Singular Behavior Of Stresses Research Articles

FEATURES OF STRESSES AT THE APEX OF AN ELASTIC WEDGE, SUPPORTED BY A THIN FLEXIBLE COATING ON THE SIDES

ObjectivesTo study the problem of determining the degree of stress at the apex of a wedge-shaped area in cases where the sides (or one of them) are covered with a thin flexible coating.MethodIt is assumed that the coating is not stretchable. On the other side of the wedge-shaped area, the same coating is assumed to be present; it is either fixed, stress-free or in smooth contact with a rigid base. Mathematically, the problem is reduced to the task of determining the roots of characteristic transcendental equations arising from the existence of a nontrivial solution to the system of linear homogeneous equations.ResultsValues for the specific characteristics of the radial component of a stress tensor are determined for different combinations of boundary conditions and solution angles. In particular, the angles at which the singular behaviour of stresses occurs are determined. The case is considered when a special boundary condition is given on the edge surface, simulating the overlay. Characteristic equations are obtained to determine the index of the degree dependency of the asymptotic solution in its vicinity for four variants of boundary conditions. In two cases, transcendental equations are obtained, which are solved numerically.ConclusionCalculations of the first positive roots of the equations depending on the angle of the edge solution and Poisson's ratio are presented. The values of the angles, at which the singular behaviour of stresses occurs, are determined. In the case of a combination of boundary conditions (III – IV), the singular stress behaviour is observed for the angle ???? = ????/8, while in the case of (III – III) this value is equal to ????/4.

Application of higher‐order strain gradient theories in the phase‐field modelling of fracture mechanics

AbstractThe simulation of crack propagation has always been a challenging task from a computational point of view. In the past, many mathematical and mechanical assumptions had to be taken into account to successfully simulate this process. The problem of tracking the sharp crack surface was among the first issues which was alleviated by the introduction of phase‐field models. The appearance of stress singularities near the crack tip in the classical continuum mechanics theory is considered to be another source of problem which makes the selection of proper mesh sizes for numerical models very cumbersome, and leads to unphysical results. In the presence of a singularity, the crack propagation starts earlier in problems with a finer discretization which is only natural considering that the strain energy density is higher and increases faster for smaller mesh sizes. Likewise, a loading which normally could not cause a crack to nucleate or propagate inside a structure, will do so in the numerical simulation due to over‐estimated stresses. Therefore, it stands to reason that removing this singular field from the results is a very crucial step in achieving physically meaningful results. The goal of the current contribution is twofold: first, existing models are shown to exhibit a singular stress behavior, and second, a possible remedy is proposed based on the application of higher‐order strain gradients.

Multi-parameter average strain energy density factor criterion applied on the sharp material inclusion problem

The tip of a sharp material inclusion (SMI) embedded in a parent material is considered as the singular stress concentrator. The SMI tip is modelled as a special case of a multi-material junction. The singular stress behaviour is caused by the material mismatch and geometrical discontinuity. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress terms exponent. The exponents are determined as a solution of eigenvalue problem. The factors are calculated by combination of analytical and numerical approaches. The terms can be either singular or non-singular depending on the stress term exponent. When approaching the concentrator, the singular terms become unbounded while the non-singular terms vanish. The non-singular terms increase precision of stress description on larger distances from the point of singularity. In some cases they provide the only means to describe the stress field well. Because of the singular stress behaviour near SMI tip, this location is prone to crack initiation. The crack initiation conditions are calculated by the average strain energy density factor (SEDF) criterion. Contrary to the case of a crack, the direction of minimum of SEDF changes with distance from the singular point. Therefore, an averaged value over specific distance is used. If the specific distance is relatively large, the employment of non-singular terms in multi-parameter criterion can significantly improve the critical parameters prediction. Thanks to that e.g. the particle composite design can be optimized.

Analytical-Numerical Determination of Stress Distribution around a Tip of Polygon-Like Inclusion

A stress distribution in vicinity of a tip of polygon-like inclusion exhibits a singular stress behaviour. Singular stresses at the tip can be a reason for a crack initiation in composite materials. Knowledge of stress field is necessary condition for reliable assessment of such composites. A stress field near the general singular stress concentrator can be analytically described by means of Muskhelishvili plane elasticity based on complex variable functions. Parameters necessary for the description are the exponents of singularity and Generalized Stress Intensity Factors (GSIFs). The stress field in the closest vicinity of the SMI tip is thus characterized by 1 or 2 singular exponents (1 - λ) where, 0<Re (λ)<1, and corresponding GSIFs that follow from numerical solution. In order to describe stress filed further away from the SMI tip, the non-singular exponents for which 1<Re (λ), and factors corresponding to these non-singular exponents have to be taken into account. Analytical-numerical procedure of determination of stress distribution around a tip of sharp material inclusion is presented. Parameters entering to the procedure are varied and tuned. Thus recommendations are stated in order to gain reliable values of stresses and displacements.

Reconstruction of a 2D stress field around the tip of a sharp material inclusion

The stress distribution in the vicinity of a sharp material inclusion (SMI) tip exhibits a singular stress behavior. The strength of the stress singularity depends on material properties and geometry. The SMI is a special case of a general singular stress concentrator (GSSC). The stress field near a GSSC can be analytically described by means of Muskhelishvili plane elasticity based on complex variable function methods. Parameters necessary for the description are the exponents of singularity and generalized stress intensity factors (GSIFs). The stress field in the closest vicinity of an SMI tip is thus characterized by 1 or 2 singular exponents (λ-1), for which 0<Re(λ)<1, and corresponding GSIFs. In order to describe a stress field further away from an SMI tip, the non-singular exponents, 1<Re(λ), and factors corresponding to these non-singular exponents have to be taken into account. For given boundary conditions of the SMI, the exponents are calculated as an eigenvalue problem. Then, by formation of corresponding eigenvectors, the stress or displacement angular functions for each stress or displacement series term are constructed. The contribution of each stress or displacement series term function to the total stress and displacement field is given by the corresponding GSIF. The GSIFs are calculated by the over deterministic method (ODM), which finds a solution of an over-determined system of linear equations by the least squares method. On the left-hand side of the system are the displacement series term functions multiplied by unknown GSIFs, while the right-hand is formed by results of finite element analysis (FEA). Thus the results of FEA, namely nodal displacements in the radial and tangential direction, are employed in order to obtain the GSIFs. In the numerical example, the stress field for particular bi-material configurations and geometries is reconstructed using i) singular terms only ii) singular and non-singular terms. The reconstructed stress field polar plots are compared with FEA results.

Micro/macro fatigue crack growth rate model for 2024-T3 aluminum panel

Purpose – Fatigue crack growth rate data for 2024-T3 aluminum are found using three parameters d*, σ* and μ* for short and long cracks for Regions I-III in conventional fatigue. Asymptotic solution of a line crack with a micro-tip is found to yield a singular stress behavior of order 0.75 in contrast to the 0.50 order known for the macrocrack. The difference is due to the micro-macro interaction effects. The three parameters account for the combined effects of load, material and geometry via the tip region. Data for short and long cracks lie on a straight with a slope of about 3.9-4.8 for R values of 0.286-0.565. The results were based on an initial crack a1 mm where a is the half length for a central crack panel. The paper aims to discuss these issues. Design/methodology/approach – The belief that specimen fatigue data could assist the design of structural components was upended when FAA discovered that the NASGRO FCGD are not valid for short cracks that are tight and may even be closed. The regular ΔK vs da/dN model was limited to long cracks. The issue become critical for short cracks connecting the long ones of a few mm to cm or even m according to da/dN for the same crack history. The danger of short/long fatigue crack growth (SLFCG) prompted FAA to introduce an added test known as Limit of Validity (LOV), a way of setting empirical limits for structural components. The dual scale SLFCG data from ΔK micro/macro provide support for the LOV tests. Findings – Data for short and long cracks lie on a straight with a slope of about 3.9-4.8 for R values of 0.286-0.565. The single dual scale relation on ΔK micro/macro can switch from microscopic to macroscopic or vice and versa. The difference is fundamental. Order other than 0.75 can be obtained for simulating different microstructure effects as well as different materials and test conditions. Originality/value – Scale shifting from short to long fatigue cracks for 2024-T3 aluminum is new. The crack driving force is found to depend on the crack tightness. The sigmoidal curve based on the regular ΔK plot disappeared. The data from ΔK micro/macro for short cracks may supplement the FAA LOV tests for setting more reliable fatigue safe limits.

Stress analysis for a cavity flow of a memory fluid

The paper deals with the plane creeping flow of a viscoelastic liquid within an idealized rotor–stator system consisting of a rotating cylindrical drum and a fixed internal plate. The rheological properties of the liquid are modelled by a single-integral constitutive equation which considers a memory but disregards shear-thinning features. It is pointed out that the flow kinematics of such a memory fluid deviates only marginally from the Newtonian kinematics whereas the stress field is much more complicated. Therefore, the Newtonian velocity field is applied which can be represented in a closed analytical form using elementary complex-valued functions. The formulation allows computing the strain history and the resulting stress state with little numerical effort. A careful asymptotic analysis close to the corners yields details of the singular stress behaviour which differs markedly from the Newtonian characteristic. Finally, a second-order approximation being valid under certain restrictions leads to explicit analytical expressions also concerning the viscoelastic stress components. Altogether, a well-founded insight as regards the complex stress distribution and the effect of Deborah number is attained.

Stress singularities and the formation of birefringent strands in stagnation flows of dilute polymer solutions

We consider stagnation point flow away from a wall for creeping flow of dilute polymer solutions. For a simplified flow geometry, we explicitly show that a narrow region of strong polymer extension (a birefringent strand) forms downstream of the stagnation point in the UCM model and extensions, like the FENE-P model. These strands are associated with the existence of an essential singularity in the stresses, which is induced by the fact that the stagnation point makes the convective term in the constitutive equation into a singular point. We argue that the mechanism is quite general, so that all flows that have a separatrix going away from the stagnation point exhibit some singular behaviour. These findings are the counterpart for wall stagnation points of the recently discovered singular behaviour in purely elongational flows: the underlying mechanism is the same while the different nature of the singular stress behaviour reflects the different form of the velocity expansion close to the stagnation point.

Simultaneity of multiscaling for macro–meso–micro damage model represented by strong singularities

The state of affairs near a crack tip is very different when viewed at the different scales, say from the macroscopic to the atomic. These different appearances have intrigued solid state physicists, mechanists and material scientists for decades. Although they are physically connected, it is not obvious what roles would their mathematical relationships play? Investigations associated with scaling in size and time, evolution of material damage, multiscaling of macro-, meso- and micro-mechanics theories are presumably topics that could provide insights to the behavior of the internal structures of matters and their interactions at the different scales. When seeking for analytical models that can address multiscaling, continuum mechanics seems to give way to the particle approach when the scale reduces to that for the atoms. It is also not clear whether the particle and continuum view points should be regarded as separate disciplines for their differences have generated much discussion with little to gain. There is the general feeling that a re-examination of what has been taken for granted may be in order. Based on the comments made earlier, a multiscale crack damage model is developed by duplicating the physics much like the capability of an optical or electronic microscope. The model contains the macroscopic, mesoscopic and microscopic damage represented by singularities. The results at the different scale levels are connected via stress and displacement compatibility conditions in continuum mechanics and they can switch from one scale to another in a discrete manner where equilibrium is assumed within the range of an arbitrarily defined scale. The sizes involved at the different scales depend on the material, load and geometry and they are determined by solving two highly non-linear equations. Simplicity is gained by using singularities to represent the geometrical discontinuities such as intergranular or transgranular failure. This discussion is limited to a particular group of material damage that can be modelled by the so-called strong singularities as compared to the r −0.5 singularity at the continuum level. The strong singularities would correspond to 0.25 < λ* < 0.5 where λ* = 0.5 corresponds to the singular stress behavior for a line crack in an elastic medium. The lowest real part of the eigenvalue λ is denoted by λ*. In passing, it should be pointed out that the 1/ r singularity (i.e. λ* = 0) corresponds to an edge dislocation which will not be included in this discussion. In order not to cluster the presentation with unnecessarily complex notations of distinguishing the difference between the macroscopic and microscopic constants, the same set of constants will be assumed. Besides, there are fundamental issues with reference to the definition of microscopic material properties whether they should be obtained from testing micro-size specimens or considered as local material properties as part of the bulk. These two contrasting views are not the same. The choice to use one or the other remains debatable. The emphasis here is placed on relating strong singularity group to the geometry of a micro-notch where one notch edge is free and the other is fixed. It is connected to a transition or mesoscopic zone. The notch angle can be varied to model different degrees of intergranular opening. The microscopic non-homogeneous features are reflected by the rotations of the lines with different notch angles in a log–log plot of the strain energy density function with distance. Keep in mind that when the same near crack tip region is viewed at the macroscopic scale, the state of affairs are homogeneous. This in essence is what a microscope does. The objective of this work is to demonstrate that such a capability can be built into an analytical model with all of the complexities of the microstructure, the description of which are made possible by the singularity representation approach.

Singular stress fields near contact boundaries in a composite bolted joint

Singular stresses arising in the neighborhood of contact surfaces introduced in laminated orthotropic plates by mechanical joining with clamp-up were investigated by using local asymptotic solutions and full-field numerical analysis. Three-dimensional B-spline approximation of the displacements and a penalty function-based contact solution was used in the numerical analysis. Recent work has shown that fracture in bolted composite joints may initiate near the outer edge of the bolt head or washer away from the hole edge, particularly if the joint is preloaded. Material and geometric discontinuities exist in these regions, resulting in singular stress behavior. Asymptotic stress analysis was performed to obtain the power of singularity in these regions as a function of the bolt-head (washer) stiffness. Frictionless contact conditions were assumed. It was found that the characteristics of the stress singularity for such practically important combinations as titanium bolt-head and carbon fiber composite plate are similar to a crack in terms of the power of singularity and uniqueness of the singular term. Coefficients of the singular terms of the asymptotic expansion were determined by comparison with the numerical solution in the close vicinity of the singular contour. Good agreement between the asymptotic and numerical solution in the transition regions was observed.

Three-dimensional stress analysis of textile composites: Part I. Numerical analysis

Three-dimensional stress analysis in a unit-cell of a plain-woven composite was performed by using B-spline displacement approximation. The spline approximation provides continuity of displacement and stress components within each yarn and matrix subregion. Two types of unit-cell problems with and without inter-yarn delamination were considered. A penalty function approach along with a contact surface characteristic function was used to obtain a full-field numerical solution for the frictionless contact problem between delaminated yarn surfaces. Yarn interfaces at yarn-crossover locations represent three-material wedge-type regions resulting in singular stress behavior. In the case of unit-cells with perfect bonding between the yarn interfaces, the numerical values of the inter-yarn normal stress did not exhibit trends typical for unbounded stress behavior, whereas the inter-yarn shear stress components displayed discontinuous behavior typical for numerical results in the vicinity of the stress singularity. In the presence of the delamination, both the inter-yarn normal and shear stress components exhibited unbounded behavior near the singularity. Notably, the inter-yarn normal stress showed signs of singular behavior in both cases of open and closed delaminations. Due to the stress singularity that exists at yarn-crossover locations containing three materials (yarn–yarn–matrix) interface intersections, the full-field numerical solution, even with high-order approximation functions, was not able to capture the directional nonuniqueness of the stress values in the vicinity of the singularity, and therefore calls for incorporation of the asymptotic singular stress analysis, which will be given in a follow-on paper [Sihn and Roy, International Journal of Solids and Structures (accepted for publication)].

Stresses and stress intensities at notches: ‘anomalous crack growth’ revisited

In this paper, a new notch stress estimation scheme is presented within the context of conventional finite element solutions. The estimation method is based on a separation of an actual notch stress state into two parts. One part is a far-field stress in the form of membrane and bending components that satisfy far-field equilibrium conditions, and the other is a self-equilibrating part that provides an effective measure of the notch stress state. The self-equilibrating part can be directly related to notch geometry and captured using a rather coarse finite element model. The singular stress behavior at an arbitrary notch is then described by a closed form solution formulated using the self-equilibrating part of the stress state for a small crack emanating from a notch. The corresponding stress intensity factor solutions are then presented by considering both the far-field stress (also called structural stress) and self-equilibrating notch stress. The stress intensity solutions are formulated using existing solutions for typical simple crack geometry. One important feature of the notch stress and stress intensity solutions is that the current solutions not only capture the singular characteristics as the notch tip is asymptotically approached, but also recover the far-field (or nominal) stress state. Finally, the effectiveness of the present notch stress estimation scheme is demonstrated by using a series of well-known short crack growth data exhibiting ‘anomalous growth’. It has been found that, for instance, the anomalous growth discussed in, for example, Fat Eng Mat Struct 6 (1983) 315; Eng Fract Mech 29 (1988) 301; as well as Surface Crack Growth: Models, Experiments, and Structures (1990) 333, can be unified with long crack data as straight lines, without resorting crack closure considerations. As a result, a two-stage crack growth model is proposed within the context of K-dominant crack growth. The first stage is dominated by the notch-induced self-equilibrating part of the stress state and the second stage is dominated by the equivalent far-field stress state or structural stresses that satisfy equilibrium conditions and can be effectively computed in a mesh-insensitive manner. The implications of these findings on fatigue growth rate data generation and fatigue life predictions will also be discussed.

309 極異方性体中の奇妙な特異応力挙動

309 極異方性体中の奇妙な特異応力挙動

On tensile cracks close to and across the interface between two welded elastic half-spaces

Tensile cracks are often employed to model magma migration in rift zones or within volcanic edifices through lateral or feeding dykes. In a crack model, the overpressure of magma with respect to the horizontal stress in the host rock is assumed to be responsible for dyke opening and propagation. In this paper the most simple heterogeneous medium is considered, made up of two welded half-spaces, characterized by different elastic parameters. The analytical solutions available for the elementary dislocation problem in such a medium are employed to set up an integral equation with generalized Cauchy kernel, representing the condition for static equilibrium. The unknown in such a problem is the dislocation density distribution, whose singular behaviour is studied near the crack tips and near the intersection with the interface between the two media. The problem for a crack entirely embedded in one of the two media is considered first (case A). The resulting Cauchy integral equation is solved by employing a truncated expansion in Chebyshev polynomials, and ;-analytical solutions are given for the displacement and stress fields provided by a constant overpressure within the crack. The problem of a crack terminating at the interface between the two half-spaces is then considered (case B) and a generalized Cauchy equation is obtained in which additional singularities, localized at the crack tip next to the interface, appear in the integral kernel. The singular stress behaviour in the proximity of the crack tip is affected considerably by the presence of the interface. Within case B, after assuming a vanishing rigidity for the upper medium, a crack open at the free surface of a half-space is obtained (case B1). The singular behaviour of the solution is found to change abruptly in this case (with respect to the case of small, non-vanishing rigidity in the upper medium). Finally, a crack crossing the interface between the two half-spaces is considered (case C). The crack is split into two interacting sections, each embedded in a homogeneous half-space, both open at the interface. A system of generalized Cauchy equations is obtained, which is responsible for the appearance of internal singularities in the dislocation density distribution, localized at the intersection between the crack plane and the interface. The order of this singularity depends only upon the elastic parameters of the media in welded contact. Crack opening, interface displacement and stress components are computed employing the method described for case A, assuming a known overpressure within the crack. It appears that layering can be responsible for stress changes, localized along the interface, which may be considerably higher than the overpressure within the dyke. These results may provide useful hints for the interpretation of induced seismicity in rift zones and in volcanic areas.

Singular Stress Behavior in a Bonded Hereditarily‐Elastic Aging Wedge. Part I: Problem Statement and Degenerate Case

Singular Stress Behavior in a Bonded Hereditarily‐Elastic Aging Wedge. Part I: Problem Statement and Degenerate Case

Singular Stress Behavior in a Bonded Hereditarily-Elastic Aging Wedge. Part I: Problem Statement and Degenerate Case

Stress singularity is investigated in a plane problem for a bonded isotropic hereditarily elastic (viscoelastic) aging infinite wedge. The general solution of the operator Lamb equations, which are partial differential equations in space co-ordinates and integral equations in time, respectively, is represented in terms of one-parametric holomorphic functions (the Kolosov-Muskhelishvili complex potentials depending on time) in weighted Hardy-type classes. After application of the Mellin transform with respect to the radial variable, the problem is reduced to a system of linear Volterra integral equations in time. By using the residue theory for the inverse Mellin transform, the stress asymptotics and strain estimates near the singular point are presented here for non-hereditary Dundurs parameters. The general case of the hereditary Dundurs operators is considered in Part 11 (see [21]).

Singular stress behavior at an adhesive interface corner

The singular stress behavior at a bimaterial wedge corner was studied analytically using the Williams' Airy stress function approach. In addition, the Finite Element Method (FEM) and the Finite Element Iterative Method (FEIM) were used to investigate the asymptotic field in adhesive butt joints of various geometries. In particular, the singular power λ and the geometry of the singular stress zones in the joints were analyzed as a function of the elastic properties of the bonded materials. It was found that the singular power λ and the singular zone geometry at the interface corners in the joints are strongly dependent on the elastic properties of the adhesive and adherend. Moreover, the numerical results of the singular powers obtained from the FEIM analysis showed excellent agreement with previously published analytical data.

Interface crack in a nonhomogeneous elastic medium

The linear elasticity problem for an interface crack between two bonded half planes is reconsidered. It is assumed that one of the half planes is homogeneous and the second is nonhomogeneous in such a way that the elastic properties are continuous throughout the plane and have discontinuous derivatives along the interface. The problem is formulated in terms of a system of integral equations and the asymptotic behavior of the stress state near the crack tip is determined. The results lead to the conclusion that the singular behavior of stresses in the nonhomogeneous medium is identical to that in a homogeneous material provided the spacial distribution of material properties is continuous near and at the crack tip. The problem is solved for various values of the nonhomogeneity parameter and for four different sets of crack surface tractions, and the corresponding stress intensity factors are tabulated.

On singular, nine-noded, distorted, isoparametric elements in linear elastic fracture mechanics

Distorted, nine-noded, Lagrangian, isoparametric elements are studied to determine if stress intensity factors can be calculated more accurately with these elements than with eight-noded quarter-point elements. To achieve square-root singular stress behavior in the nine-noded element, the mid-side nodes adjacent to the crack tip are moved to the quarter-point and the nodal point within the element is positioned along the element diagonal, at least at the quarter-point. Several optimal positions for this node are determined. Through two numerical examples, it is seen that the eight-noded element performs nearly as well, and sometimes better than the nine-noded element.

Some New Results on Edge Effect in Symmetric Composite Laminates

An improved numerical (finite element) procedure is developed to investigate the stress field in symmetrically laminated composites of finite dimensions. Emphasis is placed on assessing the singular behavior of stresses in regions close to ply-interfaces and the exposed free edge. An accurate evaluation of these stresses will help to better understand the failure process of composite laminates. Mechanical load in the form of uniaxial tension and environmental variables such as temperature and humidity exposures are considered in the formulation. Results tor lam inates under uniaxial tension are presented in this paper; thermal and hygroscopic stresses will be reported in a subsequent paper.