Constant Stress Field Research Articles

Radially transverse isotropic inclusions in isotropic elastic media: Local fields, neutral inclusions, effective elastic properties

Radially transverse isotropic inclusions in homogeneous isotropic elastic host media are considered. Mellin transform method is used for solution of the volume integral equation of the problem for an isolated inclusion subjected to a constant external stress (strain) field. The tensor structure of the solution is revealed with precision to three scalar functions of the radial coordinate, and the system of ordinary differential equations for these functions is derived. For multilayered radially transverse isotropic inclusions with constant elastic coefficients inside layers, explicit solution of these equations is obtained. An efficient numerical algorithm of solution for inclusions with an arbitrary number of the layers is proposed. Neutral inclusions that do not disturb homogeneous external fields applied to the medium are considered. It is shown that an inclusion with an isotropic core and radially transverse isotropic external layer can be weak neutral by appropriate choice of the layer elastic constants. The effective field method is used for determination of the effective elastic stiffness tensor of a homogeneous isotropic medium containing a random set of radially transverse isotropic inclusions.

Numerical simulation of fracking and gas production in shale gas reservoirs

AbstractIn this work, different stages of gas production in shale reservoirs are modelled. First, hydraulic fracturing is considered by injecting water at high pressures to crack the formation and increase the flow capacity of the reservoir. During the fluid injection, rock properties are modified and water appears in the stimulated area. Then, these changes can be detected through seismic monitoring. Finally, once the fracking stage is completed, the simulation of gas production begins. The simultaneous gas–water flow in the injection and production stages is modelled using the Black‐Oil formulation. Furthermore, a fracture criterion is applied under the hypothesis of constant temperature and constant stress field in this first analysis of the problem. The numerical simulations allow us to analyse the propagation of the fracture and the behaviour of the pore pressure and water saturation in the stimulated area. The advance of the fracturing fluid is delayed in relation to the breakdown of the rock. Besides, the presence of new fractures is detected by applying a poroviscoelastic wave propagation simulator that considers mesoscopic losses induced by heterogeneities in rock and fluids. After the fracture network is created, the injection well becomes a producer, allowing the extraction of gas and the flowback of the injected fluid. The simulated gas flow rates are compared with those obtained by a simplified single‐phase analytical solution used for practical applications, achieving optimum matching results.

Computer homogenization of porous piezoceramics of different ferrohardness with random porous structure and inhomogeneous polarization field

The article is concerned with the homogenization problems, in which the effective moduli of porous piezoceramic composites are determined taking into account the inhomogeneity of the polarization field. The homogenization problems are solved by the finite element method in the framework of the theory of effective moduli and the Hill energy principle using the ANSYS package. To this end, in static problems of electroelasticity, the displacements and electric potential, which are linear in spatial variables, are specified on the boundary of a representative volume to provide constant stress and electric induction fields for a homogeneous reference medium. After solving a set of boundary value problems under different boundary conditions and determining the volume-averaged stress components and the electric induction vector, a complete set of effective moduli for the piezoelectric composite is calculated. A representative volume of the piezocomposite is created in the form of a regular finite element mesh consisting of cubic elements. Pores in the representative volume are assumed to be filled with a piezoelectric material with extremely small moduli. Finite elements with pore properties are selected according to a simple random algorithm. The inhomogeneous polarization field is found by solving an electrostatic problem, in which the polarization process in the representative volume is modeled based on a simplified linear formulation. The local coordinate systems for individual finite elements of the composite matrix are specified by the directions of the polarization vectors. In the following, when solving the problems of electroelasticity, these local coordinate systems associated with the elements of the piezoelectric matrix allow recalculating the material properties according to the formulas of transformation of the tensor components as the coordinate systems rotate. In addition, consideration is given to different models describing the change in the moduli of the material from an unpolarized state to a polarized one as a function of the polarization vector. Computational experiments were carried out for three types of piezoceramics: soft ferroelectric piezoceramics PZT-5H, piezoceramics PZT-4 of medium ferrohardness, and piezoceramics PZT-8 with higher degree of ferrohardness. The dependences of the effective moduli on porosity are compared for different laws of polarization inhomogeneity and different kinds of piezoceramic material of the composite matrix.

Refined explicit solution for the stress field around an elliptical cylindrical tunnel with a moderate depth

We consider a horizontal elliptical tunnel buried in a medium under gravity and lateral pressure (perpendicular to the generatrix of the tunnel). The stress field in the medium is generally treated as the superposition of an initial stress field induced by the gravitational body force and lateral pressure before tunnel excavation and the perturbation stress field after the introduction of the tunnel. Closed-form solutions for the perturbation stress field are available if the interaction between the tunnel and the surface of the medium is neglected. The initial stress field is usually assumed to be constant in deriving closed-form solutions for the perturbation stress field since it is often believed that the interaction between the tunnel and the surface of the medium is negligible only for a deep-buried tunnel. However, it is indicated from the literature that the interaction between the tunnel and the surface of the medium is insignificant in determining the stress distribution around the tunnel as long as the distance between them exceeds the diameter of the tunnel’s cross section. Consequently, we are motivated to derive in this paper a refined closed-form solution for the perturbation stress field by admitting a realistic initial stress field that varies linearly in the depth direction. The refined solution may lead to more accurate results in predicting the gravity-triggered stress concentration around a moderate-depth tunnel. The refined solution is compared with the classical solution (obtained by using a constant initial stress field) in calculating the hoop stress around the tunnel via several numerical examples.

Critical analysis of generalized Maxwell homogenization schemes and related prospective problems

A linear composite medium consisting of a homogeneous matrix containing either the periodic or random set of heterogeneities is considered. Both the original and generalized Maxwell schemes are analyzed by reducing the problem to calculation of elastic fields in a finite volume of the composite embedded in the infinite homogeneous matrix medium and subjected to a constant external stress or strain field at infinity. We investigate (qualitatively and quantitatively) a correspondence between the Maxwell schemes (the original and generalized versions) and other different basic assumptions, conceptions, and methods of analytical micromechanics such as effective field method (EFM) and Mori–Tanaka method (MTM). Some deficiencies and inconsistencies of generalized Maxwell schemes in particular examples are detected and explained how the mentioned difficulties can be easily overcome by the classical micromechanical methods EFM and MTM. For the modeling of both periodic and random structure composites, the generalized Maxwell schemes taking into account the inclusion interactions inside the cluster are analyzed. The incorrectness of using of some generalized Maxwell schemes for the modeling of random structure composite materials (CMs) is shown and the directions for its improvements in the framework of the second background of analytical micromechanics are proposed.

Imposing equilibrium on experimental 3-D stress fields using Hodge decomposition and FFT-based optimization

We present a methodology to impose micromechanical constraints, i.e. stress equilibrium at grain and sub-grain scale, to an arbitrary (non-equilibrated) voxelized stress field obtained, for example, by means of synchrotron X-ray diffraction techniques. The method consists in finding the equilibrated stress field closest (in L2-norm sense) to the measured non-equilibrated stress field, via the solution of an optimization problem. The extraction of the divergence-free (equilibrated) part of a general (non-equilibrated) field is performed using the Hodge decomposition of a symmetric matrix field, which is the generalization of the Helmholtz decomposition of a vector field into the sum of an irrotational field and a solenoidal field. The combination of: a) the Euler–Lagrange equations that solve the optimization problem, and b) the Hodge decomposition, gives a differential expression that contains the bi-harmonic operator and two times the curl operator acting on the experimental stress field. These high-order derivatives can be efficiently performed in Fourier space. The method is applied to filter the non-equilibrated parts of a synthetic piecewise constant stress fields with a known ground truth, and stress fields in Gum Metal, a beta-Ti-based alloy measured in-situ using Diffraction Contrast Tomography (DCT). In both cases, the largest corrections were obtained near grain boundaries.

A 3D Full Stress Tensor Model for Oklahoma

AbstractThe stress tensor is an important property for upper crustal studies such as those that involve pore fluids and earthquake hazards. At tectonic plate scale, plate boundary forces and mantle convection are the primary drivers of the stress field. In many local settings (10–100 s of km and <10 km depth) in tectonic plate interiors, we can simplify by assuming a constant background stress field that is perturbed by local heterogeneity in density and elasticity. Local stress orientation and sometimes magnitude can be estimated from earthquake and borehole‐based observations when available. Modeling of the local stress field often involves interpolating sparse observations. We present a new method to estimate the 3D stress field in the upper crust and demonstrate it for Oklahoma. We created a 3D material model by inverting multiple types of geophysical observations simultaneously. Integrating surface‐wave dispersion, local travel times and gravity observations produces a model of P‐wave velocity, S‐wave velocity, and density. The stress field can then be modeled using finite element simulations. The simulations are performed using our simplified view of the local stress field as the sum of a constant background stress field that is perturbed by local density and elasticity heterogeneity and gravitational body forces. An orientation of N82°E, for the maximum compressive tectonic force, best agrees with previously observed stress orientations and faulting types in Oklahoma. The gravitational contribution of the horizontal stress field has a magnitude comparable to the tectonic contribution for the upper 5 km of the subsurface.

An ellipsoidal inclusion and an elliptical crack in the elastic media with ellipsoidal anisotropy

An isolated ellipsoidal inclusion and an elliptical crack in a homogeneous elastic medium with ellipsoidal anisotropy are considered. For constant external stress (strain) fields applied to the medium, the solutions of these problems are reduced to calculation of 2D-regular integrals. For materials with ellipsoidal anisotropy, the integrand functions in these integrals can be presented in explicit analytical forms that simplifies substantially construction of the numerical solutions of the problems. The stress intensity factors at the crack edge are expressed in the forms of 1D-integrals which integrand functions have also explicit analytical forms for media with ellipsoidal anisotropy.

Graphic statics in a continuum: Strut-and-tie models for reinforced concrete

This research bridges the gap between the numerical layout optimization method and the geometry-based analysis and design method of graphic statics. The study connects the two methods for the application of strut-and-tie models in reinforced concrete design. It suggests a new algorithm for the algebraic graphic statics of indeterminate trusses inspired by the layout optimization method. This research also contributes to the field of graphic statics by providing a formulation that generates form and force diagrams in a continuum, where the topology for none of the diagrams is initially provided.The generation of strut-and-tie models for reinforced concrete has extensively relied on optimization methods, which are helpful techniques for the initiation of a load path inside a continuous domain. However, the resulting truss model is a single answer, and there exists limited control to methodically modify the topology or the force distribution of the model. Furthermore, the minimized-weight truss does not guarantee a practical strut-and-tie model or an optimized performance of a reinforcement design.In contrast to the conventional optimization techniques with a single solution, the intuitive method of graphic statics provides the designer with a vast design space. It also offers explicit control over the geometry and force distribution of the generated truss models. The algorithm provided in this paper is applied to various continuous domains to systematically generate a variety of strut-and-tie models, their force diagrams, and constant stress fields. The production of the optimized truss model and its force distribution allows the direct interactive manipulations of the design while observing the changes in the stress fields and the reinforcement arrangement. The open-source repository of the implemented integrated algorithm and the examples used in this paper are also provided.

Weight Function Method for Stress Intensity Factors of Semi-Elliptical Surface Cracks on Functionally Graded Plates Subjected to Non-Uniform Stresses.

In this paper, a weight function method based on the first four terms of a Taylor’s series expansion is proposed to determine the stress intensity factors of functionally graded plates with semi-elliptical surface cracks. Cracked surfaces that are subjected to constant, linear, parabolic and cubic stress fields are considered. The weight functions for the surface, deepest and general points on the crack faces of long and deep cracked functionally graded plates are derived, which has never been done before in the literature. The accuracy of the method in this study is then validated by comparing the results with those of finite element modeling. The numerical results indicate that the derived weight functions are highly accurate and robust enough to predict the stress intensity factors for cracked functionally graded plates subjected to non-uniform stress distributions. The weight function method is therefore a time-saving technique and suitable for handling non-uniform stress fields.

Numerical simulation based upon friction coefficient model on thermo-mechanical coupling in rotary friction welding corresponding with corona-bond evolution

Numerical simulation method was employed to reveal the mechanism of manufacturing a rotary friction welding joint, in which a two-dimensional finite element model was constructed based on a comprehensive friction coefficient model to simulate the thermo-mechanical coupling during rotary friction welding. Experiments were focused on rotary friction welding of SUS304 in diameter of φ25 mm at 900 rpm under 80 MPa, of which the joint were studied on evolutions of temperature field, stress fields (i.e., both stress-Z and effective stress) and tensile strength corresponding with the corona-bond evolution. Thereafter, the simulation focused on the effect of rotation speed that varied as 500–2500 rpm, in which the minimum welding time necessary to cover a welding cycle was issued. The finite element model was shown valid by the consistency of comparison between simulation and experiment. Transition temperature fields (in contours) were verified in good agreement with transition morphologies of the joints as experimentally characterized by the corona-bonds. The results show that, rotary friction welding process can be classified into two stages: the corona-bond evolution and the plastic flow. The former stage is characterized by the corona-bond that develops and fills out the interface, in which the temperature is kept raising and the stress field kept varying; whereas the latter stage is denoted by the extrusion of flash that keeps forming at constant temperature and stress fields (i.e., both stress-Z and effective stress). The tensile test reveals that the latter stage makes negligible contribution to the joint strength, which falls in the range as 660–688 MPa as the welding time is regulated from 1.8 to 15 s. Thus, time consumed on former stage was ascertained to be the minimum welding time, which ranges in 1.5–2.9 s for SUS304 in φ25 mm of diameter welded under 80 MPa at the rotation speed of 500–2500 rpm.

Comparison of SPH and MLSM for Powder Particle Impact Modelling

In this article, SPH and MLSM methods are compared in application to particle impact modelling. It is shown, that both methods has important disadvantages because nonzero acceleration appears in the case of constant stress field with nonsymmetrical distribution of nodes in space. Some reasons of incorrect acceleration calculation are discussed.

Reversible Martensitic Transformation under Low Magnetic Fields in Magnetic Shape Memory Alloys

Magnetic field-induced, reversible martensitic transformations in NiCoMnIn meta-magnetic shape memory alloys were studied under constant and varying mechanical loads to understand the role of coupled magneto-mechanical loading on the transformation characteristics and the magnetic field levels required for reversible phase transformations. The samples with two distinct microstructures were tested along the [001] austenite crystallographic direction using a custom designed magneto-thermo-mechanical characterization device while carefully controlling their thermodynamic states through isothermal constant stress and stress-varying magnetic field ramping. Measurements revealed that these meta-magnetic shape memory alloys were capable of generating entropy changes of 14 J kg−1 K−1 or 22 J kg −1 K−1, and corresponding magnetocaloric cooling with reversible shape changes as high as 5.6% under only 1.3 T, or 3 T applied magnetic fields, respectively. Thus, we demonstrate that this alloy is suitable as an active component in near room temperature devices, such as magnetocaloric regenerators, and that the field levels generated by permanent magnets can be sufficient to completely transform the alloy between its martensitic and austenitic states if the loading sequence developed, herein, is employed.

Efficient homogenization techniques for elastic composites: Maxwell scheme vs. effective field method

The work is devoted to the comparative analysis of four homogenization techniques (the original and generalized Maxwell schemes and the one-particle and multi-particle effective field methods) in application to the problem of calculation of the effective elastic stiffness tensor of matrix composites. The generalized Maxwell scheme and multi-particle effective field method reduce the problem to calculation of elastic fields in a finite volume of the composite embedded in the infinite homogeneous matrix medium and subjected to a constant external stress or strain field. For the solution of this problem, an efficient numerical method based on Gaussian approximating functions and fast Fourier transform technique is used. The results of the methods are compared for composites with ellipsoidal inclusions which elastic moduli are much smaller or much larger than the moduli of the matrix. The case of hybrid composites with two different families of ellipsoidal inclusion is also considered. It is shown that for various shapes and elastic properties of inclusions, the generalized Maxwell scheme and multi-particle effective field method give close results. But in the case of hard inclusions of large volume fractions the results of these methods deviate substantially from the original Maxwell scheme that does not take into account interactions between inclusions.

Shape optimization of ceramic components using a failure probability based on extended multiaxial Weibull theory

For the design of components made of ceramic materials, a failure probability based on extended multiaxial Weibull theory is used. In the present paper, this failure probability is used as objective function in the framework of shape optimization performed by derivative-free and gradient-based numerical optimization. The failure probability is computed with three levels of complexity of the underlying fracture mechanics crack model: for the assumption of a locally constant or inhomogeneous stress field along the crack in a semi-infinite plane and for a locally inhomogeneous stress field along the crack in a plate. The sensitivity analysis of the failure probability is presented in detail. Finally, a numerical example assesses the effectiveness of both types of optimization algorithm.

TECTONOPHYSICAL SIGNS OF THE FORMATION OF STRONG EARTHQUAKE FOCI IN SEISMIC ZONES OF CENTRAL ASIA

Vibrations of the Earth crust and variations in the physical fields of the Earth atmosphere and ionosphere are continuously monitored by a variety of techniques and specialized facilities across the world. Nevertheless, most catastrophic earthquakes even in this century have occurred in “incidental” or “unexpected” places in “unpredicted” time. Earthquake predictions have errors as the current knowledge of focal mechanisms of strong (M≥8) earthquakes is still insufficient. It is believed today that the most common source of earthquakes is movement of rock blocks along a fault/megafracture. Such movements take place in a stepwise pattern with high or reduced friction, depending on the presence of fluids, hitches on the fault planes and other factors. Modern seismic forecasting is based on the concept of precursors. The author considers geological and geophysical settings in areas of dynamic influence of faults, wherein 8>М>7.5 earthquakes took place. Based on earthquake recurrence curves constructed for such areas, four tectonic criteria for formation of strong earthquake sources are identified: structural (large seismically active faults), kinematic (large amplitudes of the fault wing’s displacements), rheological (physical properties of the fault infill material, such as low viscosity of the intra-fault medium) and dynamic (high rates of the fault wing’s displacements) criteria. These criteria should be in the focus of quantitative studies in order to provide a solid scientific basis for long-term forecasting of strong earthquakes. Curves constructed for the criteria can show changes in the physics of earthquake foci in case of strong seismic events. With account of the tectonophysical features of faults associated with strong seismic events, the following conclusions are drawn. (1) In the continental lithosphere, catastrophic earthquakes (M≥8) occur in areas of dynamic influence of the major faults in the lithosphere in case of relatively high amplitudes of displacements of boundary blocks (i.e. fault wings). (2) In the relatively stable stress field, high amplitude displacements take place in case of reduced viscosity/quasi-viscosity of the medium comprising the internal structure of faults. (3) Reduced viscosity of the intra-fault medium is related to the physical conditions of transition of rocks in fault zones (mainly along the fault planes) in the state of quasi-plastic or plastic flow (unilateral pressure in excess of hydrostatic pressure, and relatively decreasing strength properties of the intra-fault medium with increasing length of the fault wings). (4) Reduced viscosity of the fault zone leads to an increase in the displacement rate of the fault wings in the constant stress field. The latter factor is the main one, transforming seismically active faults with M≤7.5 seismic events into faults of similar characteristics, but with earthquakes of higher energy, M≥8. Focal mechanisms of such earthquakes are associated with conditions for a potential increase of the displacement amplitude regardless of the presence of fluids, hitches on the fault planes and other poorly predictable factors. In-depth studies of the internal structure of faults with M≥8 earthquakes, their foci, conditions of the temporal regime of the seismic process before and after strong seismic events can discover a key to understanding the origin of earthquake sources, the criteria of energy release, and the occurrence of earthquakes with maximum energy. Further steps to develop the geological and geophysical (including tectonic) criteria for prediction of strong earthquakes should be focused on more detailed research of seismic zones wherein strong earthquakes were recorded.

Shape memory effect and superelasticity of NiMnCoIn metamagnetic shape memory alloys under high magnetic field

The effects of the magnetic field on the shape memory behavior of [001]-oriented metamagnetic NiMnCoIn shape memory single crystals were investigated. Thermal cycling under constant stress and magnetic field, and stress cycling under constant magnetic field tests were conducted to understand their magneto-thermo-mechanical behavior. It was observed that critical stress during the superelastic behavior increased and transformation temperatures decreased with the applied field. Under 9T applied field, compared to all magnetic shape memory alloys, an ultrahigh magnetostress level of 71MPa was observed during the stress cycling experiments and transformation temperatures were decreased by 16°C during the thermal cycling under stress.

Cave development in an uplifting fold-and-thrust belt: case study of the Tatra Mountains, Poland

Detailed structural analysis and geomorphological observations supplemented by the analysis of the distribution of karst conduit directions have been performed in 23 morphologically diverse caves in the Tatra Mountains. Based on these studies, a development scheme of vadose cave passages has been proposed for the most common geological settings in the fold-and-thrust-belt: (1) single-plain faults, (2) multiple fault cores, (3) bedding plane fractures and (4) hinge zones of recumbent folds. Results indicate that the dynamics of the massif (local gravity sliding in the nearby slope zone and regional stress fields), along with the structural pattern, influences the predisposition of structural and stratigraphic discontinuities to karst drainage. Constant tectonic stress fields affected the massif during the entire speleogenesis. This led to the rejuvenation of the same displacements in successive tectonic events, which resulted in promoting this reactivated structures in successive speleogenetic phases. Structures along which older cave levels had developed were also utilized later by vadose and phreatic drainage, leading to the intersection of the vadose passages with elevated paleo-phreatic cave levels. Independently, formation of entirely vadose caves, guided by the same group of weak and rejuvenated planes, was enabled. In the Tatras, the concentric and recumbent geometry of the main folds resulted in steep dipping of the bedding planes over a distance up to a few hundred meters which makes the bedding plane fractures subject to karst water circulation in these geologic and geodynamic settings.

Reliability assessment of germanium gate stacks with promising initial characteristics

This work reports on the reliability assessment of germanium (Ge) gate stacks with promising initial electrical properties, with focus on trap generation under a constant electric stress field (Estress). Initial Ge gate stack properties do not necessarily mean highly robust reliability when it is considered that traps are newly generated under high Estress. A small amount of yttrium- or scandium oxide-doped GeO2 (Y-GeO2 or Sc-GeO2, respectively) significantly reduces trap generation in Ge gate stacks without deterioration of the interface. This is explained by the increase in the average coordination number (Nav) of the modified GeO2 network that results from the doping.

Discontinuous Deformation Analysis Enriched by the Bonding Block Model

The discontinuous deformation analysis (DDA) has been extensively applied in geotechnical engineering owing to its salient merits in the modeling of discontinuities. However, this method assumes a constant stress field within every block and hence cannot provide reliable estimation for block deformations and stresses. This paper proposes a novel scheme to improve the accuracy of the DDA. In our method, advanced subdivision is introduced to represent a block as an assembly of triangular or quadrilateral elements, in which overlapped element edges are separated from each other and are glued together by bonding springs. The accuracy and the effectiveness of the proposed method are illustrated by three numerical experiments for both continuous and discontinuous problems.