Plane Stress Conditions Research Articles

A failure mode dependent continuum damage model for thick laminated composite plates

The fiber reinforced composite materials show stiffness degradation (termed as “Damage”) upon loading, which can be modeled mathematically using Continuum Damage Mechanics (CDM). However, the available CDM models for composites are not characterized using all the uniaxial/shear stress-strain curves, therefore, do not account for the non-linear stress-strain behavior from all uniaxial/shear material characterization tests. Therefore, the novelty of this article involves the development of a failure mode-dependent phenomenological continuum damage model involving coupling of damage in different directions for generalized plane stress conditions and the characterization of model parameters based on experimental uniaxial fiber tension-compression, matrix tension-compression, in-plane, and out-of-plane shear stress-strain curves. The finite element method with load, and displacement control is used to predict the evolution of damage in a moderately thick laminated composite plate subjected to quasi-static loading. The damage prediction capability of the developed model is compared with a failure mode independent damage model. The reported results depict the capability of the developed model to predict different damage in tension and compression in accordance with the strength values. The developed model is also tested using different coupling vectors (responsible for coupling of damage in different directions) to assess their accuracy in capturing the effect of coupling of damage in different directions.

Elastic stability of a lattice of cross-braced shear deformable beams

A plane metamaterial made up of a periodic square grid of shear deformable beams has been braced by crossing cables. This coupling, widely adopted in structural engineering, is here applied with the aim of enhancing the stability performance of a periodic material. Under suitable assumptions, analytical solutions for a rich variety of critical modes and the stability domain are determined by the Bloch wave approach and the effectiveness of our analytical results has been verified by numerical simulations. Furthermore, the experimental response of a specimen, in which hierarchical beams composed by lower order elements are modeled as equivalent shear deformable beams, has shown a satisfactory agreement with our analytical solutions. We argue that our analytical model can be efficiently used to tailor the buckling response of various new metamaterials under general plane stress states.

Theoretical solutions for 2D mode-I crack-tip stress fields in power-law plastic materials based on the stress factor derived from the developed median-energy–density equivalence method

This study presents the energy density equivalence equation, demonstrating that the strain energy density of an arbitrary representative volume element (RVE) under a complex stress state equals to that under a uniaxial stress state when volume change is neglected. Based on the developed median-energy–density equivalence method, the analytical solution for equivalent stress of the RVE at the energy density equivalence point in power-law plastic materials is derived in a new method and the concept of stress factor is proposed. Additionally, novel special functions are introduced to accurately describe the angular distributions of stresses in this study. By combining the stress factor, integral theoretical solutions for mode-I crack tip stress fields in power-law plastic materials under finite plane conditions are proposed. Finally, parameterized studies are conducted on bending and tension specimens under plane-strain and plane-stress conditions. The results of stress distributions and contour lines predicted by the integral theoretical solutions are compared with the finite element analysis and HRR solutions for five mode-I plane specimens, providing verification of the solutions’ applicability.

Damage ratio yield theory for orthotropic metals under the plane stress state

在各向同性材料损伤比强度理论研究成果的基础上, 发现脆性材料损伤比参数的定义也适用于正交异性金属材料. 首先, 根据单元体相对耗能率极值计算模型, 构建平面应力状态下正交异性金属损伤比屈服理论, 其形式简单、参数物理意义明确且能反映拉压不对称效应; 其次, 建议损伤比参数之和的取值并提出不同取向角下单轴屈服应力的计算公式. 分析结果表明: 当正交异性金属材料在单轴拉伸和双轴拉伸应力状态时, Hill-48屈服准则为损伤比强度理论的特例; 此外, 正交异性金属材料损伤比屈服准则得到镁合金、铝合金、钢材和锆的单轴屈服应力及镁合金、铝合金、钢材和Ni3Al基金属间化合物IC10合金的双轴屈服应力试验结果的验证, 计算精度与现有几类反映材料拉压非对称性的唯象屈服准则一致.

A general leading-order asymptotic theory of thin microstructural plates and uncertainty quantification of the elastic performance of composite laminates

We propose an asymptotic framework that may be used as a platform to model general thin microstructural plates, and our initiative is twofold. First, it provides an explanation to why so many tailored plate theories have been introduced for modelling plates bearing microstructures. In hypotheses-based plate theories, the microstructural details within each ply are usually smeared out. But it is shown that with such in-ply details properly taken into account, one manages to address, even only with leading-order terms, several key issues, such as transverse shear strain effects, shear locking, etc., whose rationalisation is believed to necessitate higher-order terms. To a certain degree, the proposed formulation can be used as a target for tailoring kinematic shape functions in both Reissner-Mindlin types of plate theories and zig-zag theories. Second, being slender in nature, a microstructural plate should accommodate a stress state that is close to the state of plane stress. This agrees with our findings that the (leading-order) mechanical units of a microstructural plate should be two-dimensional slices parameterised with the thickness coordinate, distinguishing the proposed framework from other plate theories based on asymptotic analysis. To demonstrate what the present framework can predict beyond existing theories, the effects on the behaviour of fibrous laminates due to manufacturing uncertainties are investigated.

A hybrid shell-beam element for straight thin-walled tubular structures

A novel beam model is proposed in order to consider deformations of its cross-section in the context of straight thin-walled tubes. For this purpose, the straight beam kinematics is enriched by addition of orthogonal shell-type displacement field of the tube section. Linear strain in terms of displacement is considered in conjunction with the non-linear coupling between local deformation of the section and its global rotation. Both Euler–Bernoulli and Love-Kirchhoff hypotheses for beam and shell kinematics, respectively, are adopted as well as the thin-walled assumption. First-order shear deformation for the radial variable and Fourier expansion in terms of the circumferential variable are also considered for the mid-surface displacement field. Then, the stress tensor is obtained under plane stress conditions. The virtual power principle is finally used to obtain the equations of motion satisfied by the corresponding generalized forces. Afterwards, an explicit updated Lagrangian Finite-Element approach using a lumped mass matrix is proposed for solving the tube governing equations and the stability condition of the time integration is given. Test-cases are then chosen to assess the present tube finite-element. Both static and dynamic problems are considered. First, the proposed model is compared to analytical solutions. Finally, a tube subjected to a distributed patch loading is studied. The influence of the number of Fourier modes, of warping and coupling terms is examined. The proposed model makes it possible to retrieve classical shell solution of the cross-section deformation with significant computational savings.

Microstructure‐Based Modeling and Estimation of Mechanical Properties of High‐Carbon Steel

Herein, attempts are made to estimate the mechanical properties using microstructure‐based finite element (FE) modeling and validate these results with the experimental results. The two high‐carbon steel specimens are hot‐rolled and air‐cooled to develop ferrite–pearlite microstructures. Different characterizations are utilized to observe microstructures as well as Vickers hardness and tensile tests are carried out to determine the mechanical properties. Two high‐resolution scanning electron micrographs are chosen for representative volume element‐based FE analyses for modeling the mechanical behavior of ferrite–cementite microstructure. Object‐oriented finite elements (OOF2) and Abaqus FEA 6.14 software are used to estimate the elastic and elastoplastic behavior assuming plane stress conditions. The correlation between cementite lamellae orientation and the predicted elastoplastic properties is investigated and compared with the experimental results. The influence of image size and mesh size on the predicted true stress–true strain behavior is discussed. The hard and brittle cementite lamellae face maximum stress while the softer ferrite matrix experiences maximum strain. It is found that strain accumulation is maximum at the interfaces of ferrite and cementite. These findings are further validated by the microvoid and crack initiation spots in the fracture surface and subsurface micrographs of broken tensile specimens.

Experimental determination of the fracture forming limits in metal forming

This paper revisits the determination of the fracture forming limits in metal forming with the purpose of providing readers understanding on the experimental procedures and identification of their pitfalls and sources of error. The presentation covers fracture in sheet and bulk metal forming under three different crack opening modes: by tension (mode I of fracture mechanics), by in-plane shear (mode II), and by out-of-plane shear (mode III). It is shown that the determination of the fracture forming limits must account for the specificities of sheet forming, which is fundamentally based on material flow under plane stress conditions, and of bulk forming, which essentially features material flow under three-dimensional conditions. The content of the paper is an attempt to merge previous work of the authors in the determination of the fracture forming limits, which is scattered through various publications, so that a comprehensive view of their accumulated experience in sheet and bulk forming can be provided in a single publication.

Peridynamics modeling of cellular elastomeric metamaterials: Application to wave isolation

We investigate how embedded geometrical imperfections can be designed to obtain significant reduction in wave amplitude and large band gap in periodic cellular metamaterials. A finite deformation peridynamics (PD) theory is developed to predict the static and dynamic responses of elastomeric metamaterials. Periodically structured cracks and volumetric geometrical imperfections, e.g., circular and elliptical holes, are embedded in elastomeric materials. Elastic wave propagation through these structured materials is studied by performing numerical simulations using PD. Two main mechanisms are exploited to enhance the energy absorption capacity compared to conventional metamaterials, viz. reflection of wave from crack surfaces and energy absorption due to local elastic instability of the cell walls. Slightly compressible neo-Hookean material model is considered, and plane strain condition is assumed. Nonlinear PD equations for quasi-static condition are solved using the Newton-Raphson method. The present model is validated against finite element solutions considering deformation of a solid elastomeric body subjected to tensile load. The model is also validated with experimental results considering rupture of a double notched elastomeric specimen assuming plane stress condition. PD predictions of nominal stress vs. strain for configurations with holes under quasi-static condition are validated against the experimental observations and numerical simulations through the finite element software ANSYS®. Newmark-beta method is employed to solve the PD equation of motion. At every time step, Newton-Raphson method is used, and convergence is established. The dynamic response obtained from the PD model is validated against finite element solutions using ANSYS® for a solid body. Various novel geometries combining cracks and holes are systematically developed guided by prediction of energy absorption capacity and band gap characteristics through numerical simulations. Wave propagation through these configurations is examined and the effects of microstructure topology on band gap and amplitude reduction are shown. Significant reduction of wave amplitude and large band gap demonstrate remarkable efficacy of the present proposal.

Machine learning-driven stress integration method for anisotropic plasticity in sheet metal forming

This study proposes a machine learning-based constitutive model for anisotropic plasticity in sheet metals. A fully connected deep neural network (DNN) is constructed to learn the stress integration procedure under the plane stress condition. The DNN utilizes the labeled training data for feature learning, and the respective dataset is generated numerically based on the Euler-backward method for the whole loading domains with one element simulation. The DNN is trained sufficiently to learn all the incremental loading paths of the input-output stress pair by using advanced anisotropic yield functions. Its performance with anisotropy is evaluated for the predictions of r-values and normalized yield stress ratios along 0–90 ° to the rolling direction. In addition, the trained DNN is then incorporated in user material subroutine UMAT in ABAQUS/Implicit. Thereafter, the DNN-based anisotropic constitutive model is tested with a cup drawing simulation to evaluate earing profile. The obtained earing profile is compatible with the one from the trained anisotropic yield function.

A comprehensive study on Hertzian contact stress behaviour of engineering thermoplastic gears using 3D finite element analysis

Two-dimensional Finite Element Analysis is frequently utilized to determine the behaviour of gear contact stress due to its computational time and higher efficiency. Based on the gear tooth-to-face width ratio, either plane stress or plane strain was selected in the investigations. However, the existing investigations failed to explain the above selection criteria. Thus, the present work attempted to do detailed contact stress investigations for all thermoplastic gears using 3D FEA and compared them with 2D FEA. Based on the criteria, the face widths of 5 and 20 mm were selected during the 3D analyses, replicating the plane stress and plane strain conditions of the 2D analysis. Contact stress variations with respect to face width were extracted at 10 distinct locations. For both 5 and 20 mm gears, higher contact stresses were obtained on gear edges, whereas lesser contact stresses were in the centre regions of the gear. For all the analysed thermoplastic gears, 5 and 20 mm face-width gears exhibited plane stress conditions. Subsurface and surface stress was detected during 3D and 2D analysis for both gears. For PP, the maximum contact stress occurred at the lowest depth value of 0.15–0.155 mm, and PC exhibited maximum contact stress at 0.275–0.28 mm depth. A dimensionless ratio of proportional limit – to – young’s modulus was introduced to predict the maximum contact stress depth for the thermoplastic gears. And it was observed that the ratio is directly proportional to the maximum contact stress depth value.

A constrained solid-shell model for the geometric nonlinear finite-element analysis of laminates with alternating stiff/soft layers. Applications to laminated glass

Solid-shell models are developed for the geometrically nonlinear analysis of multi-layered composite structures made of alternating layers with large difference in material properties. Exemplificative applications are presented for laminated glass, in which a number of stiff plies of glass are permanently shear-coupled by soft interlayers. The sectional warping due to significant transverse shear strains in the soft layers makes theories of laminated plates based on the plane-section hypothesis unreliable. The proposed approach is based on a geometrically exact solid-shell finite element model with one element per layer in the thickness direction, as alternative to solid discretization. The element approximation is based on the displacement nodal values at the top and bottom surfaces of the layers, with a natural C0 continuity. An alternative solid-shell model with fewer parameters is derived imposing the equal finite rotation of the stiff layers at each surface point by a local rotation-free re-parametrization of the nodal displacements and enforcing the plane stress condition. The approach permits an easy coupling with a fully solid discretization, e.g. to model connections, and is based on a simple strain measure quadratic in the displacement unknowns and suitable for finite strains. Extensive numerical examples for laminated glass plates and curved shells susceptible to large deflections and buckling are provided, comparing the results with those from a fully solid approach.

Problem of plane stress state of two-layer body in dynamic elastic-plastic formulation

Composite materials are widely used in industry and everyday life. Many different methods are used to calculate and develop composite materials. Many methods of calculation and design of such materials are successfully used. In this article, for the design of composite and reinforced materials, a technique for solving dynamic contact problems in more precise an elastic-plastic mathematical formulation is used. To consider the physical nonlinearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the nonlinear problem to a solution of the sequences of linear problems. The problem of a plane stress state (PStS) of a beam made from the composite reinforced double-layered material is being solved in dynamic elastic-plastic mathematical model. The reinforced or armed material consists of two layers: the upper (first) thin layer of solid steel and the lower (second) main layer of glass. This composite base is rigidly attached to an absolutely hard half-space. Rigid adhesion of the layers to each other is assumed. Glass is a very strong and very fragile material at the same time. The fragility of glass is due to the fact that there are many microcracks on the surface, and when a load is applied to the glass surface, these microcracks begin to grow and lead to the destruction of glass products. If we glue or immobilize the tops of microcracks on the surface, we will get a strong reinforced armed material that will be lighter, stronger and not subject to degradation of material properties such as aging, corrosion and creep. The impact process was modelled as a non-stationary plane stress state problem with an even distributed load in the contact area, which changes according to a linear law. The fields of the Odquist parameter and normal stresses were studied and compared to corresponding results of plane strain (PSS) problem with the same material of layers, same their thickness and size of the contact area. The upper reinforcing layer of metal or steel can be applied to the glass surface so that metal or steel atoms penetrate deeply, fill microcracks and bind their tops. The top layer can be quite thin.

Plane Strain and Stress States of Two-Layer Composite Reinforced Body in Dynamic Elastic-Plastic Formulation

For the design of composite and reinforced materials, a technique for solving dynamic contact problems in more precise an elastic-plastic mathematical formulation is used. To consider the physical nonlinearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the nonlinear problem to a solution of the sequences of linear problems. In contrast to the traditional plane strain, when one normal stress is equal to a certain constant value, for a more accurate description of the deformation of the sample, taking into account the possible increase in longitudinal elongation, we present this normal stress as a function that depends on the parameters that describe the bending of a prismatic body that is in a plain strain state. The problems of a plane strain and stress states of a beam made from the composite reinforced double-layer material is being solved. The reinforced or armed material consists of two layers: the upper (first) thin layer of solid steel and the lower (second) main layer of glass. Glass is a non-crystalline, often transparent amorphous solid that has widespread practical and technological use in the modern industry. Glass has high strength and is not affected by the processes of aging of the material, corrosion, and creep. In addition, this material is cheap and widely available. Glass can be strengthened, for example, in a melt quenching process. The reinforced composite beam is rigidly linked to an absolutely solid base and on which an absolutely solid impactor acts from above in the centre on a small area of initial contact.

Actual Stress Analysis of Small-Bore Butt-Welded Pipe by Complementary Use of Synchrotron X-Rays and Neutrons

Residual stresses in astresses in small-bore butt-welded pipe of austenitic stainless steel have never been measured. It is difficult to obtain a detailed residual stress map of the root welded part, because the gauge volume in neutron diffraction is large. The stress evaluation of the welded part by synchrotron X-rays was also difficult due to the dendritic structure. In this study, a double exposure method (DEM) with high-energy synchrotron X-rays was applied to measuring the details of the residual stress of the welded part, and we succeeded in obtaining the detailed axial and radial stress maps of the root welded part of the plate cut from the welded pipe, though the stress map was under the plane stress condition. The hoop stress map of the butt-welded pipe was obtained using the strain scanning method with neutrons under the triaxial stress state. The axial and radialradius stress maps under a triaxial stress state were made up using the complementary use of the synchrotron X-rays and neutrons. As a result, the detailed stress maps of the root welded part of the butt-welded pipe were obtained. The result of the actual stress analysis sufficiently explained the initiation and propagation of SCC.

Mesoscopic unit cell analysis of ductile failure under plane stress conditions

Ductile failure under plane stress conditions is analyzed at the meso-scale using periodic unit cell model simulations of void growth in a plastically deforming matrix. Equivalent strains to failure by the onset of plastic instability at the macro-scale are estimated using the loss of ellipticity criterion for the equilibrium equations. Failure loci obtained from the cell model simulations are compared with the predictions of an instability-based ductile failure model and the Hosford–Coulomb damage indicator model, under both proportional and non-proportional loading conditions. The instability-based model is shown to quantitatively predict the shape of the failure locus under proportional loading, including the presence of a cusp at uniaxial tension and a ductility minimum under plane strain tension, in the absence of heuristic adjustable parameters in the failure criterion. It is shown that the characteristic shape of the plane stress failure locus is primarily due to the Lode dependence of the failure criterion, and not the damage growth law as assumed in the damage indicator models. Under non-proportional loading involving a step change in loading direction at an intermediate strain, the instability-based model correctly predicts the non-linear variation of the failure strain as a function of the intermediate strain; unlike a linear variation predicted by the damage indicator models, which is not in agreement with the cell model simulations. Forming limit curves showing the strains to the onset of localized necking in thin sheets are also obtained from the cell model simulations using an appropriate modification of the macroscopic instability criterion.

Dynamic analysis of tapered symmetrically layered beams with interlayer slip

In the present contribution, for the first time the dynamic analysis of slender three-layer beams with interlayer slip, whose cross-section is tapered, is carried out. The cross-section of the middle layer is constant along the beam span, while the cross-section of the two outer layers, which varies along the beam span, is identical. The beam axis may have a small initial deflection due to intentional or unintentional imperfection. The layers are elastically bonded, resulting in an interlayer slip under load. The equations of motion and boundary conditions under planar transverse loading are derived, assuming the Euler-Bernoulli hypothesis for each layer separately. In numerous examples, the natural frequencies, eigenfunctions and the forced vibration response is computed, and the effect of different boundary conditions, initial deflection and variable cross-section is investigated. To validate this beam theory, a finite element (FE) analysis assuming a plane stress state is performed in parallel for all examples. It is demonstrated that the results of the proposed beam theory show excellent agreement with the much more computationally expensive FE analyses in all cases.

Computation of energy evolution during the dynamic fracture of elastomers using the finite viscoelastic model: Its implementation in Abaqus

Cracks propagating in elastomers consume the strain energy available in the body through viscoelastic effects and the creation of new fracture surfaces. Determining the energy consumed by the dissipative and the fracture processes explicitly helps understand the fracture mechanisms better and if they are rate-dependent. In literature, crack propagation experiments are usually performed on thin sheets of elastomers. The displacement fields in the body as the crack propagates through it during the experiments can be accessed using the Digital Image Correlation (DIC) technique. These displacement fields can then be used as boundary conditions in finite element analysis to compute the energy evolution in the body. Using an appropriate material model to describe the bulk material, the strain energy and the viscoelastic dissipation can be computed, thereby allowing to study the energy evolution in the body as the crack passes through.In the current study, the behavior of the elastomer material through which the crack passes has been described by the finite viscoelastic (FV) model. In such models, the viscoelastic effects are described by using some internal variables, whose evolution is prescribed through certain constitutive equations. This article first discusses the implementation of the FV model for plane stress conditions and applies it to study the dynamic fracture of elastomer membranes. The crack propagation through an elastomer is simulated by imposing the displacement fields extracted from the experiments along the crack faces onto the finite element model (thereby implicitly imposing the crack speed). The energy evolution in the body is studied to compute the energy release rate and the dissipation in the body as a consequence of viscoelastic effects. A significant portion of the energy has been observed to be consumed as the viscoelastic dissipation in the bulk material as the crack propagated.

On anisotropic tensile mechanical behavior of Al–Cu–Li AA2198 alloy under different ageing conditions

The anisotropic mechanical behavior of Al–Cu–Li (AA2198) alloy under different artificial ageing conditions as well as for different thicknesses of the material was examined in the present investigation. Material characterization was performed for three different sheet directions, namely, longitudinal (0°), diagonal (45°) and transverse (90°) to the rolling direction. The results showed that the grain structure did not have significant changes by applying different artificial ageing times. In T3 heat treatment condition, only δ′ (Al3Li) phase was observed, while with increasing the ageing time, the major precipitates were T1 (Al2CuLi) phases. The latter was found to increase in size with artificial aging time. Almost similar yield strength values were observed for all investigated thicknesses, while the higher thickness specimens showed higher elongation at fracture values. Anisotropy was slightly higher at T3 condition, while the lowest anisotropy degree was noticed at the peak-aged condition. The specimens extracted in the diagonal direction showed high variation in mechanical properties, when compared with the respective results at the other two investigated directions, independently from the thickness of the specimens as well as from heat treatment condition. The smaller thickness specimens presented lower elongation at fracture values and for all investigated sheet directions, since they undergo plane stress conditions. Analysis of work hardening behavior showed that Bauschinger effect, which was not observed in T3 condition, became increasingly more prominent with artificial ageing time in all directions investigated.

Reconstructing the Spatial Distribution of the Relative Shear Modulus in Quasi-static Ultrasound Elastography: Plane Stress Analysis.

Quasi-static ultrasound elastography (QSUE) is an imaging technique that mainly provides axial strain maps of tissues when the latter are subjected to compression. In this article, a method for reconstructing the relative shear modulus distribution within a linear elastic and isotropic medium, in QSUE, is introduced. More specifically, the plane stress inverse problem is considered. The proposed method is based on the variational formulation of the equilibrium equations and on the choice of adapted discretization spaces, and only requires displacement fields in the analyzed media to be determined. Results from plane stress and 3-D numerical simulations, as well as from phantom experiments, showed that the method is able to reconstruct the different regions within a medium, with shear modulus contrasts that unambiguously reveal whether inclusions are stiffer or softer than the surrounding material. More specifically, for the plane stress simulations, inclusion-to-background modulus ratios were found to be very accurately estimated, with an error lower than 3%. For the 3-D simulations, for which the plane stress conditions are no longer satisfied, these ratios were, as expected, less accurate, with an error that remained lower than 10% for two of the three cases analyzed but was around 34% for the last case. Concerning the phantom experiments, a comparison with a shear wave elastography technique from a clinical ultrasound scanner was also made. Overall, the inclusion-to-background shear modulus ratios obtained with our approach were found to be closer to those given by the phantom manufacturer than the ratios provided by the clinical system.