Distribution of stresses near V-shaped notches in the complex stressed state

By the method of singular integral equations, we obtain the solution of a two-dimensional problem of the elasticity theory for a plane containing a semiinfinite rounded V-notch under complex loading. On the basis on this solution, we establish the relationships between the stress intensity factors at the tip of the sharp V-notch and the maximum stresses and their gradients at the tip of the corresponding rounded notch. For finite bodies with V-notches, the presented solutions are obtained as asymptotic dependences for small radii of rounding of the tips. The deduced relationships can be used to perform the limit transitions and find the stress intensity factors at the tips of sharp V-notches on the basis of the solutions obtained for the corresponding rounded stress concentrators. The efficiency of the method is illustrated for the problem of determination of the stress intensity factors at the vertices of a rectangular hole in the elastic plane.     

Effect of non-singular stress on the brittle fracture of V-notched structure

The traditional brittle fracture criteria for V-notched structures are established on the base of the singular stress field at a V-notch tip where only two singular stress terms are adopted. The non-singular stress terms also play a significant role in determining the stress and strain fields around a V-notch tip, which in turn could affect the fracture character of V-notched structures predicted by the fracture mechanics criteria. In this paper, the effect of the non-singular stress on the brittle fracture properties for the V-notch problem is discussed. Firstly, the stress field around a V-notch tip is described by the Williams asymptotic expansions. At the same time, the stress field far from the V-notch tip is modeled by the conventional boundary element method since there is no stress singularity. By the combination of the Williams asymptotic expansions and the boundary integral equations, the complete stress field at a V-notch tip including several non-singular stress terms can be obtained. Then, three different brittle fracture criteria are introduced to predict the critical loading and initial crack propagation direction of V-notched structures under mixed-mode loading. Comparing with the existed experimental results, it can be found that the degree of accuracy of the predicted results when taking into account the non-singular stress terms is much higher than the predicted ones neglecting the non-singular stress.     

Analytical solution for the singular stress distribution due to V-notch in an orthotropic plate material

On the basis of general solutions of two-dimensional linear elasticity, displacement and singular stress fields near the singular point in orthotropic materials are derived in closed form expressions. According to the presented expressions, analysis formulas of displacement and singular stress fields near the tip of a V-notch under the symmetric and the anti-symmetric modes are obtained subsequently. The open literatures devoted to developing stress singularity near the tip of the V-notch in anisotropic or orthotropic materials. In this study, however, not only direct eigenequations were derived, but also the explicit solutions of displacement and singular stress fields were obtained. At the end, the correctness of the formulas of the singular stress field near the tip of the V-notch has been verified by FEM analysis.     

Evaluation of multiple stress singularity orders of a V-notch by the boundary element method

A new technique concerning the evaluation of multiple stress singularity orders of a V-notch with the boundary element method is proposed. For linear elastic material, the asymptotic displacement field in a V-notch tip region is expressed as a series expansion with respect to the radial coordinate from the V-notch tip. The series expansion of the asymptotic field is then substituted into the boundary integral equation established on the V-notch. By boundary element discretization, the boundary integral equation is transformed into an eigen-equation with respect to the stress singularity orders. Hence the eigen-values, which are the singularity orders, can be obtained simultaneously by solving the eigen-equation. An evident feature of the present method is that the real and complex singularity orders together with the non-singularity ones can be obtained at the same time.     

Asymptotic thermal and residual stress distributions due to transient thermal loads

The paper deals with the development of thermal and residual stress distributions arising from the solidification of a fusion zone near a V-notch tip. A set of numerical solutions of the problem was carried out under the hypothesis of generalized plane strain conditions by means of SYSWELD code. The intensity of the thermal and residual asymptotic stress fields at the sharp V-notch tip was studied for a given V-notch specimen geometry and a predefined fusion zone dimension after simulations on materials with different thermal, mechanical and phase transformation properties and after changing the clamping conditions at the specimen's boundary. The results were compared in terms of the elastic or elastic-plastic notch stress intensity factors giving a contribution to the interpretation of the experimental behaviour of welded joint.     

Stress intensity factors for cracks at the vertex of a rounded V-notch

The method of singular integral equations was applied to determine the stress intensity factors for a system of cracks emanating from the vertex of an infinite rounded V-notch subjected to symmetric loading. The numerical values were obtained for two cases—the case of a single crack and the case of a system of two cracks of equal length. The influence of the rounding radius of the vertex of the notch and its opening angle on the stress intensity factors at the crack tips was analyzed. The solution obtained as a result has a general nature—the stress intensity factors at the crack tip are expressed as a function of the V-notch stress intensity factor and, hence, this solution could be treated as an asymptotic relation for finite bodies with deep V-notches subjected to symmetric loads.     

爆炸载荷下边界锐V型切口尖端裂纹扩展行为

为了研究爆炸应力波作用下板条边界锐V型切口端部的动态扩展行为,采用动态焦散线实验方法,进行了爆炸载荷下板条边界锐V型切口端部裂纹扩展规律的实验研究。研究结果表明,爆炸应力波作用下,板条试件边界锐V型切口端部的扩展过程中,裂纹扩展速度、加速度和切口端部动态应力强度因子随时间是波动变化的。扩展速度最大值达到210m/s,裂纹扩展加速度最大值为9.47Mm/s2,切口端部动态应力强度因子KdⅠ最大值为0.6MN/m3/2。     

Two-dimensional fracture mechanics problems for solids with sharp and rounded V-notches

The survey of research on the problems of fracture mechanics of solids with V-shaped notches is discussed. The methods of solution of plane problems of the elasticity theory for domains with angular points on boundary contours are considered. Special attention is paid to the results of the research which authors obtained applying the unified approach to the solution of stress concentration problems in the neighborhood of the vertices of sharp and rounded V-notches. The solutions of elastostatics problems for domains with rounded V-shaped notches were obtained by singular integral equation method. Based on the relation between stress intensity factor at the sharp notch vertex and the stress concentration factor at the vertex of rounded V-notch (notch with straight edges and vertex rounded with circular arc), the stress intensity factors for V-notches in various domains were obtained. The presented examples include notch stress intensity factors for periodically placed notches in the edge of half-plane, diamond and lens-like holes in infinite plane. Through limit transition from the system of periodically placed diamond-shaped ovals to infinite double-sided notch, the stress concentration factors for rounded deep V-notch and the notch stress intensity factors for sharp notch were obtained. The comparison between calculated data and known solutions for the hyperbolic notch documents strong influence of the notch shape in the neighborhood of the rounded vertex on the stress concentration values for small vertex radii. The numerical results are presented not only in graphical form but also the numerous approximation formulas are given. Based on the solution of the nonlinear problem of rounded V-notch with plastic strip at the vertex, the new deformation fracture criterion is proposed. This criterion contains only standard strength parameters and it is intended to determine fracture toughness of the elements with rounded notches made of quasi-brittle materials.     

爆炸载荷下边界锐V型切口尖端裂纹扩展行为

为了研究爆炸应力波作用下板条边界锐V型切口端部的动态扩展行为,采用动态焦散线实验方法,进行了爆炸载荷下板条边界锐V型切口端部裂纹扩展规律的实验研究.研究结果表明,爆炸应力波作用下,板条试件边界锐V型切口端部的扩展过程中,裂纹扩展速度、加速度和切口端部动态应力强度因子随时间是波动变化的.扩展速度最大值达到210m/s,裂...     

T‐stress corrected notch stress intensity factors with application to welded lap joints

The definition, content and application of the notch stress intensity factors (NSIFs) characterizing the stress field at rounded slit tips (keyholes) is discussed. The same is done in respect of the T‐stress transferred from the corresponding pointed slit tips. A T‐stress based correction of the NSIF K_1,ρ is found to be necessary. The applicability of the T‐stress term supplemented by higher‐order terms in Williams’ solution to the slit tip stresses in tensile‐shear loaded lap joints is discussed in more detail. The role of the T‐stress in constituting the near‐field stresses of rounded slit tips is shown to cause a difference between internal and external slit tip notches. The notch stress equations for lap joints proposed by Radaj based on structural stress and by Lazzarin based on a finite element model of the rounded notch are reconsidered and amended based on the derivations above.     

Relationship between the stress intensity and stress concentration factors for sharp and rounded notches

The method of singular integral equations is used to find the solution of the plane problem of the theory of elasticity for a plane containing an infinite V-shaped rounded notch. This enables us to establish the relationship between the stress intensity factor at the tip of the sharp V-shaped notch, the stress concentration factor for the corresponding rounded notch, and the radius of curvature at the notch tip. It is shown that the indicated relationship is not unique. Indeed, for the same curvature at the notch tip, we get different dependences for different shapes of its vicinity. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 6, pp. 17–26, November–December, 2006.     

Stress field at V-notch tip in polymer materials using digital gradient sensing

The digital gradient sensing (DGS) technique was used to study the stress field at the V-notch tip of polymer materials. First, DGS governing equations at the V-notch tip were deduced using the elastic singular stress fields. Then, theoretical angular deflections of light rays at the V-notch tip were simulated, and the effect of notch angle on the angular deflections was analyzed. Finally, the DGS experiments were conducted, and the angular deflection contours were obtained. The results show that the stress intensity factors at the V-notches extracted from the angular deflections agree well with the results calculated from the finite element method.     

Anti-plane stress state of a plate with a V-notch for a new class of elastic solids

The main purpose of this study is to investigate the efficacy and usefulness of a class of recently proposed models that could be reasonable candidates for describing the response of brittle elastic materials. The class of models that are considered allows for a non-linear relationship between the linearized elastic strain and the Cauchy stress, and this allows one to describe situations wherein the stress increases while the strain yet remains small. Thus one would be in a position to model the response of brittle elastic bodies in the neighborhood of the tips of cracks and notches. In this paper we study the behavior of such models in a plate with a V-notch subject to a state of anti-plane stress. This geometrical simplification enables us to characterize the governing equation for the problem by means of the Airy stress function, though the constitutive relation is a non-linear relation between the linearized strain and the stress. We study the problem numerically by appealing to the finite element method. We find that the numerical solutions are stable. We are able to provide some information regarding the nature of the solution near the tip of the V-notch. In particular we find stress concentration in the vicinity of the singularity.     

The reflected and transmitted shadow methods for the study of sharp V-notched plates under pure bending

An experimental study for investigating the constrained zones at the tip of edge V-notches in plexiglas plates and its dependence on the ratio depth of notch to the width of the plate, under static pure bending moment is developed.The optical reflected and transmitted shadow methods were used for various angles of the V-notch under different values of bending moment and for various ratios of depth of V-notch to width of the plate.The apparatus used is of the balance type to assure that the applied forces are distributed equally and that only pure moment forces are responsible for the stress field. The apparatus is suitable for both methods, the reflected shadow and the transmitted shadow, which are also compared.Experimental evidence shows that, the constrained zone exhibits a maximum size at the one extreme of angle of V-notch zero degrees, and a minimum at the other extreme of 180°, where it vanishes. Between the two extremes a second maximum is found at about 73°, and a second minimum at about 35°. At first approximation, it is deduced that the stress distribution, as the apex of a V-notch of angle up to about 90°, can be approximated by the stress distributions at the tip of a straight crack, under the same conditions.     

Deformation Fracture Criterion for Bodies with V-Notches under Symmetric Loading

The limiting equilibrium of bodies with V-notches is extensively studied by using the force fracture criterion according to which the stress intensity factor at the notch tip is equal to its critical value. In this case, the critical stress intensity factor depends on the opening angle of the V-notch. By using the solution of the problem of plastic strips in a plate with V-notch, we develop a deformation fracture criterion based on the use of standard characteristics of the material and independent of the geometry of the specimen.     

Asymptotic characterisation of nearly-sharp notch root stress fields

A solution procedure is developed for characterising the stress state at the root of a notionally sharp notch, but possessing a small root radius, using two nested asymptotic solutions: an outer asymptote representing a sharp semi-infinite V-notch and an inner solution representing a semi-infinite rounded notch. The two asymptotes are matched to each other remote from the notch root, and to an example finite notch using a generalised stress intensity factor. It follows that the characteristic, singular, sharp-notch field diverges from the rounded-notch solution very near the root. On the other hand, the notch in a finite body diverges from the sharp semi-infinite notch in the far field. Providing that the notch root radius is sufficiently small, it follows that there is an intermediate field where the singular field does characterise the behaviour of the finite radiused notch, and this is quantified.Note that this topic has also been investigated in the very recent literature by Gomez and Elices (2004) for brittle components, using the same analytical frame presented here     

Similarities of stress concentrations in contact at round punches and fatigue at notches: implications to fretting fatigue crack initiation

A linear elastic model of the stress concentration due to contact between a rounded flat punch and a homogeneous substrate is presented, with the aim of investigating fretting fatigue crack initiation in contacting parts of vibrating structures including turbine engines. The asymptotic forms for the stress fields in the vicinity of a rounded punch-on-flat substrate are derived for both normal and tangential loading, using both analytical and finite element methods. Under the action of the normal load, P , the ensuing contact is of width 2 b which includes an initial flat part of width 2 a . The asymptotic stress fields for the sharply rounded flat punch contact have certain similarities with the asymptotic stress fields around the tip of a blunt crack. The analysis showed that the maximum tensile stress, which occurs at the contact boundary due to tangential load Q , is proportional to a mode II stress intensity factor of a sharp punch divided by the square root of the additional contact length due to the roundness of the punch, Q /(√( b − a )√ π b ). The fretting fatigue crack initiation can then be investigated by relating the maximum tensile stress with the fatigue endurance stress. The result is analogous to that of Barsom and McNicol where the notched fatigue endurance stress was correlated with the stress intensity factor and the square root of the notch-tip radius. The proposed methodology establishes a 'notch analogue' by making a connection between fretting fatigue at a rounded punch/flat contact and crack initiation at a notch tip and uses fracture mechanics concepts. Conditions of validity of the present model are established both to avoid yielding and to account for the finite thickness of the substrate. The predictions of the model are compared with fretting fatigue experiments on Ti–6Al–4V and shown to be in good agreement.     

Fictitious notch rounding concept applied to V-notches with end holes under mode 3 loading

The present technical note is aimed to provide a closed form expression for the microstructural support factor and for the fictitious notch radius in plates weakened by V-notches with root end-holes. Taking advantage of some recent closed form expressions for the stress distributions due to V-notches with end holes the fictitious notch rounding approach is applied here to mode 3 loading. The factor s for the V-notch with end holes is found to be strongly influenced by the opening angle and the new values are compared with the previous solution available in the literature and dealing with blunt V-notches. To validate the new expressions a comparison is carried out between the theoretical stress concentration factor (SCF) obtained from a rounded V-notch with a fictitiously enlarged end hole (of radius ρ_f) and the effective stress concentration factor obtained by integrating the relevant stress over the microstructural characteristic length (MCL), ρ^*, in a pointed V-notch. A sound agreement is found from the comparison. The range of validity of the present equations are limited to linear elasticity or in those cases where the plastic zone is very small with respect to the MCL of the material.     

A new form of exact solutions for mode I,II, III crack problems and implications

A new form of an exact linear elastic solution is obtained for the problem of a crack in a semi-infinite plate subjected, at infinity, to antiplane stress (Mode III) loading. The use of the conformal mapping technique results in a convenient stress and displacement solution for each point of the semi-infinite domain. For completeness, the solution technique is extended to Mode I, II problems with center-cracks as well as the Mode III V-notch problem. It is shown that the limiting case of the V-notch collapses to the stress functions independently derived for the edge-cut. At distances equivalent to 10% of the crack length away from the crack tip, the exact solutions give stresses about 7.5% greater than the one-term results. Ramifications of the exact solutions to finite-element solutions, elastic-plastic and diffusion problems are discussed.     

Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary thickness

This paper presents an analytical solution, substantiated by extensive finite element calculations, for the stress field at a notch root in a plate of arbitrary thickness. The present approach builds on two recently developed analysis methods for the in-plane stresses at notch root under plane-stress or plane strain conditions, and the out-of-plane stresses at a three-dimensional notch root. The former solution (Filippi et al., 2002) considered the plane problem and gave the in-plane stress distributions in the vicinity of a V-shaped notch with a circular tip. The latter solution by Kotousov and Wang (2002a), which extended the generalized plane-strain theory by Kane and Mindlin to notches, provided an expression for the out-of-plane constraint factor based on some modified Bessel functions. By combining these two solutions, both valid under linear elastic conditions, closed form expressions are obtained for stresses and strain energy density in the neighborhood of the V-notch tip. To demonstrate the accuracy of the newly developed solutions, a significant number of fully three-dimensional finite element analyses have been performed to determine the influences of plate thickness, notch tip radius, and opening angle on the variability of stress distributions, out-of-plane stress constraint factor and strain energy density. The results of the comprehensive finite element calculations confirmed that the in-plane stress concentration factor has only a very weak variability with plate thickness, and that the present analytical solutions provide very satisfactory correlation for the out-of-plane stress concentration factor and the strain constraint factor.