Plane Strain Formulation Research Articles

Generalized plane strain finite-element formulation for thermal and electrical buckling analysis of piezo composite beam

We develop a generalized plane strain (GPS) finite element formulation to predict the critical buckling voltage and temperature of a piezo composite beam in more generality than the cases characterized by plane strain and plane stress assumptions. This generalized plane strain formulation represents the two-dimensional finite element model as closely as possible to the three-dimensional finite element model. It is similar to the plane strain formulation that reduces a three-dimensional stress-strain relation to a two-dimensional one, but in contrast with most GPS formulations in the literature, it does not include out of plane degrees of freedom. In our formulation the reduced two-dimensional stress-strain relation incorporates the effect of allowed/applied strain0 in the dimension not included in the two-dimensional model. Further, since the goal is to deal with thermal and electrical buckling analysis, an initial strain vector is incorporated in the formulation. A finite element solver based on an eight-node quadrilateral element was developed under the new formulation, and its results show good agreement with those reported by Varelis and Saravanos (2004) and those obtained with ANSYS. The critical electrical and thermal buckling loads for examples other than those characterized by plane stress and plane strain were analyzed, and it was found that they are significantly influenced by , the parameter controlling the out-of-plane strains. Finite element analysis of smart structures has attracted much attention in recent years due to its wide range of applications. A significant amount of research has gone into the analysis of piezo composite structures. A number of finite element (FE) models have been proposed for the analysis of smart structures; and a detailed survey is given in [Benjeddou 2000]. One of the main problems addressed is the buckling analysis of smart structures. Three-dimensional beam models can be simulated using two-dimensional in-plane elements by considering only the longitudinal cross-section of the three-dimensional beam model. The boundary conditions and the loading conditions can be simulated more accurately in this two-dimensional model than in the one-dimensional beam element model. Still, the two-dimensional analysis of the beam is based on either the plane stress or plane strain assumption, and cases outside these assumptions cannot be handled by two-dimensional in-plane elements. The generalized plane strain formulation can be used to model cases other than plane stress and plane strain at the cost of additional degrees of freedom. The generalized plane strain formulation has been discussed extensively in the literature and has been used for several applications. Most composite problems are generalized plane strain in nature, and they

Thermal expansion behaviour of aluminum matrix composites with densely packed SiC particles

Abstract The coefficient of thermal expansion (CTE) of Al-based metal matrix composites containing 70 vol.% SiC particles (AlSiC) has been measured based on the length change from room temperature (RT) to 500 °C. In the present work, the instantaneous CTE(T) of AlSiC is studied by thermo-elastic models and micromechanical simulation using finite element analysis in order to explain abnormalities observed experimentally. The CTE(T) is predicted according to analytical thermo-elastic models of Kerner, Schapery and Turner. The CTE(T) is modelled for heating and cooling cycles from 20 °C to 500 °C considering the effects of microscopic voids and phase connectivity. The finite element analysis is based on a two-dimensional unit cell model comparing between generalized plane strain and plane stress formulations. The thermal expansion behaviour is strongly influenced by the presence of voids and confirms qualitatively that they cause the experimentally observed decrease of the CTE(T) above 250 °C.

Real time discrete techniques for distortion and microstructure of long product sections

A combined suite of in house finite element (FE) and finite prism (FP) models has been developed to address real time predictions of temperature, distortion and structure during cooling of long product sections. The present paper also describes various applications of cooling, and compares the developed approaches with conventional three-dimensional and generalised plane strain formulation elements and experimental results. These models have also been used to simulate the effectiveness of alternative cooling strategies (e.g. spray cooling) in reducing levels of thermal distortion.

A combined necking and shear localization analysis for aluminum sheets under biaxial stretching conditions

A combined necking and shear localization analysis is adopted to model the failures of two aluminum sheets, AA5754 and AA6111, under biaxial stretching conditions. The approach is based on the assumption that the reduction of thickness or the necking mode is modeled by a plane stress formulation and the final failure mode of shear localization is modeled by a generalized plane strain formulation. The sheet material is modeled by an elastic-viscoplastic constitutive relation that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity, and the softening due to the nucleation, growth, and coalescence of microvoids. Specifically, the necking/shear failure of the aluminum sheets is modeled under uniaxial tension, plane strain tension and equal biaxial tension. The results based on the mechanics model presented in this paper are in agreement with those based on the forming limit diagrams (FLDs) and tensile tests. When the necking mode is suppressed, the failure strains are also determined under plane strain conditions. These failure strains can be used as guidances for estimation of the surface failure strains on the stretching sides of the aluminum sheets under plane strain bending conditions. The estimated surface failure strains are higher than the failure strains of the forming limit diagrams under plane strain stretching conditions. The results are consistent with experimental observations where the surface failure strains of the aluminum sheets increase significantly on the stretching sides of the sheets under bending conditions. The results also indicate that when a considerable amount of necking is observed for a sheet metal under stretching conditions, the surface failure strains on the stretching sides of the sheet metal under bending conditions can be significantly higher.

Relaxation of Thermal Stress by Dislocation Motion in Passivated Metal Interconnects

The development and relaxation of stress in metal interconnects strained by their surroundings (substrate and passivation layers) is predicted by a discrete dislocation analysis. The model is based on a two-dimensional plane strain formulation, with deformation fully constrained in the line direction. Plastic deformation occurs by glide of edge dislocations on three slip systems in the single-crystal line. The substrate and passivation layers are treated as elastic materials and therefore impenetrable for the dislocations. Results of the simulations show the dependence of the stress evolution and of the effectiveness of plastic relaxation on the geometry of the line. The dependence of stress development on line aspect ratio, line size, slip plane orientation, pitch length, and passivation layer thickness are explored.

Wellbore Stability Analysis: From Linear Elasticity to Postbifurcation Modeling

This paper reviews some of the progress made on wellbore stability modeling during the last two decades. First we demonstrate the improvement made on mud-pressure predictions by moving from analytic elastic solutions to finite element elastoplastic modeling. We show this progress, presenting a finite element model based on a generalized plane strain formulation for analyzing efficiently the three-dimensional problem of stability in deviated wellbores. On a more research oriented work, we present results from two advanced theories capable of modeling localization of deformation in shear bands, which causes borehole breakouts. The first theory is based on a more established approach, the Cosserat continuum. The second theory, called gradient elastoplasticity, is being developed to resolve some of the drawbacks of gradient plasticity theories. Gradient elastoplasticity is a unified theory where both elastic and plastic parts are of gradient type. We demonstrate that both theories, in addition to localization, can also model the scale effect observed in thick-walled cylinder tests

Numerical simulation of stress evolution during electromigration in IC interconnect lines

A finite element simulation of stress evolution in thin metal film during electromigration is reported in this paper. The electromigration process is modeled by a coupled diffusion- mechanical partial differential equations (PDEs). The PDEs are implemented with a plane strain formulation and numerically solved with the finite element (FE) method. The evolutions of hydrostatic stress, each component of the deviatoric stress tensor, and Von Mises' stress were simulated for several cases with different line lengths and current densities. Two types of displacement boundary conditions are considered. The simulation results are compared with Korhonen's analytical model and Black and Blech's experimentalesults.

Discrete dislocation analysis of size effects in thin films

A discrete dislocation plasticity analysis of plastic deformation in metal thin films caused by thermal stress is carried out. The calculations use a two-dimensional plane-strain formulation with only edge dislocations. Single crystal films with a specified set of slip systems are considered. The film-substrate system is subjected to a prescribed temperature history and a boundary value problem is formulated and solved for the evolution of the stress field and for the evolution of the dislocations structure in the film. A hard boundary layer forms at the interface between the film and the substrate, which does not scale with the film thickness and thus gives rise to a size effect. It is found that a reduction in the rate of dislocation nucleation can occur abruptly, which gives rise to a two-stage hardening behavior.

Plasticity in Polycrystalline Thin Films: a 2D Dislocation Dynamics Approach

AbstractThermal stress evolution in polycrystalline thin films is analyzed using discrete dislocation plasticity. Stress develops in the film during cooling from a stress-free configuration due to the difference in thermal expansion coefficient between the film and its substrate. A plane strain formulation with only edge dislocations is used and each grain of the polycrystal has a specified set of slip systems. The film–substrate interface and the grain boundaries are impenetrable for the dislocations. Results are presented for two film thicknesses, with higher hardening seen for the thinner films.

A mechanistic analysis of the correlation between overall strength and indentation hardness in discontinuously reinforced aluminum

It is well established that the indentation hardness of metallic alloys shows a reasonable correlation with their yield strength or ultimate strength. Experiments illustrate that such a unique correlation is nonexistent for discontinuously reinforced metal matrix composites, even when the indentation size is much greater than the reinforcement size. For aluminum alloys reinforced with silicon carbide particles, the same composite yield strength and tensile strength with different reinforcement fractions do not lead to similar hardness, or vice versa. Finite element analyses are carried out to rationalize the experimental findings. The modeling utilizes a two-dimensional plane-strain formulation. Discrete particles are included in the material model, and the overall stress-strain response and the indentation response are numerically simulated. The results confirm the lack of unique correspondence between the composite hardness and strength. The alteration of local heterogeneity in the composite is found to affect the indentation response. Effects of the geometrical arrangement of particles and thermal residual stresses on the indentation response are also investigated numerically.

Simulations of Dislocation Dynamics in Aluminum Interconnects

AbstractA discrete dislocation simulation of plastic deformation in metallic interconnects caused by thermal stress is carried out. The calculations are carried out using a two dimensional plane strain formulation with only edge dislocations. A boundary value problem is formulated and solved for the evolution of the thermal stress field and the evolution of the dislocation structure in the cross-section of the line as cooling proceeds. For lines with a small cross section (height or width less than 1 μm), the local concentration of stresses due to dislocation patterning strongly affects the overall stress build up and relaxation. The results show a clear dependence of the transverse stress development on the line aspect ratio.

A numerical model of transformation superplasticity for iron

A numerical model of transformation superplasticity for an elastic, ideally plastic material is presented using a two-dimensional plane-strain formulation considering both temperature and displacement. The evolution of temperature, stresses and strains during the α/γ phase transformation of iron is computed for different values of the applied stress. For low stresses, the numerical model predicts a linear relationship between the uniaxial applied stress and the uniaxial plastic strain increment accumulated after crossing the phase transformation range. For high stresses, the relationship becomes non-linear: the strain increments tend to infinity as the applied stress approaches the yield stress. Both of these trends are in qualitative agreement with existing analytical solutions for transformation superplasticity in an ideally plastic material. Furthermore, upon introducing plane-strain specific equivalent quantities for the transformation mismatch and the yield stress, the numerical model is in good quantitative agreement with both analytical predictions and experimental data for pure iron.

Evolution of stresses in passivated and unpassivated metal interconnects

This paper discusses computational simulations of the evolution of stresses and deformation in unpassivated and SiO2-passivated Al lines on Si substrates. The finite element model accounts for elastic-plastic deformation in the Al lines during etching, passivation, and subsequent thermal cycling, by recourse to a generalized plane strain formulation within the context of a unit cell with appropriately constrained boundary conditions. The effects of different controlled variations in thermal history, and in the width, height, spacing, and yield behavior of the Al lines are analyzed; all these factors are seen to have potentially strong effects on the evolution of stresses within the lines. The predictions of the computations presented in this work are amenable for direct comparisons with experiments of curvature evolution along and perpendicular to the lines upon patterning, passivation, and thermal loading. The predicted stresses in metal interconnects can be directly used for reliability modeling purposes.

Non-Newtonian Creep Into a Two-Dimensional Cavity of Near-Rectangular Shape

The plane-strain formulation for the steady-state closure of a near-rectangular, single isolated opening in an indefinite viscoelastic medium is presented. A power creep law describes the creep behavior of the viscous medium. Because of the highly nonlinear nature of the creep, an analytic solution is not possible for the proposed opening geometry, hence an approximation method based upon the minimum principle for velocities is used. The analytic function is used to describe the shape of opening (circular, elliptical, and rectangular with rounded corners).

Sensitivity analysis in plane stress elasto-plasticity and elasto-viscoplasticity

Development of computational models for plane stress elasto-plasticity is known to make some difficulties non-existent in 3D and plane strain formulations. Tedious derivation of plane stress algorithmic tangent matrices which cannot be directly obtained as a special case of 3D matrices may serve here as an example. In this paper specific features of plane stress computations are discussed in the context of parameter sensitivity analysis. Elasto-plastic material model with isotropic/kinematic hardening as well as two viscoplastic models are considered. Explicit expressions for sensitivity gradients are derived. Instructive examples solved by FEM illustrate the paper.

Stresses, curvatures, and shape changes arising from patterned lines on silicon wafers

Experimental and numerical results are presented on the evolution of stresses and the accompanying changes in the overall curvatures due to the patterning of silicon oxide lines on silicon wafers and subsequent thermal loading. The finite element analysis involves a generalized plane strain formulation, which is capable of predicting the wafer curvatures in directions parallel and perpendicular to the lines, for both the patterning and thermal cycling operations. The predictions compare reasonably well with systematic curvature measurements for several different geometrical combinations of the thickness, width and spacing of the patterned lines. The non-uniform stress fields within the fine lines and the substrate are also analyzed. It is shown both experimentally and theoretically that certain geometries of patterned lines on the substrate induce dramatic shape changes and reversals of curvature in the direction perpendicular to the lines. The mechanistic origin of this effect is identified to be the Poisson effect arising from the anisotropic strain coupling in the patterned structure.

A plane strain formulation of the elastic-plastic constitutive law for hardening von mises materials

Elastic-plastic (path dependent but time-independent) materials, linear and isotropic in the elastic range and governed by the von Mises yield condition and the relevant associated flow rule, are considered. Both kinematic and isotropic hardening are included. Starting from previous, more general but partial, results and exploiting the purely deviatoric nature of plastic flow, a plane strain version of the constitutive law is formulated in terms of in-plane variables only. The actual plastic strains are replaced by equivalent in-plane measures and the effects of transverse yielding appear as a fictitious, directional kinematic hardening. On this basis, the plane strain elastic-plastic problem can be formulated in a way fully analogous to the equivalent plane stress one: as in the elastic case, differences are confined to the expressions of the constitutive parameters.

Thermal stresses during cooling of prismatic bodies

The plane strain formulation of the thermal stress problem was used to obtain the thermal stresses in a rapidly cooled prismatic body for various aspect ratios and external conditions. The maximum principal stress in a body exposed to a constant ambient temperature during rapid cooling increased with Biot number. However, the thermal stress showed an extremum with change in aspect ratio. Controlling the ambient temperature to decrease exponentially can be used to decrease the maximum principal stress during cooling. General results are presented for various aspect ratios and cooling rates and may be applied to the design of process conditions for the cooling of sintered bodies, in particular, to avoid thermal fragmentation.

Coupled thermal and stress-strain models for the continuous casting of steels

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Newton‐like methods with limited storage, for the solution of elasto‐viscoplasticity problems

AbstractQuasi‐Newton and Secant‐Newton methods have been successful in solving problems of elasto‐plasticity. In this paper these methods are applied to elasto‐viscoplastic analyses, in an algorithm coupling the global solution for displacement increments with local Newton‐Raphson stress iteration at each sampling point. The familiar rank‐two DFP and BFGS methods are generalized to the Broyden family of algorithms, and this generalization is also used to produce a Secant‐Newton algorithm corresponding to Hoshino's method. Consideration is given to the most effective strategy when only a limited number of Quasi‐Newton iterates may be stored; overwriting, re‐start or continuation in Secant‐Newton form. Results are presented for a tunnelling problem using both plane strain and axisymmetric formulations.