Uniform Stress Field Research Articles

Need the Uniform Stress Field Inside Multiple Interacting Inclusions Be Hydrostatic?

Since the pioneering work of Eshelby on a single ellipsoidal inclusion embedded in an infinite space, much attention has been devoted in the literature to the question of the uniformity of the stress field inside inclusions surrounded by an elastic matrix. Over the last decade or so, researchers have established the existence of multiple (interacting) inclusions enclosing uniform internal stress distributions when embedded in an infinite elastic matrix subjected to a uniform far-field loading and identified a variety of shapes of such inclusions. In the design of multiple inclusions with uniform internal stresses, it is customary to assume that the uniform stress field inside each inclusion (each with different shear modulus distinct from that of the matrix) is hydrostatic. In this paper, we examine whether this assumption is actually necessary to ensure the required existence of multiple inclusions enclosing uniform stresses. By establishing several theorems in the theory of functions of a complex variable, we prove rigorously that for any collection of multiple inclusions enclosing uniform stresses in an infinite isotropic plane subjected to uniform remote in-plane loading, the internal uniform stress field must indeed be hydrostatic if the corresponding inclusion’s shear modulus is distinct from that of the matrix.

A circular Eshelby inclusion with linear eigenstrains interacting with a coated non-parabolic inhomogeneity with internal uniform anti-plane stresses

We establish criteria which guarantee the uniformity of stresses inside a coated non-parabolic open inhomogeneity located in the vicinity of a circular Eshelby inclusion undergoing linear anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. The associated inverse problem in anti-plane elasticity which identifies the required geometry of the constituents of the composite is successfully solved using conformal mapping techniques and analytic continuation. Three conditions on the remote loading in the matrix, uniform eigenstrains imposed on the non-parabolic inhomogeneity and linear eigenstrains imposed on the circular Eshelby inclusion for given geometric and material parameters of the composite are found in order to ensure the uniformity of internal stresses. The internal uniform stress field is found to be independent of the specific open shapes of the inhomogeneity-coating and coating-matrix interfaces, the shear modulus of the coating and also of the existence of the nearby circular Eshelby inclusion. However, the open shapes of the two interfaces are significantly influenced by the presence of the circular Eshelby inclusion. We also discuss the general case in which arbitrary polynomial eigenstrains are imposed on the circular Eshelby inclusion.

Evaluation of local constitutive properties of Al2024 friction stir-welded joints using digital image correlation method

The intense applied thermal gradients during the welding process leads to the variation in the properties of the weld zone and its surrounding areas. In this regard, determining the local mechanical properties and evolved microstructures of the weld and its surrounding zones are essential in the evaluation of welded structures. In this study, the local mechanical properties of Al2024 friction-stir-welded joints were precisely examined. Digital image correlation (DIC) technique using uniform stress (USM) and virtual field methods (VFM) were utilized to evaluate the local mechanical properties. The local stress-strain curves were plotted for different weld regions using local strain from the DIC technique. It was observed that the advancing side of the thermo-mechanically affected zone (TMAZ) had the lowest values of Young’s modulus and yield strength (YS), approximately 9% and 31% of the base metal, respectively. Effects of welding parameters, such as tool rotational and traverse speeds, were also taken into account. The plotted local stress-strain curve for the fractured region of welded specimens at lower rotational speed illustrated a higher strength and elongation. Furthermore, lower rotational and also higher traverse speeds resulted in the reduction of the grain size.

A hole of irregular shape interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

We use conformal mapping techniques together with analytic continuation to show that a non-parabolic open elastic inhomogeneity continues to admit a state of uniform internal stress when a hole with closed curvilinear traction-free boundary is placed in its vicinity and the surrounding matrix is subjected to uniform remote anti-plane stresses. The internal uniform stress field inside the inhomogeneity is found to be independent of the existence of the nearby hole and the specific non-parabolic shape of the inhomogeneity. In contrast, the non-parabolic shape of the inhomogeneity is influenced solely by the existence of the nearby hole.

Altered stress field of the human lens capsule after cataract surgery

The lens capsule of the eye is important in focusing light onto the retina during the process of accommodation and, in later life, housing a prosthetic lens implanted during cataract surgery. Though considerable modeling work has characterized the mechanics of accommodation, little has been done to understand the mechanics of the lens capsule after cataract surgery. As such, we present the first 3-D finite element model of the post-surgical human lens capsule with an implanted tension ring and, separately, an intraocular lens to characterize the altered stress field compared to that in a model of the native lens capsule. All finite element models employed a Holzapfel hyperelastic constitutive model with regional variations in anisotropy. The post-surgical lens capsule demonstrated a dramatic perturbation to the stress field with mostly large reductions in stresses (except at the equator where the implant contacts the capsule) compared to native, wherein maximal changes in Cauchy stress were −100% and −145% for the tension ring and intraocular lens, respectively. However, implantation of the tension ring produced a more uniform stress field compared to the IOL. The magnitudes and distribution of the perturbed stress field may be an important driver of the fibrotic response of inhabiting lens epithelial cells and associated lens capsule remodeling after cataract surgery. Thus, the mechanical effects of an implant on the lens capsule could be an essential consideration in the design of intraocular lenses, particularly those with an accommodative feature.

Displacement of surrounding rock in a deep circular hole considering double moduli and strength-stiffness degradation

The problem of cavity stability widely exists in deep underground engineering and energy exploitation. First, the stress field of the surrounding rock under the uniform stress field is deduced based on a post-peak strength drop model considering the rock’s characteristics of constant modulus and double moduli. Then, the orthogonal non-associative flow rule is used to establish the displacement of the surrounding rock under constant modulus and double moduli, respectively, considering the stiffness degradation and dilatancy effects in the plastic region and assuming that the elastic strain in the plastic region satisfies the elastic constitutive relationship. Finally, the evolution of the displacement in the surrounding rock is analyzed under the effects of the double moduli characteristics, the strength drop, the stiffness degradation, and the dilatancy. The results show that the displacement solutions of the surrounding rock under constant modulus and double moduli have a unified expression. The coefficients of the expression are related to the stress field of the original rock, the elastic constant of the surrounding rock, the strength parameters, and the dilatancy angle. The strength drop, the stiffness degradation, and the dilatancy effects all have effects on the displacement. The effects can be characterized by quantitative relationships.

Sequential damage study induced in fiber reinforced composites by shear and tensile stress using a newly developed Arcan fixture

This work presents the application of both uniaxial and bi-axial in-plane loads to unidirectional carbon fiber reinforced polymer laminates using a newly developed Arcan fixture, which is a reliable experimental set-up to obtain a uniform shear stress field in a butterfly specimen. The set-up can be used for both damage model validation and parameters identification at various fiber orientations while using the same specimens. A sequential damage study was completed to highlight the influence of diffused damage induced in pure shear on the fiber direction tensile behavior of the laminate. This was accomplished by applying load on the specimens in two steps: (i) the pure shear step and then unloading at approximately 70% of the shear failure strength, (ii) in the tensile step until final failure. A clear drop in the tensile behavior of the laminate was observed by the diffused damage induced in the first loading step of pure shear. The experimental study is also supplemented with numerical simulations using a nonlinear elasto-plastic coupled damage constitutive law by employing Puck’s failure theory for mesodamage activation. In addition to the damage pattern, the non-linear mechanical behavior in shear is predicted and found in good correspondence with the experimental results.

Uniform hydrostatic stresses inside a coated non-parabolic inhomogeneity in the vicinity of a concentrated couple

We establish the existence of an internal uniform hydrostatic state of stress inside a coated non-parabolic open elastic inhomogeneity when the surrounding matrix is subjected to a concentrated couple and uniform remote in-plane stresses. The proof of existence is accomplished through analytic continuation and the introduction of a specialized form of the conformal mapping function containing an infinite number of first-order poles. The complex constants appearing in the mapping function can be uniquely determined from a resulting set of non-linear recurrence relations for given material, geometric and loading parameters. The convergence criterion for the series in the mapping function is rigorously established. A simple condition on the remote loading in the matrix is found for given elastic constants of the inhomogeneity and the matrix in order to ensure that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. The analysis indicates that the internal uniform hydrostatic stress field, the constant mean stress in the coating and the constant hoop stress along the inhomogeneity-coating interface on the coating side are all independent of the specific open shapes of the two interfaces and are also independent of the existence of the nearby concentrated couple. However, the non-parabolic open shapes of the two interfaces are significantly influenced by the concentrated couple. We also accomplish the design of an uncoated non-parabolic harmonic elastic inhomogeneity in the presence of an arbitrary number of concentrated couples applied in the matrix.

Research on Displacement Failure Characteristics of Deep Roadway Surrounding Rock

In order to explore the displacement failure characteristics of deep roadway surrounding rock, the surrounding rock stress field model of the uniform stress field in the circular tunnel is simplified. It analyzes the stress evolution features of deep coal rock and discusses the mechanism of zonal disintegration. It observes the deep displacement surrounding rock of Qujiang mine n the depth of 1000m, the displacement shows that the wave crests and troughs appears alternately; wave crest position for damage area, trough part for non-damage zone. The surrounding rock of roadway exists zonal disintegration phenomenon, each roadway section is a total of four fracture zones. Due to different positions of roadway section tunnel excavation time rheological effect, the longer the tunnel is excavated, the more serious the surrounding rock is damaged.

Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couple.

Using conformal mapping techniques, we design novel lamellar structures which cloak the influence of any one of a screw dislocation dipole, a circular Eshelby inclusion or a concentrated couple. The lamellar structure is composed of two half-planes bonded through a middle coating with a variable thickness within which is located either the dislocation dipole, the circular Eshelby inclusion or the concentrated couple. The Eshelby inclusion undergoes either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. As a result, the influence of any one of the dislocation dipole, the circular Eshelby inclusion or the concentrated couple is cloaked in that their presence will not disturb the prescribed uniform stress fields in both surrounding half-planes.

Estimation of the heterogeneity of stress fields using misfit angles in focal mechanisms

The stress field of the Earth's crust is spatially heterogeneous on various scales. The focal mechanism of an earthquake provides important information associated with the stress state at the seismogenic depths. These focal mechanisms are affected by the pore fluid pressure and/or friction coefficient. In this study, we focused on misfit angles, particularly the difference in the angle between the slip direction in an observed focal mechanism and the resolved shear stress direction that can be obtained using the directions of the principal stress axes and the stress ratio of a reference stress field. Resolved shear stress does not depend on the fault friction or pore fluid pressure, allowing us to use misfit angles to evaluate the heterogeneity of stress fields if the effect of an error in a focal mechanism has been properly removed. By statistically evaluating the error in focal mechanisms using a Bayesian approach, we evaluated the misfit angles of the aftershocks of the 2000 Western Tottori Earthquake, in which numerous focal mechanisms could be precisely determined using dense seismic observation. The spatial distribution of the misfit angles of the aftershocks suggested that the stress field in the aftershock region, particularly in the southern portion of the region, was in a heterogeneous state. We also demonstrated that this spatial variability cannot be explained solely by errors in the focal mechanisms. If we assume a uniform stress field before the mainshock, the observed cumulative curve of misfit angles can be explained by coseismic stress changes and the background stress field's depth gradient of differential stress of 3.2 MPa/km.

A circular inhomogeneity interacting with an open inhomogeneity designed to admit internal uniform hydrostatic stresses

We use analytic continuation and conformal mapping techniques to prove the existence of an internal uniform hydrostatic stress field inside a two-dimensional non-parabolic open inhomogeneity interacting with a nearby circular elastic inhomogeneity. The circular inhomogeneity and the surrounding matrix have distinct Poisson's ratios but identical shear moduli. The single complex constant appearing in the conformal mapping function is determined analytically. A simple condition is found that ensures that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. We also establish the existence of an internal uniform hydrostatic stress field inside a non-parabolic open inhomogeneity near an arbitrary number of circular elastic inhomogeneities. The circular inhomogeneities and the matrix have distinct Poisson's ratios but a common shear modulus. All of the complex constants in the conformal mapping function are uniquely determined by solving a set of non-linear equations via iterations.

Modal identification experimental investigations and calculation method for biaxial tensioned membranes

Abstract The utilization of membrane structures in space engineering has attracted considerable attention in recent decades due to ultra-light weight characteristics. The measurement and calibration of dynamic performance on ground is challenging due to air effects and equipment limitation. In this paper, a novel vacuum-level vibration-testing device is developed to realize an environment of adjustable air pressure. An improved membrane tension system is utilized to guarantee simply-supported boundary conditions and a uniform stress field, which is more consistent with theoretical and simulation requests. The modal identification method is modified based on only four measuring points because of limited space of the vacuum tank, which is a combination of FFT (Fast Fourier Transform) and SSI (Stochastic Subspace Identification) methods. The effects of non-uniform stress field and air pressure on membrane modal are investigated based on the proposed testing device and method. The effect of the non-uniform stress field on membrane frequency is analyzed by theoretical, numerical and experimental methods, and the results show that the non-uniform stress field resulting from uneven tensions has little effect on the frequency of biaxial tensioned membranes. Meanwhile, an empirical formula is proposed to predict the first three structural frequency of a specific biaxial tensioned membrane under different air pressures.

Activation of optimally and unfavourably oriented faults in a uniform local stress field during the 2011 Prague, Oklahoma, sequence

SUMMARYThe orientations of faults activated relative to the local principal stress directions can provide insights into the role of pore pressure changes in induced earthquake sequences. Here, we examine the 2011 M 5.7 Prague earthquake sequence that was induced by nearby wastewater disposal. We estimate the local principal compressive stress direction near the rupture as inferred from shear wave splitting measurements at spatial resolutions as small as 750 m. We find that the dominant azimuth observed is parallel to previous estimates of the regional compressive stress with some secondary azimuths oriented subparallel to the strike of the major fault structures. From an extended catalogue, we map ten distinct fault segments activated during the sequence that exhibit a wide array of orientations. We assess whether the five near-vertical fault planes are optimally oriented to fail in the determined stress field. We find that only two of the fault planes, including the M 5.7 main shock fault, are optimally oriented. Both the M 4.8 foreshock and M 4.8 aftershock occur on fault planes that deviate 20–29° from the optimal orientation for slip. Our results confirm that induced event sequences can occur on faults not optimally oriented for failure in the local stress field. The results suggest elevated pore fluid pressures likely induced failure along several of the faults activated in the 2011 Prague sequence.

An analysis of the factors that control fault zone architecture and the importance of fault orientation relative to regional stress

AbstractThe moment magnitude 7.2 El Mayor–Cucapah (EMC) earthquake of 2010 in northern Baja California, Mexico produced a cascading rupture that propagated through a geometrically diverse network of intersecting faults. These faults have been exhumed from depths of 6–10 km since the late Miocene based on low-temperature thermochronology, synkinematic alteration, and deformational fabrics. Coseismic slip of 1–6 m of the EMC event was accommodated by fault zones that displayed the full spectrum of architectural styles, from simple narrow fault zones (< 100 m in width) that have a single high-strain core, to complex wide fault zones (> 100 m in width) that have multiple anastomosing high-strain cores. As fault zone complexity and width increase the full spectrum of observed widths (20–200 m), coseismic slip becomes more broadly distributed on a greater number of scarps that form wider arrays. Thus, the infinitesimal slip of the surface rupture of a single earthquake strongly replicates many of the fabric elements that were developed during the long-term history of slip on the faults at deeper levels of the seismogenic crust. We find that factors such as protolith, normal stress, and displacement, which control gouge production in laboratory experiments, also affect the architectural complexity of natural faults. Fault zones developed in phyllosilicate-rich metasedimentary gneiss are generally wider and more complex than those developed in quartzo-feldspathic granitoid rocks. We hypothesize that the overall weakness and low strength contrast of faults developed in phyllosilicate rich host rocks leads to strain hardening and formation of broad, multi-stranded fault zones. Fault orientation also strongly affects fault zone complexity, which we find to increase with decreasing fault dip. We attribute this to the higher resolved normal stresses on gently dipping faults assuming a uniform stress field compatible with this extensional tectonic setting. The conditions that permit slip on misoriented surfaces with high normal stress should also produce failure of more optimally oriented slip systems in the fault zone, promoting complex branching and development of multiple high-strain cores. Overall, we find that fault zone architecture need not be strongly affected by differences in the amount of cumulative slip and instead is more strongly controlled by protolith and relative normal stress.

Elastic-Plastic-Damaged Zones around a Deep Circular Wellbore under Non-Uniform Loading

Wellbores are largely constructed during coal mining, shale gas production, and geothermal exploration. Studying the shape and size of the disturbed zone in surrounding rock is of great significance for wellbore stability control. In this paper, a theoretical model for elastic-plastic-damage analysis around a deep circular wellbore under non-uniform compression is proposed. Based on the elastoplastic softening constitutive model and Mohr-Coulomb strength criterion, the analytical expressions of stresses in the elastic, plastic and damaged zones around a circle wellbore are derived. Further, the boundary line equations among the three zones are obtained according to the conditions of stress continuity. Then, the influence rules of non-uniform in-situ stress and mechanical parameters on the stress distribution and plastic zone size in surrounding rock mass are analyzed. The plastic and the damaged zones are both approximately elliptical in shape. When the lateral stress coefficient of the in-situ stress field takes the value 1, the model degenerates into the Yuan Wenbo’s Solution. If the brittleness coefficient of the surrounding rock is 0, the model degenerates into the Kastner’s Equation. Finally, the results are compared with those under two special cases (in the elastoplastic softening rock under a uniform stress field, in the ideal elastoplastic rock under a non-uniform stress field) and a common approximation method (the perturbation method).

Locally exact asymptotic homogenization of periodic materials under anti-plane shear loading

A new implementation of the asymptotic homogenization theory for anti-plane shear loading is constructed based on locally exact elasticity solutions of unit cell problems at different orders in the asymptotic field expansion. The approach is a generalization of the previously developed locally exact homogenization theory for the determination of homogenized elastic moduli under uniform stress fields. The new theory has been developed for periodic materials in the presence of stress gradients and non-vanishing microstructural scale relative to the structural dimensions. The local interior unit cell problems are solved exactly up to the third order using Fourier series representations of the microfluctuation functions. The exterior unit cell periodic boundary value problems at different orders are tackled by an asymptotic extension of the previously introduced balanced variational principle for periodic materials. The effectiveness of the theory is verified by solving a structural problem with varying number of inclusions directly, and comparing the local fields with those reconstructed by the locally exact asymptotic homogenization approach. The theory is also employed to assess the accuracy of an uncoupled homogenization-localization approach in the reconstruction of local stress fields in periodic materials with different levels of microstructural refinement in the presence of stress gradients. The present contribution provides the foundation for further extension of the new locally exact asymptotic homogenization theory to in-plane loading problems of unidirectionally reinforced periodic materials.

Uniform fields inside two interacting non-parabolic and non-elliptical inhomogeneities

With the aid of conformal mapping for a doubly connected domain, we prove the existence of internal uniform stress fields inside two interacting elastic inhomogeneities: one of non-parabolic open shape and the other of non-elliptical closed shape, when the matrix is subjected to uniform remote anti-plane and in-plane stresses. The uniformity property is unconditional for anti-plane elasticity but conditional for plane elasticity. The internal uniform stress fields are independent of the specific open and closed shapes of the two inhomogeneities. Typical numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed theory.

A circular Eshelby inclusion interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

Using conformal mapping techniques and analytic continuation, we prove that when subjected to anti-plane elastic deformations, a non-parabolic open inhomogeneity continues to admit an internal uniform stress field when a circular Eshelby inclusion is placed in its vicinity and the surrounding matrix is subjected to uniform remote stresses. Explicit expressions for the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion are obtained. The internal uniform stress field is independent of the shape of the inhomogeneity and the presence of the circular Eshelby inclusion, whereas the existence of the circular Eshelby inclusion exerts a significant influence on the shape of the non-parabolic open inhomogeneity as well as on the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion itself.

Design of a New Arcan Fixture for In-plane Pure Shear and Combined Normal/Shear Stress Characterization of Fiber Reinforced Polymer Composites

Engineering elastic constants and strength parameters shall be known to the designer in order to predict the structural response of composite materials. These properties shall be determined in various directions as per the designated standards. Testing of composite materials under the combined effect of in-plane shear and normal stress is of great importance. In this work, a new Arcan fixture has been designed, and then manufactured from aluminum alloy AA7075-T73. This test fixture can be used for uniaxial as well as biaxial testing of fiber reinforced polymers (FRPs) laminates in pure shear, and combined normal/shear stress states. Nevertheless, the fixture can be used for other materials where combined in-plane testing is required including metals and polymers. The special butterfly shape Arcan specimen was optimized based on the parametric study in ABAQUS environment by a Python scripting. The objective of the parametric study was to get a uniform shear stress field in the notch section of the specimen which is the cross-sectional area between the two opposite V-notch roots. The domain of the uniform shear stress-strain distribution in the notch section must be greater enough than the size of the strain gauge rosettes where these can be effectively installed for strain acquisition. The experimental set-ups for the realization of pure shear stress, normal stress, and combined normal/shear stress scenarios are schematically elaborated. The pure shear stress-strain response based on the Arcan test was compared with the tensile testing of ±450 symmetric laminate.