Plane Strain Conditions Research Articles

A quasi-2D exploration of optimum design settings for geotextile-reinforced sand in assistance with PIV analysis of failure mechanism

Many earlier studies were focused on testing different types of geosynthetics to investigate effect of reinforcement on bearing capacity, but the effect of tensile strength on the failure mechanism has not been examined sufficiently. Within this scope, a test setup was prepared to apply strip loads on densely compacted reinforced sand under the plane strain condition. The tank containing the reinforced sand was a rectangular prism with perfect transparency, and its interior dimensions were 960 × 200 × 650 mm3. Firstly, optimum values of design variables (depth of first sheet, length and number of sheets, space between sheets, tensile strength of sheets) for the woven geotextile reinforced sand were determined experimentally. Then, the failure mechanisms of the soil, which were reinforced with geotextiles of different tensile strengths, were observed and analyzed with particle image velocimetry (PIV) technique. Consequently, the failure mechanism of the sand with a single geotextile reinforcement was similar to general shear failure of unreinforced soil. Contrarily, the failure surfaces were deeper and longer. Additionally, the deep-footing mechanism reached out large depth in the case of four reinforcement layers. The failure mechanism converted into a punching type due to a hypothetic increase in the bearing depth of reinforcement.

Dynamic response of elastic anisotropic solid with nanoinclusions

AbstractThe finite elastic anisotropic solid with multiple elastic isotropic/anisotropic nanoinclusions of arbitrary shape, size, number and geometrical configuration subjected to time‐harmonic or transient loads is examined under plane strain conditions. The proposed mechanical model is based on a hybrid usage of elastodynamic theory for the bulk elastic anisotropic solid under non‐classical boundary conditions (BC), supplemented with a localized constitutive equation for the solid‐nanoinclusion interface in the framework of the Gurtin‐Murdoch theory of surface elasticity. The developed and verified numerical scheme for solution of the transient problem is based on the hybrid usage of direct and inverse Fourier transform (FFT), boundary element method (BEM) in conjunction with closed form frequency dependent fundamental solution of elastodynamic equation for anisotropic materials. The proposed model is flexible, numerically efficient and has virtually no limitations regarding the type of material anisotropy, nanoinclusions’ shape, size, number and geometrical configuration. The numerical results show a marked dependence of the wave fields on the surface elasticity effects, on the size, number and position of the nanoinclusions, on the type of material anisotropy, on the type and properties of the dynamic load, on the mutual interactions between nanoinclusions and between them and the external solid's boundary.

Dynamic Fracture Resistance under Plane Strain Conditions of High-Density Polyethylene Nanoclay Composites.

Polymer nanoclay composites have received significant attention due to their substantially enhanced mechanical, thermal and barrier properties. However, the effect of these nanoclays on the dynamic fracture resistance of a polymer matrix during fast fracture events has not been documented. In this study, the effect of nanoclay addition on the rapid crack propagation (RCP) resistance of high-density polyethylene (HDPE) was investigated through the high-speed double torsion test. Results showed that the addition of 1, 3, and 5% of nanoclays improved the dynamic fracture resistance under the plane strain conditions (Gd1) of HDPE up to 65%. An increase in the storage and loss modulus, and a decrease in crystallinity and melt flow index with nanoclay content was also found. Although the presence of agglomerates can hinder the enhancement of Gd1 as it promotes agglomerate fracture and debonding, the increase in energy consumption through fibrillation and crazing promoted by the nanoclay prevails, suggesting that the nanoclay's toughening effect that has been extensively reported under quasi-static and impact tests, is also present under RCP conditions, and that the HDPE nanocomposites could be used in applications in which RCP must be prevented.

Effect of retained austenite on the fracture behavior of a novel press-hardened steel

• Effect of RA on both the tensile and bending properties of the PHS has been studied. • RA with various mechanical stabilities and volume fraction are obtained via tuning the degree of auto-tempering of martensite during die quenching. • The tensile performance is improved with increasing of volume fractions of RA in the novel PHS. • The mechanical stability, rather than volume fraction of RA, has significant impact on the bendability of the PHS. Tensile and bending properties are two critical attributes of press hardened steels (PHSs) for automotive body structural components. However, the research on these properties of PHSs with retained austenite (RA), which is introduced to improve the mechanical properties, has not been well reported. In this study, the effect of RA on the quasi-static uniaxial tensile and three-point bending behaviors has been systematically investigated for a newly developed Cr and Si alloyed PHS. RA with various degrees of mechanical stabilities was obtained by tuning the auto-tempering of martensite through the control of the die contact pressure during the press hardening process. Mechanically stable RA provides the optimal combination of tensile and bending performance, while the PHS with RA of poor mechanical stability has a deteriorated bending toughness as manifested by a reduction of bending angle from approximately 61.6° to 56.8°. However, its tensile properties, in contrast, are the best in terms of the product of ultimate tensile strength and total elongation (15.9 GPa %). This is mainly attributed to the unstable RA on the outermost surface of the bending sample. The unstable RA can easily transform into brittle martensite under local plane strain and strain gradient conditions in the bending test compared with uniaxial stress in the tensile test, promoting crack initiation and propagation. Furthermore, the effect of martensitic auto-tempering or contact pressure on the quantity and stability of RA are also discussed. This study would provide a useful reference to guide hot stamping process optimization for the newly developed Cr and Si-containing press-hardened steel.

Undrained Stability of Opening in Underground Walls in Anisotropic Clays

A two-dimensional (2D) finite-element limit analysis (FELA) was performed under plane strain conditions to investigate the undrained stability of tunnel launch with a rigid wall in anisotropic clays. The anisotropic undrained shear (AUS) failure criteria were applied to identify anisotropic soil behaviors. The average bound solutions were employed to compute the stability load factor. Different sensitivity analyses were carried out to determine the effect of cover–depth ratio (C/D), overburden factor (γD/suTC), and anisotropic strength ratio (re) on the load factor (N) of the opening in tunnel launching with a rigid wall. The failure patterns that were predicted by certain parameters were explored and investigated. To adopt a practical approach, some new design charts for the undrained stability of tunnel launching with a rigid wall in anisotropic clays were devised.

Full Field Model Describing Phase Front Propagation, Transformation Strains, Chemical Partitioning, and Diffusion in Solid–Solid Phase Transformations

AbstractA novel mathematical formulation is presented for describing growth of phase in solid‐to‐solid phase transformations and it is applied for describing austenite to ferrite transformation. The formulation includes the effects of transformation eigenstrains, the local strains, as well as partitioning and diffusion. In the current approach the phase front is modeled as diffuse field, and its propagation is shown to be described by the advection equation, which reduces to the level‐set equation when the transformation proceeds only to the interface normal direction. The propagation is considered as thermally activated process in the same way as in chemical reaction kinetics. In addition, connection to the Allen–Cahn equation is made. Numerical tests are conducted to check the mathematical model validity and to compare the current approach to sharp interface partitioning and diffusion model. The model operation is tested in isotropic 2D plane strain condition for austenite to ferrite transformation, where the transformation produces isotropic expansion, and also for austenite to bainite transformation, where the transformation causes invariant plane strain condition. Growth into surrounding isotropic austenite, as well as growth of the phase which has nucleated on a grain boundary are tested for both ferrite and bainite formation.

Progressive Failure Analysis of Soil Slope with Strain Softening Behavior Based on Peridynamics

In this study, the nonlocal, nonordinary state-based peridynamics (NOSBPD) is introduced as a novel regularization technique to study the strain localization and progressive failure of soil slopes due to strain softening behavior. The NOSBPD formulations are introduced first, and then the strain-softening Mohr–Coulomb model is incorporated into the nonlocal framework. The implicit formulations of the nonlinear NOSBPD model under the plane strain condition combined with the energy dissipation-based arc-length method to avoid the limitations of the force and displacement control in the nonlinear analysis are given. Numerical examples, including the plane strain biaxial experiments and slope models, validate the effectiveness of the proposed method in terms of alleviating mesh sensitivity. The results show that this nonlocal method can remove the mesh dependence, and the localized shear band is related to the nonlocal parameter δ. The proposed method involves being able to capture strength mobilization in different parts of the sliding surface during the progressive failure process of the soil slope. It can avoid the dilemma that the peak strength is too dangerous and the residual strength is too conservative due to the rigid body assumption in the limit equilibrium method.

Response of 2D and 3D crystal plasticity models subjected to plane strain condition

The plane strain assumption is generally applied in crystal plasticity finite element (CPFE) simulations in a 2D space to characterize the macroscopic material response considering microstructural features. However, the reliability and accuracy of 2D approximations need to be addressed. In this paper, crystal plasticity finite element simulations of 2D and 3D RVEs are performed with local and averaged plane strain assumptions in Abaqus/Standard. Plane strain postulation is implemented via plane strain elements in 2D and zero average thickness strain in 3D. Irregularly shaped RVEs are generated using the open-source software library Voro++. A conforming mesh is rendered to assign periodic boundary conditions on geometrically periodic RVEs. Periodic boundary condition (PBC) is applied using a prescribed macroscopic deformation gradient tensor. A rate-independent finite strain crystal plasticity model is employed as the user-defined material behavior in finite element simulations. A discrepancy is observed between macroscopic flow curves of 2D and 3D RVEs. The comparison was made for three cases of latent hardening in the crystal plasticity model. In all cases, 3D flow curves exceed 2D results. The results indicate that the deviation is caused by out-of-plane slip activation in 3D simulations, which proves to be an additional hardening source.

Study of stamping of unequal channels. Part 7. Kinematic and stress states of the workpiece under constrained extrusion. 1. Plastic area under the resulting thick wall

Based on the complete system of equations of the theory of plastic flow, the beginning of solving the problem of determining the kinematic and stressed states of the workpiece under constrained extrusion of channels under conditions of plane strain with a misaligned location of the punch and matrix is outlined. The flow rates and stresses in the area of plastic deformation under the forming thick channel wall are determined. Formulas are obtained for determining the technological parameters of extrusion of unequal channels with a thin horizontal bridge. Keywords: die forging, extrusion, misalignment of punch and matrix, plane strain, plastic flow rates, stresses. mt13@bmstu.ru

A STUDY OF DEFORMATION AND FRACTURE OF THE EUTECTIC IN AN ADDITIVELY MANUFACTURED AL-SI COMPOSITE ALLOY

The structure of AlSi12 alloy fabricated by wire-feed electron beam additive manufacturing was studied experimentally. A numerical study was performed to examine the thermomechanical behavior and fracture of a eutectic fragment at the scale of several microns. Boundary dynamic problems were solved under plane strain conditions. The composite structure of the eutectic system consisting of an aluminum matrix and silicon particles was taken into account explicitly in the calculations. Isotropic models of the thermoelastic-plastic matrix and elastic-brittle particles were implemented in ABAQUS/Explicit. Strains in the composite were calculated both with and without allowance for residual stresses caused by cooling after fabrication. It was shown that after the cooling of the eutectic, silicon particles are compressed, and the aluminum matrix is under both bulk compressive and tensile as well as under pure shear stresses. It was found that residual stresses play a negative role at the stages of intense deformation of the composite. The fracture strain of the eutectic strongly depends on the yield stress of the matrix, while the ultimate fracture stress varies little. Favorable morphology of silicon particles was determined which prevents early fracture of the eutectic.

3D effects of soil-atmosphere interaction on infrastructure slope stability

It has long been established that pore water pressure (pwp) variations affect the stability and serviceability of slopes. Pwp variations may be due to consolidation/swelling processes within soils of low permeability or may be due to seasonal evapotranspiration and precipitation processes. The simultaneous study of such phenomena and of hydro-mechanical coupling in sloping ground requires use of advancednumerical methods. Often, infrastructure slopes are considered in plane strain (two-dimensional) conditions for simplicity and to date this is the case for most numerical analyses considering soil-atmosphere interaction. However, this approach makes it impossible to study the longitudinal extent of a possible slip surface forming following vegetation removal. Three-dimensional, fully-coupled numerical analyses of acut slope were performed herein to study the effect of vegetation clearance on stability and explore numerically ways of implementing effective vegetation management.

On energy release rate for an electrically permeable interface crack between two different 1D hexagonal piezoelectric QCs

A bi-material composed of two 1D piezoelectric hexagonal quasicrystals with a crack along the material interface was considered. Phonon and phason remote loading providing plane strain conditions were applied at infinity and an open crack model was adopted. Because electromechanical fields have an oscillating singularity at the crack tip, the energy release rate (ERR) is the most important fracture parameter in this case. The main purpose of this study is to define ERR in an analytical form. Using the obtained earlier asymptotic presentations of the phonon and phason fields at the crack tip and the crack closure integral approach, the desired formula is obtained in a simple form. The eligibility of the formula is verified by means of a crack in the bi-material composed of two 1D piezoelectric hexagonal quasicrystals and by considering a crack in homogeneous materials of this type. An additional comparison of the results obtained for a particular case of an isotropic material with known values confirmed the validity of the obtained formula.

Двопараметрична модель зони передруйнування у квазіпружному матеріалі біля вершини міжфазної тріщини

A model of the pre-fracture zone, which under plane strain conditions extends from the tip of the interface crack at an angle to the plane interface of two different homogeneous isotropic materials, has been developed. The pre-fracture zone is represented by the displacement discontinuity line, on which the strength condition of the Mises-Hill type is satisfied. The parameters of the model are the tensile resistance and shear resistance of the material in which the zone develops. The model takes into account brittle and ductile fracture mechanisms and can be used to describe the quasi-brittle fracture of a piece-homogeneous body with an interface crack. The possibility of a transition between brittle and ductile fracture mechanisms is shown when the ratio of tensile and shear resistance of the material changes.

Sound synthesis approach based on the elastic stress analysis of a wrinkled thin film coating

This paper examines the ability to employ the model of a wrinkled thin film coating under plane strain conditions for the generation and manipulation of sounds. It is assumed that the film-on-substrate structure represents a multilevel system consisting of four phases with different elastic properties such as surface layer, film material, interphase and substrate material. The undulation of surface and interface geometry leads to the complex stress state of the film coating resulting from the superposition of two perturbed stress fields evolved near the curved surface and interface. The analysis of the stress state reveals the periodic distribution of the longitudinal stresses smoothly changing from surface to interface. This facilitates the creation of a sound synthesis technique similar to the timbre morphing method which provides the transition between two waveforms while creating new intermediate waveform during this process. The perspective of using the wrinkled thin film model for sound generation is gained by its complex behavior, where the influence of one parameter on the stress distribution is affected by other parameters, which in turn reflects in the rich sound morphologies during the mapping of the stress oscillations onto sound.

Calibration of a large shaking table equipment for testing geotechnical physical models under plane strain conditions

Experimental tests carried out on physical models under laboratory-controlled conditions can provide a valuable insight into the seismic performance of geotechnical systems and are extremely valuable to check the predictions of numerical analyses.In this vein a new shaking table apparatus with a large flexible soil container has been set up in the laboratory of Geotechnical Engineering of the University of Messina.The apparatus consists of a shaking table connected to a servo-hydraulic actuator, a large shear stack container for the soil and an automated system for soil deposition.The actuator can excite the low friction shaking table by applying horizontal displacements reproducing real and artificial seismic motions characterized by amplitude, frequency and energy contents varying in wide intervals.The large soil container consists of a laminar shear box which has been assembled to accurately model a soil layer of infinite lateral extent; under horizontal shaking the container deforms in plane strain conditions according to a shear beam mode, thus reproducing a free-field condition under vertically propagating shear waves.The soil deposition system consists of a hopper that can be moved back and forth above the soil container allowing sand pluvial deposition; the velocity of the hopper, the width of its lower opening and the soil falling height can be set to attain a desired relative density in the soil model.The paper describes this new experimental facility pointing out its peculiarities and provides the results of some static and dynamic calibration tests demonstrating that: (i) the plane strain conditions can be satisfactorily assumed for the soil specimen in the laminar box; (ii) the servo-hydraulic control system is capable to accurately reproduce a prescribed acceleration time-history at the shaking table platform, regardless the amplitude, frequency and energy content of the desired motion; (iii) no significant boundary effects arise in the portions of the soil specimen placed near the end-walls of the laminar box; (iv) the shaking table tests carried out with the new apparatus are repeatable.

A meso-scale numerical modelling method for the stability analysis of tunnels in sandy cobble stratum

This paper presents a meso-scale numerical modelling method to investigate the stability of tunnels in sandy cobble stratum. Using Monte Carlo simulation, Python routine is implemented to randomly generate and filter the location information of blocks including center coordinates, sizes and orientation angles, then extract and import them into CAD modelling software in batches to mesh. The problem of overlapping detection for elliptical and ellipsoidal blocks is solved based on affine transformation, achieving higher efficiency and applicability than the existing methods. Then, using the proposed numerical modelling method, convergence-confinement analysis of circular tunnels with different cover depths is performed in plane strain condition. Meanwhile, face stability analysis of shield tunnels is carried out in three-dimensional condition. Results show that sandy cobble stratum may not be in accordance with the scale-independent soil-rock mixture. Blocks with small sizes are geotechnically significant and contribute much to the overall strength. When the cover depth ratio reaches a certain level, limit support pressure remains basically unchanged due to the strong arching effect above tunnel crown, while permissible support pressure increases approximately linearly to control deformation. Compared with the multi-scale model, hardening soil model considering the overall strength is suitable for the stability analysis of tunnels in sandy cobble stratum at the macro-scale with high precision, while the macro homogeneous model ignoring the presence of blocks provides a conservative (i.e. uneconomical) prediction for the design of tunnel stability.

New insights into the bond-based and ordinary state-based models in Peridynamics

Twenty years in since its introduction, Peridynamics established by Silling provides an attractive theory for modeling deformation and fracturing in solid materials. The paper aims to revisit the well-established bond-based Peridynamic (BPD) and ordinary state-based Peridynamic (OSPD) models and attempts to provide new insights into them. To this aim, a novel decomposition of the Peridynamic bond length change is developed, and the bond length change components are used to define a new micro-potential function. The macroscopic nonlocal elastic strain energy density (NESED) is computed through the integral of the micro-potential over the horizon. The constitutive relation is obtained by the Frechet derivative of the NESED with respect to the deformation state. The NESEDs and constitutive bond force densities are discussed with specific influence functions. The main foundings of the present work are as follows: (1) An explicit expression of bond micro-potential is proposed, and the NESED obtained by the integral of the micro-potential over the horizon can retrieve the ad hoc NESED in the OSPD model. It could pave the way for the development of micro-potential based bond failure criterion for the OSPD model. (2) It is proven that the constitutive relation in the OSPD model can be recovered by the present constitutive relation with specific influence functions in the case where the deformation is small. (3) As the Poisson ratio is 1/3 in plane stress condition or 1/4 in plane strain condition, the present Peridynamic model with specific influence functions is reduced to the BPD model in the case of small deformation, and thus the BPD model can be considered as a reduced form of an OSPD model. Moreover, numerical investigations on a single edge-notched plate under mode II dominated loading and three-point bending tests are conducted to illustrate the descriptive and predictive capabilities of the present Peridynamic model.

Uncertainty quantification and global sensitivity analysis of seismic metabarriers

Seismic metabarriers consist of an array of locally resonant elements (i.e., mechanical resonators) installed over the soil surface, whose design is rationally engineered to reduce ground-induced vibrations and shield vulnerable structures against seismic surface waves. Successful design and implementation of seismic metabarriers require a comprehensive knowledge and characterization of the role played by the model parameters (and their associated uncertainty) governing soil-barrier dynamic interaction. In this context, sensitivity analysis techniques allow assessing the response of a given model through the quantification of the influence and action of model inputs (and model input uncertainties) concerning a target model output. This study relies on global sensitivity analysis techniques to investigate the influence that the uncertainty associated with three key mechanical parameters of a metabarrier (i.e., soil density, soil shear modulus, and mass of mechanical resonators) has on its seismic isolation performance. The latter is measured in terms of transmission coefficient (TC). We do so by employing a two-dimensional wave finite element model developed under the plane-strain conditions to evaluate the dispersion relation and transmission coefficient of a metabarrier interacting with Rayleigh waves in the low-frequency regime (i.e., frequencies between 2 Hz and 7 Hz). Our results suggest that the shear modulus is the uncertain parameter with the most significant influence on the transmission coefficient of the metabarrier across the entire frequency range of interest. Otherwise, the resonator mass plays a substantial role in the frequency range close to the metabarrier resonant frequency.

Effect of Loading Rate on Fracture Behaviour and Shear Lips Formation of Al6061 Alloy

Aluminiumalloys are widely used in many engineering industries due to their mechanical and material properties. It is used in many applications, especially automotive parts, like automobile cylinder heads, engine blocks, and chassis. It is important to obtain the behaviour of fracture of this material. Several materials testing has been conducted on aluminium 6061 alloy to fulfil the requirement in industries. However, the effect of specimen thickness on shear lip ratio and plastic zone size on fracture specimens for aluminium 6061 under the three-point bending test is hardly studied. Also, the fracture behaviour of this material has only sometimes been clarified. The objective of this paper is to study the effect of the thickness of the sample as well as the loading rate on the formation of shear lips for aluminium 6061. To find out whether shear lips occur on the fracture surface of aluminium 6061, a three-point bending test was performed on a Single Edge Notched Bending (SENB) specimen. Analysis has been made to determine the shear lips ratio dependence on specimen thickness and loading rate. Based on the result of the analysis, the shear lips ratio decreases as the specimen's thickness increases. Besides that, the shear lips ratio decreases as the loading rates increases. A high loading rate and specimen thickness resulted in low plasticity behaviour because the stress states are in plane strain conditions where the formation of shear lips is minimum. From the three-point bending test, a good agreement of data gains from the dependency of shear lips ratio on specimen thickness and loading rates. Overall, aluminium 6061 is fractured in a ductile manner with significant shear lips formation.

Research of Mechanical Properties of the Aluminium Alloy Amag 6000 Under the Plane Stress State Conditions

Today's demands directly encourage to utilize the light alloy materials in industrial production. In the automotive industry, there is an increasing tendency to reduce the total weight of the car and thus also the weight of the individual car-body parts. A correct description of the material deformation behaviour with respect to the different stress states is very necessary for the own possibility to use these materials in the production. This paper describes a measurement of the mechanical properties and deformation behaviour of aluminium alloy Amag 6000 under the plane strain conditions. Thus these properties are monitored and evaluated by means of the plain strain tensile test. In this case, the plane strain is achieved by the shape concept of the testing specimen, which is loaded by the uniaxial tensile stress state using a standard testing device. The determined mechanical properties and the resulting stress-strain curves of the material under the plain stress state can be further used at defining material models for a more accurate description of the material deformation process during the sheet metal forming process.