Material Length Scale Effects Research Articles

Decoupling toughness and strength through architected plasticity

We investigate the elastic–plastic fracture of architected materials through experiments and theory, with a focus on understanding the combined effects of material length scale and geometry, using a pillar array as a model structure. We show that load sharing across the pillars, and hence toughness, can be controlled by changing the spatial distribution and height of the pillars. A simple relation is presented to relate the extent of the plastic fracture process zone to the pillar array structure and the resulting toughness. This relation allows for quantitative prediction of failure loads of specimens with plasticity localized to a structured region and reveals that strength and toughness can be decoupled through architecture. Our findings establish a foundation for design of architected materials with enhanced fracture toughness.

Benchmark solutions for the material length scale effect in flexoelectric nanobeam using a couple stress theory

This paper is concerned with the derivation of the exact solutions for investigating the effect of material length scale on the static responses of the simply supported flexoelectric nanobeams considering a couple stress theory. The material length scale parameter estimates the coefficient linearly relating the rotational strain gradient to the couple stress in terms of the shear modulus of the beam material. Exact solutions for the displacements, stresses, electric displacements and the electric potential are derived while the beam is under closed circuit and open circuit conditions. Optimum value of the length parameter below which the response of the beam becomes independent of the couple stress effect has been determined while the beam acts as an actuator. The effect of this length parameter on the performance of the beam as an energy harvester has also been investigated for finding the optimum value of this length parameter. The benchmark results presented here may be useful for verifying further research and suggests that the flexoelectric nanobeams may be effectively exploited for developing advanced smart nanosensors and nanoactuators.

Bending, buckling, and vibration analysis of third-order shear deformation nanoplate based on modified couple stress theory

In this paper a third order shear deformation rectangular nanoplate with simply supported boundary conditions is developed for bending, buckling and vibration analysis. In order to consider the small-scale effects, the modified couple stress theory, with one length scale parameter, is used. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load, the dimensionless critical force under a uniaxial surface force in x direction and dimensionless frequencies are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The governing equations are numerically solved. The effect of material length scale, length, width and thickness of the nanoplate on the bending, buckling and vibration ratios are investigated and the results are presented and discussed in detail.

Vibration Analysis of Rectangular Kirchhoff Nano-Plate using Modified Couple Stress Theory and Navier Solution Method

In this study, the characteristics of rectangular Kirchhoff nano-plate vibrations are investigated using a modified couple stress theory. To consider the effects of small-scale, the modified couple stress theory proposed by Young (2002) is used as it has only one length scale parameter. In modified couple stress theory, the strain energy density is a function of the components of the strain tensor, curvature tensor, stress tensor, and symmetric part of the couple stress tensor. After obtaining the strain energy, external work, and kinetic energy equation and inserting them in the Hamilton principle, the main and auxiliary equations of nano-plate are obtained. Then, by applying the boundary and force conditions in the governing equations, the vibrations of the rectangular Kirschhof nano-plate with the thickness are investigated with simple support around. The solution method used in this study is the Navier method and the effects of material length scale, length and thickness of the nanoplate on the vibration are investigated and the results are presented and discussed in details.

Transverse wave propagation in viscoelastic single-walled carbon nanotubes with surface effect based on nonlocal second-order strain gradient elasticity theory

Nowadays, the viscoelastic carbon nanotubes (CNTs) have been used as one of the most promising candidates for nanodevices (or nanoelectronics) and super-strong reinforcement fibers in polymer nanocomposites. In recent years there have been many papers that considered the effects of material length scales in the investigations of the material constants related on the structural parameters on the wave propagation in the viscoelastic CNTs. By using different approaches, for examples, Eringen’s differential nonlocal model, modified couple stress theory, and strain gradient theories, the stiffness softening or strengthening effect can be well predicted. However, the influence of the rapid stiffness enhancing effect on the wave characteristics of the viscoelastic CNTs is still not reported. To deal with such problem, this work aims to investigate the transverse wave propagation in viscoelastic single-walled carbon nanotubes (SWCNTs) adhered by surface material based on the nonlocal second-order strain gradient elasticity theory. The characteristic equation of wave motion of the viscoelastic SWCNTs with surface effect is systematically formulated to achieve the analytical dispersion relation between the wave frequency (or phase velocity) and the wave number. In the end, some most important conclusions are concluded.

Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory

The bending of the Euler-Bernoulli micro-beam has been extensively modeled based on the modified couple stress (MCS) theory. Although many models have been incorporated into the literature, there is still room for introducing an improved model in this context. In this work, we investigate the thermoelastic vibration of a micro-beam exposed to a varying temperature due to the application of the initial stress employing the MCS theory and generalized thermoelasticity. The MCS theory is used to investigate the material length scale effects. Using the Laplace transform, the temperature, deflection, displacement, flexure moment, and stress field variables of the micro-beam are derived. The effects of the temperature pulse and couple stress on the field distributions of the micro-beam are obtained numerically and graphically introduced. The numerical results indicate that the temperature pulse and couple stress have a significant effect on all field variables.

Stability of Vibrating Functionally Graded Nanoplates with Axial Motion Based on the Nonlocal Strain Gradient Theory

Considering both the nonlocal scale and material length scale effects, we investigate vibration and stability behaviors of functionally graded nanoplates with axial motion in order to model the two-dimensional nanobelt in nanoengineering. The nonlocal strain gradient theory is applied and the differential nonlocal strain gradient constitutive relation is adopted. Using the physical neutral plane of a functionally graded thin plate, we derive the governing equation of motion for the functionally graded nanoplate with axial motion via Hamilton’s principle, where the kinematic characteristics are introduced into the dynamic behaviors. The governing equation is numerically solved using the Galerkin method. Effects of the nonlocal scale and material length scale parameters, axial velocity, gradient index, biaxial pre-tensions and aspect ratio are discussed. The results demonstrate that complex frequencies of the functionally graded nanoplate with axial motion decrease with an increase of axial velocity in the subcritical region, while the moving nanoplate experiences a divergent instability or flutter instability in the supercritical region. Natural frequency and critical speed decrease with the increase of the nonlocal scale parameter while increase with the increase of the material length scale parameter, reflecting the nonlocal softening and strain gradient hardening mechanisms, respectively. Besides, natural frequency and critical speed increase with the increase of the biaxial pre-tensions and aspect ratio, but decrease with the increase of the gradient index. In particular, the influences of the gradient index and size or weight of functionally graded nanoplates on the critical speed are explored.

An exact solution on gold microbeam with thermoelastic damping via generalized Green-Naghdi and modified couple stress theories

The vibration of the gold microbeam resonator considering the effects of the material length scale under the temperature effect is investigated in this article. The modified couple stress theory is utilized to show the material length scale effects. The effects of temperature considering the thermoelastic damping are taken into account based on Green-Naghdi thermoelasticity theory. The Green-Naghdi coupled thermoelasticity and modified couple stress governing equations for Euler-Bernoulli beam are derived based on Hamilton’s principle. The Laplace method is utilized to solve the obtained equations. The inverse Laplace method as Talbot method is used to obtain the lateral deflection and temperature response of the microbeam in the time domain. The effects of the various parameters such as the nondimensional material length scale parameter, the thickness of the microbeam, and the initial temperature conditions, on the lateral deflection as well as temperature responses of the microbeam are studied to investigate the results. Raising the material length scale parameter and width to length ratio makes the lateral deflection of the beam decrease in the time domain. Increase in temperature shock and Poisson’s ratio makes the lateral deflection increase, and both have a direct effect on the lateral deflection in the time domain, except that this increase is linear for temperature shock. The temperature of the microbeam decreases by an increase in Poisson’s ratio and progressing in the axial direction. For the final observation, it should be noted that the couple stress has an inverse effect on the lateral deflection.

Structural analysis of size-dependent functionally graded doubly-curved panels with engineered microarchitectures

Advances in multi-material 3D printing technologies have opened a new horizon for design and fabrication of architected multi-materials in multiple length scales from nano-/microscale to meso-/macroscales. In this study, we apply modified couple stress and first-order shear deformation theories for a size-dependent structural analysis of 3D printable functionally graded (FG) doubly-curved panels where their microarchitecture can be engineered to improve their structural performance. This non-classical model incorporates the microstructure-dependent size effects for the structural performance through the introduction of a length scale in the kinematics of deformation. The volume fraction of matrix in the dual-phase (inclusion and matrix) FG size-dependent panels varies continuously through the thickness. The microarchitecture of inclusion and matrix in FG panels is engineered to show its effect on the structural responses. We implement the standard mechanics homogenization technique via finite element simulation to accurately predict the effective mechanical properties of FG materials for different topologies of engineered microarchitecture to show the significance of selecting appropriate micromechanical modeling for analyzing FG structures. Governing equations derived by variational Hamilton’s principle are solved by applying the Galerkin method for different sets of boundary conditions. We investigate the effects of material length scale, material composition, heterogeneous material distribution, particulate topology, length-to-thickness ratio, and panel curvature on the structural performance. It is found that the fundamental frequencies of size-dependent two-phase FG doubly-curved panels with square-shape inclusions are higher than for those with other topologies, which sheds lights on the engineering of the inclusion shape in advanced architected materials to optimize their structural performance.

Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation

In this article, a nonlocal strain gradient cylindrical shell model is developed to study vibration analysis and instability of a single-walled carbon nanotube conveying viscous fluid. The fluid flow is modeled by modified Navier-Stokes equation considering slip boundary condition and Knudsen number. The obtained governing equations of motion and corresponding boundary conditions are discretized using generalized differential quadrature method. Simplifying the nonlocal strain gradient theory, the results of this theory are compared to those of classical, strain gradient, and nonlocal theories. The effects of material length scale and nonlocal parameters on natural frequency and critical flow velocity are further investigated. Also, for the first time, the effect of fluid flow on vibration behavior of the carbon nanotube is studied by molecular dynamics simulation. In the simulations, TIP4P/2005 water model is used, which accurately considers thermo-physical properties of water such as viscosity. Moreover, in order to apply carbon–carbon interaction, modified Tersoff potential is used, with Lennard-Jones potential being employed for other interactions. In this study, size-dependent parameters of nonlocal strain gradient theory are calibrated and variation of calibrated values under the effect of flow velocity is explored.

Geometrically nonlinear isogeometric analysis of functionally graded microplates with the modified couple stress theory

In this study, a new and efficient computational approach based on isogeometric analysis (IGA) and refined plate theory (RPT) is proposed for the geometrically nonlinear analysis of functionally graded (FG) microplates. While the microplates’ size-dependent effects are efficiently captured by a simple modified couple stress theory (MCST) with only one length scale parameter, the four-unknown RPT is employed to establish the displacement fields which are eventually used to derive the nonlinear von Kámán strains. The NURBS-based isogeometric analysis is used to construct high-continuity elements, which is essentially required in the modified couple stress and refined plate theories, before the iterative Newton-Raphson algorithm is employed to solve the nonlinear problems. The successful convergence and comparison studies as well as benchmark results of the nonlinear analysis of FG microplates ascertain the validity and reliability of the proposed approach. In addition, a number of studies have been carried out to investigate the effects of material length scale, material and geometrical parameters on the nonlinear bending behaviours of microplates.

Static instability of beam-type NEMS subjected to symmetric electrostatic actuation based on couple stress theory and Timoshenko beam theory

The behavior of nanoelectromechanical systems subjected to symmetrical electrostatic actuation and symmetrical Casimir intermolecular force has been investigated. Two different phenomena (i.e. electromechanical bifurcation and electromechanical buckling) have been considered to explore the static electromechanical instability of such systems. The Timoshenko beam model has been employed to find the effect of shear deformation on these systems. Modified couple stress theory has been used to investigate size-dependency. Besides, the compressive and tensile residual stresses of nanobridges have been measured on the basis of electromechanical buckling. The governing equations and corresponding boundary conditions have been obtained by means of the principle of minimum potential energy. Finally, following validation of results, the effects of material length scale, length, shear deformation, beam geometry, and gap distance on the symmetric electromechanical behavior have been discussed and examined.

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.

Improvement of orthogonal cutting simulation with a nonlocal damage model

In this paper, the implementation of a nonlocal gradient damage formulation is described which aims to improve the reliability of the numerical prediction of orthogonal cutting simulations. The major concerns are accuracy of computational results, independence of element size, modeling of failure phenomenon for cutting process simulation and proposing reliable and stable separation criteria for orthogonal metal cutting. To avoid pathological localization and mesh dependence and to incorporate length scale effects due to microstructure evolution, the damage growth is driven by a nonlocal variable with a second order partial differential equation. Two governing equations, i.e. equilibrium and nonlocal averaging equation, are solved simultaneously. The results are presented to show the effect of the nonlocal damage model on separation criterion, cutting force, width of shear band, and effect of material length scale to make results mesh independence. The Johnson–Cook damage criterion is used to compare local and nonlocal model results. Numerical simulations are validated through comparison the cutting force in the nonlocal damage model with the cutting force measured by experiment results taken from literature.

Numerical simulation of fabric anisotropy and strain localization of sand under simple shear

AbstractThis paper studies the effects of initial fabric anisotropy of dry sand in simple shear deformation. The effects of anisotropy are taken into consideration through the modification of the mobilized friction in the Mohr–Coulomb‐type yield surface as a function of a fabric parameter. In addition, the constitutive model uses a gradient term that directly incorporates the effects of material length scale. The constitutive formulation is implemented into ABAQUS finite element code and used to simulate shearing of the dry sand under various conditions of simple shear. The numerical simulations show that while the shear stress response is affected by fabric anisotropy, its effects on strain localization in simple shear are minimal. This is in contrast to other devices such as the biaxial shear. The strain localization in simple shear is controlled more by the imposed boundary conditions. The use of material length scale is shown to remove the effects of strain localization in the shearing response. Copyright © 2009 John Wiley & Sons, Ltd.

On the formulations of higher-order strain gradient crystal plasticity models

Recently, several higher-order extensions to the crystal plasticity theory have been proposed to incorporate effects of material length scales that were missing links in the conventional continuum mechanics. The extended theories are classified into work-conjugate and non-work-conjugate types. A common feature of the former is that existence of higher-order stresses work-conjugate to gradients of plastic strain is presumed and an extended principle of virtual work involving such an additional virtual work contribution is formulated. Meanwhile, in the latter type, the higher-order stress quantities do not appear explicitly. Instead, rates of crystallographic slip are influenced by back stresses that arise in response to spatial gradients of the geometrically necessary dislocation densities. The work-conjugate type and the non-work-conjugate type of theories have different theoretical backgrounds and very unlike mathematical representations. Nevertheless, both types of theories predict the same kind of material length scale effects. We have recently shown that there exists some equivalency between the two approaches in the special situation of two-dimensional single slip under small deformation. In this paper, the discussion is extended to a more general situation, i.e. the context of multiple and three-dimensional slip deformations.

System-Level Reliability Assessment of Mixed-Signal Convergent Microsystems

The next-generation convergent microsystems, based on system-on-package (SOP) technology, require up-front system-level design-for-reliability approaches and appropriate reliability assessment methodologies to guarantee the reliability of digital, optical, and radio frequency (RF) functions, as well as their interfaces. Systems approach to reliability requires the development of: i) physics-based reliability models for various failure mechanisms associated with digital, optical, and RF Functions, and their interfaces in the system; ii) design optimization models for the selection of suitable materials and processing conditions for reliability, as well as functionality; and iii) system-level reliability models understanding the component and functional interaction. This paper presents the reliability assessment of digital, optical, and RF functions in SOP-based microsystems. Upfront physics-based design-for-reliability models for various functional failure mechanisms are presented to evaluate various design options and material selection even before the prototypes are made. Advanced modeling methodologies and algorithms to accommodate material length scale effects due to enhanced system integration and miniaturization are presented. System-level mixed-signal reliability is discussed thorough system-level reliability metrics relating component-level failure mechanisms to system-level signal integrity, as well as statistical aspects.

Size effects in the Charpy V-notch test

Issues related to the size dependence of the upper shelf energy (USE) and the ductile-to-brittle transition temperature (DBTT) in the Charpy V-notch test are investigated. Emphasis is placed on the interplay between inertial, strain rate hardening, strain hardening, thermal softening and material length scale effects. Geometrically similar specimens are considered first. For such specimens, the ductile-to-brittle transition temperature is found to increase with specimen size, with the amount of the increase depending on the material properties. To model available experiments, calculations are also carried out for Charpy specimens where only the ligament size is varied and two classes of pipe steels are considered. For a relatively high strength pipe steel, the experimental results exhibit no size dependence of the DBTT. On the other hand, a significant shift in the DBTT is obtained for a low strength steel. The numerical studies are used to understand the difference between these two classes of steels. The extent to which the size effect is material dependent is investigated.