Plane Stress Strain Research Articles

Analysis of the Results of Obtaining Thermomechanical Diagrams Based on the Hypotheses of the Plane Stress–Strain State and Plane Deformation of Parts in the TMC Rings and Cylinders

Analysis of the Results of Obtaining Thermomechanical Diagrams Based on the Hypotheses of the Plane Stress–Strain State and Plane Deformation of Parts in the TMC Rings and Cylinders

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.

Stress Field and Crack Pattern Interpretation by Deep Learning in a 2D Solid

ABSTRACTA nonlinear variational auto‐encoder (NLVAE) is developed to reconstruct the plane strain stress field in a solid with embedded cracks subjected to uniaxial tension, uniaxial compression, and shear loading paths. Latent features are sampled from a skew‐normal distribution, which allows encoding marked variations of the features of the stress field across the load steps. The NLVAE is trained and tested based upon stress maps generated with the finite element method (FEM) with cohesive zone elements (CZEs). The NLVAE successfully captures stress concentrations that develop across the loading steps as a result of crack propagation, especially when enhanced disentanglement is emphasized during training. Some latent variables consistently emerge as significant across various microstructure descriptors and loading paths. Correlations observed between the evolution of fabric descriptors and that of their significant stress latent features indicate that the NLVAE can capture important microstructure transitions during the loading process. Crack connectivity, crack eccentricity, and the distribution of zones of highly connected opened cracks versus zones with no cracks are the fabric descriptors that best explain the sequences of latent features that are the most important for the reconstruction of the stress field. Notably, the distributional shape, tail behavior, and symmetry of microstructure descriptor distributions have more influence on the stress field than basic measures of central tendency and spread.

Establishment of ductile fracture locus of 2219 aluminum alloy at cryogenic temperature

Abstract Cryogenic forming has been used to manufacture intricate curved surface aluminum components. The special geometry and mold constraints cause the stress state to continuously evolve. Coupling with the cryogenic temperature makes the deformation process more complicated. Especially for the final fracture, which greatly determines the forming quality of the components. Thus, the cryogenic fracture behavior of three typical stress states was obtained by conducting tensile experiments. Fracture strains and stress paths were obtained using a hybrid numerical-experimental method. Finally, the cryogenic fracture locus was obtained based on the MMC ductile fracture criterion. The results indicate that the AA2219-O exhibits superior ductile fracture properties at cryogenic temperature. The fracture strain of AA2219-O was increased by 52.6% at the uniaxial tensile stress state, and by up to 80.0% at under plane strain stress state. This research has a guiding significance for cryogenic forming.

An analytical model for predicting residual stress in shot peening with strain energy method

In this study a model by novel analytical approach is developed and experimentally verified for shot peening residual stress distribution. The residual stress field induced by single shot impact is calculated by using Glinka–Molski energy-based method and kinematic hardening model. The formulation of the compressive residual stress (CRS) distribution is often based on plane strain or plane stress. It can be determined from the derived relation presented in this paper, the final residual stress in the full coverage conditions is the average of the two strain and stress plane expressions proposed by previous researchers. The distribution of residual stress is one of the key differences between the profiles produced by the results of the current model. There is a significant distinction between surface residual stress and maximum CRS, because the CRS profile near the surface is more curved compared to profiles obtained in earlier analytical models. The experimental data obtained by XRD analysis indicate the correctness and precision of the current model. Another goal of this study is to increase the fatigue life of GTD-450 stainless steel by shot peening at two different peening intensities. The fatigue life of samples were obtained by rotary bending test. Analytical results that confirmed by experimental findings shown bigger maximum compressive residual stresses occurred in higher shot peening intensities. This incident can improved fatigue life by deeper plastically deformed layer.

Моделирование упругопластического разрушения компактного образца

The strength of a compact specimen under normal fracture (fracture mode I) is studied within the framework of the Neuber-Novozhilov approach. A model of an ideal elasto-plastic material with limiting relative elongation is chosen as the model of the deformable solid. This class of materials includes low-alloy steels that are used in structures operating at temperatures below the cold brittleness threshold. The crack propagation criterion is formulated using the modified Leonov-Panasyuk-Dugdale model, which uses an additional parameter, i.e., the diameter of the plasticity zone (the width of the prefracture zone). In the case of stress field singularity occurring in the vicinity of the crack tip, a two-parameter (coupled) criterion for quasi-brittle fracture is developed for type I cracks in an elastoplastic material. The coupled fracture criterion includes the deformation criterion attributed to the crack tip and the force criterion attributed to the model crack tip. The lengths of the original and model cracks differ by the length of the prefracture zone. Diagrams of the quasi-brittle fracture of the specimen under plane strain and plane stress conditions are constructed. The constitutive equations of the analytical model are analyzed in terms of the characteristic linear dimension of the material structure. Simple formulas applicable to verification calculations of the critical fracture load and pre-fracture zone length for quasi-brittle and quasi-ductile fractures are obtained. The parameters in the presented model of quasi-brittle fracture are analyzed. It is proposed to select the parameters of the model according to the approximation of the uniaxial tension diagram and the critical stress intensity factor.

Finite Element Formulation and Computation of Superplastic Metal Forming Processes with Optimized Rate of Deformation Control

Superplastic forming (SPF) is a material forming technique that uses superplastic exceptional elongations and deformation characteristics to form superplastic materials into certain shapes. The combination of superplastic forming with diffusion bonding (SPF/DB) gives rise to an almost unlimited extension of superplastic forming since more integral lightweight cellular structural components can be manufactured. This paper discusses numerical modelling of the mechanism of superplasticity in metallic materials. The SPF computational method based on the finite element technique augmented with the controlling rate of deformations is developed to examine a range of design or operating conditions leading to more economical forming processes. The non-Newtonian ‘viscous flow’ material is used to model the constitutive of superplastic material during the forming period. The contact mechanics between the sheet material and the mold surface and the intersheet material contact mechanics are imposed using the penalty control method, in which the sticking contact boundary conditions are employed. The space discretization is carried out using the membrane element under plane strain and axisymmetric flow stress conditions, while the implicit time integration technique is utilized to follow the shape changes of the formed sheet material. The validation of the SPF finite element formulation was performed by comparing it with the available analytical solution of Hydraulic Free Bulging of Thin strips. The SPF of a hemispherical dome made of 7475 aluminum sheet alloy was performed to demonstrate the forming process as well as to validate the results obtained between the SPF finite element numerical simulation and the experimental results. The SPF/DB of the multicell component section is considered in the final part.

Computational and Experimental Technique for Constructing a Diagram for Thermomechanical Joints of Rings in a Plane Stress–Strain State

Computational and Experimental Technique for Constructing a Diagram for Thermomechanical Joints of Rings in a Plane Stress–Strain State

Crack tip plastic zone shape for anisotropic material subjected to mode -I loading

This paper discusses the effect of anisotropy in terms of anisotropy ratio on ductile materials subjected to mode-I type of loading. Thin plate with plane stress condition is considered for this work. σ3 is taken as zero. For considering the effect of Poisson’s ratio, σ3 = ν (σ1 + σ2) is considered. Isotropic plane strain and anisotropic plane stress condition is considered to develop mathematical relations for stress triaxiality and plastic zone around the crack tip by using Hill – von Mises’s yield criterion. The effect of poisons ratio, anisotropy ratio and stress triaxiality is shown on the size of plastic zone around the crack tip. The results were generated and are compared in the form of polar plot for plane strain condition at different Poisson’s ratio. In addition to this, the effect of anisotropy ratio on stress triaxiality is also investigated. It was observed that increase in Poisson’s ratio causes significant reduction in the size of plastic zone around the crack tip. The increase in anisotropy ratio also shows the same trend, especially for its higher values. It is observed that anisotropy plane stress problem of fracture is almost equivalent to isotropic fracture in plane strain condition.

Nonlocal thermomechanical coupled modeling method for two-dimensional rolling contact using a peridynamic approach

The development of thermomechanical coupled simulations for rolling contacts is challenging when the adjacent material is temperature-dependent or when there are cracks or other discontinuities on the contact surface. This paper proposes a novel thermomechanical coupled modeling method for rolling contacts using a nonlocal peridynamic (PD) approach. Fully coupled, ordinary state-based PD thermomechanical equations for the plane strain and plane stress cases are derived based on the strain-energy density, including thermal coupling terms, and detailed information on each parameter is provided. Subsequently, an algorithm for rolling thermal contacts in PD is proposed based on the penalty method. This study further provides a detailed modeling process and an algorithm for the proposed method based on a multi-rate time-integration scheme for numerical implementation. A typical two-dimensional (2D) wheel–rail rolling contact model is established to evaluate the performance of the proposed contact modeling method in wheel–rail thermal contact and to analyze the influence of temperature-dependent material and rail-surface cracks on the thermomechanical response. It is found that the contact stress and rail-surface temperature rise in the case of small creepages and temperature-independent materials are in good agreement with the analytical solution. In the case of large creepages, the mechanical and temperature results of using temperature-dependent materials and temperature-independent materials are quite different, and the frictional heat generated can cause martensitic transformation of adjacent materials. Rail-surface cracks cause a sharp increase in the contact stress and temperature rise near the cracks, and the high temperature also promotes crack growth.

Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in the Reissner Approximation

Elastic bending stresses and strains in the lithosphere are typically calculated based on the Kirchhoff–Love (thin) plate theory. The criterion for its applicability is the smallness of plate thickness to length ratio. In oceanic plates, due to the buoyant force of the mantle, the main deformations are not uniformly distributed along the plate but are concentrated in the vicinity of the subduction zone. Therefore, the effective length of the bending part of the plate is a few fractions of the actual length, and the criterion of a thin plate is partially violated. In this paper, we analyze the possibility of applying thick plate bending equations. The existing variational theories of 3D bending of thick plates are much more complicated than the Kirchhoff–Love theory, as they involve solving three differential equations instead of one, and have limited application due to their complexity. Since geophysical applications frequently use 2D models, in this paper we analyze in detail the potential and accuracy of the thick plate bending theory for 2D models. After the conversion to the 2D plane strain and plane stress approximation, the original 3D Reissner thick plate bending equations are written out in the form similar to the Kirchhoff equations with additive corrections and are supplemented with the explicit formulas for longitudinal displacement. The comparison of the analytical solutions of the 2D Reissner equations with the exact solutions shows that the 2D approximation only provides a correction for the plate bending function. However, this correction refines the Kirchhoff–Love theory by almost an order of magnitude. At the same time, the solution of the equations in this case turns out to be almost as simple as the solution of the thin plate equations.

СТРІЧКОВИЙ ФУНДАМЕНТ З ПОВЗДОВЖНІМ ВИРІЗОМ ПО ПІДОШВІ МАСИВНОЇ ПІДПІРНОЇ СТІНИ

The paper analyzes the designs of traditional strip foundations with a flat bottom, the load from which causes the plane strain stress state of the soil base, and other variations of conventionally strip (continuous) foundations, which due to their shape (configuration) of the contact area with the base change its stress state, which enables designing more sustainable foundations for continuous buildings and structures. Proceeding from the solutions to the mixed problem of the theory of elasticity and plasticity using the Mohr-Coulomb criterion strength criterion, analytical studies of the development of plastic zones in the base of a strip foundation with a longitudinal cut-out have been conducted, which show that the limit state always occurs first in the foundation’s edge zones, that is, underneath the outer edges of the foundation. It is also noted, however, that the design resistance of soil decreases when there is no additional load in the area of the cut-out; therefore, a patented design has been proposed of a strip foundation with a longitudinal cut-out in the bottom, where the cut-out with the height is filled with low-modulus material to improve the design resistance of soil. Based on experimental and theoretical studies, methods have been proposed for calculating the soil base (design resistance , settlement and inclination ) for a strip foundation of a massive retaining wall with a longitudinal cut-out in the bottom. A real example shows that the total width of the strip foundation with a cut-out is shorter by 1.5 m in comparison with the continuous shape of the bottom of the foundation, which has a significant economic effect on every linear meter of the wall foundation. Overall, the proposed methods make it possible to reasonably design effective eccentrically loaded foundations with cut-outs in the bottom and to improve the permissible vertical pressure on the base in comparison with a continuous bottom, all other things being equal. Keywords: strip foundation, massive retaining wall, eccentricity, bottom, cut-out, soil base, calculation procedure.

Numerical modelling of fatigue crack closure and its implication on crack front curvature using ΔCTODp

One of the widely used approaches to characterize fatigue crack propagation is the use of the effective stress intensity factor range ΔKeff, which relies on determination of crack closure value Kcl. The used models of crack closure are most frequently-two-dimensional, however, for real cracks, the 3D effects should be taken into account. The paper presents 3D finite element analyses of the influences of crack front shape, inserted cycles (loading cycles between node releases) and crack closure on the crack driving force in terms of ΔKeff and ΔCTODp (plastic part of crack tip opening displacement range). Numerically obtained crack closure depended on the simulation strategy. In the case of inserted cycles, crack closure disappeared in the internal part of the specimen and remained only near the free edges. The use of ΔCTODp had the advantage of a well-defined parameter in situations where ΔKeff was problematic, namely at the corner points, which did not allow finding of equalized crack driving force along the whole crack front using ΔKeff. Equalized crack driving force in terms of ΔCTODp was found for crack front curvature with the edge angle 15.4° in simulation with crack closure, which was in good agreement with the experimentally measured value of 16°. Loops produced by loading and unloading branches of the force vs CTOD diagrams helped to describe the crack closure process and magnitude. Actual values of CTOD did not agree with the classical idea of 2D solutions under plane strain and plane stress. CTOD was larger in the internal part of the specimen than at the free edges, even in simulations with no crack face contact.

Cyclic stress–dilatancy relations and plastic flow potentials for soils based on hypothesis of complementarity of stress–dilatancy conjugates

Soil, rocks and rock masses dilate or compact when sheared, i.e., distortion necessitates volume change. This coupling between distortional strains and volumetric strains, described by stress–dilatancy theories, endows soils with manifestation of peculiar characteristics when they are subjected to shear. Stress–dilatancy theories have become central in describing the mechanical energy dissipation mechanism and further establishing flow rules in constitutive modelling of soils. The classical stress–dilatancy theories, such as Taylor's and Rowe's, are endowed with simplicity and descriptive power, but they were developed for describing the dilatancy behaviour of soils subjected to loading in shear (mobilizing away from isotropic stress state) and needed to be extended for describing plastic dissipation and shear-induced volumetric changes when soils are subjected to cyclic shear. In this paper, hypothesis of complementarity of stress–dilatancy conjugates is proposed as a unifying hypothesis for deriving stress–dilatancy relations for both loading in shear and unloading in shear. Then, plastic potential functions are derived based on the resulting stress–dilatancy relations. In so doing, the resulting stress–dilatancy relations and plastic potential functions are rendered with a quality to be used for the modelling of deformation behavior of soils subjected to both monotonic and cyclic shearing. The theoretical framework is applied first for plane strain and axisymmetric stress–strain conditions; and then extended for the general stress condition considering the Lode angle dependency of the shear strength of soils, using the multilaminate framework and applying the Matsuoka–Nakai spatial mobilized plane.

Integral‐based averaging with spatial symmetries for non‐local damage modelling

AbstractSimulations utilizing local constitutive equations for strain‐softening materials are known to be pathologically mesh‐dependent. The rational solution to this problem is to use nonlocal material models. In this paper, we discuss the application of the integral‐based approach to the simulation of non‐local damage accumulation and fracture. Various combinations of non‐localities with common spatial symmetries are analysed analytically and numerically. The symmetries include the plane strain and the axisymmetric cases, as well as the presence of symmetry planes and cyclic symmetries. Although not a symmetry, the practically important case of thin plates is also analysed. We show that the delocalization procedure should be executed using symmetry‐adapted averaging kernels. For the considered spatial symmetries, analytical closed‐form expressions are obtained. Moreover, a new easy‐to‐use averaging kernel is suggested, the same for 3D, plane strain and plane stress applications. To showcase the delocalization procedures, we consider a ductile damage model, based on the multiplicative decomposition of the deformation gradient, as well as hyperelastic relations between stresses and strains. FEM solutions for a series of problems are presented, including graduate damage accumulation, crack initiation and fracture.

The Effect of Hydrogen on Failure of Complex Phase Steel under Different Multiaxial Stress States

The demand for advanced high-strength steel (AHSS) in the automotive industry has increased over the last few years. Nevertheless, it is known that AHSSs are susceptible to hydrogen embrittlement. Therefore, the influence of hydrogen on the localization and damage behavior of a CP1000 steel sheet was investigated in this work. The sheet metal was electrochemically charged to a hydrogen content of about 3 ppm (by weight). Tensile tests were performed at different nominal strain rates between 0.00004 s−1 and 0.01 s−1 to investigate the effects of strain rates on their susceptibility to hydrogen embrittlement. Nakajima tests were utilized to investigate the hydrogen effects on the steel’s formability under different stress states. Three different Nakajima specimen geometries were employed to represent a uniaxial stress state, a nearly plane strain stress state, and an equibiaxial stress state. Further, forming limits were evaluated with the standardized section line method. Hydrogen embrittlement, during tensile testing, occurred independent of the strain rate, unlike the Nakajima test results, which showed hydrogen effects that were strongly dependent on the stress state.

Finite element analyses of crack extensions in curved compact tension specimens of irradiated Zr-2.5Nb pressure tube materials by nodal release method and cohesive zone modeling approach

Crack extensions in thin CCT specimens of irradiated and hydrided irradiated Zr-2.5Nb pressure tube materials are simulated using the nodal release method and cohesive zone modeling (CZM) approach. The load vs displacement curves from the 3-D FE analyses with consideration of plastic orthotropy using the nodal release method match with the experimental data. The changing separation work rates of the 3-D FE analyses using the nodal release method and the experimental J-R curves are adopted as references for the changing cohesive energies in 2-D plane strain and plane stress FE analyses with the CZM approach. The simulation results with the CZM approach match well with the experimental data.

Towards three dimensional aspects of plasticity-induced crack closure: A finite element simulation

The mutual interactions between intrinsic (damage) and extrinsic (shielding) mechanisms are essential for understanding fatigue crack growth in ductile materials. In the latter case, plasticity-induced crack closure is the dominating retardation mechanism in the Paris regime. The transition between plane strain and plane stress states leads to a locally different fracture surface contact, which is difficult to access during experiments. In this work, plasticity-induced crack closure under constant amplitude loading of an AlCu4Mg (AA2024) aluminium sheet material is studied from a three-dimensional perspective. An elastic–plastic 3D finite element model of an M(T)160 specimen with a bilinear isotropic hardening model is used to study the evolution of plasticity during fatigue crack growth. Cyclic crack propagation is simulated with the releasing-constraint method.In the results, plasticity-induced crack closure is present up to a load ratio R = 0.3. Detailed investigations of the contact pressure distributions indicate a strong dependence on the stress intensity factors and the sheet thicknesses. The contact pressure distributions on the fracture surface can be divided into three characteristic regions. Near the plastic zone, plastic energy is still induced during the unloading process after the crack is already closed. However, the crack surface contact does not affect the shape of the plastic zone. Additionally, further plastic energy accumulates in the plastic wake region during crack closure within subsequent load cycles, described here as cyclic plastic wake. In summary, the finite element model can reproduce the major features of fatigue crack closure like Kop or Kcl and the three-dimensional and load-dependent evolution of plasticity-induced crack closure.

The simulation of two-dimensional plane problems using ordinary state-based peridynamics

Abstract The ordinary state-based peridynamics (OSB PD) model is an integral nonlocal continuum mechanics model. And the three-dimensional OSB PD model can deal with linear elastic solid problems well. But for plane problems, the calculation results of existing models have large deviations. In this paper, a set of OSB PD models for plane problems is established by theoretical derivation. First, through the strain energy density function equivalence of peridynamics and classical continuum mechanics, the equivalent coefficients of the plane strain and plane stress problems of OSB PD are deduced. Then, consider the cantilever beam deformation simulation under concentrated load. The simulation results show that the maximum displacements are in good agreement with the corresponding analytical solutions in all directions. Finally, in the simulation of the slab with a hole, the two cases of uniform displacement and uniform load are considered, respectively. The simulation results are consistent with the ANSYS analysis results, and the deviation is small, which verifies the validity of the model.

Characterisation of the crack tip plastic zone in fatigue via synchrotron X‐ray diffraction

AbstractThis paper describes a new methodology for characterising the plastic zone ahead of a fatigue crack. This methodology is applied to a set of experimental data obtained by synchrotron X‐ray diffraction on a bainitic steel compact tension specimen. The methodology is based on generating the equivalent Von Mises strain field from the X‐ray experimental elastic strain maps. Based on the material response, a threshold is then applied on the equivalent strain maps to identify the size and shape of the plastic zone. The experimental plastic zone lies between the plane strain and plane stress Westergaard's bounds but closer to the plane strain theoretical prediction confirming that the volume analyzed is predominantly subjected to plane strain conditions. However, the observed plastic zone has a somewhat flatter shape, extending further from the crack plane but less extended in the crack growing direction.