Plane Stress Conditions Research Articles

Exceptional cryogenic-to-ambient impact toughness of a low carbon micro-alloyed steel with a multi-heterogeneous structure

A low-carbon micro-alloyed (LCMA) steel with a body-centered cubic (bcc) crystal structure suitable for extremely low temperatures was developed by overcoming the intrinsic ductile-to-brittle transition in bcc alloys at cryogenic temperatures. The excellent cryogenic-to-ambient impact toughness in the LCMA rolled plate results from its heterogeneous microstructure, which gradually changes from bamboo-like ultrafine grains (∼ 1.1 μm) on the surface to relatively equiaxed coarse grains in the core (∼ 3.4 μm), accompanied by a distinct texture gradient variation. The heterostructured LCMA steel displays a cryogenic impact toughness of ∼200 J/cm2 at 77 K, which is 24 times higher than the coarse-grained LCMA steel. Such high impact toughness of heterostructured LCMA arises from the coordinated deformation mechanisms over different length-scales coupled with delamination toughening. At 77 K, the heterostructured steel plate deforms by forming cellular sub-structures at the core to the surface, which refines the microstructure and promotes hetero-deformation induced (HDI) hardening to improve intrinsic toughening. Moreover, the subsequent delamination process induces extrinsic toughening by shielding and blunting the cracks, with the local plane-stress conditions induced by delamination promoting ductile fracture of the coarse grains in the core regions. This low alloy steel with its heterogeneous microstructure exhibits extraordinary impact toughness at cryogenic temperatures highlights the possibility of materials design strategies for sustainable development.

Deformation state of a graphene sheet within the framework of the continuum moment-membrane theory of elasticity

The paper proposes an approach to finding the stress-strain state (SSS) of structures containing graphene, a novel nanomaterial that has currently found а wide range of practical applications in nanoelectromechanical systems. Graphene is a 2D basic building block for other carbon structures such as membranes, sheets, nanotubes, etc. To describe the SSS of a graphene sheet, the phenomenological continuum moment-membrane theory of plates is used, from which, due to the fact that graphene is an ultrathin material, the concept of thickness is excluded. The physical elasticity relationships of a graphene sheet are expressed through its rigidity characteristics, determined using the harmonic potential of interatomic interactions in carbon. A differential formulation and the corresponding variational formulation are given for the problem of static deformation and determination of the natural frequencies and modes of vibration of a graphene sheet. The variational formulation is based on the Lagrange principle and is implemented numerically using the finite element method. Finite element relations are constructed taking into account moment effects of the behavior of a graphene sheet. For approximation, a 4-node rectangular finite element is used. Numerical solutions to several problems of static deformation of a graphene sheet under conditions of a plane stress state and transverse bending are presented, and the analysis of its natural vibrations is also performed. Good convergence of numerical simulation results in all considered problems is demonstrated. The obtained numerical solutions are essential in designing and calculating resonators in which ultrathin nanostructures are used. The establishment of the fact that a graphene sheet has a high intrinsic frequency falling in the GHz region (for example, quartz resonators are characterized by megahertz frequencies) opens up new prospects for using graphene itself as an ultrasensitive nanomechanical resonator for detecting small masses and ultrasmall displacements.

Characterisation and damage modelling of oxide/oxide composites considering temperature dependence

Continuous fibre reinforced ceramic matrix composites (CMCs) are increasingly being employed in safety critical applications. As a result, their use in these applications requires the adoption of appropriate design methodologies that are able to consider their complex non-linear mechanical behaviour. In this paper, a simple but robust formulation for continuum damage model, thermodynamically consistent and written under plane stress condition, is proposed for a laminated 2D woven composite material. This model is implemented via the Classical Laminate Theory (CLT) extended to non-linear behaviour to predict the behaviour of different stacking sequences. In-plane tensile tests were performed on an oxide/oxide composite up to 1000 °C to fully identify the model. The procedure is described and used on tests at room temperature before being extended higher temperature tests. Finally, the law and its coupling with the CLT are validated by means of tests on quasi-isotropic and cross-ply laminates at different temperatures.

Plane-Stress Deformation Behavior of CoCrFeMnNi High-Entropy Alloy Sheet under Low Temperatures.

High-entropy alloys are promising candidates expected to be applied in transportation equipment serving in extreme environments due to their excellent properties. CoCrFeMnNi high-entropy alloy is a typical representative of them, and its low temperature performance is excellent. In this study, to evaluate the feasibility of forming HEA shells, the deformation behavior of CoCrFeMnNi under a plane-stress state at lower temperatures was thoroughly studied. Firstly, a thin-walled HEA tube was fabricated using hot extrusion and further formed into a thin shell for uniaxial tensile and biaxial bulging tests. Subsequently, uniaxial tensile tests at cryogenic temperatures were conducted. Both the strength and the ductility improves as the temperature decreases from -160 °C to -196 °C. Then, a systematic low-temperature bulging test was performed using isothermal dome tests and the thickness uniformity analysis of the bulged specimens was carried out. In addition, grain microstructural observation using EBSD was characterized analyze the possible deformation mechanism at the cryogenic temperature under the biaxial stress state. This study, for the first time, investigated the biaxial deformation behavior of HEA. Considering the plane-stress state deformation is the dominant type in the thin-walled shell deformation, this study enables us to provide direct guidance for various sheet-forming processes of HEAs.

Investigation of the effect of anisotropy on Generalized Forming Limit Diagram

In contemporary industrial practices, various methods are employed to subject raw metal sheets to deformation in order to fabricate requisite components. These sheets exhibit a defined capacity for deformation during the forming process. Over recent decades, a plethora of experimental and numerical methodologies have emerged to ascertain these forming limits. Initially, a forming limit diagram (FLD) was devised, predicated on the phenomenon of necking, under the assumption that forming takes place under plane-stress conditions. However, in certain complex processes like hydroforming and incremental forming, necking can manifest at sites where normal and through-thickness shear stresses act upon the sheet in addition to the in-plane stresses, rendering the plane-stress assumption inadequate for predicting forming limits in such scenarios. Thus, it becomes imperative to derive a diagram that can accurately forecast forming limits in these processes. This study aims to establish a Generalized Forming Limit Diagram (GFLD) through numerical means. GFLDs were constructed utilizing two distinct yield functions, namely Von-Mises and Hill48, for isotropic and anisotropic states, respectively. The findings reveal that normal compressive stress and through-thickness shear strain augment the formability of sheet metals. Furthermore, the outcomes illustrate that accounting for anisotropy introduces variances between diagrams in some regions of the FLD curve while the discrepancies are minor within the central regions.

Practical fracture limit function considering data flexibility under plane-stress conditions in Lagrangian formulation

Ductile fracture limits have been extensively studied from the perspectives of material science and mechanics. While these studies have provided scientifically significant insights into ductile fracture limits, the influence of the evolution of anisotropy observed in actual experimental data suggests a need to account for diversity in the experimental data. To address this, it may be possible to further develop the model by investigating more influences from the field of materials science. However, from an engineering perspective, it can be effective to represent the diversity of acquired experimental data as practical functions for use, rather than solely focusing on the material science aspect. This paper presents mathematical formulations representing the strain- and stress-based fracture limit surfaces of ductile materials, to consider diverse data set for the evolution of anisotropy, under plane-stress conditions within the framework of the Lagrangian formulation. The model introduces a strain-based fracture limit surface (FLS), which has an additional anisotropic profile function based on the commonly used two-dimensional strain-based fracture limit curve (FLC). The anisotropic profile function is capable of capturing the evolving anisotropic profile based on the deformation mode and principal loading direction to represent more flexible shapes. With this function, a three-dimensional strain-based FLS, which can present the variation in fracture strains in the space of the deformation mode and principal loading direction, can be constructed. Then, the proposed model transformed this strain-based FLS into a stress-based FLS by converting the presented state vector in the former into the corresponding vector in the latter in the Lagrangian formulation. To achieve this, this study developed a formulation that considers the nonlinear evolution of stress anisotropy independent of the strain anisotropy during stress evolution. The two proposed models were validated by experimental results on a dual-phase steel. These models provide flexible fracture limit functions for ductile materials under varying stress and strain conditions in engineering applications.

Analysis of finite creep strains in aluminum plates at static and cyclic loading

The formulation of the problem and the basis of the method for solving two-dimensional creep problems for the case of finite strains are presented. The method is based on the Arbitrary Lagrange-Euler (ALE) approach in numerical modeling using FEM. The method is implemented in the form of additional units to the creep calculation software. The plane stress state is considered. The results of an experimental study of the creep of AlMg6 alloy specimens under static and cyclic loading are described. The obtained creep curves were used to verify the calculation method. It is shown that for a material with viscous creep properties, calculations using finite strains quite satisfactorily describe the tertiary, accelerated section of the creep curve. The creep of square plates in tension by static and cyclically varying traction was considered, data on their shape change and stress redistribution were obtained. A significant intensification of creep due to the addition of a cyclic component to the static loading was established.

Initiation and growth of fatigue cracks in sheets with U-shaped notches in the first and mixed modes of fracture

Initiation and growth of fatigue cracks in sheets with U-shaped notches in the first and mixed modes of fracture

Improving the performance of damage repair in thin-walled structures with analytical data and machine learning algorithms

In the last four decades, bonded composite repair has proven to be an effective method for addressing crack damage propagation. On the other hand, machine learning (ML) has made it possible to employ a variety of approaches for mechanical and aerospace problems and such significant approach is the repair mechanism and hence ML algorithms used to enhance in the present work. The current work investigates the effect of the single-sided composite patch bonded on a thin plate under plane stress conditions. An analytical model was formulated for a single-sided composite patch repair using linear elastic fracture mechanics and Rose's analytical modelling. From the analytical model, the stress intensity factors (SIF) were calculated by varying all possible parameters of the model. Next, ML algorithms were selected, and comparative studies were conducted for the best possible performance and to identify the parametric effects on optimum SIF. Also, the analytical model is validated with existing work, and it shows good agreement with less than 10% error. This study is particularly important for designing the single-sided composite patch repair method based on analytical modelling. Also, it is important to compare ML algorithms with analytical solutions in regression applications.

ВИНИКНЕННЯ І РОЗВИТОК ТРІЩИНОУТВОРЕННЯ В ГІРСЬКИХ ПОРОДАХ ПРИ СТАТИЧНОМУ НАВАНТАЖЕННІ

The paper examines the occurrence and development of cracks in rocks under static loads, various conditions and factors of crack initiation and development. For comparison, such rocks as concrete, granite, sandstone, plasterboard are taken. This is said to help the cracks in the rock spread in the right direction. The technology of crack formation with the help of insert plates under the action of non-explosive mixtures is considered. When modeling, the minor and major axes of the ellipse around the insert of the plate with holes are taken. Construct isolines of rock destruction at a point in the form of ellipses. Determine the most likely direction of cracks during rock destruction. The conditions for the development of cleavage cracks relative to the axes of anisotropy with maximum and minimum strength are considered. The theoretical and experimental justifications of the rational technological parameters of mining stone blocks by the hole method by the formation of a separation crack in the rock in a given direction have been studied. The determination of the plane stress state around a hole with a 5- and 8-mm-thick insert plate is studied, and equivalent stress diagrams are given. The methods of directional destruction and the occurrence of cracks in rocks when pouring non-explosive mixtures into wells and inserting cartridges with a destructive substance into wells are considered. It clearly shows the greater efficiency of using cartridges that guarantee the appearance of cracks in a precisely specified direction. The influence of the load scheme on the development of a splitting crack in holes with stress concentrators is presented. The model for the analysis of crack formation at the tops of stress concentrators is considered. Non-explosive methods of directional destruction of monoliths using special structures are described. Considered patents for static rock splitting. The results of destruction over a certain time are given. In addition, non-explosive mining of gabbroid rock blocks using destructive mixtures together with surface-active substances is considered.

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

Effect of fiber orientation-based composite lamina on mitigation of stress intensity factor for a repaired plate: a finite element study

The bonded composite repair has proven to be an effective method for addressing crack damage propagation. Numerous studies have employed experimental and simulation techniques to demonstrate the repair performance through the composites. These studies have explored various parameters related to bonded composites, such as size and properties, to enhance repair effectiveness. However, one aspect that has not been thoroughly investigated is the impact of fiber orientation within the composites. Therefore, the current work investigates the effect of the fiber direction of the composite patch bonded on a thin plate under plane stress conditions. Three types of fiber orientation of composite patch have been considered. In this investigation, the finite element method was used to determine the stress intensity factor using the ANSYS commercial code. The research findings showed that the fiber direction influenced the mitigation of stress intensity factor. This study is particularly important for designing the composite patch based on the fiber direction.

A parametric study on the elastic limit stresses of rotating variable thickness orthotropic disk

This paper investigates the stress fields at the onset of plastic yield of variable thickness rotating orthotropic disk, which is rigidly fixed on an inclusion. In the analytical modeling of the problem, two different analytical solution methods have been displayed where small deformations have been considered with the application of plane stress conditions. Well-known power law is considered for the disk's thickness variation, and Hill’s yield criterion is applied to obtain the elastic limits. Four parameters have been utilized while analyzing the limit fields: geometric parameter to manipulate the disk thickness, orthotropy parameter from the ratio between Young's modulus in radial and tangential directions, and two parameters owing to the applied yield criteria. The effects of these parameters on the limit fields have been comprehensively examined in the numerical examples, and possible outcomes have been discussed. Additionally, using Autodesk Inventor Nastran, finite element solution of the disk is generated, analytical and numerical results have been compared, and consequently, closely matching results have been achieved.

Subduction plate interface shear stress associated with rapid subduction at deep slow earthquake depths: example from the Sanbagawa belt, southwestern Japan

Abstract. Maximum shear stress along an active deformation zone marking the subduction plate interface is important for understanding earthquake phenomena and is an important input parameter in subduction zone thermomechanical modeling. However, such maximum shear stress is difficult to measure directly at depths more than a few kilometers and is generally estimated by simulation using a range of input parameters with large associated uncertainties. In addition, estimated values generally represent maximum shear stress conditions over short observation timescales, which may not be directly applicable to long-timescale subduction zone modeling. Rocks originally located deep in subduction zones can record information about deformation processes, including maximum shear stress conditions, occurring in regions that cannot be directly accessed. The estimated maximum shear stress is likely to be representative of maximum shear stress experienced over geological timescales and be suitable to use in subduction zone modeling over timescales of millions to tens of millions of years. In this study, we estimated maximum shear stress along a subduction plate interface by using samples from the Sanbagawa metamorphic belt of southwestern (SW) Japan, in which slivers of mantle-wedge-derived serpentinite are widely distributed and in direct contact with metasedimentary rocks derived from the subducted oceanic plate. These areas can be related to the zone of active deformation along the subduction plate interface. To obtain estimates of maximum shear stress at the subduction interface, we focused on the microstructure of quartz-rich metamorphic rocks – quartz is the main component of the rocks we collected and its deformation stress is assumed to be roughly representative of the stress experienced by the surrounding rock and plate interface deformation zone. Maximum shear stress was calculated by applying deformation temperatures estimated by the crystallographic orientation of quartz (the quartz c-axis fabric opening-angle thermometer) and the apparent grain size of dynamically recrystallized quartz in a thin section to an appropriate piezometer. Combined with information on sample deformation depth, estimated from the P–T (pressure–temperature) path and deformation temperatures, it is suggested that there was nearly constant maximum shear stress of 15–41 MPa in the depth range of about 15–30 km, assuming plane stress conditions even when uncertainties related to the measurement direction of thin section and piezometer differences are included. The Sanbagawa belt formed in a warm subduction zone. Deep slow earthquakes are commonly observed in modern-day warm subduction zones such as SW Japan, which has a similar thermal structure to the Sanbagawa belt. In addition, deep slow earthquakes are commonly observed to be concentrated in a domain under the shallow part of the mantle wedge. Samples showed the depth conditions near the mantle wedge, suggesting that these samples were formed in a region with features similar to the deep slow earthquake domain. Estimated maximum shear stress may not only be useful for long-timescale subduction zone modeling but also represent the initial conditions from which slow earthquakes in the same domain nucleated.

Plane Stress Fracture Toughness Testing of Freestanding Ultra-Thin Nanocrystalline Gold Films on Water Surface.

Fracture toughness, which is the resistance of a material to crack propagation, is a critical material property for ensuring the mechanical reliability of damage-tolerant design. Recently, damage-tolerant design is introduced to flexible electronics by adopting micro-cracked ultra-thin nanocrystalline (NC) gold films as stretchable electrodes in a plane stress state. However, experimental investigation of the plane stress fracture toughness of those films remains challenging due to the intrinsic fragility from their sub-100nm thicknesses. Here, a quantitative method for systematically evaluating the plane stress fracture toughness of freestanding ultra-thin NC gold film on water surface platform is presented. After effectively fabricating single-edge-notched-tension samples with femtosecond laser, mode I stress intensity factors are measured in the plane stress state on water surface. Moreover, investigation regarding the effect of notch length, notch sharpness, and notch tip plasticity validates this method based on linear elastic fracture mechanics theory. As a demonstration, the thickness-dependent plane stress fracture toughness of ultra-thin NC gold films is qualitatively unveiled. It is revealed that the thickness confinement effect on grain boundary sliding induces a transition in fracture behavior. This method is expected to further clarify the fracture-related properties of various ultra-thin films for next-generation electronics.

Fatigue Crack Growth on Several Materials under Single-Spike Overloads and Aircraft Spectra during Constraint-Loss Behavior

Abstract The phenomenon of flat-to-slant crack growth has been studied by many in the fracture mechanics community. At low stress-intensity factors, a fatigue-crack surface is flat (tensile mode) and the crack-front region is under plane-strain conditions (high constraint). As the crack grows with higher stress-intensity factors, a 45° shear lip occurs through the thickness of the sheet or plate. This behavior is the shear mode, which is under low constraint or plane-stress conditions. In 1966, Schijve found that the transition from flat-to-slant crack growth on a 2024-T3 Alclad aluminum alloy over a wide range in stress ratios (R) occurred at a “constant” crack-growth rate. Also, Newman and Hudson showed the same behavior on 7075-Temper-6 and Ti–8Al–1Mo–1V alloys, validating Schijve’s observation that crack-growth rate was the key parameter for flat-to-slant crack-growth behavior. The materials considered herein are 2024-T3, 7075-T6, and 9310 steel. Crack-growth behavior during single-spike overloads and simulated aircraft spectrum loading are presented. The fatigue structural analysis (FASTRAN) crack-closure based life-prediction code was used to correlate the constant-amplitude (CA) crack-growth-rate data over a wide range in stress ratios (R = Smin/Smax) and rates from threshold to near fracture, and to calculate or predict the crack-growth behavior on single-spike overload tests. Crack-closure behavior is strongly dependent upon the level of constraint. The main objective was to see if the constraint-loss region is the primary reason for crack-growth delays after single-spike overloads. Also, crack-growth analyses are presented on tests that were conducted by Wanhill on 2024-T3 Alclad aluminum alloy under the transport wing standard (TWIST) spectrum. Crack-growth analyses using crack-closure theory without constraint loss was “unable” to predict crack growth under spike overloads or simulated aircraft spectra. However, predicted crack length against cycles with constraint-loss behavior compared reasonably well with all tests.

The Influence of Reverse Yielding on the Plastic Conditioning of Interference Fits in Power Transmission Engineering

Interference fits are very common shaft–hub connections due to their low manufacturing costs and excellent technical properties. The Plastic Conditioning of this machine element is a new and not very well-known method. During the development of this method, it was discovered that Reverse Yielding occurs in certain applications and has a negative impact on the result. This paper examines the effects of Reverse Yielding on the technology of Plastic Conditioning of interference fits in Power Transmission Engineering. Based on the Shear Stress Hypothesis (SH), the Plane Stress State (PSS), and the ideal plastic behavior of materials, established stress–mechanical relationships are used to find the influencing parameters of Reverse Yielding on the technology of Plastic Conditioning and their limits. As a result, a new computational concept is developed that allows the user to maximize Plastic Conditioning while avoiding Reverse Yielding. Analytical calculation suggestions and diagrams for practical application are provided. Furthermore, the deviations in the obtained results, taking into account other material models such as the Von Mises Yield Criterion (VMYC) and material hardening, as well as the Bauschinger effect, are examined in comparison with our own numerical results from the development of Plastic Conditioning, and the resulting need for further research is defined. In addition, the method of Plastic Conditioning of interference fits is introduced and its basic principles are briefly explained.

A Label-Free Measurement Method for Plane Stress States in Optical Isotropic Films with Spectroscopic Ellipsometry

A Label-Free Measurement Method for Plane Stress States in Optical Isotropic Films with Spectroscopic Ellipsometry

Unified theoretical solutions for describing the crack-tip stress fields of mode-II cracked specimens under the fully plastic condition

Potential novel singular solutions to the crack-tip stress fields beyond the asymptotic solution frame are discussed in this study. It is believed that one of them is not only suitable for the crack-tip stress fields of mode-I cracked specimens but also for those of mode-II cracked specimens. The crack-tip stress fields of mode-II cracked specimens are investigated based on the singular solution and the stress factor of mode-II cracked specimen derived from energy density equivalence principle. Further, special functions for describing the coefficients of normalized stress distributions of mode-II cracked specimens are proposed. The method to determine the parameters in the theoretical solutions are introduced. To give an example, one set of parameters for compact shear specimen under plane-strain conditions are determined by the finite element analysis of a few typical materials and geometries conditions. Finally, comparisons of radial and circumferential stress distributions and contour lines of equivalent stress are completed for the compact shear specimens and Arcan-shaped shear specimens of power-law plastic materials under plane-strain and plane-stress conditions. And the results show that the applicability and reliability of the proposed unified theoretical solutions are verified.

Presentation of a new 2D fast and straightforward version for the Lemaitre’s ductile damage model

Damage is an undesirable and progressive process that ultimately leads to complete failure of materials. Obtaining a robust tool to precisely and reliably predict damage evolution behavior in ductile sheet metals is among the most crucial challenges for researchers and engineers. In this paper, first, based on the continuum damage mechanics (CDM) and the original Lemaitre’s ductile damage model, a new 2D fast and straightforward version specialized for the plane stress conditions as well as a novel numerical integration algorithm are fully presented. Additionally, a modified numerical approach, founded on the bisection method is suggested for determining the material-dependent ductile damage parameter of the model. Then, an explicit subroutine is developed and implemented to numerically predict damage evolution behavior in different ductile sheet metals and shapes. Finally, the numerical simulation results of damage initiation, growth, and crack onset are attained and validated. The numerical simulation predicted results reveal that the novel offered version of the Lemaitre’s ductile damage model in conjunction with the enhanced numerical approach is able to more rapidly predict the material-dependent ductile damage parameter and also the evolution of damage in ductile sheet metals under various loading and boundary conditions.