Plane Strain Conditions Research Articles

Closed-form solution of the deflection of a prestressed multilayer cantilever

Abstract In this work, a closed-form solution for the deflection of a multilayer cantilever with initial residual stress in each layer is presented. In particular, this model aims to take into account the relaxation of stresses due to the curvature of the cantilever and to calculate the curvature radius and the bending moment due to stresses. Stress-induced deflections and force & stress-induced deflections of cantilever with different shapes (rectangular, triangular, truncated triangular, V-shaped and truncated V-shaped cantilevers) are obtained from the Euler-Bernoulli beam theory. Taking a SiN/SiO2/Si multilayer as an example, the calculated stress-induced profiles do not depend on the shape of the beam and are in good agreement with finite element results. A comparison with results obtained from a model under generalized plane strain condition is discussed. Stress \& Force-induced deflections are also successfully compared to finite element results. Finally, a new formula is proposed to accurately determine the stress in a thin film material from the study of the curvature of a multilayer cantilever. As illustrated by the example of a virtual SiO2/Si bi-layer rectangular cantilever, the initial stress value which is an input parameter of the finite element analysis is calculated back from the simulated curvature with an accuracy of 0.2%.

A Riemann-based SPH formulation for modelling elastoplastic soil behaviour using a Drucker–Prager model

The present paper aims at proposing and investigating a Riemann-based SPH formulation to simulate the elastoplastic behaviour of soils undergoing large deformations, using a Drucker–Prager model. Basing on the pioneer work from Parshikov and Medin (Parshikov and Medin, 2002), a Riemann solver is used to maintain regular fields while being free of tuning parameters. By contrast to the work in Parshikov and Medin (2002) where piecewise constant reconstructions were employed, piecewise linear reconstructions are preferred in this work to reduce the numerical diffusion. A Particle Shifting Technique (PST) is used to maintain regular particle distributions and consequently accurate SPH interpolations. To the best of the author’s knowledge, the use of a Riemann solver specific to solid mechanics with a pressure-dependent elastoplastic Drucker–Prager yield surface to model the behaviour of the material represents a novelty with respect to the existing literature. A Boundary Integral Method (BIM) initially derived for fluid dynamics (Ferrand et al., 2013; Chiron et al., 2019) is adapted to solid mechanics in order to handle complex geometries. It allows to deal with wall treatment without using fictitious particles, and shows satisfactory results even in sharp angle regions. The ability of the proposed Riemann-based formulation to simulate accurately elastoplastic problems and its robustness are examined through several test cases in plane strain conditions. Attention is paid to the capacity of the formulation to mitigate the occurrence of Tensile Instability (TI) with respect to other schemes, for which additional treatment is required to treat this issue, such as the additional artificial stress method (Gray et al., 2001).

Uplift capacity analysis of inclined strip anchors considering spatial variability of undrained shear strength: RAFELA and ANN

This study investigates the uplift resistance of inclined strip anchors embedded in clays where the undrained shear strength varies spatially. Four input parameters are considered, namely the inclination angle (α), the cover-depth ratio (H/B), the coefficient of variation (COV), and the dimensionless correlation length (Θ) of the undrained shear strength. The random field is modeled using the Random Adaptive Finite Element Limit Analysis (RAFELA) and Monte Carlo simulation in OPTUM G2 under plane strain conditions. The results indicate that COV and Θ significantly affect the shape of probability density function (PDF) and cumulative distribution function (CDF) charts. The probability of failure (PoF) depends evidently on COV and Θ, while H/B and α have lower impacts. The correlation between four inputs and the mean of stability factor (μNc) is depicted through parametric studies. Additionally, Artificial Neural Network (ANN) is implemented to propose a regression model to predict the mean and standard deviation of the stability factor (μNc and σNc). Based on the optimal ANN structure, Permutation Feature Importance (PFI) is applied for the sensitivity analysis, showing that H/B is the most important feature, followed by COV, Θ, and α. The results from this study significantly contribute to the research on the pull-out behavior of strip anchors, particularly in clays with spatial variations in undrained shear strength.

Triaxial tests and numerical simulations on frozen silty clays under different stress paths and minor principal stresses

To investigate the mechanical characteristics of frozen silty clay under complex stress paths, using the true triaxial instrument for permafrost, tests were carried out under triaxial compressive and plane strain stress states using the true triaxial instrument for permafrost to analyze deformation characteristics and strength evolution law under different stress paths and minor principal stresses (σ 3) and establish strength criterion under plane strain conditions. PFC3D numerical simulation results were compared to test results and meso-crack evolution law was discussed. The results showed that stress-strain curves were characterized by strain hardening. Destructive strength showed a gradual increase with the increase of σ 3 and the values obtained from plane strain tests were higher than those of triaxial compression tests. Volume strains basically showed shear shrinkage characteristics and all σ 3 directions were expansion deformation. Strength at damage under plane strain state was approximated based on generalized Mises and Lade-Duncan plane strain strength criterion using generalized plane strain strength criterion. Stress-strain curves obtained from numerical simulation tests in PFC3D basically agreed well with those obtained from indoor test results. The number of tensile and shear cracks in the developed numerical model under various stress paths were increased with generalized shear strain.

Numerical investigation of soil arching and integral abutment bridge in a pile-supported railway embankment

ABSTRACT Rail transport plays a crucial role in promoting sustainable mobility and tackling environmental challenges. However, the construction of a railway track on soft soil poses significant challenges due to the inherent instability and potential for excessive settlement. To address these issues, this study numerically assessed a pile-supported embankment and an integral abutment bridge. Two different numerical models are simulated in two-dimensional (2D) plane strain conditions. Findings from numerical analyses of the pile-supported embankment revealed that soil arching is significantly influenced by embankment height and pile spacing. The maximum settlement of the embankment decreases with an increase in embankment modulus and friction angle. Furthermore, a correlation among soil arching indicators was established. Conversely, the analysis of integral abutment railway bridges (IARBs) indicates that the performance of the transition zone is significantly influenced by factors such as approach slab geometry and multiple axle passages.

Physics-informed neural network solution for thermo-elastic cavity expansion problem

ABSTRACT This paper proposes a physics-informed neural network (PINN) to solve the thermo-elastic coupled cavity expansion problem under plane strain conditions. The solid is modelled as a linear thermo-elastic material, and cavity expansion is driven by a constant temperature and a radial displacement subjected to the inner cavity wall. Governing partial differential equations (PDEs) for the problem is firstly presented, and an analytical solution is available to help benchmark the PINN approach. Then, the PDEs are normalised into dimensionless forms and neural network details for solving the coupled PDEs are demonstrated. Finally, the performance of the PINN approach is shown by comparison with the analytical benchmark solution and it is found that the PINN can predict thermo-elastic cavity expansion behaviour with high accuracy. The PINN approach and the analytical solution can also serve as a benchmark for solving thermo-elastic coupled problems in geotechnical engineering by more advanced PINN.

A computationally efficient Element Edge point numerical integration scheme in the meshless method framework for solving fracture problems

The fracture problem is modeled using the Radial Point Interpolation Meshless Method (RPIM) to solve for displacements. Further, the stresses and Stress Intensity Factor (SIF) are calculated using obtained displacements. An effective numerical integral quadrature called the Element Edge point(EE) scheme is used as an alternative to conventional Gauss Quadrature to improve computational efficiency. A comparative study based on two numerical integration schemes, the Element Edge point scheme and Gauss Quadrature, is conducted on four benchmark problems of thick, cracked plates owing to plane strain conditions. The study reveals that the proposed Element Edge point (EE) scheme is computationally efficient and works well in the meshless method framework.

Stress triaxiality and Lode angle parameters driven phase field coupled finite deformation plasticity formulation of ductile fracture

The effect of stress triaxiality and Lode angle parameters on crack initiation and propagation is investigated using the phase field coupled elasto-plasticity formulation. To accomplish this, a multi-invariants, i.e., first invariant of stress tensor and second and third invariants of deviatoric stress tensor, dependent finite deformation hyperelasto-plasticity is coupled with phase field theory of ductile fracture. The novel ideas include a proposition of a phase field coupled multi-invariants dependent yield function by including the Hosford equivalent stress in the Drucker–Prager yield function in order to accurately predict the ductile response of the material prior to initiation of the failure and the postulation of the threshold energy in the phase field evolution equation defining a measure of the ductility of materials. The measure of ductility is reported in the form of damage initiation surface in terms of threshold plastic energy as a function of stress triaxiality and normalized Lode angle parameters using the Mohr–Coulomb fracture initiation criterion. The model parameters are calibrated based on the available experimental results in the literature considering the stress triaxiality and Lode angle parameters dependent responses of different specimens. The crack initiation and propagation paths predicted by the proposed model under the different states of stress triaxiality and Lode angle parameters, such as axisymmetric tension, plane strain condition, and axisymmetric compression, match with those predicted experimentally in the literature.

3D strain heterogeneity and fracture studied by X-ray tomography and crystal plasticity in an aluminium alloy

Strong correlations between measured strain fields and crystal plasticity finite element (CP-FE) predictions based on the real microstructure are found for a plane strain tensile specimen made of 6016 T4 aluminium alloy. This is achieved using multimodal X-ray lab tomography giving access to both the initial grain structure and the strain evolution. The real microstructure of the central region of interest (ROI) of the undeformed specimen is obtained non destructively using lab-based diffraction contrast tomography (DCT) and meshing. An in situ tensile test, using absorption contrast tomography (ACT) is then performed for twelve loading increments up to fracture. Taking advantage of the plane strain condition, the evolution of the internal strain field is measured by two-dimensional digital image correlation (DIC) in the material bulk using the natural speckle provided by intermetallic particles. Early strain heterogeneities in the form of slanted bands, that are spatially stable over time, are revealed and the fracture path – determined from the post mortem scan – is found to coincide with the bands exhibiting maximum strain. CP-FE simulations are performed on the meshed microstructure of the specimen acquired by DCT and are compared with image correlation measurements. The measured strain fields are well described by 3D CP-FE predictions, whilst it is shown that neither a macroscopic anisotropic plasticity model nor a CP-FE simulation with random grain orientations could reproduce the measurements.

Creasing instability of polydomain nematic elastomers in compression

Polydomain liquid crystalline (nematic) elastomers exhibit unique mechanical properties such as soft elasticity, where the material largely deforms at nearly constant stress, due to microstructural evolution. In this paper, we numerically study the effect of such remarkable soft behavior on the surface instability of a half-space polydomain nematic elastomer, which is uniformly compressed parallel to the interface under a plane-strain condition. We compare the creasing instability of nematic elastomers with that of neo-Hookean elastomers by presenting bifurcation diagrams, stress and strain development in the elastomers, energy relaxation, and surface morphology at the creased state. Our results reveal that soft elasticity stabilizes nematic elastomers in plane-strain compression. Remarkably, the critical strain and stress at which the crease nucleates depend nonlinearly on the degree of anisotropy in nematic elastomers. Moreover, we find that the morphology of the creased surface in nematic elastomers exhibits the universal cusp shape previously observed in neo-Hookean elastomers.

Understanding a new wrinkling behavior of annular grooved panel during flexible free incremental sheet forming

Wrinkling behavior of sheet metal by inappropriate process parameters and the resultant variation of thickness and microstructure during incremental sheet forming (ISF) have significant impact on forming stability and geometric profile. The wrinkling-related mechanism is yet to be well understood, which make it difficult to effectively control wrinkling defects during ISF. In the present work, the wrinkling behavior and related mechanism of a typical annular grooved panel are investigated during a novel flexible free ISF process. Experiments demonstrate that during this ISF process, the sheet material sequentially undergoes 4 deformation modes: contacting state under plane strain condition, contracting state with only elastic deflection, thickening state under uniaxial compression condition, and wrinkling state where the circumferential compressive stress exceeds the critical wrinkling stress. The evolution of these deformation modes is dominantly caused by the increasing of edge contraction and circumferential stress, which are induced by the bending moment from the forming tool and auxiliary sheets. Based on the mechanism, an analytical model is developed to predict the wrinkling behavior, by which the corresponded sheet thickness, bending moment, circumferential compressive stress and critical wrinkling stress during the wrinkling process could be comprehensively calculated. Experimental results validate that the analytical model can accurately predict the wrinkling behavior under different process parameters, material properties and auxiliary sheet materials. Through microstructural characterization, numerical simulation and analytical modeling, the influence of process parameter and material property on wrinkling behavior is systematically investigated, along with the variation of sheet thickness and microstructure under different wrinkling modes. This work enhance a deep understanding of wrinkling behavior during ISF process, thereby providing effective methods for process optimization and quality improvement.

The role of the interfaces and cross-links on the mechanical behavior of mineralized collagen fibrils. A numerical approach

Mechanical properties of bone tissue are highly dependent on its hierarchical structure. The presence of microcracks and diffuse damage in lamellar bone is correlated with the failure of the collagen-mineral interface in mineralized collagen fibrils (MCF). The main goal of this work is to evaluate the mechanical behavior of the interfaces and quantify the stiffness loss of the MCF associated with different failure mechanisms, under controlled in-plane displacement. Additionally, we aim to study the role of the cross-links on the fibril mechanical response, beyond the interface failure. Inter- and intra-microfibrilar cross-links are analyzed. In order to address the first issue, a detailed representative volume of the MCF is analyzed by means of the finite element method, under the assumption of plane strain and periodic boundary conditions. In this model the interfaces between constituents are modeled with an exponential cohesive law. Enzymatic cross-links, located at the molecular terminals connecting each 4D (D=67nm) staggered molecules, are represented by non-linear springs. Three in-plane controlled deformations are applied. The results of this work provide the anisotropic stiffness loss of the tissue involved in the different failure mechanisms at the nano-scale length. The initiation of microcracks and the presence of damage zones are compatible with the failure mechanisms observed at interfaces. Interface failure entails a progressive stiffness loss, bringing a non-linear behavior of bone. The strength obtained for the longitudinal maximum deformation is more than 20 times the transverse strength and 3.5 times the shear strength. The quantification of the reduction percentage in the elastic moduli and the shear stiffness when the fibril is damaged, has a potential application in improving failure criteria based on degradation of elastic constants. When longitudinal elongation is applied, the mechanical contribution of the cross-links in delaying the failure initiation of the interface is shown. Likewise, results of this work confirm the scarce influence of the cross-links in the strain range analyzed. Additionally, a three-dimensional numerical model of several microfibrils is defined with the aim of analyzing the mechanical relevance of inter- and intra-microfibrilar cross-links, beyond the interface failure. Results confirm that cross-links transfer the load when strain increases, being highlighted the mechanical competence of the trivalent cross-links.

Comparison of elements and state-variable transfer methods for quasi-incompressible material behaviour in the particle finite element method

AbstractThe Particle Finite Element Method (PFEM) is attractive for the simulation of large deformation problems, e.g. in free-surface fluid flows, fluid–structure interaction and in solid mechanics for geotechnical engineering and production processes. During cutting, forming or melting of metal, quasi-incompressible material behaviour is often considered. To circumvent the associated volumetric locking in finite element simulations, different approaches have been proposed in the literature and a stabilised low-order mixed formulation (P1P1) is state-of-the-art. The present paper compares the established mixed formulation with a higher order pure displacement element (TRI6) under 2d plane strain conditions. The TRI6 element requires specialized handling, involving the deletion and re-addition of edge-mid-nodes during triangulation remeshing. The robustness of both element formulations is analysed along with different state-variable transfer schemes, which are not yet widely discussed in the literature. The influence of the stabilisation factor in the P1P1 element formulation is investigated, and an equation linking this factor to the Poisson ratio for hyperelastic materials is proposed.

On-bottom stability of a radial fin integrated surface-laid pipe-section against vertical penetration

This study focuses on the on-bottom stability of surface-laid submarine pipelines during vertical penetration under thermal-induced buckling. Submarine pipelines are susceptible to buckling due to their high slenderness ratio, and the resistance to such displacement relies on pipeline stiffness and soil resistance. To investigate this, laboratory model tests were conducted using a scaled-down pipe section placed on a clay bed. The clay bed was prepared with remoulded Kaolin clay, ensuring uniform moisture content, and the model pipe was scaled down from a typical industry prototype. These tests were conducted in 2D plane strain conditions within a tank, equipped with observation windows to monitor pipe movement and soil behaviour. A significant improvement in the pipe section's penetration resistance and on-bottom stability was observed after incorporating fin in the pipe. The addition of fins altered the earth pressure distribution beneath the vertically loaded pipe, leading to a larger failure mechanism in the adjacent soil. Furthermore, the soil-surface-heaving progressed towards the tank wall with increasing angular distance between the radial fins. Thus, the research demonstrated the effectiveness of radial fins in enhancing the stability and resistance of submarine pipelines against vertical penetration, shedding light on potential improvements in submarine pipeline design and deployment.

On flexural vibrations of orthotropic strips of variable thickness taken into account of transverse effects with elasticaly built in edges in the presence of axial force

Abstract The problem of bending vibrations of orthotropic strips of variable thickness is considered, taking into account transverse effects elastically built in edges in the presence of axial force. The problem was solved using the numerical collocation method. A numerical analysis of the problem was carried out. The problem of Euler stability of orthotropic plate-strips of variable thickness under plane strain conditions is considered, taking into account transverse shear and rotational inertia. The resulting equations make it possible to determine both the frequencies of natural vibrations and the critical values of the axial force at which the plate-strip loses stability. The problems are solved using a modified collocation method under various boundary conditions at the ends and values of geometric and physical parameters characterizing a plate-strip of variable thickness.

Collapse Pattern in Gypseous Soil using Particle Image Velocimetry

Abstract Gypseous soil is prevalent in arid and semi-arid areas, is from collapsible soil, which contains the mineral gypsum, and has variable properties, including moisture-induced volume changes and solubility. Construction on these soils necessitates meticulous assessment and unique designs due to the possibility of foundation damage from soil collapse. The stability and durability of structures situated on gypseous soils necessitate close collaboration with specialists and careful, methodical preparation. It had not been done to find the pattern of failure in the micromechanical behavior of gypseous sandy soil through particle image velocity (PIV) analysis. This adopted recently in geotechnical engineering to track the motion of soil grains and using tracer particles by applying digital particle image analysis. It has also been used to study the displacement distribution in some cases of granular materials. Therefore, the goal of this study is to find out how gypseous sand medium moves when in contact with a rigid strip foundation that is under static stress and plane strain conditions. The experimental model would focus on two common types of wetting, namely water table rise and dry conditions. The PIV showed that the collapse pattern under the footing is of the type of punching shear failure. The predominant mechanism of soil deformation was the vertical compression of the gypseous granular soil. The results showed that understanding gypseous sandy grain displacement and failure patterns at the local scale is crucial for enhancing the design of foundations under static stress conditions.

Mobility of dislocations in carbon nanotube bundles

Horizontally aligned carbon nanotubes (CNTs) have a unique combination of mechanical and physical properties and find numerous applications. The shear deformation of a horizontally aligned CNT bundle is studied under plane-strain conditions using the method of molecular dynamics. The CNTs in the bundle are not perfectly packed, and it is important to see how defects alter the mechanisms of their deformation under applied load. The defects in the form of edge dislocations are introduced into the bundle. It is found that the dislocation core structure depends on the CNT diameter, as the deformability of the CNT cross section increases with the diameter. Different deformation mechanisms (dislocation sliding or cracking) are revealed in the bundles with CNTs of different diameters. The results presented describe how the dislocation core structure affects the dislocation mobility in CNT bundles and the mechanisms of their deformation.

Crease instability in Gent-Gent hyperelastic materials

We study the crease instability of an incompressible Gent-Gent elastomer under general loading conditions. The Gent-Gent model has two distinct characteristics: the logarithmic nonlinearity triggered by I2 energy term and the strain hardening due to limiting chain extensibility. By performing finite element analysis combined with an evaluation of surface tension, we investigate the effects of the two characteristics on the critical creasing onset and the creasing onset delayed by surface tension. The results show that uniaxial compression induces logarithmic nonlinearity that accelerates the critical creasing onset, whereas equibiaxial compression hastens the initiation of strain hardening that delays the critical creasing onset. In addition, logarithmic nonlinearity mitigates the delay in creasing onset caused by surface tension under the uniaxial condition, and strain hardening suppresses creasing in cooperation with surface tension under the equibiaxial condition. Conversely, under the plane strain condition, two characteristics have little effect on the creasing onset delayed by surface tension.

Modeling and characterization of contact behavior of asperities with irregular shapes

This paper introduces a model for characterizing the contact behavior of irregular asperities, transforming it into a superposition of sinusoidal asperity contact behaviors. A new sinusoidal asperity model is developed for bilinear hardening under plane strain conditions. Empirical equations are proposed, considering geometric shapes, tangent modulus, and Young’s modulus. The frequency of asperity height is extracted through Fourier transform for irregular asperities. Contact area and pressure are predicted using the sinusoidal asperity model, and the behavior of irregular asperities is obtained by superimposing those with the first three frequencies. Experimental validation is conducted with milling and knurling-formed asperities, showing good alignment between the model and experimental results. In rough surface models, the proposed irregular asperity model exhibits greater accuracy in predicting contact behavior than a single sinusoidal asperity when interference exceeds 10% of the amplitude.

Shear Properties of Fissured Clay with Different Fissure Orientations under Plane Strain Conditions

Shear Properties of Fissured Clay with Different Fissure Orientations under Plane Strain Conditions