Surface stress effects on the free vibration behavior of nanoplates

A new frontier of research in the area of computational nanomechanics is to study the behavior of structures at very small length scales. As the dimensions of a structure approach the nanoscale, the classical continuum theories may fail to accurately predict the mechanical behavior of nanostructures. Among these nanostructures, nanobeams are attracting more and more attention due to their great potential engineering applications. One of the most important factors that influence the behavior of such submicron-sized structures is surface stress effect because of their high surface to volume ratio. In this paper, a non-classical solution is proposed to analyze bending and buckling responses of nanobeams including surface stress effects. Explicit formulas are proposed relevant to each type of beam theory to evaluate the surface stress effects on the displacement profile and critical buckling load of the nanobeams. Numerical results are presented to demonstrate the difference between the behaviors of the nanobeam predicted by the classical and non-classical solutions which depends on the magnitudes of the surface elastic constants.     

Higher‐order extended FEM for weak discontinuities – level set representation,quadrature and application to magneto‐mechanical problems

In this article, we present the application of bilinear and biquadratic extended FEM (XFEM) formulations to model weak discontinuities in magnetic and coupled magneto‐mechanical boundary value problems. For properly resolving the location of curved interfaces and the discontinuous physical behaviour, the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher‐order XFEM formulations. In order to reduce the complexity of the representation of curved interfaces, an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed. Integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains. Eventually, a coupled magneto‐mechanical demonstration problem is modelled and solved by XFEM. For demonstration purposes, a magneto‐mechanical coupling due to magnetic stresses is considered. Errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface. Copyright © 2012 John Wiley & Sons, Ltd.     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

Delamination of FRP-reinforced concrete by means of an extended finite element formulation

The present analysis focusses on the finite element modeling of delamination tests in FRP-reinforced concrete blocks using a continuous–discontinuous XFEM approach, which takes into direct consideration the mechanical properties of concrete, adhesive and FRP. Both FRP and adhesive layers are considered as linear elastic materials, while the constitutive behavior of the concrete is governed by an elasto-damaging constitutive law. As soon as a critical damage threshold is reached in the concrete, additional degrees of freedom, representative of the displacement discontinuity corresponding to the delamination process, are added. The proposed approach only requires the use of material parameters of direct mechanical meaning, which can be obtained by standard testing procedures. Hence the use of one-dimensional bond–stress–slip laws, whose parameters must be identified through a cumbersome procedure, appears to be unnecessary. Validation of the proposed approach is carried out by comparison of the computed results with two sets of experimental results. In both cases, after adoption of a simple damage criterion, structural behavior, slip, strain and stress profiles along the reinforcement are satisfactorily reproduced. Moreover, bond–stress–slip laws, which agree with those assumed in previous works based on the use of interface elements, were obtained retrospectively.     

Linear smoothed extended finite element method

The extended finite element method was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to be split by a discontinuity, strong or weak, and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes. A variety of approaches have been proposed to facilitate these special types of numerical integration, which have been shown to have a large impact on the accuracy and the convergence of the numerical solution. The smoothed extended finite element method (XFEM), for example, makes numerical integration elegant and simple by transforming volume integrals into surface integrals. However, it was reported in the literature that the strain smoothing is inaccurate when non‐polynomial functions are in the basis. In this paper, we investigate the benefits of a recently developed Linear smoothing procedure which provides better approximation to higher‐order polynomial fields in the basis. Some benchmark problems in the context of linear elastic fracture mechanics are solved and the results are compared with existing approaches. We observe that the stress intensity factors computed through the proposed linear smoothed XFEM is more accurate than that obtained through smoothed XFEM.     

Delamination of FRP-reinforced concrete by means of an extended finite element formulation

The present analysis focusses on the finite element modeling of delamination tests in FRP-reinforced concrete blocks using a continuous–discontinuous XFEM approach, which takes into direct consideration the mechanical properties of concrete, adhesive and FRP. Both FRP and adhesive layers are considered as linear elastic materials, while the constitutive behavior of the concrete is governed by an elasto-damaging constitutive law. As soon as a critical damage threshold is reached in the concrete, additional degrees of freedom, representative of the displacement discontinuity corresponding to the delamination process, are added. The proposed approach only requires the use of material parameters of direct mechanical meaning, which can be obtained by standard testing procedures. Hence the use of one-dimensional bond–stress–slip laws, whose parameters must be identified through a cumbersome procedure, appears to be unnecessary. Validation of the proposed approach is carried out by comparison of the computed results with two sets of experimental results. In both cases, after adoption of a simple damage criterion, structural behavior, slip, strain and stress profiles along the reinforcement are satisfactorily reproduced. Moreover, bond–stress–slip laws, which agree with those assumed in previous works based on the use of interface elements, were obtained retrospectively.     

Mechanism modelling of shot peening effect on fatigue life prediction

A new mechanism modelling is proposed in this paper to explain the shot peening effect on fatigue life predictions of mechanical components. The proposed methodology is based on the crack growth analysis of shot peened specimens, which are affected by the interaction of surface roughness and residual stress produced during the shot peening process. An asymptotic stress intensity factor solution is used to include the surface roughness effect and a time‐varying residual stress function is used to change the crack tip stress ratio during the crack propagation. Parametric studies are performed to investigate the effects of surface roughness and the residual stress relaxation rate. Following this, a simplified effective residual stress model is proposed based on the developed mechanism modelling. A wide range of experimental data is used to validate the proposed mechanism modelling. Very good agreement is observed between experimental data and model predictions.     

Accuracy improvement of XFEM using Wilson's incompatible element technique

Based on the extended finite element method (XFEM) and Wilson incompatible element technique, an incompatible extended finite element method (IXFEM) is presented to deal with the cracked bending problems in this study. The additional displacement modes of Wilson's incompatible element are introduced into the approximation of the XFEM. The internal degrees of freedom induced by additional models are condensed to improve efficiency. Derivation of the equilibrium equations of the IXFEM is performed. Several examples are employed to validate the capabilities of the proposed method. The influence of bending degree on the performance of proposed method is studied by changing the crack length. The numerical results indicate that the stress and displacement solutions of the IXFEM show a slight oscillation. The IXFEM achieves better convergence rate and computational accuracy compared with the traditional XFEM for the bending problems.     

Analysis of viscoelastic Bernoulli–Euler nanobeams incorporating nonlocal and microstructure effects

The present study introduces an analytical–computational model to simulate the effects of different simultaneous aspects on the behavior of nanobeams. The first one deals with the space nonlocality interaction and taking into account the microstructure effects, which has been formulated by using the nonlocal couple-stress elasticity. The second factor deals with the memory-dependent effect and has been investigated in the framework of linear viscoelasticity theory. It is the first time to apply the coupled effects of the microstructure and long-range interactions between the particles, to reflect the size-dependency of viscoelastic structures. Bernoulli–Euler nanobeam is taken as a vehicle to present the details of the proposed model. Eringen nonlocal elasticity and the modified couple-stress theory are used to formulate the two phenomena of long-range cohesive interaction and the microstructure local rotation effects, respectively. Boltzmann superposition viscoelastic model, endowed by Wiechert series, is used to simulate the linear behavior of isotropic, homogeneous and non-aging viscoelastic materials. The extended Hamilton’s principle is applied to formulate the analytical model of mechanical behavior of the nonlocal couple-stress nanobeam. The model has been verified and some results are compared with those published in the literature and a good agreement has been obtained. It is shown that the material-length scale parameter, nonlocal parameter, viscoelastic relaxation time and length-to-thickness ratio have a significant effect on the bending response of viscoelastic nanobeams with various boundary conditions.     

A semi‐Lagrangean time‐integration approach for extended finite element methods

Many computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi‐Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapolated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one‐dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two‐dimensional and a three‐dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche‐type formulation for interface problems with embedded strong discontinuities. Copyright © 2014 John Wiley & Sons, Ltd.     

Interface handling for three‐dimensional higher‐order XFEM‐computations in fluid–structure interaction

Three‐dimensional higher‐order eXtended finite element method (XFEM)‐computations still pose challenging computational geometry problems especially for moving interfaces. This paper provides a method for the localization of a higher‐order interface finite element (FE) mesh in an underlying three‐dimensional higher‐order FE mesh. Additionally, it demonstrates, how a subtetrahedralization of an intersected element can be obtained, which preserves the possibly curved interface and allows therefore exact numerical integration. The proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their ‘eXtended axis‐aligned bounding boxes’. The intersection method is applied to a highly reduced number of intersection candidates. The resulting linearized interface is used as input for an elementwise constrained Delaunay tetrahedralization, which computes an appropriate subdivision for each intersected element. The curved interface is recovered from the linearized interface in the last step. The output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element. Application of the interface algorithm currently concentrates on fluid–structure interaction problems on low‐order and higher‐order FE meshes, which may be composed of any arbitrary element types such as hexahedra, tetrahedra, wedges, etc. Nevertheless, other XFEM‐problems with explicitly given interfaces or discontinuities may be tackled in addition. Multiple structures and interfaces per intersected element can be handled without any additional difficulties. Several parallelization strategies exist depending on the desired domain decomposition approach. Numerical test cases including various geometrical exceptions demonstrate the accuracy, robustness and efficiency of the interface handling. Copyright © 2009 John Wiley & Sons, Ltd.     

Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behavior of an FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

A simple and efficient XFEM approach for 3-D cracks simulations

In this work, a simple and efficient XFEM approach has been presented to solve 3-D crack problems in linear elastic materials. In XFEM, displacement approximation is enriched by additional functions using the concept of partition of unity. In the proposed approach, a crack front is divided into a number of piecewise curve segments to avoid an iterative solution. A nearest point on the crack front from an arbitrary (Gauss) point is obtained for each crack segment. In crack front elements, the level set functions are approximated by higher order shape functions which assure the accurate modeling of the crack front. The values of stress intensity factors are obtained from XFEM solution by domain based interaction integral approach. Many benchmark crack problems are solved by the proposed XFEM approach. A convergence study has been conducted for few test problems. The results obtained by proposed XFEM approach are compared with the analytical/reference solutions.     

Multi‐scale modeling of surface effect via the boundary Cauchy–Born method

In this paper, a novel multi‐scale approach is developed for modeling of the surface effect in crystalline nano‐structures. The technique is based on the Cauchy–Born hypothesis in which the strain energy density of the equivalent continua is calculated by means of inter‐atomic potentials. The notion of introducing the surface effect in the finite element method is based on the intrinsic function of quadratures, called as an indicator of material behavior. The information of quadratures is derived by interpolating the data from probable representative atoms in their proximity. The technique is implemented by the definition of reference boundary CB elements, which enable to capture not only the surface but also the edge and corner effects. As the surface effect is important in small‐scale simulation, the relative number of boundary CB elements increases which leads to predomination of boundary effects in the model. In order to implement the equivalent continua in boundary value problems, the updated‐Lagrangian formulation of nonlinear finite element is derived. The numerical simulation of the proposed model together with the direct comparison with fully atomistic model indicates that the technique provides promising results for facile modeling of boundary effects and investigating its effect on the mechanical response of metallic nano‐scale devices. Copyright © 2010 John Wiley & Sons, Ltd.     

An extended finite element method applied to levitated droplet problems

We consider a standard model for incompressible two‐phase flows in which a localized force at the interface describes the effect of surface tension. If a level set method is applied then the approximation of the interface is in general not aligned with the triangulation. This causes severe difficulties w.r.t. the discretization and often results in large spurious velocities. In this paper we reconsider a (modified) extended finite element method (XFEM), which in previous papers has been investigated for relatively simple two‐phase flow model problems, and apply it to a physically realistic levitated droplet problem. The results show that due to the extension of the standard FE space one obtains much better results in particular for large interface tension coefficients. Furthermore, a certain cut‐off technique results in better efficiency without sacrificing accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

In nano‐structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano‐scale level is much higher than that of the macro‐scale level. In this paper, a novel temperature‐dependent multi‐scale model is presented based on the modified boundary Cauchy‐Born (MBCB) technique to model the surface, edge, and corner effects in nano‐scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo‐mechanical finite element model. The temperature‐related Cauchy‐Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

Contact fatigue of automotive gears: evolution and effects of residual stresses introduced by surface treatments

Helical gears from an automotive gearbox, previously subjected to the surface treatments of carbo‐nitriding and shot‐peening, were submitted to contact fatigue tests. The X‐ray diffraction technique was used to characterize the evolution of different mechanical and metallurgical parameters as a function of gear damage. Particular attention was paid to residual stress relief. A numerical model was developed to predict residual stress relaxation and estimate the most likely localization of contact fatigue crack initiation. The stress–strain laws of the surface‐treated layers were determined by means of two separate experimental methods, based on locally measured parameters. The Dang Van multiaxial fatigue criterion was used to analyse the failure of the gears, taking into account the effects of friction and roughness.     

Multiscale analysis method for thermo‐mechanical performance of periodic porous materials with interior surface radiation

This study develops a novel multiscale analysis method to predict thermo‐mechanical performance of periodic porous materials with interior surface radiation. In these materials, thermal radiation effect at microscale has an important impact on the macroscopic temperature and stress field, which is our particular interest in this paper. Firstly, the multiscale asymptotic expansions for computing the dynamic thermo‐mechanical coupling problem, which considers the mutual interaction between temperature and displacement field, are given successively. Then, the corresponding numerical algorithm based on the finite element‐difference method is brought forward in details. Finally, some numerical results are presented to verify the validity and relevancy of the proposed method by comparing it with a direct finite element analysis with detailed numerical models. The comparison shows that the new method is effective and valid for predicting the thermo‐mechanical performance and can capture the microstructure behavior of periodic porous materials exactly.s Copyright © 2015 John Wiley & Sons, Ltd.     

A finite element model for shape memory behavior

This paper deals with a finite element implementation concerning the shape memory behavior. Shape memory behavior is usually driven by temperature changes. This model allows the simulation of problems integrating complex mechanical loading effects under random temperature variations. According to the relationship between stress and strain, the shape fixation during cooling phases and the memory effect during heating phase are modelised through a hereditary behavior needing incremental formulation developments. The step by step process introduces an additional fixed stress. Simulations request, for complex geometries including boundary conditions, a finite element approach. Thermodynamic developments are presented in order to define energetic balance and dissipations. In this paper, we propose to generalize this dependence of elastic modulus variations. A formulation for random mechanical loading and temperature variations is proposed. An experimental validation is proposed about shape memory alloy polymer DP5.     

An XFEM‐based numerical strategy to model mechanical interactions between biological cells and a deformable substrate

Contractile cells are known to constantly probe and respond to their mechanical environment through mechanosensing. Although the very mechanisms responsible for this behavior are still obscure, it is now clear that cells make full use of cross‐talks between mechanics, chemistry, and transport to organize their structure and generate forces. To investigate these processes, it is important to derive mathematical and numerical models that can accurately capture the interactions between cells and an underlying deformable substrate. The present paper therefore introduces a computational framework, based on the extended FEM (XFEM) and the level set method, to model the evolution of two‐dimensional (plane stress) cells lying on an elastic substrate whose properties can be varied. Cells are modeled with a continuum mixture approach previously developed by the authors to describe key phenomena of cell sensing, such as stress fiber formation, mechanosensitive contraction, and molecular transport whereas cell–substrate adhesion is formulated with a linear elastic cohesive model. From a numerical viewpoint, cell and substrate are discretized on a single, regular finite element mesh, whereas the potentially complex cell geometry is defined in terms of a level set function that is independent of discretization. Field discontinuities across the cell membrane are then naturally enforced using enriched shape functions traditionally used in the XFEM formulation. The resulting method provides a flexible platform that can handle complex cell geometries, can avoid expensive meshing techniques, and can potentially be extended to study cell growth and migration on an elastic substrate. In addition, the XFEM formalism facilitates the consideration of the cell's cortical elasticity, a feature that is known to be important during cell deformation. The proposed method is illustrated with a few biologically relevant examples of cell–substrate interactions. Generally, the method is able to capture some key phenomena observed in biological systems and displays numerical versatility and accuracy at a moderate computational cost.Copyright © 2012 John Wiley & Sons, Ltd.