the present model deals with some typical contacts involving different indenter shapes and multi-indenters, which is difficult to be implemented in the case of two-dimensional contact.

Micropolar plasticity—Part I: modeling based on curvature tensors related by mixed transformations

When formulating finite deformation micropolar plasticity, the structure of the theory strongly depends on the choice of the variables describing the deformation kinematics. This holds true even for classical plasticity. However, in contrast to classical plasticity, the set of kinematical variables in micropolar plasticity includes, besides strain tensors, so-called micropolar curvature tensors. There are only a few investigations addressing such aspects, so the aim of the paper is to highlight the effect of a specific micropolar curvature kinematics on the structure of a micropolar plasticity. We do this by developing a general finite deformation micropolar plasticity, which relies upon a class of micropolar curvature tensors related to each other by mixed transformations. That means, the pull-back and push-forward transformations characterizing the class involve both deformation gradient and micropolar rotation tensors. The curvature kinematics is discussed by using geometrical methods developed previously. The plasticity theory is based on the assumption that the yield function and the flow rules are functions of specific micropolar Mandel’s stress tensors. The definition of the Mandel’s stress tensor is suggested by the adopted curvature kinematics and reveals a characteristic feature of the resulting plasticity. Moreover, the presence of curvature variables in plastic arc length gives reason to introduce a characteristic internal material length, which in turn seems to urge the form of the formulation of the theory. A specific version of von Mises micropolar plasticity with kinematic and isotropic hardening, derived in the theoretical context of the present paper, is elaborated in Part II.

A generalisation to cohesive cracks evolution under effects of non-uniform stress field

The aim of the present work is to study the stabilizing effect of the non-uniformity of the stress field on the cohesive cracks evolution in two-dimensional elastic structures. The crack evolution is governed by Dugdale's or Barenblatt's cohesive force models. We distinguish two stages in the crack evolution: the first one where all the crack is submitted to cohesive forces, followed by a second one where a non cohesive part appears. Assuming that the material characteristic length dc associated with the cohesive model is small by comparison to the dimension L of the body, we develop a two-scale approach, and using the complex analysis method, we obtain the entire crack evolution with the loading in a closed form for the Dugdale's case and in semi-analytical form for the Barenblatt's case. In particular, we show that the propagation is stable during the first stage, but becomes unstable with a brutal jump of the crack length as soon as the non cohesive crack part appears. We discuss also the influence of all the parameters of the problem, in particular the non-uniform stress and cohesive model formulations, and study the sensitivity to imperfections.

A displacement-based formulation for interaction problems between cracks and dislocation dipoles in couple-stress elasticity

Interaction problems of a finite-length crack with plane and antiplane dislocation dipoles in the context of couple-stress elasticity are presented in this study. The analysis is based on the distributed dislocation technique where infinitesimal dislocation dipoles are used as strain nuclei. The stress fields of these area defects are provided for the first time in the framework of couple-stress elasticity theory. In addition, a new rotational defect is introduced to satisfy the boundary conditions of the opening mode problem. This formulation leads to displacement-based hyper-singular integral equations that govern the crack problems, which are solved numerically. It is further shown that this method has several advantages over the slope formulation. Based on the obtained results, it is deduced that in all cases the cracked body behaves in a more rigid way when couple-stresses are considered. The effect of couple-stresses is highlighted in a small zone ahead of the crack-tip and around the dislocation dipole, where the stress level is significantly higher than the classical elasticity prediction. Further, the dependence of the energy release rate and the configurational force exerted on the defect on the characteristic material length and the distance between the defect and the crack-tip is discussed. In the plane problems, couple-stress theory predicts either strengthening or weakening effects while in the antiplane mode a strengthening effect is predicted.

Interaction problems between cracks and crystal defects in constrained Cosserat elasticity

In this work, interaction problems between a finite-length crack with plane and antiplane crystal defects in the context of couple-stress elasticity are presented. Two alternative yet equivalent approaches for the formulation of crack problems are discussed based on the distributed dislocation technique. To this aim, the stress fields of climb and screw dislocation dipoles are derived within couple-stress theory and new ‘constrained’ rotational defects are introduced to satisfy the boundary conditions of the opening mode problem. Eventually, all interaction problems are described by single or systems of singular integral equations that are solved numerically using appropriate collocation techniques. The obtained results aim to highlight the deviation from classical elasticity solutions and underline the differences in interactions of cracks with single dislocations and dislocation dipoles. In general, it is concluded that the cracked body behaves in a more rigid way when couple-stresses are considered. Also, the stress level is significantly higher than the classical elasticity prediction. Moreover, the configurational forces acting on the defects are evaluated and their dependence on the characteristic material length of couple-stress theory and the distance between the defect and the crack-tip is discussed. This investigation reveals either a strengthening or a weakening effect in the opening mode problem while in the antiplane mode a strengthening effect is always obtained.