Determination Of Stress Field Research Articles

Direct spatio-temporal stress field determination combining full-field deformation measurements and explicit finite element method: Concept verification

High strain rate loading of materials with spatially heterogeneous properties produces non-homogeneous and transient stress-strain states. Such stress-strain states are atypical of classical split Hopkinson pressure bar (SHPB) analysis that assumes homogeneous deformation and dynamic force equilibrium. To determine spatio-temporal stress fields in heterogeneous materials, a novel concept combining digital image correlation (DIC) full-field deformation measurements and explicit dynamic finite element method (EFEM) is described in this paper. The concept utilizes measured boundary forces and DIC measurements including both displacements and accelerations as input. The measured input is used to calculate internal forces, and hence the stress fields, within each continuum finite element by solving the governing equations of motion using the framework of EFEM. The DIC+EFEM concept procedure is verified using simulated synthetic data considering one-dimensional stress wave propagation in a bar. The reconstructed axial stress fields match identically with the exact EFEM solution for both homogeneous and heterogeneous materials that are elastic or elastic-plastic. The influence of DIC measurement noise on the reconstructed stresses is also studied.

Atomistic determination of stress fields in samples with defects

The main goal of the study is the atomistic determination of the static stress-strain state at the crack tip in isotropic and anisotropic linearly elastic materials by the molecular dynamics method implemented in the open source LAMMPS (Large-scale Atomistic/Molecular Massively Parallel Simulator). An extensive class of computational experiments has been carried out for a single-crystal copper specimen with a central crack in the LAMMPS. The circumferential distributions of the stress tensor components obtained by molecular dynamics are compared with the angular distributions of brittle fracture continuum mechanics, i.e. with Williams’ analytical solution to the problem of an infinite plate loaded with a central crack in a linearly elastic isotropic material. The comparison of the angular distributions of the stress tensor components gained in the framework of atomistic modeling and the angular distributions obtained from the classical solution of continuum mechanics has shown that, at the nanoscale level, the stress fields are in good agreement with their macroscopic values.

Stress analysis of a circular hole in an orthotropic plate with finite width using complex potential method

AbstractOpen holes are essential design features in technical structures. However, stress concentrations arise and for a safe and lightweight optimal design, precise stress analysis means are crucial. Scope of this paper is the stress field determination of an open hole in an orthotropic rectangular plate of finite width using complex potential method. To derive results of high precision, the stress boundary conditions at the straight edges as well as along the hole boundary need to be fulfilled. In doing so, the stress field of the corresponding infinite‐dimensions problem is supplemented with auxiliary functions, which are based on a novel periodic arrangement technique and enable to take into account the stress‐free straight edges of the finite‐domain problem.

Nonuniform Stress Field Determination Based on Deformation Measurement

Abstract In this investigation, we demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain (currently in 2D) and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same directions. The particular measures for strain and stress chosen in this study are the Lagrangian (or Green-Lagrangian) strain and the second Piola–Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola–Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola–Kirchhoff stress is then determined. The Cauchy stress is related to the second Piola–Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. Such a relationship is valid for finite deformations. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses, σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The stresses determined using this technique obviously satisfy the equation of equilibrium, in contrast to those obtained from a constitutive model with input from deformation measurement. The technique and the associated numerical scheme are verified and validated through two virtual test cases representative of the simply-connected and multiply-connected domains, where exact solutions are available. The technique is then applied to an experimental case of nonuniform deformation of a polyvinyl chloride (PVC) sheet with a circular hole and subject to uniaxial tension. In this case, the associated stress field is also determined through a constitutive model of hyperelasticity, the generalized neo-Hookean (GNH) model, calibrated for the particular PVC sheet. Limitations and restrictions of the technique and the associated numerical scheme, as well as possible extensions will be discussed.

A Comprehensive Experiment-Based Approach to Generate Stress Field and Slip Lines in Cutting Process

Abstract This study proposes a comprehensive experiment-based method to determine stress field and slip lines in metal cutting process. The chip geometry and workpiece's strain and strain rate fields are determined using an in situ imaging technique. The two-dimensional (2D) heat transfer problem for the steady-state cutting process is solved to derive the cutting temperature, and the flow stresses of work material in the main deformation zone are calculated based on the plasticity theory. Furthermore, the stress field is comprehensively determined to satisfy the stress equilibrium, friction law along the tool-chip interface, and traction-free boundary condition along the uncut chip surface. In addition, slip lines in the main deformation zone are derived according to the direction of maximum shear stress without the assumption of perfect rigid-plastic material. The proposed method is validated by comparing the cutting forces calculated based on the obtained stress field with the experimental results.

Closed‐form stress solution of open holes in a finite domain

AbstractOpen holes are essential design features in many technical structures. For a lightweight optimal design precise stress analysis tools are vital. Focus of this contribution is the stress field determination of an open hole in a rectangular plate of finite width by analytical means, particularly the Airy stress function. This is beneficial in terms of computational effort. To obtain results of high precision, the stress boundary conditions at the straight edges as well as along the hole boundary must be fulfilled. This is reached by supplementing the field modelling the corresponding infinite dimensions problem – the Kirsch solution – with auxiliary functions.

The present‐day stress field in the Zagros Fold-Thrust Belt of Iran, from inversion of focal mechanisms

The knowledge about spatial variation of stress field is an important step in the identification of geodynamic and seismotectonic characteristics. Inversion of focal mechanism solutions is one of the most common methods for determination of stress field and the shape ratio in a tectonic region. In this paper, Iterative Joint Inversion method is used to determine principal stress axes from earthquake focal mechanisms in the Zagros continental collision zone of Iran. In order to determine the influence of dependent events on the inversion results, the declustering process is carried out by the Urhammer's method. The area is divided into seven subzones according to the stress field heterogeneity in the Zagros region. In each subzone, the trends of principal stress axes, the orientation of maximum horizontal stress, stress regime and the shape ratio are determined. The results show that the compressional stress direction averages NNE-SSW that accord with the current convergence between the Arabian plate and Iranian micro-continent. A gradual counterclockwise rotation of the compressional axis is also concluded which is compatible with spatial changes in the Arabian–Eurasian convergence paths along the Zagros. The stress regime of most of the subzones is deduced as the Transpression that reflects oblique convergence. The orientation of the maximum principal stress (σ1) show the lowest uncertainty in respect to the intermediate (σ2) and minimum (σ3) principal stress axes. The shape ratio error in all zones are less than 6 and 7.2 percent before and after implementing exclusion respectively, which indicate this error increases with removing the dependent events in all zones except for zone 3.

Photoelastic evaluation of stress fields and notch stress intensity factors for blunt V-notches

Fracture loads of engineering components with blunt notches cannot be evaluated by the linear elastic fracture mechanics (LEFM) due to no stress singularity. In this study, an approach to determining stress fields and notch stress intensity factors (NSIFs) at blunt V-notches is developed using the photoelastic technique. To solve the nonlinear governing equations of the NSIFs derived from the Filippi’s stress equation, the overdeterministic least squares method combined with the Newton-Raphson method is applied. The NSIFs for a series of three-point-bending specimens with a blunt V-notch up to fracture are evaluated for different notch angles, tip radii, and notch lengths under mode I loading conditions. It is observed that the NSIFs obtained from photoelastic measurements agree well with the numerical values, and the notch fracture toughness is accordant to the mean stress (MS) criterion. The experimental results will be useful for the fracture characterization and strength evaluation of engineering components with complex notches.

Determination of nonuniform stress field and rheological properties by caliper log measurements

The authors formulate and solve the problem on finding nonuniform stress field and effective viscosity in the plane oriented orthogonally to a borehole axis drilled in rocks with rheological properties by the data of caliper logging using a viscoelastic model. From the numerical research with artificial input data, the effective viscosity range is determined, and the testing time such that the problem has the unique solution is found.

Simplified stress field determination for an inclined crack and interaction between two cracks under tension

Multiple initiated cracks are frequently occurred on structural components, and they have significant effects on fatigue life due to interaction between several cracks. The objective of the present study was to investigate the inclination of a crack and the interaction effect between two cracks. By analyzing the stress distribution around a single crack and various results from the available literature on the stress intensity factors of the interaction between two cracks, an estimation method was introduced to evaluate the inclination effect and the crack interaction between two cracks in a variety of positions subjected to tension loads under Mode I loading conditions. The effect of cracks length and cracks distance on stress distribution was examined. The study results showed that crack interaction obtained by proposed method was in good agreement with previous findings from the stress intensity factor handbook. Finally, an approximate procedure for estimating the crack interaction in an arbitrary position for two- and three-dimensional cases was proposed.

Damage Zone Size Limit During the Crack Propagation

The principal goal of this work is to limit the damage zone length during the crack propagation in brittle materials. This study is based on the determination of stress fields by varying the distance between a semi-infinite crack and a neighboring dislocation. The model suggested is a rectangular element (a dish), having a semi-infinite crack in one of its ends and a dislocation located in the vicinity of a crack-tip, subjected to a tensile stress on mode I. The problem is treated numerically using Finite Element Method.For each distance between the two cracks (semi-infinite crack and dislocation), stress fields are given. On the basis of these stress fields, a limiting damage zone length is obtained.

Free-edge stress fields in cylindrically curved cross-ply laminated shells

This paper discusses a semi-analytical approach to the determination of three-dimensional stress fields in cylindrically curved orthotropic symmetrically or unsymmetrically laminated composite shells under bending load under consideration of free-edge effects. The analysis method incorporates a plane-strain analysis in the innermost regions of the shell (the so-called inner solution) in conjunction with a free-edge solution which enables the accurate determination of the interlaminar stress field at the free shell edges. The free-edge solution employs a discretization of the laminated shell into a number of mathematical layers with respect to the thickness direction wherein in each layer interface approximate and a priori unknown displacement functions are introduced. Lagrangian shape functions are used to interpolate between the displacement functions in the interfaces, and various orders of shape functions are investigated in this paper. The differential equations and boundary conditions that describe the current stress problem are derived from the principle of minimum elastic potential of the laminated shell. The resultant differential equations allow for an exact solution for the displacement functions whereas the boundary conditions at the free shell edges are satisfied in an integral sense. The results of the developed analysis method are verified by comparison with detailed finite element computations, and it is found that the semi-analytical approach works with comparable accuracy, however only at a fraction of the required computational effort.

Earthquake relocation, focal mechanism and stress field determination in central Brazil

In this work, we perform a seismological investigation in the central Brazil region for focal mechanisms and stress. This large area (910.000 km2) is surrounded by several geological provinces: Eastern Amazon craton; southwestern Parnaíba basin; western São Francisco craton; northern Paraná basin and almost the whole Tocantins province. The area is crossed, in an SW-NE direction, by the continental-scale discontinuity - Transbrasiliano lineament (LTB). In the Brazilian seismic catalog (1720–2019), it is possible to distinguish a seismic belt following the LTB lineament and some dispersed events in the northern, western and eastern areas. The study area is characterized mainly by low magnitude events (M < 4); only 11 events reported in the catalog with M > 4 occurred, and among them just one with magnitude 5. Based on quality criteria a set of 118 events from 338 present in the catalog were selected for a thorough analysis. This work was divided in three parts: (i) Events relocation with the iLoc code and Regional Seismic Travel Time (RSTT) velocity model; (ii) Focal mechanisms determination of 10 events using waveform envelopes inversion and polarities, and; (iii) Inversion of focal mechanisms for stress field. The study shows that the seismicity of the area is mainly concentrated in two relatively narrow belts and that the major compressional stress axis of the whole zone is well resolved with azimuth about 133° which is compatible with previous studies of stress in central Brazil.

Analytic solution of angle-ply laminated plates under extension, bending, and torsion

This paper develops a Hamiltonian state space approach for analytic determination of deformation and stress fields in multilayered monoclinic angle-ply laminates under the combined action of extension, bending, and torsion. The present solution satisfies the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the four edge surfaces of the rectangular section, and the interfacial continuity conditions in multilayered laminates. The proposed method only requires the solutions of matrix and eigen equations, regardless of the number or lamination of the layers. The finite element analyses are used to validate the accuracy of the analysis. The analytical solution and the numerical solutions are in excellent agreement.

Stress field determination based on digital image correlation results

The aim of this work was to determine the stress distribution during plastic deformation, based on the displacement field obtained using the digital image correlation (DIC) method. To achieve stress distribution, the experimentally measured displacement gradient and the elastoplastic material model with isotropic hardening were used. The proposed approach was implemented in the ThermoCorr program. The developed procedure was used to determine stress fields for uniaxial tension and simple shear processes, carried out on samples made of austenitic steel 304L. Both material parameters, such as the Young's modulus, Poisson's ratio, yield stress, and parameters of the hardening curve, were acquired experimentally. The macroscopic force obtained from the DIC-based stresses and its finite element analysis (FEA) equivalent were compared with that measured during the experiment. It was shown that the DIC-based approach gives more accurate results with respect to FEA, especially for a simple shear test, where FEA significantly overestimates the value of experimentally obtained macroscopic force.

On the Radial Multipliers Method in the Gradient Elastic Fracture Mechanics

A non-singular solution of the gradient elastic fracture mechanics for the cracks of Modes I and II is constructed in this paper. Previously, a similar problem was solved only for cracks of Mode III. A generalized theory of elasticity is used, in which the governing equation in displacements is represented as a product of the Lame operator and the Helmholtz operator, and the classical boundary value problem for the total stress is completely distinguished in the problems of crack mechanics. As a result, the determination of local stress fields reduces to solving the Helmholtz equations with the known right-hand side of the equations. The Papkovich-Neuber representation in a complex form is used to construct a solution of the mechanics of cracks. We used also the method of radial multipliers, which allows us to construct fundamental solutions of the Helmholtz equations corresponding to analytical functions with a fractional exponent and, as a result, to find solutions that compensate for the singularities.

Effect of crystal orientation on indentation-induced residual stress field: Simulation and experimental validation

Experimental determination of local residual stress fields beneath a spherical indent on a ⟨001⟩ copper single crystal revealed the presence of a significant tensile residual stress zone. This finding was adequately reproduced by Crystal Plasticity Finite Element simulations. Further simulations of one spherical indent on many other crystal orientations showed a strong variation of the residual stress field with crystal orientation. Accumulative effects of five superposing indents were simulated on two extreme orientations ⟨001⟩ and ⟨111⟩. The simulations showed that plastic anisotropy is responsible for potentially uneven compressive residual stresses after surface mechanical treatments.

Moment tensor inversion of small-to-moderate size local earthquakes in Egypt

Well resolved moment tensor solutions reveal information about the sources of seismic waves. In this article, a waveform inversion technique, which uses waveform data from near-source seismic stations that belong to the Egyptian National Seismological Network (ENSN) is applied in order to estimate both focal mechanism and moment magnitude of small and moderate local earthquakes that occurred after 2010 till 2014 in and around Egypt. Fifty nine earthquakes with local magnitude (ML) ranging between 3.5 and 5.5 are analyzed. The earthquake focal mechanism obtained by waveform inversion was fundamentally validated by means of the basic technique of first onset polarities. Based on the obtained results, we identified several seismic sources in and around Egypt (i.e. Gulf of Aqaba, northern Gulf of Suez, southern Gulf of Suez-northern Red Sea, Aswan zone, Dahshour zone, and the passive continental margin). A combination of the newly estimated focal mechanisms and the previously published ones of earthquakes occurred before 2010 are used in determination of stress field for each relevant seismic source. A new scaling relationship between local magnitude (ML) and the estimated moment magnitude (Mw) has been developed using a linear regression. The results of this study may be used to improve the ENSN's stations configuration and the efficiency to record teleseismic and regional events.

The Use of Optical Methods in the Analysis of the Areas with Stress Concentration

Abstract Anchoring parts of technical systems are often damaged due to the heavy forces acting on these systems during their operation. For that reason, various modifications are suggested and created on the anchoring screws, which should help to reduce the mechanical stress values in a place of the first load-bearing thread of a female screw. For the determination of stress fields on the surface of a plane model of a threaded joint, two non-contact optical methods were used – conventional transmission photoelasticimetry and modern digital image correlation.

On the determination of stress fields and displacements in a thin elastoplastic plate containing elastic inclusion - a shim

The paper is devoted to the determination of the stress-strain state of a mechanical structure, which consists of a thin infinite elastoplastic plate with a hole and a continuous fine elastic inclusion. The complexity of this problem lies in the fact that the shape of the boundary between the elastic and plastic zones in the plate is not known in advance. The small parameter method is used as the solution method, while the small parameter characterizes the deviation of the shape of the contour from the circle and the perturbation of external static boundary conditions. As the zero solution, the axisymmetric elastoplastic state of the plate with inclusion is chosen. Two variants of inclusion fixation in a plate are considered: inclusion is enclosed with tension or soldered. As a result of solving the problem within the framework of ideal plasticity, the distribution of the stress and displacement fields and the shape of the elastoplastic boundary are obtained. To illustrate the case of a plane-stressed state, the results of a numerical experiment on the mathematical model obtained are presented.