Continuum Finite Element Model Research Articles

Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes (SWCNTs) is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.

Study of architectural responses of 3D periodic cellular materials

The functional properties of periodic cellular solids such as photonic and phononic crystals, nanocrystal superlattices and foams may be tuned by an applied inhomogeneous mechanical strain. A fundamental methodology to analyse the structure of periodic cellular materials is presented here and is compared directly with indentation experiments on three-dimensional microframed polymer photonic crystals. The application of single-continuum-scale finite-element modelling (FEM) was impossible due to the numerous cells involved and the intricate continuum geometry within each cell. However, a method of dual-scale FEM was implemented to provide stress and displacement values on both scales by applying an upper scale continuum FEM with reference to the lower scale continuum FEM to provide coarse-grained stress–strain relationships. Architecture and orientation dependences of the periodic porous materials on the macro-/microscopic responses were investigated under different loading conditions. Our study revealed a computational tool for exploring elastic strain engineering of photonic crystals and, more broadly, may help the design of metamaterials with mechanical controllability.

A novel approach to estimate trabecular bone anisotropy using a database approach

Continuum finite element (FE) models of bones have become a standard pre-clinical tool to estimate bone strength. These models are usually based on clinical CT scans and material properties assigned are chosen as isotropic based only on the density distribution. It has been shown, however, that trabecular bone elastic behavior is best described as orthotropic. Unfortunately, the use of orthotropic models in FE analysis derived from CT scans is hampered by the fact that the measurement of a trabecular orientation (fabric) is not possible from clinical CT images due to the low resolution of such images. In this study, we explore the concept of using a database (DB) of high-resolution bone models to derive the fabric information that is missing in clinical images. The goal of this study was to investigate if models with fabric derived from a relatively small database can already produce more accurate results than isotropic models.A DB of 33 human proximal femurs was generated from micro-CT scans with a nominal isotropic resolution of 82µm. Continuum FE models were generated from the images using a pre-defined mesh template in combination with an iso-anatomic mesh morphing tool. Each element within the mesh template is at a specific anatomical location. For each element within the cancellous bone, a spherical region around the element centroid with a radius of 2mm was defined. Bone volume fraction and the mean-intercept-length fabric tensor were analyzed for that region. Ten femurs were used as test cases. For each test femur, four different models were generated: (1) an orthotropic model based on micro-CT fabric measurements (gold standard), (2) an orthotropic model based on the fabric derived from the best-matched database model, (3) an isotropic-I model in which the fabric tensor was set to the identity tensor, and (4) a second isotropic-II model with its total bone stiffness fitted to the gold standard. An elastic-plastic damage model was used to simulate failure and post failure behavior during a fall to the side.The results show that all models produce a similar stress distribution. However, compared to the gold standard, both isotropic-I and II models underestimated the stress/damage distributions significantly. We found no significant difference between DB-derived and gold standard models. Compared to the gold standard, the isotropic-I models further underestimated whole bone stiffness by 26.3% and ultimate load by 14.5%, while these differences for the DB-derived orthotropic models were only 4.9% and 3.1% respectively.The results indicate that the concept of using a DB to estimate patient-specific anisotropic material properties can considerably improve the results. We expect that this approach can lead to more accurate results in particular for cases where bone anisotropy plays an important role, such as in osteoporotic patients and around implants.

A finite element model for the bending and vibration of nanoscale plates with surface effect

A continuum finite element model for the nanoscale plates considering the surface effect of the material is developed. Governing equations for Kirchoff and Mindlin nanoplates are derived by using the Galerkin finite element technique. The model is verified by comparing the results with available analytical solutions. The results indicate that, depending on the boundary conditions, the deflections and frequencies of the plate have a dramatic dependence on the residual surface stress and surface elasticity of the plates. The present model is an efficient tool for the analysis of the static and dynamic mechanical behaviors of nanoscale plates with complex geometry, boundary and loading conditions and material properties.

Impact response and damage evolution of triaxial braided carbon/epoxy composites. Part II: finite element analysis

A series of ballistic tests on triaxial braided carbon/epoxy composites were described in Part I of this paper. In this part, numerical simulations were carried out to investigate the impact response, damage evolution, and penetration mechanisms of these composites. A continuum finite element model was developed to obtain the time history of the projectile velocity, displacement, penetration resistance force, and energy absorption during the impact process. By fitting the numerical data, the ballistic limit velocity can be obtained. Good agreements were achieved between numerical results and experimental results. Numerical predictions for ballistic tests of the composites indicate that, based on variations of damage mechanisms and failure features, the impact process can be subdivided into three stages, i.e. phase I – shock compression, phase II – bulge deformation, and phase III – penetration process, among which phase II consumes most of the projectile kinetic energy. The differences between a blade-like projectile and a cylindrical projectile in the impact process are also addressed in detail.

Significance of geotechnical loads on local buckling response of buried pipelines with respect to conventional practice

Long-term large deformation geohazards can impose excessive deformation on a buried pipeline. The ground displacement field may initiate pipeline deformation mechanisms that exceed design acceptance criteria with respect to serviceability requirements or ultimate limit states. The conventional engineering approach to define the mechanical performance of pipelines has been based on combined loading events for in-air conditions. This methodology may be conservative, as it ignores the soil effect that imposes geotechnical loads, and also provides restraint, on buried pipelines. The importance of pipeline–soil interaction and load-transfer mechanisms that may affect local buckling of buried pipelines is not well understood. A three-dimensional continuum finite element model, simulating the local buckling response of a buried pipe, using the software package ABAQUS/Standard was developed and calibrated. A comprehensive parametric study was previously conducted to investigate the effect of several parameters on local buckling response of pipelines buried in firm clay. A new strain criterion for local buckling of buried pipelines in firm clay through response surface methodology was developed. In this paper, the new criterion is compared with several existing in-air criteria to study the effect of soil restraint on the local buckling response of buried pipelines. The criterion developed in this study predicts greater characteristic critical strain capacity than in-air based criteria that highlights the influence of soil restraint.

A new approach to determine the accuracy of morphology–elasticity relationships in continuum FE analyses of human proximal femur

Continuum finite element (FE) models of bones are commonly generated based on CT scans. Element material properties in such models are usually derived from bone density values using some empirical relationships. However, many different empirical relationships have been proposed. Most of these will provide isotropic material properties but relationships that can provide a full orthotropic elastic stiffness tensor have been proposed as well. Presently it is not clear which of these relationships best describes the material behavior of bone in continuum models, nor is it clear to what extent anisotropic models can improve upon isotropic models. The best way to determine the accuracy of such relationships for continuum analyses would be by quantifying the accuracy of the calculated stress/strain distribution, but this requires an accurate reference distribution that does not depend on such empirical relationships. In the present study, we propose a novel approach to generate such a reference stress distribution. With this approach, stress results obtained from a micro-FE model of a whole bone, that can represent the bone trabecular architecture in detail, are homogenized and the homogenized stresses are then used as a reference for stress results obtained from continuum models. The goal of the present study was to demonstrate this new approach and to provide examples of comparing continuum models with anisotropic versus isotropic material properties.Continuum models that implemented isotropic and orthotropic material definitions were generated for two proximal femurs for which micro-FE results were available as well, one representing a healthy and the other an osteoporotic femur. It was found that the continuum FE stress distributions calculated for the healthy femur compared well to the homogenized results of the micro-FE although slightly better for the orthotropic model (r=0.83) than for the isotropic model (r=0.79). For the osteoporotic bone also, the orthotropic model did better (r=0.83) than the isotropic model (r=0.77). We propose that this approach will enable a more relevant and accurate validation of different material models than experimental methods used so far.

Non-oscillatory continuous FEM for transport and shallow water flows

The paper presents a sign-preserving continuous finite element model (FEM) for coupled transport equations, and in particular for the depth-integrated shallow water equations. The model incorporates an enhanced flux corrected transport method (FCT) with a high order solution given by a generalization of the characteristic based split finite element method (CBS), and a predictor positive algorithm founded in a finite element upwind methodology. Numerical experiments explore the efficacy of the procedure, with emphasis in the simulation of flows with evolutionary boundaries determined by null values of any flow property, and in severe conditions of propagation such as dam-breaking waves.

Study on the Nonlinear Vibration of Complex Rotor System under the Effect of Temperature Fields

In this paper, the flow and heat model of high pressure rotor (HPR) has been established in condition of nature convection at different parking times. The flow and temperature fields of the HPR were studied by using CFD. Moreover, the continuous finite element model with multiple nodes of the rotor system was established considering the asymmetric temperature effect. Comparison of different models between ignoring and considering temperature fields was employed in detail. The calculated results show that the vibration amplitudes are increased and the orbits become disordered when the parking temperature field is considered for the HPR. In the frequency spectrum, there are abundant frequent components caused by temperature and rubbing force and the vibration characteristics considering temperature effect are more complicated than ignoring it for the HPR.

Influence of building process on stiffness: numerical analysis of a masonry vault including mortar joint shrinkage and crack re-closure effect

The assessment of masonry structures is more and more useful to characterize the behaviour of masonry walls, tower and bridges. The owner of these constructions would like to understand the mechanical performance of their structures. Experiments and computational methods are two essential elements to accurately study and determine the sensitive points of the edifices. This paper presents the numerical modelling of a vault in undamaged and damaged domain with a macro-homogenisation analysis. The importance of considering the building phases in order to determine the actual stiffness of the structure is highlighted. The numerical tool is a continuous finite element model based on an original anisotropic damage model considering the masonry as a homogenised continuous media. The model is implemented in a massive finite element for three dimensional formulation. It computes the localized damage and allows indicating zones of privileged crack opening in the structure. It is able to predict the cracking zones in large masonry constructions without pre-positioning joint elements. It uses a regularization method ensuring an independence of the numerical response from the mesh. The model includes also a re-closure function for cracks. This re-closure function allows simulating the progressive stiffness restitution during the crack re-closure process. The paper focuses on this aspect in relation with the crack induced by restrained shrinkage of joint mortar during the building history. First, a state of the arts is proposed, then the damage model is presented. In order to validate the model and to propose a global methodology, the damage model is applied to analyse a thin masonry vault previously tested in a laboratory. The numerical results are discussed regarding the experimental ones.

Constitutive modeling of fiber composites with a soft hyperelastic matrix

This paper presents an experimental and numerical study of unidirectional carbon fiber composites with a silicone matrix, loaded transversally to the fibers. The experiments show nonlinear behavior with significant strain softening under cyclic loading. The numerical study uses a plane-strain finite element continuum model of the composite material in which the fiber distribution is based on experimental observations and cohesive elements allow debonding to take place at the fiber/matrix interfaces. It is found that accurate estimates of the initial tangent stiffness measured in the experiments can be obtained without allowing for debonding, but this feature has to be included to capture the non-linear and strain-softening behavior.

Investigating pipeline–soil interaction under axial–lateral relative movements in sand

This paper presents results from an experimental and numerical study on the axial–lateral interaction of pipes with dense sand. A series of centrifuge tests were conducted, with a rigid pipeline displaced in the horizontal plane in a cohesionless test bed. The relative pipe–soil interaction included axial, lateral, and oblique loading events. A three-dimensional continuum finite element model was developed using ABAQUS/Standard ( Hibbitt et al. 2005 ) software. The numerical model was calibrated against experimental results. A parametric study was conducted, using the calibrated finite element model to extend the investigations. The ultimate axial and lateral soil loading was found to be dependent on the angle of attack for relative movement between the pipe and soil. Two different failure mechanisms were observed for axial–lateral pipeline–soil interaction. This study confirms and improves on a two-part failure criterion that accounts for axial–lateral coupling during oblique soil loading events on buried pipelines.

A Combined Molecular Dynamics and Finite Element Analysis of Contact and Adhesion of a Rough Sphere and a Flat Surface

A combined molecular dynamics and finite element model and simulation of contact and adhesion between a rough sphere and a flat surface has been developed. This model uses the results of molecular dynamics (MD) simulations, obtained using an embedded atom potential, of a nanoscale Ru-Ru asperity contact. A continuum finite element model of an elastic–plastic microscale Ru-Ru contact bump is then created. In this model, the surface roughness is represented by a system of nanoscale asperities, each of which is represented by a nonlinear hysteretic force vs. distance relationship. The nonlinear hysteretic character of these relations is determined from curve-fits of the MD results. Load vs. interference and contact area vs. interference are determined using this two-scale model for loading and unloading. Comparisons with a single-scale continuum model show that the effect of the nanoscale asperities is to reduce both the adhesion and the real area of contact. The choice of Ru as the material for this work is due to its relevance in microswitches.

Fracture prediction of welded steel connections using traditional fracture mechanics and calibrated micromechanics based models

Fracture resistance is of primary concern in the seismic design of beam-to-column connections in steel moment resisting frames (SMRFs). Micromechanics based fracture models such as the void growth model (VGM) and the stress modified critical strain (SMCS) model provide alternative approaches for ductile fracture prediction by relating micro-mechanisms of void nucleation, growth and coalescence to macroscopic stresses and strains. In this study, the VGM and SMCS models were calibrated for Q345 structural steel and the corresponding weld, through smooth notched tensile (SNT) tests and complementary continuum finite element models (FEMs). A series of seven local connections representing beam-to-column connections in SMRFs were tested under monotonic tensile loading and the specimen elongations at fracture critical point were obtained. The traditional J-integral based fracture mechanics and micromechanics based fracture models (VGM and SMCS) were applied to predict fracture in each tested local connection through refined three-dimensional FEM. Comparisons between these numerical approaches and experimental observations in prediction of fracture critical displacement, indicated that the VGM and SMCS models were able to predict fracture of welded connection with good accuracy, while the J-integral based approach resulted in quite conservative fracture prediction. This paper has bridged the gap between small-scale material tests and large-scale structural experiments in fracture evaluations.

An efficient C0 FE model for the analysis of composites and sandwich laminates with general layup

A C0 continuous finite element model is developed to model the refined higher order shear deformation theory. The proposed element is an upgraded version of an element based on higher order shear deformation theory. The C0 continuity of the present element is compensated in the stiffness matrix calculations. The computational efficiency is achieved by the C0 continuous finite element model by satisfying the inter-laminar shear stress continuity at the interfaces and zero transverse shear stress conditions at plate top and bottom. The performance of the upgraded element is illustrated with many numerical examples.

Dynamic Characteristic Analysis of Large Space Deployable Articulated Mast

Equivalent continuum beam model and finite element model of space deployable articulated mast are developed. Dynamic characteristics of the mast without and with tip load are analyzed based on two models respectively. Frequencies and mode shapes of the mast are calculated and simulated by Euler beam theory and finite element simulation, and theoretical solutions are compared with simulation results. Influence of mast structural parameters on mast natural frequencies is analyzed, the sensitivities of mast bending, torsional, axial frequencies to structural parameters are calculated. The measure for improving mast dynamic characteristic is proposed through sensitivity analysis.

Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 Torr to 625 Torr

This paper reports on experimental and computational investigations into the thermal performance of microelectromechanical systems (MEMS) as a function of the pressure of the surrounding gas. High spatial resolution Raman thermometry was used to measure the temperature profiles on electrically heated, polycrystalline silicon bridges that are nominally 10 μm wide, 2.25 μm thick, and either 200 μm or 400 μm long in nitrogen atmospheres with pressures ranging from 0.05 Torr to 625 Torr (6.67 Pa–83.3 kPa). Finite element modeling of the thermal behavior of the MEMS bridges is performed and compared with the experimental results. Noncontinuum gas effects are incorporated into the continuum finite element model by imposing temperature discontinuities at gas-solid interfaces that are determined from noncontinuum simulations. The results indicate that gas-phase heat transfer is significant for devices of this size at ambient pressures but becomes minimal as the pressure is reduced below 5 Torr. The model and experimental results are in qualitative agreement, and better quantitative agreement requires increased accuracy in the geometrical and material property values.

Improved FE model for simulation of friction stir welding of different materials

One of the most relevant aspects of friction stir welding is the possibility to weld different materials. In the present paper, the authors present an improved continuum finite element model for the simulation of friction stir welding processes aimed to obtain T joints, made of a stringer in AA7175-T73511 and of a skin in AA2024-T4. The model, taking into account the thermomechanical behaviours of the two different materials, is utilised to study the occurring material flow and residual stress state. Numerical results are compared with experimental observations: the model is able to predict the material flow, obtaining important information on the joint failure mode.

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are an exciting development in computational physics. For example, the force-based quasicontinuum approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we analyze the stability of the force-based quasicontinuum approximation. We then use our stability result to obtain an optimal order error analysis of this coupling method that provides theoretical justification for the high accuracy of the force-based quasicontinuum approximation -- the computational efficiency of continuum modeling can be utilized without the loss of significant accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact (which we prove) that the linearized quasicontinuum operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the $W^{1,p}$-$W^{1,q}$ "duality pairing" with $1/p+1/q=1$, if $1 \leq p < \infty$. We must therefore derive an inf-sup stability condition for a discrete version of the $W^{1,\infty}$-$W^{1,1}$ "duality pairing" which then leads to optimal order error estimates in a discrete $W^{1,\infty}$-norm.

Characterizing the fracture parameters of a graphene sheet using atomistic simulation and continuum mechanics

The fracture behavior of a graphene sheet, containing a center crack (length of 2 a ) was characterized based on the atomistic simulation and the concept of continuum mechanics. Two failure modes, i.e., opening mode (Mode I) and sliding mode (Mode II), were considered by applying remote tensile and shear loading, respectively, on the graphene sheet. In the atomistic simulation, the equilibrium configurations of the cracked graphene subjected to applied loadings, before and after the crack extension of 2 Δ a , were determined through molecular dynamics (MD) simulation, from which the variation of the potential energy and the strain energy release rate of the discrete graphene sheet because of crack extension was calculated accordingly. It is noted that because of the discrete attribute, there is no stress singularity near the crack tip, and thus, the concept of stress intensity factor that is generally employed in the continuum mechanics may not be suitable for modeling the crack behavior in the atomistic structures. For the sake of comparison, the continuum finite element model with the same geometric parameters and material properties as the atomistic graphene sheet was constructed, and the corresponding strain energy release rate was calculated from the crack closure method. Results indicated that the strain energy release rates obtained from the continuum model exhibit good agreement with those derived from discrete atomistic model. Therefore, it is suggested that the strain energy release rate is an appropriate parameter, which can be employed in the atomistic model and the continuum model for describing the fracture of covalently bonded graphene sheet.