Two-dimensional Axisymmetric Finite Element Model Research Articles

Simulations of a stress and contact model in a chemical mechanical polishing process

A two-dimensional axisymmetric quasi-static contact finite element model for the chemical mechanical polishing process (CMP) was established. The von Mises stress on the wafer surface was investigated. The findings indicate that the profile of the von Mises stress correlated with that of the removal rate. The larger the elastic modulus of the pad or the smaller the elastic modulus of the carrier film, the larger is the maximum von Mises stress. The thicker the pad or the thinner the film, the smaller is the maximum von Mises stress. The larger the load exerted on the carrier, the greater is the maximum von Mises stress.

Finite element modeling, characterization, and optimization design for the polymer-typed capacitive micro-arrayed ultrasonic transducer

This study constructs a two-dimensional axisymmetric finite element model (FEM) of the polymer-typed capacitive micromachined ultrasonic transducers (P-CMUT). The electromechanical FE model is created using the APDL programming technique. The ANSYS multiphysics solver with sequential approach and the Physics environment files are applied for the solution of the electrostatic-structural coupled-field analysis. Simulations are performed to investigate the operational characteristics, such as collapse voltage and the resonant frequency of the P-CMUT. The numerical results are found to be in good agreement with experimental observation. Having confirmed validity and accuracy of the proposed numerical model, the study of influence of each defined parameter on the collapse voltage and resonant frequency of the P-CMUT are also presented. To solve some conflict problems taken place in diverse physical fields, an integrated multi-objective design method involving electrical and mechanical characterization is developed to optimize the geometric parameters and material properties of the P-CMUT. The optimization search routine conducted using the genetic algorithm (GA) is connected with the commercial FEM software ANSYS to obtain the best design variable using multi-objective functions. The results show that the optimal parameter values satisfy the conflicting objectives of the design procedures, namely to minimize the collapse voltage while simultaneously maintaining a high resonant frequency. Overall, the presented results confirm that the combined FEM/GA optimization approach provides an efficient and versatile means for optimization design of the P-CMUT.

Influence of a Retaining Ring on Strain and Stress in the Chemical Mechanical Polishing Process

In this article, a two-dimensional axisymmetric quasi-static finite element wafer scale model for chemical mechanical polishing (CMP) process involved in the wafer carrier, the carrier film, the wafer, the pad, and the retaining ring was developed to investigate the effect of a retaining ring surrounding the wafer carrier to the strain, stress, and nonuniformity of the wafer surface for the purpose of improving edge exclusion of wafer and preventing the wafer sliding from the carrier while grinding. Considering the same revolutions of the wafer and the pad and the axisymmetric distributed force forms of the wafer carrier and the retaining ring, and applying the principle of minimum potential energy, a two-dimensional axisymmetric quasi-static finite element model for CMP process was established. Following the developed model, the effects of the retaining ring on the strain components, the stress components, the von Mises stress, and the wafer's nonuniformity were investigated. The findings indicated that a retaining ring installed in the conventional CMP mechanism could reduce the variation of the von Mises stress distribution to reach the lower wafer's nonuniformity effectively, improve the over-grinding phenomenon and prevent the wafer sliding from the carrier while grinding.

Modelling of hardening behaviour of cold expanded holes in medium-carbon steel

Cold expansion is a well-known approach for the enhancement of the fatigue life of fastener holes in structural components by introducing residual compressive circumferential stresses around them. In this work, the hardening of a medium-carbon structural steel is studied experimentally and numerically from the point of view of the material behaviour during the cold expansion process of holes in structural steel components. Six material hardening models obtained on the basis of symmetric strain-controlled experiments and half-cycle test data from unidirectional tension experiments have been used in a two-dimensional axisymmetric finite element model of the process. Parallel with this an x-ray diffraction analysis of the residual stresses at the entrance and exit faces of the structural component has been carried out to prove the finite element result’s authenticity. It is established that the nonlinear kinematic hardening model obtained by a strain-controlled cyclic test to achieve a stabilized cycle for this steel secures finite element results close to the experimental ones. The constitutive hardening model obtained can be applied to a corresponding finite element model for stress analysis of steel structural component with cold expanded holes, subjected to an external load.

A study on the mechanical stress relieving and safety assessment without post-weld heat treatment

For full welded body valve, the temperature of grommet cannot exceed 150 °C in order to prevent it from damaging and assure the tightness and the service life of valve. Therefore, post-weld heat treatment (PWHT) cannot be used to relieve the residual stresses. In this study, the effect of the mechanical stress relieving (MSR) treatment on the residual stresses was studied by the finite element method and experimental work. A pressure and time diagram of MSR treatment was established. A two-dimensional axisymmetric finite element model was used to simulate the residual stresses field. Before and after MSR treatment, the residual stresses on the outer surface were measured by the blind hole drilling method. Finally, the fracture toughness behaviors of weld zone (WZ) and heat affected zone (HAZ) were investigated in terms of crack tip opening displacement (CTOD) according to BS7448 and DNV-OS-C401 fracture toughness tests standards. The safety of the valve in active service was assessed without PWHT. Through comparison and analysis, the axial residual stresses and the hoop residual stresses on the outer surface of valve are mainly tensile. The peak value of tensile stress occurs nearer to the outer surface of the valve. MSR treatment can decrease the peak value of axial residual stresses and hoop residual stresses on the outer surface obviously and make the residual stresses distribution more uniform. The safety of the valve in active service is reliable without PWHT.

Study of the Residual Strain in Lap Joints

Both experiments and finite element analysis were carried out to study the residual strain (stress) in the riveted lap joint for a better understanding of the fatigue life of fuselage lap joints. A force-controlled riveting process was used to apply a constant load ramp to install the rivets. Strain variations on the joint surface were measured using microstrain gauges during the riveting process. Because neutrons are known to penetrate through many centimeters of aluminum alloys, neutron diffraction was used to provide a nondestructive technique to determine strains at certain depths in the joint. Parallel to the experimental testing, a two-dimensional axisymmetric finite element model was developed to simulate the riveting process. Both material and geometric nonlinearities, as well as nonlinear contact boundary conditions, were used in this numerical model. Comparisons between the numerical simulations and experimental results focused on the rivet driven head deformations and strain variations, and the results showed that the current two-dimensional axisymmetric finite element model using the proper boundary conditions can reliably be used to determine the residual strains (stresses) present in joints that were induced during the riveting process. Nomenclature C = material parameter D = rivet shank diameter Dmax = maximum rivet shank diameter after riveting d = spacing between lattice plane d0 = stress-free lattice spacing E = Young’s modulus m = material parameter r = distance to fastener hole centre e = normal strain etrue = true strain θ = diffracted neutrons’ angle λ = wavelength of the incident neutron beam ν = Poisson’s ratio σ = normal stress σtrue = true stress beyond the initial yield stress σy = initial yield stress

Analysis of retaining ring using finite element simulation in chemical mechanical polishing process

During the chemical mechanical polishing process (i.e., CMP for short), it is expected to attain the requirement of global planarization. However, the stress concentration, which occurred when approaching the wafer edge, has resulted in over-grinding. The increasing material removal rate has also contributed to the wafer’s nonuniformity. In this paper, a retaining ring surrounding the wafer carrier was added to the conventional CMP mechanism in order to improve the over-grinding phenomenon and avoid the wafer sliding from the carrier as much as possible. The revolutions of the wafer and the pad were assumed to be the same, and the force forms of the carrier and the retaining ring were axisymmetric uniformly-distributed. In addition, when the principle of minimum potential energy was applied, a two-dimensional axisymmetric quasi-static finite element model for CMP including the carrier, the retaining ring, the film, the wafer and the pad could be established. Following this model, the von Mises stress distribution of the wafer surface without a retaining ring was analyzed to verify the model. The effects of the gap between the ring and the pad and the ratio of the ring load and the carrier load on the stress and the wafer’s nonuniformity were investigated. The results indicated that the von Mises stress distribution of the wafer surface was almost uniform near the wafer center, and the maximum value appeared near the edge. The value decreased as the edge was approached, but it rose again very close to the edge. Besides, the wafer’s nonuniformity would be prohibited while the gap and the load were designed within the certain range to supply the retaining ring.

A multi-coupled finite element analysis of resistance spot welding process

A two-dimensional axisymmetric finite element model is developed to analyze the transient thermal and mechanical behaviors of the Resistance Spot Welding (RSW) process using commercial software ANSYS. Firstly a direct-coupled electrical-thermal Finite Element Analysis (FEA) is performed to analyze the transient thermal characteristics of the RSW process. Then based on the thermal results a sequential coupled thermo-elastic-plastic analysis is conducted to determine the mechanical features of the RSW process. The thermal history of the whole process and the temperature distribution of the weldment are obtained through the analysis. The mechanical features, including the distributions of the contact pressure at both the faying surface and the electrode-workpiece interface, the stress and strain distributions in the weldment and their changes during the RSW process, the deformation of the weldment and the electrode displacement are also calculated.

Deformation of press-fitted metallic resurfacing cups. Part 2: Finite element simulation

The deformation of metallic acetabular cups employed for metal-on-metal hip resurfacing procedures was considered theoretically using the finite element method in the present study, following on the experimental investigation reported in Part 1. Three representative cups, characterized by the cup wall thickness as thin, intermediate, and thick, were considered. For the intermediate cup, the effects of both the size and the diametral interference on the cup deformation were investigated. Both two-dimensional axisymmetric and three-dimensional finite element models were developed to examine the important parameters during and after the press-fit procedure, and in particular the deformation of the metallic cup. The theoretical prediction of the cup deformation was in reasonable agreement with the corresponding experimental measurement reported in Part 1. The most significant factor influencing the cup deformation was the cup wall thickness. Both the size and the diametral interference were also shown to influence the cup deformation. It is important to ensure that the cup deformation does not significantly affect the clearance designed and optimized for tribological performances of metal-on-metal hip resurfacing prostheses. Furthermore the contact parameters at the cup and bone interface associated with the press fit were also discussed.

Strain and stress analysis of the oxide film surface in the chemical mechanical polishing process

A two-dimensional axisymmetric quasi-static finite element model for a chemical mechanical polishing (CMP) process of 12″ wafer was established. The strain, stress and non-uniformity of oxide film surface are examined. The findings indicate that the axial strain was the maximum in strain components and its peak occurred at the end, the axial stress had a similar trend as the axial strain, and the elastic modulus and the thickness of the pad and carrier load would significantly affect the axial strain, the von Mises stress and the non-uniformity, but those of the film have little effect.

Numerical simulation of mechanical stress relieving in a multi-pass GTA girth welded pipe–flange joint to reduce IGSCC

Welding of piping components produces highly non-uniform residual stresses in the weldment which consequently affect material structural response under loading and also expedite origination/propagation of defects such as stress corrosion cracking and brittle fracture. For welding-induced residual stresses, mechanical stress relieving (MSR) is one of the mitigation techniques to improve the service life of pressure vessel components. However, its application to piping systems is rarely reported in the literature. This paper presents a two-dimensional axisymmetric finite element model of a pipe–flange joint subjected to a multi-pass girth welding followed by an MSR process. Sequentially coupled non-linear transient thermo-mechanical analysis for multi-pass gas tungsten arc welding is first performed to calculate welding residual stresses. Subsequently separate parametric studies for three different types of MSR load including internal pressure, external pressure and axial pull are performed on the pre-stressed model, and stress relieving behaviour is studied. It is concluded that both the internal pressure and axial pull have a significant effect on residual stresses on the inner surface of the joint.

Modeling of chemical mechanical polishing process using FEM and abductive network

In this paper, modeling of a two-dimensional axisymmetric quasic-static finite element model in conjunction with an abductive network for chemical–mechanical polishing process (CMP) was established. Three prediction models can be achieved, i.e., model for von Mises stress at wafer center, model for maximum von Mises stress and model for nonuniformity on wafer surface under various combinations of process parameters. The data of von Mises stress and nonuniformity on wafer surface can be first achieved under different conditions of the carrier load, pad's elastic modulus and thickness by using the developed finite element model for CMP. Next, an abductive network was applied to synthesize the data sets from the FE simulation. It is a self-organizing adaptive modeling tool that establishes the mathematical relationship between input and output variables based on abductive modeling technique and it can automatically synthesize the optimal network structure, including the optimal network structure, the number of layers and the form of functional nodes. Finally, the results from the three developed abductive networks with test data are compared with those from FE simulation to confirm the feasibility of this approach. The findings verified that the results confirm the feasibility and the proposed prediction models for CMP are acceptable.

Optimization of the Closure-Weld Region of Cylindrical Containers for Long-Term Corrosion Resistance Using the Successive Heuristic Quadratic Approximation Technique*

Welded cylindrical containers are susceptible to stress corrosion cracking (SCC) in the closure-weld area. An induction coil heating technique may be used to relieve the residual stresses in the closure-weld. This technique involves localized heating of the material by the surrounding coils. The material is then cooled to room temperature by quenching. A two-dimensional axisymmetric finite element model is developed to study the effects of induction coil heating and subsequent quenching. The finite element results are validated through an experimental test. The container design is tuned to maximize the compressive stress from the outer surface to a depth that is equal to the long-term general corrosion rate of the container material multiplied by the desired container lifetime. The problem is subject to several geometrical and stress constraints. Two different solution methods are implemented for this purpose. First, an off-the-shelf optimization software is used. The results however were unsatisfactory because of the highly nonlinear nature of the problem. The paper proposes a novel alternative: the Successive Heuristic Quadratic Approximation (SHQA) technique. This algorithm combines successive quadratic approximation with an adaptive random search within varying search space. SHQA promises to be a suitable search method for computationally intensive, highly nonlinear problems.

Modelling creep deformation in floating ice

The ability to predict the response of an ice sheet to long-term loading is important in many situations. In northern regions, ice sheets have been used as construction platforms, drilling rig platforms, airfields, parking lots, and festival platforms. Numerical models can be used to predict the deflection of an ice sheet over time and, together with a failure criterion based on allowable deflection, can facilitate the safe use of an ice cover under long-term loading situations. In this study, a two-dimensional axisymmetric finite element model was developed to model the behaviour of a homogeneous, transversely isotropic, infinite ice cover under long-term loading. The model was validated using 33 sets of long-term load test data from large-scale field experiments performed on lake, bay, and reservoir ice and was shown to be capable of reliably predicting deflections under a variety of loading scenarios.Key words: bearing capacity of ice, long-term loading, ice platforms.

Artificial neural network and finite element modeling of nanoindentation tests

This study first presents two-dimensional (2-D) axisymmetric and three-dimensional (3-D) finite element (FE) models of nanoindentation tests. Calculated load-displacement curves from the FE models are compared with the load-displacement curves from nanoindentation measurements on annealed copper. Numerical parametric studies are also performed to examine the effect of indenter geometry and the material’s stress-strain behavior on the load-displacement response. The 2-D and 3-D FE load-displacement curves compare well with the measured results on annealed copper. The second aspect of this study introduces a new modeling approach for indentation tests using artificial neural networks (ANN). In this approach, ANN models are generated to approximate the FE load-displacement curves for a wide range of material and geometric parameters. The ability of the ANN models to predict the indentation response is examined against other FE results not used as part of the training data. These models are shown to accurately predict the load-displacement behavior of a nonlinear homogeneous material as well as one with a hard film, such as an oxide film, on a relatively soft substrate. It is shown that the monotonic indentation load-displacement response during loading contains ample information for the ANN model to extract material flow properties of the indented material.

Width-Wise Variation of Magnetic Tape Pack Stresses

A model is developed for predicting the stress and displacement fields within a magnetic tape pack, where those quantities are allowed to vary in both the pack’s radial and transverse (cross-tape) directions. As has been the case in previous analyses based upon one-dimensional wound roll models, the present approach accounts for the anisotropic and nonlinear constitutive properties of the layered tape, and the incremental manner in which the pack is wound. Further, such widthwise variation effects as differential hub compliance and nonuniform winding tension, which can be significant in data cartridge design, are also treated in the model. The pack is analyzed through a two-dimensional axisymmetric finite element model that couples individual representations of the hub/flange and layered tape substructures. The bulk radial elastic modulus of the tape, which depends on the in-pack radial stress, is measured for a variety of media samples, and a reduced-order model is developed to capture the nonlinear modulus-stress correlation. The stiffness matrix of the hub/flange at its interface with the media provides a mixed boundary condition to the tape substructure. In this manner, design-specific hubs can be readily analyzed, and criteria for their optimization explored. Simulations of several cartridge designs are presented, and the roles of hub compliance and wound-in tension gradient in setting the pack’s stress field and cross-tape width change are discussed.

Parametric finite element analysis of verterbral bodies affected by tumors

The vertebral column is the most frequent site of metastatic involvement of the skeleton. Due to the proximity to the spinal cord, from 5% to 10% of all cancer patients develop neurologic manifestations. As a consequence, fracture risk prediction has significant clinical importance. In this study, we model the metastatically involved vertebra so as to parametrically investigate the effects of tumor size, material properties and compressive loading rate on vertebral strength. A two-dimensional axisymmetric finite element model of a spinal motion segment consisting of the first lumbar vertebral body (no posterior elements) and adjacent intervertebral disc was developed to allow the inclusion of a centrally located tumor in the vertebral body. After evaluating elastic, mixed, and poroelastic formulations, we concluded that the poroelastic representation was most suitable for modeling the metastatically involved vertebra's response to compressive load. Maximum principal strains were used to localize regions of potential vertebral trabecular bone failure. Radial and axial vertebral body displacements were used as relative indicators of spinal canal encroachment and endplate failure. Increased tumor size and loading rate, and reduced trabecular bone density all elevated axial and radial displacements and maximum tensile strains. The results of this parametric study suggest that vertebral tumor size and bone density contribute significantly to a patients risk for vertebral fracture and should be incorporated in clinical assessment paradigms.

Numerical modelling of the driving response of thin‐walled open‐ended piles

AbstractThe driving response of thin‐walled open‐ended piles is studied using numerical simulation of the wave propagation inside the soil plug and the pile. An elastic finite element analysis is carried out to identify the stress wave propagation in the vicinity of the pile toe. It is found that the shear stress wave has the highest magnitude above the bottom of the soil plug. Below the bottom of the soil plug, the vertical stress wave has the highest magnitude. Although the shear stress wave propagating in the radial direction is similar in magnitude to the vertical stress wave at the bottom of the soil plug, it decays rapidly while travelling downwards. The highest vertical stress at the bottom of the soil plug appears after the vertical stress wave interacts with the shear stress wave travelling in the radial direction. Initially, the vertical stress wave propagates with the dilation wave velocity in both the radial and vertical directions. After it interacts with the shear stress wave, the vertical stress wave starts to propagate with the shear wave velocity in the radial direction and with the axial wave velocity downwards. It is concluded that at the bottom of the soil plug, the interaction between the waves travelling in radial and vertical directions is important.The capabilities of several one‐dimensional pile‐in‐pile models to reproduce the driving response given by a two‐dimensional axisymmetric finite element model is studied. It is seen that when the base of the soil plug fails, a one‐dimensional pile‐in‐pile model can be used to achieve results in agreement with the finite element model. However, when the pile is unplugged, where the base of the soil plug does not fail, a reduced finite element mesh that permits the radial wave propagation inside the soil plug must be used. Copyright © 2001 John Wiley & Sons, Ltd.

Fracture testing and finite element modeling of pure titanium

This paper presents the results of fracture experiments and corresponding finite element analyses (FEA) of pure titanium. This investigation was motivated by the desire to develop a J– R testing protocol and numerical procedures that are applicable to a titanium/titanium boride layered functionally graded material. Tensile tests and a two-dimensional axisymmetric finite element model were used to determine the plasticity data for the titanium. Crack growth experiments were conducted in three-point bending using single edge notched bend specimens. Three-dimensional FEA of crack growth initiation and two-dimensional FEA with automatic crack propagation were performed. Two crack propagation conditions based on experimental data were used: (a) crack length versus load-line displacement and (b) crack length versus crack mouth opening displacement. The subsequent predictions of the non-linear finite element models are in reasonable agreement with the measured value of J at initiation and with the rising J– R data during crack propagation.

A TWO-DIMENSIONAL AXISYMMETRIC NUMERICAL ANALYSIS OF A BAR–BAR TENSILE IMPACT APPARATUS BY ELASTOPLASTIC FEM

At present, a bar–bar Tensile Impact Apparatus (BTIA) is widely used to measure the dynamic tensile properties of solid materials. A BTIA consists of two long bars, between which a specimen is connected. Hence, there are several physical and geometric discontinuous sections in a BTIA. In the present paper, a two-dimensional axisymmetric elastoplastic finite element model for a BTIA with a dumb bell-shaped cylindrical specimen is established. The numerical solution for the model is solved by ADINA. In the scope of elastoplastic theory, the numerical analysis confirms the validity of the 1D experimental measuring principle of BTIA, so long as certain foundations are satisfied. The effects of the specimen geometry and strain rate are also discussed. The matching relation between the specimen and the BTIA system has been studied.