Finite Element Analysis Solutions Research Articles

J-integral expressions for semi-elliptical cracks in round bars

A finite element analysis (FEA) using the J-integral crack tip parameter has been employed to study 3-D semi-elliptical crack growth in round bars for a strain hardening material subject to uniaxial fatigue loading in both linear-elastic and elastic–plastic low cycle fatigue (LCF) conditions. The bar and crack geometries as well as the loading conditions are based on experimental observations. Similar FEA can be found in the literature, but they are limited to linear elastic conditions where a J-integral analysis is not necessary. To extend the analysis to incorporate plasticity, material Ramberg–Osgood hardening constants are input into the models. A generalized form of the stress intensity correction factor for the elastic and plastic parts of Δ J is obtained by manipulating the elastic and plastic solutions for the J-integral and utilizing the FEA solutions for estimating Δ J under elastic and elastic–plastic conditions. The correction factor for the plastic part of Δ J is derived to be dependent on the bar and crack geometry and also strongly dependent on the Ramberg–Osgood hardening exponent, which conflicts with previous results for 2-D and 3-D crack geometries.

Experimental, analytical, and finite element analyses of nanoindentation of multilayer PZT/Pt/SiO2 thin film systems on silicon wafers

The mechanical properties of lead zirconate titanate (PZT) multilayer systems are needed to model and design micro-electromechanical systems (MEMS) devices. Nanoindentation is a promising tool for obtaining the elastic properties of thin films. However, no means exist to obtain the elastic modulus of the lead zirconate titanate (PZT) in the multilayer system. The indentation modulus versus a/t behavior of the multilayered PZT/Pt/SiO2 film system on a silicon substrate was investigated and compared with finite element models and a new analytical solution. Six different PZT film thicknesses were indented (100, 140, 400, 700, 1500, and 2000 nm), using 5-, 10-, and 20-μm radius indenters. Good agreement was shown between the finite element analysis (FEA) and analytical solutions, and the experimental data. However the behavior of multilayer systems is complex, making deconvolution of properties difficult for films of less than a micron thick.

Approximate analysis for buckling of compression loaded long rectangular plates with flexural/twist anisotropy

Approximate expressions for buckling loads of long rectangular plates with flexural/twist anisotropy and simply supported edges are derived. The novelty of the current approach lies in the use of bounded non-dimensional quantities to reduce the complexity of formulation to such a state that closed form solutions are possible. Such expressions are validated with respect to detailed finite element analysis and numerical solutions found in the literature. A number of studies are conducted using a composite material with extremely large ply stiffness in the fibre direction and whose layers are arranged into extreme stacking sequences. Such an arrangement maximizes anisotropic effects and so challenges the validity of the solutions in a most testing manner. Despite such extreme studies excellent agreement is found.

Finite Element Analysis of Arc-Welding Process by Parallel Computation

2) introduced the method for the analysis of welding processes. The subjects of the FEA (finite element analysis) of arcwelding processes may be categorized into two in general. The first is on the modeling of welding process itself, and it has been already established regarding how to model heat input, what constitutive equation to use, and how to simulate the filling of filler material, etc. The other is on the numerical analysis and the associated techniques for the fast solution of the threedimensional FEA for welding process, for the three-dimensional FEA of welding process requires an enormously great deal of computing time and computer memory. Three-dimensional FEA of welding process, which particularly includes heat transfer, metallurgical phase transformation and transformation plasticity, needs a huge amount of computation time and, then it limits practical applications of the method for industrial welding problems. Application of parallel computation is a general alternative to reduce the computation time without any loss of accuracy of the results. In this paper, we explore on the effectiveness of the application of parallel computation for arc welding problems. The outline of the paper is as follows. Firstly, a brief review of finite element formulation for arc-welding process is given in Section 2, and the implementation of a parallel frontal solver developed for FEA of arc-welding process is given in Section 3. In Section 4, a numerical test for butt-welding process is provided to demonstrate the effectiveness of parallel computation.

Application of plastic framing requirements for polar ships

A new International Association of Classification Societies (ICAS) standard for Polar Ship design, in the form of a Unified Requirement is being developed by an international committee with representatives from many classification societies and with the active participation of many polar nations. The framing structural requirements have been developed by a combination of analysis of existing rules and ships, finite element analysis and analytical solutions of plastic collapse mechanisms. This paper describes the use of the rule formulae in designs. The formulae contain interactions between shear and bending and give the designer options and flexibility in satisfying the strength requirements. Effective design requires knowledge of the effects, and the possible approaches to reach a satisfactory design.

Derivation of plastic framing requirements for polar ships

A new International Association of Classification Societies standard for polar ship design, in the form of a unified requirement is being developed by an international committee with representatives from many classification societies and with the active participation of many polar nations. The framing structural requirements have been developed by a combination of analysis of existing rules and ships, finite element (FE) analysis and analytical solutions of plastic collapse mechanisms. This paper describes the derivation of the 3-hinge and asymmetrical shear plastic collapse mechanisms using work–energy principles. Energy methods are robust and well suited for developing design standards. The results are shown to compare well with non-linear FE analyses of frame strength.

Crack-tip force method for computing energy release rate in delaminated plates

A new method called the crack-tip force method (CTFM) is derived for computing the energy release rate in delaminated beams and plates. In this method the delaminated plate is divided into two laminates on either side of the plane of delamination. The interaction forces, called crack-tip forces, between the sub-laminates at the crack-tip are computed. The energy release rate is expressed as a quadratic function of the crack-tip forces and the plate compliance coefficients. The CTFM is compared to the virtual crack closure technique (VCCT) as well as to a previously derived method called the strain energy density method using double cantilevered beam specimens as examples. The CTFM is found to be very efficient as the crack-tip forces are part of the solution of finite element analysis of delaminated plates, and they can be readily used to compute the point-wise energy release rate along the delamination front.

Wrinkling of sandwich column: comparison between finite element analysis and analytical solutions

Wrinkling analysis of sandwich columns was carried out. Two methods were used, namely finite element methods and analytical solutions. In the finite element methods, shell elements based on Reissner–Mindlin were employed across the beam thickness. A very fine element was used to model both the skin and the core across the beam thickness. The buckling mode was increased up to 100 in order to be able to model the wrinkling mode. The analytical solutions used energy methods and Raleigh–Ritz solutions developed earlier by the author. Finally, comparisons were made with experimental data taken from published results. Good agreements were achieved between the experimental results and both the finite element and analytical solution. The difference was less than 10%. Both finite element and analytical solutions were able to predict both symmetrical and asymmetrical wrinkling.

Elastomer biaxial characterization using bubble inflation technique. II: Numerical investigation of some constitutive models

AbstractAn elastomeric material was investigated with a bubble inflation rheometer, and its mechanical behavior was modeled as a rubbeer‐like solid. Classical strain energy functions were considered and the hyper‐elastic were calculated by a direct identification procedure from simple uniaxial and equibiaxial extension test data, and the reults are compared against those obtained by an inverse method from matching the meaured response to a finite element to a finite element analysis solution, which dependent on the unknown material parameters. The optimised employed the Levenberg‐Marquardt algorithm and Abaqus software to compute the cost function and its gradients. The constants so obtained were further used in finite element analysis, and the numerical results were compared with experiments. This study showed that the inverse method, used to estimate the material parameters, is a good alternative to the direct identification, especially since the latter often requires homogeneous strain state, which is very difficult to obtain.

Effect of Environment on Fracture Toughness and Debond Growth of Aluminum/FM®73/Boron-Epoxy Adhesively Bonded Joints

Tests were conducted to determine the fracture toughness of aluminum/FM®73/boron-epoxy adhesively bonded joints. Double cantilever beam (DCB) specimens and end-notched flexure (ENF) specimens were exposed to one of the following environments: hot/wet (71° C(160° F) and >90% RH) conditions for 5000 h, thermal cycles between 71° C (160° F) and −54° C (−65° F), or an outdoor “rooftop” environment for 10 000 h. The environmentally exposed specimens were tested along with as-received (or unexposed specimens). The different adherend materials had different coefficients of thermal expansion (CTE). The mismatch of CTE between adherends caused residual thermal stresses in the adhesive joints at every temperature other than the elevated cure temperature. The residual stresses could not be accounted for with closed form solutions. Finite element analysis (FEA) of the specimens was required. Using FEA solutions, it was determined that the fracture toughness of the environmentally exposed specimens was lower than the as-received specimens for every environment tested for both DCB and ENF specimens.

An illustration of analytical/numerical matching with finite-element analysis for structural vibration problems

Analytical/numerical matching (ANM) is an accurate and efficient method for solving many types of problems with discontinuities. The method separates local and global effects, and solves separate subproblems using high resolution around the discontinuity and low resolution away from the discontinuity. The work presented in this manuscript demonstrates a methodology for applying ANM to a dynamic structure using finite-element analysis (FEA) for the solution of the high-resolution (local) and the low-resolution (global) subproblems. The ANM method is illustrated on a thick, two-dimensional beam having several displacement constraints attached to its lower surface. Ordinarily (and here, for verification purposes) this problem would be solved using two-dimensional plane elements due to the local discontinuities around the constraints and the thickness of the beam. Using ANM, these discontinuities and through-thickness effects are modeled in the geometrically compact local problem using a high-resolution mesh of two-dimensional eight-node plane elements. The much larger global problem contains no discontinuities and is reduced to the solution of a low-resolution finite-element mesh of two-node Bernoulli–Euler beam elements. A third subproblem (matching) is solved analytically (no computational overhead). The agreement between the ANM solution and the purely FEA solution is excellent, and the computational savings are significant.

Comparison of a Higher Order Theory for Sandwich Beams with Finite Element and Elasticity Analyses

Sandwich construction is known to provide high stiffness with light weight. However the question of analysis methods for the deformation and stresses in sandwich construction is still not completely settled. A higher order theory for sandwich beams has recently been presented in the literature. This theory is compared with a two-dimensional finite element analysis and elasticity solution in the present work. The results show that both the higher order theory and the finite element analysis predict strain concentrations in the faces adjacent to support and concentrated load regions, and overall good agreement is obtained between these methods. Classical and first order shear deformation theories are not capable of predicting these strain concentrations. Stress and strain distributions in the core are not in as good agreement in these regions, due to the simplifying assumptions made by the higher order theory. The results show that the higher order theory is an effective tool in predicting the stresses in the faces and the interface between the face and core, although the shear stress distributions in the core are not as well predicted.

Experimental, numerical and analytical results for bending and buckling of rectangular orthotropic plates

Solid unstiffened, sandwich and hat-stiffened rectangular orthotropic fiber reinforced plastic (FRP) plates were tested for buckling by in-plane compression and for stresses and deflections under uniform out-of-plane pressure. The solid unstiffened and hat-stiffened plates were 154 × 77 cm (1 × w) (72 × 36 in), while the sandwich plates were 102 × 77 cm (1 × w) (48 × 36 in). Balsa core was used in the sandwich plates and in the hat-stiffeners. The two short edges of the unstiffened and sandwich plates were clamped, while the two long edges were simply supported. The two long edges of the hat-stiffened plates were free, while the short edges were clamped. The buckling load, as well as stresses and deflections from the tests, were then compared to those from finite element analysis (FEA) and analytic solutions. There was reasonably good agreement between FEA, analytic, and experimental buckling stresses for the unstiffened solid plates. There was reasonable agreement in buckling stresses between FEA and experimental results for the hat-stiffened plate. There was poor agreement between FEA, analytic, and experimental elastic buckling results for the sandwich plates because they failed in local buckling prior to global buckling. Under out-of-plane uniform pressure, FEA and analytic solutions of the stresses and deflections for the unstiffened solid plates agreed well with experimental results. There was poor agreement between FEA and experimental results for stresses and deflections of the hat-stiffened or sandwich plates. Experimental error could be traced, in part,to plate fabrication, the method of applying out-of-plane pressure, edge support, and instrumentation accuracy.

Estimating the plastic collapse of pressure vessels using plasticity contours

This investigation concerns the use of non-linear finite element analysis (FEA) to predict the plastic collapse of pressure vessels with and without defects. Computational non-linear plastic analysis of pressure vessels to estimate the collapse pressure is an iterative process. As a consequence, if the models are large and complex, considerable computer resources and time are required to generate solutions. As normal materials tend not to be ideally elastic–plastic (no sudden plastic collapse), a number of techniques are used to estimate the collapse pressure based on the FEA solution. These techniques often rely on plotting the pressure–strain curve; however, other techniques are also possible and are considered here. The aim of this paper is to assess the different methodologies in conjunction with FEA. Firstly, to consider their accuracy and secondly, to ascertain whether a particular technique would result in a reduction in the analysis time necessary to recover the plastic collapse pressure. Of the methods analysed, one based on the measurement (as a percentage) of the leading edge of the effective plastic strain (EPS) contour band showed promise. Under increasing pressures, regardless of geometry or material yield curve, the %EPS contour area coverage displays linear behaviour before through-wall plasticity is achieved. Based on this it is possible to extrapolate from low levels of plastic deformation up to plastic collapse pressures without requiring additional non-linear analyses, resulting in significant cost reductions in computational analysis times.

Mixed Finite Element Analysis and Numerical Solitary Solution for the RLW Equation

In this paper a mixed finite element (MFE) formulation is proposed for the 1-D regularized long wave (RLW) equation $u_t+auu_x-\delta u_{xxt}=0$. The existence of its mixed generalized solution and semidiscrete and fully discrete mixed finite element solutions is proved, error estimates based on energy methods are given, and finally, results of numerical experiments comparing this formulation with a Galerkin method based on cubic splines are also given.

Modelling crack propagation in structural adhesives under external creep loading

Abstraet-Bulk adhesive compact tension specimens have been subjected to creep loads which range from 45% to 80% of the initial static failure load. Measurements of crack growth, load-line displacement, and crack tip profile were taken. Tensile creep tests were carried out on the same material and the results were used to generate material models for the analytical solution of steady-state power law (SSPL) creep and the numerical (finite element analysis) solution which incorporates the transient and primary creep material behaviour. Standard solutions for SSPL creep of compact tension specimens were used to calculate fundamental parameters such as the C* integral, creep zone growth, and transition time. From these data it would seem likely that the crack growth is controlled by creep, rather than elastic, singular fields. Crack driving parameters from both elastic and creep fields appear equally valid but the latter appear more discriminating. Finite element analyses were carried out for both a stationary crack and a propagating crack. The former are relevant to the incubation period and allow the redistribution of stress and accumulation of creep strain to be assessed. The latter enable the predicted conditions in the domain surrounding the propagating crack to be investigated for appropriate crack driving parameters. There is strong evidence to suggest that the adhesive uni-axial creep rupture strain can be successfully used to predict creep crack growth.

Theory of beam-induced substrate heating

In all electron, ion, and photon beam systems, the beam unavoidably deposits heat in the target substrate. Resist sensitivity varies with temperature, causing unwanted linewidth variation. Substrate heating has been studied by finite element analysis solution of the heat equation. The four-dimensional nature of the problem, three spatial dimensions together with time, leads to prohibitively long computation time. This limits our ability to obtain a physical understanding of the problem. Alternatively one can superimpose elementary analytic solutions, which have simple space-time dependency, to construct cases of practical interest. The purpose of this article is to describe mathematically the basic building blocks of a general analytic theory of beam heating, with the aim of efficiently and quickly computing temperature rise due to beam-induced substrate heating. Beam heating is most critical in writing of quartz reticles. Because of the poor thermal conductivity of quartz, the temperature rise at 10 μC/cm2, 30 A/cm2, and 50 kV is 45–85 °C, depending on pattern density. Heating is also significant for direct writing on silicon. At 10 μC/cm2, 50 A/cm2, and 100 kV, the temperature rise is 0.4–48 °C, depending on pattern density. Beam heating depends directly on the incident energy per unit area, which is given by the dose times the beam voltage. The dose is, in turn, proportional to the beam voltage, due to the increased transparency of resist layers at high voltage. Beam heating therefore increases with the square of the beam voltage. These trends suggest fast resist and low voltage as the best means of containing the effect. The latter conflicts with the need for high resolution, which requires high beam voltage. This suggests an optimum beam voltage for a given set of conditions.

Fracture mechanics approaches to coating strength evaluation

The incidence of mechanical failure of flame sprayed molybdenum coating on piston rings prompted development of reliable methods of characterising and improving bond strengths. A method of using the ASTM specimen with a small modification, to estimate fracture toughness properties of the coating is proposed. Flame spraying process parameters are optimised to produce coatings of high adhesive strength using Taguchi techniques by using the modified ASTM specimen. Experiments are conducted using four point bend specimen, to study the coating behaviour under complex stress conditions. The effect of coating thickness on the fracture resistance of the interface is studied. Coating strength is characterised in terms of fracture mechanics parameters fracture toughness and J crit; using modified ASTM and four point bend specimens, respectively. The J crit values are determined by virtual crack extension technique, in conjunction with finite element analysis solutions. Theoretical investigations by finite element analysis are conducted on the test specimens to gain information about the stresses responsible for failure initiation and the failure initiation zones.

Analysis of thermally enhanced SOIC packages

The thermal performance of a number of thermally enhanced small outline integrated circuit (SOIC) package designs is evaluated by means of a thermal model utilizing finite element analysis (FEA) and compared to that of a standard design. A number of enhancements are explored: reduced pad-inner lead tip spacing, fused leads, padless chip-on-lead (COL) construction, and heat slugs. In most of these cases the impact of replacing the standard epoxy molding compound (MC) with a thermally enhanced molding compound was investigated. A mathematical technique is demonstrated whereby the die (junction) temperature can be calculated at an arbitrary number of power levels and values of air velocity using the results of a small number of FEA solutions. This technique is particularly efficient when dealing with nonlinear boundary conditions involving buoyancy-driven convection and radiation. The predictions of the model were verified for one package type and shown to be in good agreement with experiment. Postprocessing of the FEA solution was performed for each design to calculate the thermal resistance of the principal heat flow paths and the partitioning of the heat dissipated by the IC among these paths. It is shown that the primary heat flow path is die-to-lead-to-circuit board-to-air and that a low junction-to-lead thermal resistance is the primary indicator of high thermal performance. The heat slug package and the COL package (when all leads have direct contact to the die) were shown to have the lowest junction-to-lead thermal resistance and best thermal performance. The use of an enhanced molding compound was shown to have a significant effect on the thermal performance only when the heat flowing from the junction to the leads flows through the molding compound. As such, it did not benefit the heat slug and optimal COL designs.

Combination of finite element analysis and analytical crack tip solution for mixed mode fracture

A finite element program was developed which combines the analytical crack tip solution with a conventional finite element analysis and evaluates various crack tip parameters as part of the solution. This program was used to analyze cracked specimens subjected to mixed mode loading. The importance of retaining the second term of the series expansion for local stress, a contribution which is independent of the distance from the crack tip, was demonstrated. It was first shown analytically that the presence of a load applied parallel to the crack reveals itself only through this constant second term, which vanishes only for specific loading conditions. The results of the numerical analysis demonstrate that the stress intensity factor K I is independent of the load applied parallel to the crack only when this term is included in the analytical crack tip solutions. Failure to include the constant term has the effect that K I varies with the horizontal load. The parameter K 11 is independent of this load in both cases. This indicatesonce again that it is this constant term which accounts solely and entirely for the presence of a load applied parallel to the crack.