Elastic Stress Field Research Articles

Influence of Wear Profile Geometry on Critical Plane Fatigue Crack Initiation Criteria in Plane and Axisymmetric Elastic Fretting Contacts

The subsurface elastic stress fields in plane and axisymmetric contacts with friction under oscillating tangential loading are calculated via a very robust, high-precision method, which operates with appropriate superpositions of analytic solutions for the respective Hertzian contact problems. Based on the stress fields, two critical plane fatigue crack initiation criteria—the Smith-Watson-Topper (SWT) parameter and the Findley parameter—are evaluated for three types of contact profile geometries: (unworn) parabolic contact, the partial slip limiting wear profile of an initially parabolic contact, and truncated parabolic contact. Appropriate scaling laws are introduced to formulate a general solution in terms of non-dimensional variables. The crack initiation criteria are determined in the full subsurface loading plane. It is found that the truncated profile—which may originate from sliding wear—has practically the same local distribution of crack initiation criteria as the unworn profile, despite the (weak) stress singularity at the edge of the flat face. The partial slip limiting wear profile, on the other hand, exhibits a strong edge stress singularity at the boundary of the permanent stick zone, the crack initiation criteria are drastically increased (and theoretically infinite). Also, while for the unworn and truncated profiles high values of the crack initiation criteria are extremely localized around “hotspots” at the surface, for the partial slip limiting wear profile they reach much deeper into the subsurface material. This offers a new explanation for the dominance of fatigue failure in the partial slip regime of fretting. The differences between plane and axisymmetric cases are generally small. The SWT parameter is generally more localized than the Findley parameter.

Macroscale stress induced stabilization of coherent precipitates

In systems with 'moderate' misfit, on nucleation and in the initial stages of the growth, precipitate crystals may be coherent with the matrix. This reduces the interfacial chemical energy; but a higher cost is paid in terms of the energy stored in the long range strain fields. On the growth beyond a critical size (r*), the energy of the system is minimized via the formation of interfacial misfit dislocation loops. It has been shown that the coherent state is stable in two separate size regimes of the precipitate in finite bodies (i.e. shows a re-entrant interface character transition) and further the coherent state can be fully stabilized below a critical size of the domain (R*). The presence of stress fields in the material, either due to internal sources or due to externally applied loads, is expected to alter the magnitude of the r*. In the current work, finite element simulations are used to study the effect of elastic stress fields on r*. The Cu-2wt.%γFe alloy is used as a model system towards this end. A map is generated demarcating the regions of stability of the coherent and semi-coherent states; with normalized r*, domain size (R) and elastic stress (σyy) as axes. We conclude the following. (i) Stresses of the order of GPa leads to a change in r*. (ii) The re-entrant interface character transition to the coherent state is suppressed by tensile stress. (iii) The coherent state is stabilized due to compressive stress. (iv) Compressive stress expands the phase field of the coherent state, while tensile stress has the opposite effect. (v) In 'very small' domains the coherent state can be stabilized in spite of being subjected to tensile stress.

Elastic and plastic fields induced by a screw dislocation in a nanowire within Mindlin’s second strain gradient theory

Due to the wide applications of nanowires as a main component of nano-scale electronic devices, a lot of attentions have thus far been paid to study their relevant problems. Knowing that the presence of dislocations in nanowires affects their mechanical properties, this paper is dedicated to the determination of analytical expressions for the elastic displacement, elastic strains, and plastic distortion developed in a nanowire due to a screw dislocation, as well as the image force acting on the dislocation core, within the framework of Mindlin’s second strain gradient theory. Expecting that the size effect plays an important role in such a problem, this theory seems an appropriate continuum approach for its addressing. Furthermore, utilizing the adopted theory leads to the removal of all classical singularities in the elastic displacement, strain, stress, and plastic distortion fields, as well as the image force. It is, in addition, seen that the elastic double- and elastic triple-strains are regularized. Furthermore, the results evidently illustrate the size effect, i.e. the effect of the ratio between the radius of the nanowire and the characteristic lengths of its constituent material.

Combined notch and size effect modeling of metallic materials for LCF using plasticity reformulated critical distance theory

Fatigue assessment of notched parts plays imperative performance in structural integrity and reliability of major equipment. The theory of critical distance (TCD) involves determining the fatigue strength of notched parts from the linear elastic stress field surrounding the notch apex. In addition, the fatigue strength of engineering part is also size dependent. Accordingly, by coupling a plasticity reformulated TCD with the weakest link theory, this study elaborates a novel scheme for probabilistic low cycle fatigue (LCF) analysis under notch and size effects. Results of fatigue tests and complementary FE analysis with the proposed model is utilized to compute highly stressed volume (HSV). Furthermore, test data of titanium alloys TC4 (Ti-6Al-4V) and TC11 (Ti-6.5Al-3.5Mo-1.5Zr-0.3Si) as well as aluminum alloy 7075-T6 are utilized for model verification. Results specify that the model predicted P–S–N curves are in good agreement with experimental data.

When do corners behave like cracks in lap-shear geometries?

A cohesive-zone analysis is used to explore the question of whether a crack is needed to determine the toughness of an interface using a lap-shear geometry. The analysis shows that, if the cohesive-length scale is large enough, the work done against tractions at the corner of a bonded ligament is identical to what would be given by the J-integral for the same geometry with a crack at the interface. This means that the lap-shear strength is insensitive to the details of the corner under the appropriate conditions. From a practical perspective, this means that one may not have to introduce a flaw into the interface when using a lap-shear geometry to measure the toughness of an interface. Furthermore, it has been shown that, for the special case of a symmetrical lap-shear geometry and a large cohesive-length scale, the partition of the work done against the corner tractions into shear and normal components is independent of geometry. This is probably a general result, although the large-scale phase angle for a crack is different from the LEFM value.If the cohesive-length scale is small, the details of the corner become very important. The corner work depends on both the corner angle and the cohesive-length scale. A dimensional argument suggests that the relationship between the corner work and the cohesive length depends on the singularity of the elastic stress field. The concept of a critical stress-intensity factor modified for corners, Kc, is valid, but it must incorporate the cohesive length of the interface to relate it to the material property of toughness. However, Kc cannot be considered to be an interface property in itself, because it is linked to the toughness by the geometry of the corner. In the special case of a crack, Kc is independent of the cohesive-length scale. Conversely, in the continuum limit, where the cohesive length goes to zero, Kc→∞ for all geometries other than a crack.

Combined approach for fatigue crack characterisation in metals

In this work a comprehensive characterisation of the fatigue crack growth of an Al 2024 alloy is presented. A powerful Christopher-James-Patterson (CJP) model is used for multi-parameter characterisation of the crack state at a few different locations through the fatigue crack growth. The CJP model of crack tip displacements and stress fields was proposed in order to better capture the influences on the applied elastic stress field of the plastic enclave that is generated around a growing fatigue crack. The model does this through a set of elastic stresses applied at a notional elastic-plastic boundary, and it has been shown to accurately model plastic zone shape and size, whilst its ability to predict the effective range of stress intensity factor during a fatigue cycle has been verified. Thus, the model can predict the driving force component of the crack and additional components that account for different shielding mechanisms. CJP model is combined with full-field non-contact Digital Image Correlation (DIC) so that the model can be fed at any moment with real experimental information. The results are then correlated with Paris law data for Al 2024 alloy through Scanning Electron Microscopy (SEM) observations of the fracture surface. In this work the effect of crack length and load level are thoroughly studied and validated with fatigue crack growth rate measurements.

FEATURES OF MECHANICALLY STIMULATED CHANGES IN THE ELECTRICAL CONDUCTIVITY OF P-Si CRYSTALS AFTER THE ACTION OF A MAGNETIC FIELD

The research investigated mechanically stimulated changes in resistance ( R/R 0 = f(σ) ) of p -type conductivity single-crystal silicon samples that underwent long-term (~ 600 days) magnetic treatment. The permanent magnetic field induction was В = 0.354 Т . It was established that the action of the magnetic field leads to the appearance of a characteristic maximum on the curves R/R 0 = f(σ) at the initial stage of uniaxial elastic deformation. A similar effect was observed after small doses of X-irradiation ( D = 312 Gy ). It was found that the position and magnitude of the characteristic maximum on the R/R 0 = f(σ) curves significantly depend on the time that has passed since the previous cycle of deformation (compression-decompression). In particular, it is shown that the characteristic maximum is not observed if this time does not exceed 20 hours. It was also found that with an increase in time (which has passed since the previous deformation cycle), the value of the maximum increases, and its position shifts towards higher mechanical stresses σ . At the same time, the amount of residual resistance at the beginning of the next deformation cycle also decreases. So, it can be seen that the relaxation processes, which are accompanied by a change in the value of the residual resistance of the crystal, do not have a clear time periodicity, but there is a clear tendency to its increase in value. The change of the defective background in the crystal under the action of the load and the magnetic field affects the concentration and mobility of the charge carriers, which are related to the electrical conductivity of the semiconductor. A characteristic feature of dislocations in silicon crystals is the presence around them of areas with an increased concentration of point defects (Cottrell clouds). It has been confirmed that the long-term effect of a constant magnetic field on experimental p -Si crystals leads to the disintegration of defects, such as hydrogen-containing and oxygen-containing complexes (Si–O–Si, Si–H 2 , O–Si–O, Si–O–C, Si –CH 3 , H–OH, H 2 O, Si–OH, etc.). As a result of such decay, the formed hydrogen can migrate in the elastic stress fields of the near-surface layer, passivating the acceptor bonds. This, accordingly, leads to a decrease in the surface electrical conductivity of the experimental samples. Key words : silicon, magnetic field, dislocations, uniaxial elastic deformation.

Оценка вкладов различных механизмов в сопротивление деформированию монокристаллов сплава Cu–12 ат.% Al

The evolution of the structure in the process of plastic deformation by compression of Cu – 12 at.% Al single crystals is investigated. Data on the stress-strain curve, deformation stages and the relationship of stages with the types of forming substructures are presented. Quantitative measurements of various parameters characterizing the substructures were carried out. The results are presented in the form of a diagram of the dependence of the fraction of substructures on the density of dislocations. Dependences of the distances between non-reactive intersections of dislocations, inter-dislocation reactions, obstacles of different types, and dislocation lines on the degree of strain are shown. Particular attention is paid to the quantitative assessment of the contributions to the flow stress from local obstacles and long-range elastic stress fields, as well as the contributions to strain associated with the development of slip and twinning.

A new idea of modeling shear band in metallic glass based on the concept of distributed dislocation

Shear bands are closely linked with the plasticity and fracture behaviors of metallic glasses (MGs). This work proposes a new idea to predict shear bands by continuously distributed dislocations. The dislocations are unreal and used to model the plastic deformation of shear band. The possible positions of shear band initiation are determined based on the elastic stress field, and the direction and length of shear band propagation are determined by the distributed dislocation technique. Finite element simulations based on constitutive model are carried out to compare with the theoretical modeling. Two examples are considered, i.e., shear bands near a void and a notch under tensile loading. The results show that the theoretically predicted shear band morphology is well consistent with the finite element simulations, which verifies the validation of using distributed dislocations to predict shear bands. This work provides a new way to model shear band, and it has potential applications in predicting shear band morphology and fracture behaviors in MGs.

Deep learning prediction of stress fields in additively manufactured metals with intricate defect networks

In context of the universal presence of defects in additively manufactured (AM) metals, efficient computational tools are required to rapidly screen AM microstructures for mechanical integrity. To this end, a deep learning approach is used to predict the elastic stress fields in images of defect-containing metal microstructures. A large dataset consisting of the stress response of 100,000 random microstructure images is generated using high-resolution Fast Fourier Transform-based finite element (FFT-FE) calculations, which is then used to train a modified U-Net style convolutional neural network (CNN) model. The trained U-Net model more accurately predicted the stress response compared to alternative CNN architectures, exceeded the accuracy of low-resolution FFT-FE calculations, and was generalizable to microstructures with complex defect geometries. The model was applied to images of real AM microstructures with severe lack of fusion defects, and predicted an increase of maximum stress as a function of pore fraction, and higher stress compared to comparable microstructures with circular holes. Together, the proposed CNN offers an efficient and accurate way to predict the structural response of defect-containing AM microstructures.

A semi-analytical method for elastic-plastic analysis of a deep-buried elliptical tunnel

The elastic–plastic problem of an elliptical tunnel is of great significance in engineering. Previous studies were mainly focused on the circular tunnel. An elastic–plastic analysis is performed for determining the plastic zone around a deep-buried elliptical tunnel subjected to far-field uniform in-situ stresses. The yielding of the rock mass obeys the Mohr-coulomb criterion. The elastic stress field is given by the complex variable method and the plastic stress field is numerically determined by the slip line theory. The solution is obtained according to the stress continuity condition along the elastic–plastic interface. Correspondingly, a set of nonlinear equations involves the mapping function coefficients is established and solved with the help of differential-evolution algorithm. The validity of the method is verified by comparing the results with the numerical simulation of FLAC3D software. Estimates of the size and shape of the plastic zone are then presented for varying parameters including in-situ stress, cohesion, internal friction angle and axis ratio.

Macroscopic elastic stress and strain produced by irradiation

Using the notion of eigenstrain produced by the defects formed in a material exposed to high energy neutron irradiation, we develop a method for computing macroscopic elastic stress and strain arising in components of a fusion power plant during operation. In a microstructurally isotropic material, the primary cause of macroscopic elastic stress and strain fields is the spatial variation of neutron exposure. We show that under traction-free boundary conditions, the volume-average elastic stress always vanishes, signifying the formation of a spatially heterogeneous stress state, combining compressive and tensile elastic deformations at different locations in the same component, and resulting solely from the spatial variation of radiation exposure. Several case studies pertinent to the design of a fusion power plant are analysed analytically and numerically, showing that a spatially varying distribution of defects produces significant elastic stresses in ion-irradiated thin films, pressurised cylindrical tubes and breeding blanket modules.

The role of a shear planar mesodefect in the nucleation of a crack at a grain junction due to athermal grain boundary sliding

Ductile fracture of polycrystalline metals is usually preceded by a long deformation stage, during which grains of polycrystal are gradually divided into mutually misoriented regions (fragments) separated by strain-induced grain boundaries. The size of these regions usually does not exceed 0.2 – 0.4 mm. At such small sizes of fragments, classical models of crack nucleation based on the concept of lattice dislocation pile-ups retarded by grain boundaries become incorrect. In recent years, models have been developed to describe the nucleation of cracks under the action of elastic stress fields of rotational and shear-type mesodefects formed at the grain junctions and grain boundaries due to inhomogeneous plastic deformation in the ensemble of polycrystal grains. In this paper we consider the possibility of the nucleation of microcracks at the grain junction due to athermal sliding along the grain boundary, containing a planar mesodefect of the shear-type induced by strain. It is assumed that a planar mesodefect, represented in the initial state by uniformly distributed glissile components of orientational misfit dislocations, loses its stability when the external stress exceeds a certain threshold value. As a result of the strain-induced grain boundary sliding and plastic shear retarding, a stress concentrator arises near the triple junction of grains, creating conditions for the appearance of a Zener crack. The dependences of the critical external stress for the nucleation of microcracks on the length of the mesodefect, its strength and the threshold stress of athermal sliding are obtained. It is shown that the presence of a mesodefect at the grain boundary can lead to a significant decrease of the crack nucleation stress in comparison with the case of pure grain boundary sliding. It is concluded that the proposed model can be considered as one of the possible mechanisms for the nucleation of microcracks in materials with a fragmented structure.

Nonlinear inclusion theory with application to the growth and morphogenesis of a confined body

One of the most celebrated contributions to the study of the mechanical behavior of materials is due to J.D. Eshelby, who in the late 50s revolutionized our understanding of the elastic stress and strain fields due to an ellipsoidal inclusion/inhomogeneity that undergoes a transformation of shape and size. While Eshelby’s work laid the foundation for significant advancements in various fields, including fracture mechanics, theory of phase transitions, and homogenization methods, its extension into the range of large deformations, and to situations in which the material can actively reorganize in response to the finite transformation strain, is in a nascent state. Beyond the theoretical difficulties imposed by highly nonlinear material response, a major hindrance has been the absence of experimental observations that can elucidate the intricacies that arise in this regime. To address this limitation, our experimental observations reveal the key morphogenesis steps of Vibrio cholerae biofilms embedded in hydrogels, as they grow by four orders of magnitude from their initial size. Using the biofilm growth as a case study, our theoretical model considers various growth scenarios and employs two different and complimentary methods – a minimal analytical model and finite element computations – to obtain approximate equilibrium solutions. A particular emphasis is put on determining the natural growth path of an inclusion that optimizes its shape in response to the confinement, and the onset of damage in the matrix, which together explain the observed behavior of biofilms. Beyond bacterial biofilms, this work sheds light on the role of mechanics in determining the morphogenesis pathways of confined growing bodies and thus applies to a broad range of phenomena that are ubiquitous in both natural and engineered material systems.

The role of hydrogen in the edge dislocation mobility and grain boundary-dislocation interaction in [formula omitted]-Fe

The atomistic mechanisms of dislocation mobility depending on the presence of hydrogen were investigated for two edge dislocation systems that are active in the plasticity of α-Fe, specifically ½<111>{110} and ½<111>{112}. In particular, the glide of the dislocation pile-ups through a single crystal, as well as transmission of the pile-ups across the grain boundary were evaluated in bcc iron crystals that contain hydrogen concentrations in different amounts. Additionally, the uniaxial tensile response under a constant strain rate was analyzed for the aforementioned structures. The results reveal that the presence of hydrogen decreases the velocity of the dislocations – in contrast to the commonly invoked HELP (Hydrogen-enhanced localized plasticity) mechanism -, although some localization was observed near the grain boundary where dislocations were pinned by elastic stress fields. In the presence of pre-exisiting dislocations, hydrogen-induced hardening was observed as a consequence of the restriction of the dislocation mobility under uniaxial tension. Furthermore, it was observed that hydrogen accumulation in the grain boundary suppresses the formation of new grains that leads to a hardening response in the stress-strain behaviour which can initiate brittle fracture points.

Effect of plastic deformation on the elastic stress field near a crack tip under small-scale yielding conditions: An extended Irwin's model

The effective stress intensity factor derived from Irwin’s model is greater than the linear elastic counterpart. This is in contradiction to the shielding effect of crack-tip plastic zone. Besides, the classic Irwin’s model underestimates the size of the crack-tip plastic zone compared with Dugdale’s model and FE analysis. In this paper, an extended Irwin’s model for elastic perfectly-plastic material under small-scale yielding conditions is developed through imposing static equivalence between the plasticity-affected stress field and linear elastic stress field in the plasticity-affected zone. The extended Irwin’s model is capable of quantifying the effect of the crack-tip plastic zone on the crack-tip elastic stress field. It reduces to the classic Irwin’s model in the limiting case. This new version of Irwin’s model provides new insights in elucidating the effect of the crack-tip plastic zone on fracture behavior.

Formation of complex radiation defects in irradiated materials

The principles of formation of the complex vacancy defects (V-clusters), their ensembles and patterns of formation of superlattices of the V-clusters are determined. The inclusion of the drift component of the elementary defects into the field of elastic stresses of the V-cluster in the analysis allowed describing its genesis and development adequately. The mechanisms of motion of the V-clusters in the material are described in detail, considering their interaction with each other. The authors have developed the original physical and mathematical formalism within which it has become possible to describe the order-disorder phase transition when an ensemble of clusters chaotically distributed in the irradiated solid transforms into an ordered coherent superlattice. The critical point of the phase transition and the parameters of the defect lattice itself are determined. They are confirmed by the experimental results. The ordering process in this system is understood as the motion of the undamped wave of order parameter through the material, while other configuration states of the V-cluster ensemble constitute rapidly damping fluctuations. The article also shows the mechanism of linking the symmetry of the V-cluster superlattice to the symmetry of the initial crystal.

Hysteretic response of bulk magnetostrictive material employing a novel hyperbolic vector generalized magneto-thermoelastic constitutive model

Taylor series expansion of Gibbs free energy density function for elastic deformation, the physics of demagnetization and the Weiss molecular field coupled with the thermodynamic relations have been used to develop a generalized nonlinear hysteretic thermo-magneto-elastic vector model with potential application for the development of sensors and actuators. The nonlinear elastic strains in the magnetostrictive material are induced due to the magnetic domain rotation caused by tensile and compressive stresses in giant magnetostrictive materials. A vector function of the hyperbolic tangent is defined to take into account this nonlinear characteristic validating the elastic stress field boundary condition. The vector generalized Jiles-Atherton hysteresis model with modified Langevin equation is employed to incorporate the pinning of magnetic domain walls in the mathematical model. Thus, derived macroscopic model is put to the test for 1D, 2D and 3D nonlinear magnetostriction, magnetization and elastic constitutive relation with Terfenol-D (giant magnetostrictive materials) as a potential candid material. Finite element analyses for rods and films have been conducted using the general H(B) relation coded in the C programming language, as shown in the solution algorithm (Fig. 2). Employing the calibrated physical and experimental parameters (Tables 2 and 3), the confidence of the model is checked with existing analytical and experimental models. The magnetization and magnetostriction hysteresis responses have been evaluated considering the variation of a broad range of the prestress (up to 110 MPa), temperature (0 °C–80 °C) and applied magnetic field (0 kA/m–193.2 kA/m). The numerical simulation of this fully coupled phenomenological model demonstrates good agreement with the experimental data. The theoretical modelling error obtained from averaging the normalized root mean square error for each experimental data set is found to be maximum of 2.8 % and hence vindicates the efficacy of the present model.

A Methodology to Determine the Effective Plastic Zone Size Around Blunt V-Notches under Mixed Mode I/II Loading and Plane-Stress Conditions

The determination of the ductile failure behavior in engineering components weakened by cracks and notches is greatly dependent on the estimation of the plastic zone size (PZS) and, particularly, the effective plastic zone size (EPZS). Usually, time-consuming complex elastic–plastic analyses are required for the determination of the EPZS. Such demanding procedures can be avoided by employing analytical methods and by taking advantage of linear elastic analyses. In this sense, this work proposed a methodology for determining the PZS around the tip of blunt V-notches subjected to mixed mode I/II loading and plane-stress conditions. With this aim, firstly, existing approximate mathematical expressions for the elastic stress field near round-tip V-notches reported in the literature are presented. Next, Irwin’s approach (fundamentally proposed for sharp cracks) and a yield criterion (von Mises or Tresca) were applied and are presented. With the aim of verifying the proposed methodology, elastic–plastic finite element analyses were performed on virtual AISI 304 steel V-notched specimens. It was shown that the analytical formulations presented cannot estimate the complete shape of the plastic zone. However, the EPZS, which is crucial for predicting the type of ductile failure in notched members, can be successfully estimated.

A screw dislocation near a semi-infinite crack partially penetrating an elastic inhomogeneity of arbitrary shape

A novel yet simple method is proposed to derive an analytical solution to the interaction problem of a screw dislocation inside an elastic inhomogeneity of arbitrary shape partially penetrated by a semi-infinite crack. The two analytic functions characterizing the elastic fields of stresses and displacement in the two-phase composite are readily obtained. Furthermore, we derive explicit expressions for the mode III stress intensity factor at the crack tip and the image force acting on the screw dislocation. Several specific examples are presented to demonstrate the obtained solution. An analytical solution to the case in which the screw dislocation is located in the cracked matrix is similarly derived.