Plane Strain Finite Element Research Articles

Extended formulation of macro-element based modelling – Application to single-lap bonded joints

An extended formulation of the macro-element (ME) based models, representing for both adherends and adhesive along the entire overlap in only one four-node element, is presented. Compared to earlier modelling, continuum ME (CME) and discrete ME (DME) based models, the adherend parts are also modelled as plane continuum media, for which high order displacement fields are freely supposed. Both extended CME (ECME) and extended DME (EDME) based modelling can have their displacement field orders of each continuum layer be set individually, and can be enriched using springs for interfaces or boundary conditions modelling, allowing the stresses to vanish at the free edges. The methodology and formulation of the stiffness matrix for both the adherend without adhesive and the bonded overlap is presented. The assessment is performed for a single-lap joint geometry by comparison results from a plane strain finite element (FE) model with extended ME-based models. Good agreements are shown for both thin and thick adhesive case studies.

Scalar- and vector-valued seismic fragility assessment of segmental shield tunnel lining in liquefiable soil deposits

Seismic fragility analysis can quantitatively evaluate the seismic performance of structures from a probabilistic viewpoint and accurately characterize the relationship between the degree of structural damage and ground motion intensity. This study investigates the seismic fragility of shield tunnels in three different liquefiable and non-liquefiable soils. A plane-strain finite element model of the saturated soil and shield tunnel is established via the OpenSees computational platform employing the multi-yield surface elastoplastic PressureDependMultiYield and PressureIndependMultiYield models to simulate the constitutive behaviour of liquefiable and non-liquefiable soils. The developed model is utilized to conduct nonlinear dynamic effective stress time history analyses to generate the seismic fragility curves and surfaces based on the incremental dynamic analysis method. Meanwhile, appropriate scalar- and vector-valued intensity measures are identified based on their correlation, efficiency, practicality and proficiency. Compared with the fragility curves based on scalar-valued intensity measures, the fragility surfaces based on the vector-valued intensity measures can better describe the effect of ground motion characteristics on the structural seismic demand, and thus can more accurately assess the structural seismic performance. The seismic damage probabilities derived from the fragility curves and surfaces reveal that the seismic damage risk of the shield tunnel in sandwiched liquefiable soil deposit is higher than that of the tunnel structure located in entirely liquefiable or non-liquefiable soil profiles. This finding underscores the importance of carefully evaluating the seismic safety of shield tunnels situated in sandwiched liquefiable soil deposits.

Analytical and Finite Element Analysis of the Rolling Force for the Three-Roller Cylindrical Bending Process.

In the roll bending process, the rolling force acting on the roller shafts is one of the most important parameters since, on the one hand, it determines the process settings including the pre-loading, and, on the other hand, its distribution and size may affect the integrity of both the bending system and the final product. In this study, the three-roller bending process was modeled using a two-dimensional plane-strain finite element method, and the rolling force was determined as a function of plate thickness, upper roller diameter, and yield strength for various API steel grades. Based on the numerical simulation results, a critical bending angle of 41° was identified and the rolling systems were divided into two categories, of less than or equal to, and greater than 41°, and an analytical model for predicting the maximum rolling force was developed for each category. To determine the optimal pre-tensioning force, two optimization formulations were proposed by minimizing the maximum equivalent stress and the absolute maximum displacement. The rolling forces predicted by the analytical models were found to be in good agreement with the numerical simulation results, with relative errors generally less than 10%. The predictive analytical models developed in this study capture well the complex deformation behavior that occurs during the roll bending process of steel plates, providing guidelines and predictions for industrial applications of this process.

Analysis of mode I crack propagation in glassy polymers under cyclic loading using a molecular dynamics informed continuum model for crazing

Craze and crack propagation in glassy polymers under cyclic mode I loading are investigated by employing a recently developed continuum-micromechanical model for crazing. This model accounts for the local morphology change from microvoids to fibrils during craze initiation, viscoplastic drawing of bulk material into fibrils, and viscoelastic creep recovery of the fibrillated craze matter during unloading. To ensure consistency between the bulk and craze model parameters, the material parameters of the craze model are normalised and calibrated based on a hybrid approach integrating experimental findings from the literature and molecular dynamics results. This yields a generic, yet representative glassy polymer response.In the framework of 2D plane strain finite element simulations, we study brittle as well as ductile glassy polymers and assess the results by drawing comparisons to the experimental and numerical literature. For brittle materials, characterised by a purely elastic bulk behaviour, the model reproduces craze characteristics such as the craze opening contour, the craze length-to-width ratio, a double stress peak at the craze and crack tip, and a non-proportional stress redistribution during loading-unloading cycles. In ductile glassy polymers, the interaction of shear yielding in the bulk and crazing along the ligament is analysed. In particular, shear bands emanate from the crack tip in each loading cycle and arch forward towards the craze. This plastic zone shares resemblance to the so-called epsilon-shaped deformation zone. The current simulations capture normal fatigue crack propagation, where craze and crack growth occur near the peak load in every cycle and the craze length remains relatively constant across the loading cycles. Moreover, findings from this study suggest that plasticity-induced unloading of the craze adjacent to the crack tip impedes crack growth.

Deformation in the wrinkle–crease transformation

We investigate deformation in the wrinkle–crease transformation to reveal the missing link between the wrinkling and creasing instabilities, which occur on the surface of elastomers under compression. Plane-strain finite element (FE) analysis is performed by combining a nonlinear perturbation approach to find two bifurcated solutions for the flat surface in a metastable state. The first solution is the stable deformation path for crease evolution, and the second solution is the previously unknown unstable deformation path from wrinkling to creasing. The two deformation paths are not independent, and they are connected at the critical strain for creasing εC. The resulting deformation at εC acts as a precursive deformation to cause creasing. Although the unstable path is initiated by the wrinkling instability as a bifurcation, the creasing instability (i.e., εC) is interpreted as the singular point at which the unstable path switches to the stable path. A chain reaction model is developed to describe the mechanism and deformation of the wrinkle–crease transformation, and it shows good agreement with the FE prediction within an inevitable limitation of FEs.

Correction of Plane Strain Analyses for Corrugated Metal Culverts Subjected to Patch Loads

Today, culverts are often analyzed and designed using plane strain finite element analyses in which a transverse slice of the structure is simulated and assumed equal to any other slice. Although this works well for soil loads, in reality live loads are discrete pressure loads from tires, and the out-of-plane effects can be significant, especially when cover depths are relatively low. While current design specifications define minimum cover that precludes significant localized pressure effects on culverts, many culverts are currently in service that were constructed under very shallow cover depths. In this study, analyses of corrugated metal culverts under truck load were carried out using two-dimensional (2D) (plane strain) and three-dimensional (3D) finite element analyses to assess the difference between structural responses predicted by these two approaches. Thirty-six culvert models with different shapes, spans, and cover depths were created using both 2D and 3D meshes. A significant difference was found between the results of these two analysis methods. Specifically, the thrust from the 3D models under shallow cover can be as much as three times the thrust results from the 2D model, which is unconservative from a safety standpoint. An innovative explanation was provided to explicate the differences. Finally, to address the issue, a reduction ratio method was developed to correct the 2D results to match the corresponding 3D results so that the relatively simple plane strain approach can be used.

Deterministic and probabilistic analysis of great-depth braced excavations: A 32 m excavation case study in Paris

The Fort d’Issy-Vanves-Clamart (FIVC) braced excavation in France is analyzed to provide insights into the geotechnical serviceability assessment of excavations at great depth within deterministic and probabilistic frameworks. The FIVC excavation is excavated at 32 m below the ground surface in Parisian sedimentary basin and a plane-strain finite element analysis is implemented to examine the wall deflections and ground surface settlements. A stochastic finite element method based on the polynomial chaos Kriging metamodel (MSFEM) is then proposed for the probabilistic analyses. Comparisons with field measurements and former studies are carried out. Several academic cases are then conducted to investigate the great-depth excavation stability regarding the maximum horizontal wall deflection and maximum ground surface settlement. The results indicate that the proposed MSFEM is effective for probabilistic analyses and can provide useful insights for the excavation design and construction. A sensitivity analysis for seven considered random parameters is then implemented. The soil friction angle at the excavation bottom layer is the most significant one for design. The soil-wall interaction effects on the excavation stability are also given.

An enriched finite element for the simplified stress analysis of an entire bonded overlap: Continuum macro-element

A methodology for the formulation of the stiffness matrix of an enriched finite element, called continuum macro-element (CME), representing for the full length of a bonded overlap and both the adhesive and the adhesive in only one four-node elements, is presented. Compared to earlier macro-elements modelling the bonded overlap as beams on elastic foundation, the CME supposed the adherends and the adhesive as plane continuum media, for which higher-order displacement fields is supposed for the adhesive. The formulation of the stiffness matrix for the adherend outside of the overlap is presented as well to address the stress analysis of single-lap bonded joint for assessment purpose. The assessment is performed by comparisons with the results from (i) a plane strain finite element and (ii) two recent papers by Nguyen and Le Grognec (2021) and Methfessel and Becker (2022). Good agreements are shown.

Micromechanics of kink-band formation in bimetallic layered composites

Bimetallic layered composites with ultrathin layers possess high strength and hardness, and excellent shock resistance and radiation damage tolerance. However, they are prone to deformation localization and readily form kink-bands when subjected to layer parallel compression. Following this, we carried out extensive plane strain finite element finite deformation analyses to understand and rationalize the experimentally observed kink-band formation in these composites. In the calculations, both constituent materials were assumed to follow a rate-independent isotropic elastic-plastic constitutive relation with an ad-hoc thickness dependent yield strength. Our results demonstrate that in these composites, layer refinement, together with the strength differential between the layers of the constituent materials, is sufficient to trigger kink-banding. Importantly, this phenomenon occurs even in the absence of elastically stiff layers, geometrical imperfections (such as waviness of the layers), and extreme plastic anisotropy within the layers. Additionally, we performed parametric studies to investigate the individual effects of layer thickness, strength differential between the two constituent materials, and strain-hardenability of the materials on kink-band formation. The outcomes of our parametric studies reveal that the strain-hardenability of the constituent materials stabilizes the formation of kink-bands. Furthermore, it leads to a transition from the abrupt formation of through-width kink-bands to the onset and propagation of stable inclined wedge-shaped kink-bands, similar to the experimental observations.

Study of plane strain compression of composite sheets

In the plane strain compression of multi-layers metal, each layer of metal shows a different degree of deformation due to the differences in material properties. Also, the deformation in different positions of a metal layer is always inhomogeneous. In this study, the effects of material ratios, friction between the metal layers and the amount of compression on the deformation behaviors of a bi-layer metal strip under plane strain compression are investigated. 2D plane strain finite element simulation models of AISI-1008/AL6061-T6 bi-layer metal strips are used. The results indicate that the strips show different deformation behaviors under different conditions, and non-smooth interfaces caused by the inhomogeneous material flow during the deformation process are detected in most cases.

Recent advanced thermal interfacial materials: A review of conducting mechanisms and parameters of carbon materials

Understanding the effect of process-induced residual stresses on the final shape of fiber-reinforced polymer composites is critical due to their widespread use in structural applications. It is crucial for more dependable composite production, as residual stresses change the internal stress level of the composite component during service life, and residual shape distortions may prevent the part from achieving the required geometrical tolerances. Residual strains are the result of many interplays among physical processes, such as heat transfer, material flow, polymerization, crystallization, or heat transfer. Numerical process models must be constructed instead of time-consuming and inefficient trial-and-error procedures to design and optimize the composite manufacturing process digitally. This study aims to review the model creation and applications for predicting residual stresses and form distortions in composite manufacturing. Uses for both thermoset and thermoplastic composites are investigated in detail. Predictions of the process-induced deformations of laminated composite structures have been made using elastic constitutive models, such as a plane strain finite element model with a cure-hardening, instantaneous linear elastic constitutive model. to learn more about the factors that lead to shape distortion and how they can be reduced, such as by adjusting cure cycles, tool surfaces, geometry, or lay-up.

On seismic isolation of soil-meta-foundation-structure systems

Meta-materials are engineered composites with physical properties that can introduce frequency band gap regions suitable for wave manipulation and filtering. This study investigates the seismic behavior of soil–structure systems with a meta-foundation designed using alternatively repeated layers of concrete and rubber. First, the frequency band gaps of the meta-foundation were derived using the Floquet-Bloch theory. The frequency band gaps for different meta-foundation configurations with infinite and finite numbers of unit cells were presented. Then, a two-dimensional (2D) plane-strain finite element simulation was proposed to study the isolation performance of meta-foundations on or embedded in soil half-space. Finally, two four-story steel structures with different dynamic characteristics were modeled under different input motions. The results show that direct embedment of meta-foundations in soil may break periodicity, requiring special treatments on the side boundaries of the foundation. Moreover, the numerical results show considerably enhanced structural performance for the meta-foundation system with a reduction up to 95.1% and 97.9% for the maximum horizontal acceleration and inter-story drift, respectively, compared to the non-isolated system. The results also indicate enhanced soil response under the meta-foundation and acceptable structural performance outside the band gap frequencies compared to the non-isolated system.

Response of Skirted Strip Footing Resting on Layered Granular Soil Using 2-D Plane-Strain Finite Element Modeling

Response of Skirted Strip Footing Resting on Layered Granular Soil Using 2-D Plane-Strain Finite Element Modeling

Effectiveness of soil improvement for deep excavation in under-consolidated soil: A case study

The main objective of this article is to investigate the effectiveness of soil improvement methods, such as jet grouting and cement deep mixing, for a deep excavation in under-consolidated soil. A well-documented case history, located in Zhuhai, China, was used for analysis. The analyses were conducted using two-dimensional plane-strain finite element analysis. The studies included an examination of the effect of wall length on lateral wall deformation, the effect between the degree of consolidation and lateral wall deformation, and the influence of soil improvement on lateral deformation and settlement. The deformations induced by under consolidating states are greater than those caused by normally consolidated states. A similar trend was found with or without soil improvement. The greater the degree of consolidation is, the smaller the deflection of the wall. In this case, the retaining wall's length is well designed and stable, but the analysis results showed that the wall length can be shorter than the constructed length. Massive jet grouting was used behind the left wall to successfully reduce wall deflection and ground surface settlement. Finally, deep cement mixing has only a small effect on reducing wall deflection and ground surface settlement.

Stability Charts for Limiting Horizontal Normal Pressure on Cohesive-frictional Backfill for Deep Contiguous Piled Walls

Background: This study presents a numerical solution for determining the limiting uniform normal pressure acting horizontally behind cohesive-frictional backfill material in a deep contiguous piled wall. At this limiting pressure, the soil tends to flow out in gaps between a series of vertical piles placed at a certain uniform horizontal spacing. Methods: The lower and upper bound plane strain finite element limit analysis (FELA) has been carried out for this purpose. The Mohr-Coulomb failure criterion, using an associated flow rule, was employed to impose the yield condition in the soil mass. Results: A parametric study was carried out to obtain the magnitude of the non-dimensional limiting lateral resistance (F/cD) as a function of normalized pile spacing (S/D), the friction angle of soils (ϕ), and the adhesion factor (δ) at the soil-pile interface; here, F refers to the lateral normal resistance (force) per unit length offered by the pile, S forms the clear spacing between piles, D is the diameter of the pile and c is soil cohesion. The impact of the different parameters on the failure mechanisms has been examined comprehensively. Conclusion: The lateral resistance (F/cD) offered by the piles increases generally with a decrease in the spacing between the piles. The magnitude of F/cD increases further with an increase in the values of ϕ and δ.

Finite Element Simulation of the Effect of Phase Transformation on Residual Stress in a Thick Section T-Joint

It has long been known that residual stresses can profoundly affect the integrity of engineering components. Evidence has recently emerged to confirm that solid-state phase transformations in steels can significantly influence the as-welded residual stresses. A two-dimensional thermo-mechanical, generalized plane strain finite element model was created to simulate the effect of phase transformation on residual stress during the multi-pass weld process in a thick section T-joint. The effect of phase transformation and martensite start temperatures was investigated. The results showed that phase transformation generated compressive residual stress underneath the last bead to be deposited for this multi-pass weld model. However, some constraints around the bead were essential to provide that stress. Tensile residual stress was generated in the bulk of the weld area when phase transformation was considered. Therefore, phase transformation may be helpful for single pass and other groove welds but may be unhelpful in the case of the T-joint examined here. The effect of the martensite start temperature is small compared with the main difference between having a phase transformation and not having one.

Building blocks that govern spontaneous and programmed pattern formation in pre-compressed bilayers

Surface wrinkling in stiff-film/soft-substrate bilayers is a common phenomenon in biological systems and is increasingly being exploited in thin-film technology. While the onset of surface wrinkling in end-compressed bilayers is well understood, questions remain with regards to the evolution of the wrinkling pattern in the intermediate and deep post-wrinkling regimes, especially when the substrate is strongly pre-compressed. Here, we explore the bifurcation landscape of end-compressed bilayers with strongly pre-compressed substrates, using hyperelastic, plane strain finite-elements and generalized path-following algorithms. After bifurcating from a flat into a sinusoidally wrinkled state, bilayers undergo further n -tupling bifurcations into stable wrinkling patterns of longer wavelength whose periodicity n = { 4 , … , 8 } is a function of overall bilayer length. These five n -tupling wrinkling patterns are shown to be independent localizations of the deformation mode and are accordingly identified as stable ‘building blocks’ that govern the intermediate post-wrinkling regime. Additional end-shortening into the deep post-wrinkling regime then leads to further period doubling and coalescence of the building blocks. Beyond a certain length threshold, a bilayer can form a combinatorial side-by-side arrangement of the five building blocks. In the limit of an infinitely long bilayer, this leads to the phenomenon known as spatial chaos with the emergence of an infinite set of possible wrinkling patterns. In reality, though, the precise side-by-side arrangement of the building blocks is governed by the initial conditions. We show that the morphological evolution of the wrinkling pattern can be programmed by a judicial placement of manufactured dents in the thin film, creating new manufacturing capabilities for textured bilayers.

Effectiveness of Stone and Deep Mixing Lime Columns on Stability of Embankments Constructed on Soft Consolidating Soil

One of the most common, effective and innovative ways to improve soft soil is to use stone columns or lime columns. An extensive study has been conducted to find the effectiveness of lime or stone columns to improve the settlement of embankments constructed over soft consolidating soil. There are very few studies to evaluate the effects of columns on embankment safety factor during foundation soil consolidation. This study examined the effectiveness of stone and lime columns on the embankment stability, both in terms of safety factor and settlement, which are determined at different intervals of time, during the foundation soil consolidation. Further, it was also investigated the effectiveness of stone and lime column composite system. In this numerical study, two-dimensional plane strain finite element method is used. This study shows that, stone columns are more effective than lime columns in reducing post-construction settlement and to accelerate consolidation process, whereas lime columns are observed to be more effective than stone columns in improving the safety factor after the embankment construction indicating that the usage of stone or lime columns is significantly affected by the purpose for which columns are used. In addition, it is also observed that the column effectiveness on safety factor is strongly influenced by the relative shear strength of embankment and foundation soils. Further, provision of stone and lime columns as a composite system performs better than providing only stone or lime columns.

Numerical analyses of coupled thermo-mechanical behaviors of cement sheath in geothermal wells

The stability of cement sheath under high temperature and high pressure is one of the most critical issues for the durability of geothermal well systems. In this study, a two-dimensional plane-strain finite element code was developed to investigate the coupled thermo-mechanical behaviors of the casing-cement-formation system. Different from previous linear elastic analyses, a thermoelasto-plastic constitutive model based on the thermodynamic theory was adopted for the cement sheath. It is shown that the finite element simulations using the proposed model provide a more accurate and realistic prediction of stress–strain responses of the cement sheath under high temperature. The results demonstrate that the radial stress concentration and the tensile strain concentration occur at both the cement–casing interface and the cement–formation interface, where the cement sheath is most likely to fail. High strength and low stiffness in the cement sheath and the formation are preferred for the integrity of the system. Both large thermal cycles and large differences between the internal fluid pressure and the external pressure should be avoided during operation. The new code is an alternative tool for guiding the geothermal well design. The finite element framework described herein is universal for other thermo-mechanical applications, such as energy foundations and energy tunnels.

Plasticity-induced damage and material loss in oscillatory contacts

Knowledge of plasticity-induced damage leading to the loss of material in oscillatory contacts is of paramount importance to various electromechanical systems comprising contact-mode elements exposed to high-frequency vibrations. However, experimental investigation of the wear behavior of devices experiencing microscopic oscillatory contact (fretting) is complex, time consuming, and expensive. More importantly, the progression of critical damage processes, such as the decrease of the material’s strength with the accumulation of plastic deformation in the vicinity of the contact interface and the removal of material in the form of microscopic wear debris, cannot be tracked in real time, necessitating cumbersome and costly post-testing microanalysis. Alternatively, computational wear modeling is more effective than experiments and can provide valuable insight into the effect of important parameters, such as load, coefficient of friction, oscillation amplitude, and material behavior, on the loss of material during oscillatory sliding contact. Accordingly, the principal objective of this study was to introduce a computational approach, which can be used to analyze the loss of material due to plasticity-induced damage in oscillatory mechanical components. To achieve this aim, a plane-strain finite element model of a rigid cylinder in reciprocating sliding contact with an elastic-plastic half-space exhibiting isotropic strain hardening was used to study how the evolution of damage due to the progression of plasticity leads to the loss of material. A quasi-static, isothermal damage model based on a ductile failure criterion was implemented in the finite element analysis to simulate the removal of the fully damaged elements. Numerical results illuminate the effects of the load and coefficient of friction on the development of plasticity, cumulative damage, and loss of material with accruing oscillation cycles. The deviation of the wear behavior from classical wear theory in the high-load simulations is explained by the plastic shear strain distribution and less slip at the contact interface encountered at high loads. The novelty of this study is the development of a computational methodology, which sheds light into the evolution of plasticity, damage, and loss of material in reciprocating sliding contacts and provides an effective computational capability for assessing the effects of load, friction, and material behavior on the mechanical performance of components operating in oscillatory contact mode.