Non-singular Stress Research Articles

Stress fields due to inclined notches and shoulder fillets in shafts under torsion

Closed-form expressions of the stress fields in notched rounded bars subjected to torsion are obtained. The notch profile is mathematically described according to Neuber’s conformal mapping z = ( u + iv) q, which gives parabolic and hyperbolic profiles depending on q. The notch axis is inclined with respect to the rounded bar axis. This condition results in two eigenvalue functions: the former is associated with the antisymmetric stress field and gives a stress distribution of asymptotic nature when the notch radius is equal to zero. Conversely, the latter is associated with the symmetric stress field and results in a non-singular stress distribution. This specific condition has been noted in the literature only rarely, where only the antisymmetric part of the stress field is discussed in detail. Theoretical results are compared with numerical data as determined from two models weakened by a parabolic notch and a hyperbolic notch, both notches having the local axis inclined to 45° with respect to the bar longitudinal axis. The agreement is found to be satisfactory.

Significance of K-dominance zone size and nonsingular stress field in brittle fracture

Stress intensity factor has been used to characterize the fracture toughness of a brittle material. This practice is apparently based on the assumption that the singular stress alone at the crack tip is responsible for fracture and that the nonsingular part of the near tip stress has no effect on fracture. In this study, mode I fracture experiments were conducted on a brittle material (PMMA) with four different specimen configurations. The result indicated that fracture toughness cannot be described by stress intensity alone and that a second parameter representing the influence of the nonsingular stress is needed. A two-parameter fracture model was proposed and validated with the experimental result. This two-parameter model was shown to be able to account for various effects created by specimen configurations, crack sizes, and loading conditions, on the fracture behavior of brittle materials.

Fracture analysis of V-notched components – Effects of first non-singular stress term

The effect of the first non-singular stress term on the fracture behavior of notched structures was investigated under symmetric geometry and loading conditions. According to the Williams series expansion, for a large domain of notch angles the non-singular stress terms of sharp notches are functions of complex eigenvalues and their corresponding complex coefficients. Hence, a new representation of stress field near the notch tip was developed in which the higher order terms are expressed as several explicit functions of real and imaginary parts of both the complex eigenvalues and their complex coefficients. A critical stress criterion was then applied to the new stress formulations to assess the influence of the first non-singular stress term on the apparent fracture toughness. Several finite element analyses were also performed on two laboratory specimens in order to show the effects of first non-singular term on the near field stress distribution of notched specimens. The results demonstrated that neglecting the first non-singular stress term could lead to significant errors in predicting the apparent fracture toughness of notched components.

Stress distribution and crack propagation under biaxial low cyclic loading

Biaxial tension-tension cyclic loading tests were conducted with cruciform shaped specimens using AZ31 Mg alloy sheet, which produced by warm rolling. The residual stress of the sample surface can be obtained by in-situ measurement of x-ray under biaxial low cyclic loading. The surface fracture morphology of the cyclic load test specimens was examined by scan electron microscopy (SEM) and optical microscope. From the results, it was found that the compressive residual stress occurs at crack tip in Y-axis direction. Crack nucleation is earliest in the load ratio r=1. In addition, the crack initiation angle is different due to the coupling effect of KI and KII in mixed modes I and II for 45° case. The crack propagation rate was significantly influenced by the non-singular stress parallel to the crack when it was correlated to the stress intensity range.The surface of the case of 90° is similar to that of the case of 45°, and striation-like-patterns and twin boundaries were observed in the specimens.

Fatigue behavior of aluminum alloys under biaxial loading

The biaxiality effect, especially the effect of non-singular stress cycling, on the fatigue behavior was studied, employing cruciform specimens of aluminum alloys 1100-H14 and 7075-T651. The specimens, containing a transverse or a 45 o inclined center notch, were subjected to in-phase (IP) or 100% out-of-phase (hereinafter referred to as “out-of-phase or OP”) loading of stress ratio 0.1 in air. The biaxiality ratio λ ranged from 0 to 1.5, and 3 levels of stress were applied. It was observed that: (1) at a given λ, a lower longitudinal stress induced a longer fatigue life under IP and OP loading, and the fatigue life was longer under IP loading, (2) the fatigue crack path profile was influenced by λ, phase angle (0 o or 180 o), and initial center notch (transverse or 45 o inclined); (3) the fatigue crack path profiles, predicted analytically and determined experimentally, had similar features for the specimens with a transverse center notch under IP loading; and (4) the fatigue crack growth rate was lower and the fatigue life longer for a greater λ under IP loading, whereas it changed little with change in λ under OP loading. These results demonstrate that non-singular stress cycling affects the biaxial fatigue behavior of aluminum alloys 1100-H14 and 7065-T651under IP and OP loading.

Evaluation of first non-singular stress term in bi-material notches

The influence of the first non-singular stress term on the stress field near the bi-material notches has been evaluated. An extensive study has been carried out on the higher order eigenvalues and their correlation with the notch angle and the material combination of the bi-material joints. The coefficients of singular and non-singular terms of the stress field were then calculated for two laboratory specimens with different notch angles and material configurations of the bi-material joints using the displacement field obtained from finite element analysis. It was shown that considering only the two singular terms of a bi-material notch in order to characterize the near stress field may lead to significant errors. Adding the first non-singular term called I-stress, however, could improve the accuracy of the stresses near the notch tip efficiently. Therefore, a three-parameter model including the two singular terms and the I-stress term for predicting the onset of brittle fracture in the bi-material notched components is likely to be more accurate than the classical models which are based only on the singular terms.

Non-Singular Stresses in Gradient Elasticity at Bi-Material Interface with Transverse Crack

Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is well-known, singularities in the stresses appear when classical elasticity is used. Moreover, the nature of the singularity depends on the difference in elastic constants of the two materials. In this paper, the gradient elasticity theory of Aifantis is used to remove these singularities. This is demonstrated for a range of ratios between the two Young’s moduli.

Fracture mechanics approaches in the analysis of strains and fractures of bodies with notches and scotches

Fracture mechanics models and criteria for bodies with cracks have been developed to analyze strains and fractures of bodies with notches and scotches. Criterion equations and corresponding crack growth resistance diagrams for a body with a notch taking into account the changes in the degree of strain constraint at the notch apex due to finiteness of its apex radius rounding and a nonsingular stress component (T-stresses) have been considered. Analytical relations for calculating the J-integral in the case of bodies with blunt U-and V-notches under elastic and plastoelastic loading have been given. Capabilities of the method of separable functions for the experimental study of plastoelastic crack growth resistance for nonstandard samples with scotches have been presented.

Brittle fracture analysis of the offset-crack DCDC specimen

Applications of fracture mechanics in the strength analysis of ceramic materials have been lately studied by many researchers. Various test specimens have been proposed in order to investigate the fracture resistance of cracked bodies under mixed mode conditions. Double Cleavage Drilled Compression (DCDC) specimen, with a hole offset from the centerline is a configuration that is frequently used in subcritical crack growth studies of ceramics and glasses. This specimen exhibits a strong crack path stability that is due to the strongly negative T-stress term. In this paper the maximum tensile stress (MTS) criterion is employed for investigating theoretically the initiation of brittle fracture in the DCDC specimen under mixed mode conditions. It is shown that the T-stress has a significant influence on the predicted fracture load and the crack initiation angle. The theoretical results suggest that brittle fracture in the DCDC specimen is controlled by a combination of the singular stresses (characterized by KI and KII) and the non-singular stress term, T-stress.

Fundamental Research on Fatigue Properties of Closed-Packed Hexagonal Lattice Metal under Biaxial Stress

Assisted living instruments and medical implants, such as wheelchairs and joint prostheses are usually subjected to biaxial or three-axial stresses instead of uniaxial stress. So, authors already developed a servo biaxial fatigue-testing machine, and clarified about the performance evaluation. Moreover, closed-packed hexagonal lattice metal, such as magnesium and titanium, is frequently used for assisted living instruments or medical implants. In this research, fatigue crack propagation tests of magnesium alloy AZ31B and pure titanium TP340C were conducted under conditions of biaxial and uniaxial loading by using a cruciform specimen in a bi-axial fatigue machine, in order to investigate the effect of non-singular stress cycling on the fatigue crack growth properties ⊿K-da/dN. From these comprehensive experiments, in the magnesium alloy, the re-markable effect was found in the specific biaxial load stress ratio on ⊿K-da/dN relation. When biaxial load stress ratio was 0.5, it turned out that the fatigue crack propagation rate of a magnesium alloy becomes very slow. On the other hand, in the titanium, it was confirmed that there is a little influence of a biaxial load stress ratio on ⊿KⅠ -da/dN relation.

Finite Element Analysis of an Improved Center Crack Specimen

A new crack specimen called the diagonally loaded square plate (DLSP) containing an angled central crack is analyzed using the finite element method. The stress intensity factors and the T-stress are calculated for the new specimen in the full range from pure mode I to pure mode II. Unlike the conventional center cracked specimen under uniaxial loading, the improved specimen is able to provide pure mode II. It is shown that the effect of non-singular stress component T in the DLSP specimen is not ignorable relative to the singular stress terms, particularly for mode II dominated loading conditions.

Effect of compressive transverse normal stress on mode II fracture toughness in polymeric composites

Effect of transverse normal stress on mode II fracture toughness of unidirectional fiber reinforced composites was studied experimentally in conjunction with finite element analyses. Mode II fracture tests were conducted on the S2/8552 glass/epoxy composite using off-axis specimens with a through thickness crack. The finite element method was employed to perform stress analyses from which mode II fracture toughness was extracted. In the analysis, crack surface contact friction effect was considered. It was found that the transverse normal compressive stress has significant effect on mode II fracture toughness of the composite. Moreover, the fracture toughness measured using the off-axis specimen was found to be quite different from that evaluated using the conventional end notched flexural (ENF) specimen in three-point bending. It was found that mode II fracture toughness cannot be characterized by the crack tip singular shear stress alone; nonsingular stresses ahead of the crack tip appear to have substantial influence on the apparent mode II fracture toughness of the composite.

On the full set of elastic T-stress terms of internal circular cracks under mixed-mode loading conditions

Analytical expressions for all non-singular stress terms of the asymptotic crack tip field, the so-called T-stresses of internal mixed mode circular (penny shaped) cracks in infinite homogeneous and isotropic elastic solids are addressed. To solve the problem the mixed mode crack problem is divided into sub-problems using the superposition method, and each sub-problem is then solved for the asymptotic stress field. Considering the expansion of the local stress field at the crack front, the elastic T-stress terms are derived for each sub-problem. The results are superimposed to give the analytical expressions of the so far missing elastic T-stresses of internal mixed mode penny shaped cracks. The effect of the T-stresses on the size and shape of the plastic zone at the crack tip is discussed, and analytical results are compared to the ones from finite element analyses, both for the T-stress components and the size of the plastic zone. For an accurate prediction of the plastic zone all the singular terms and the constant terms ( T-stresses) in the stress expansion formulae should be considered. It is observed that negative T-stresses increase the size of the plastic zone, while positive ones reduce it.

Fracture toughness study for a brittle rock subjected to mixed mode I/II loading

A semi-disk specimen containing an angled edge crack has been used in the past for conducting fracture tests on a brittle rock named Johnstone [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212]. The test specimen is appropriate for investigating brittle fracture when the rock samples are subjected to the combined effects of tension and shear along the crack line. However, the experimental results reported in Lim, Johnston, Choi, Boland [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212.] are inconsistent with all of the well-known theoretical criteria available for predicting mixed mode brittle fracture. In this paper, a modified criterion is used to provide accurate predictions for the reported experimental results. The modified criterion makes use of a three-parameter model (based on K I, K II and T) for describing the crack tip stresses. It is shown that the non-singular stress term T has a significant role when the rock fracture tests are conducted on the semi-disk specimens.

Wedge and twist disclinations in second strain gradient elasticity

The aim of this paper is to study disclinations in the framework of a second strain gradient elasticity theory. This second strain gradient elasticity has been proposed based on the first and second gradients of the strain tensor by Lazar et al. [Lazar, M., Maugin, G.A., Aifantis, E.C., 2006. Dislocations in second strain gradient elasticity. Int. J. Solids Struct. 43, 1787–1817]. Such a theory is an extension of the first strain gradient elasticity [Lazar, M., Maugin, G.A., 2005. Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184] with triple stress. By means of the stress function method, the exact analytical solutions for stress and strain fields of straight disclinations in an infinitely extended linear isotropic medium have been found. An important result is that the force stress, double stress and triple stress produced by wedge and twist disclinations are nonsingular. Meanwhile, the corresponding elastic strain and its gradients are also nonsingular. Analytical results indicate that the second strain gradient theory has the capacity of eliminating all unphysical singularities of physical fields.

Fatigue Crack Propagation of Magnesium Alloy and Aluminum Alloy in Biaxial Stress Fields

In this research, fatigue crack propagation tests of magnesium alloy AZ31B and aluminum alloy 2024T3 were conducted under conditions of biaxial and uniaxial loading by using a cruciform specimen in a biaxial fatigue machine, in order to investigate the effect of non-singular stress cycling. From these comprehensive experiments, in the magnesium alloy, the remarkable effect was found in the specific biaxial load stress ratio RB (= σx 0/σy 0) on KI-da/dN relation. On the other hand, in the aluminum alloy, it was confirmed that there is no influence of a RB on KI-da/dN relation.

Plastic Zone Coalescence and Edge Break Conditions of Internal Cracks in a Semi-infinite Sheet

The problem of two cracks in a semi-infinite sheet is analyzed. The critical conditions when adjacent plastic zones just coalesced are obtained. Also, the conditions when a plastic zone just reached the sheet edge are obtained. Assuming the crack and plastic zones as a fictitious crack, the integral equations are formulated in terms of surface traction, nonsingular stress, and zero crack face displacement at the coalescent point or at the sheet edge. By solving the equations, critical remote stress, plastic zone sizes, and crack tip opening displacements are obtained. Numerical results are presented.

A criterion study for non-singular stress concentrations in brittle or quasi-brittle materials

In engineering applications, it is difficult to avoid the non-singular stress concentrations that often play an important role in structure designs. The simplest engineering strength criteria are in general not appropriate due to their incapacity in dealing with important size effect induced by stress gradients. In this paper, we present first a simple experimentation, which consists of plates with a central hole under uniaxial tensile loading, showing important size effect. Second, numerous criteria, including commonly used engineering criteria, crack initiation criteria based on the finite fracture mechanics, or cohesive criteria, were adapted to fit the experimental results. We found that most of these criteria, including criteria with a single material parameter and those with two material parameters, are not suitable for fracture prediction of materials under non-singular stress concentrations. It seems that three material parameters would be the minimum to establish an adequate fracture criterion for arbitrary stress concentrations. By analysing the energy dissipation of micro-crack bands under different stress concentrations, we proposed a new fracture criterion with three material parameters based on the finite fracture mechanics. It is shown that this criterion can provide accurate critical remote loads comparing with experimental data. We believe that the three parameter concept is physically reasonable and can be used in establishing fracture criteria in both the cases of singular and non-singular stress concentrations.

Surface crack and cracks networks in biaxial fatigue

Semi-elliptical fatigue crack growth in 304 L stainless steel, under biaxial loading, was investigated. Compared to those of through-cracks under uniaxial loading, the growth rate of surface cracks is increased by a non-singular compressive stress and reduced by a tensile stress, when R = 0. Plasticity-induced crack closure under biaxial loading was investigated through 3D finite element simulations with node release. Roughness and phase-transformation-induced closure effects were also discussed. The interactions in two-directional crack networks under biaxial tension were investigated numerically. It appears that the presence of orthogonal cracks should not be ignored. The beneficial influence of interaction-induced mode-mixities was highlighted.

Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity

The aim of this paper is to study the elastic stress and strain fields of dislocations and disclinations in the framework of Mindlin’s gradient elasticity. We consider simple but rigorous versions of Mindlin’s first gradient elasticity with one material length (gradient coefficient). Using the stress function method, we find modified stress functions for all six types of Volterra defects (dislocations and disclinations) situated in an isotropic and infinitely extended medium. By means of these stress functions, we obtain exact analytical solutions for the stress and strain fields of dislocations and disclinations. An advantage of these solutions for the elastic strain and stress is that they have no singularities at the defect line. They are finite and have maxima or minima in the defect core region. The stresses and strains are either zero or have a finite maximum value at the defect line. The maximum value of stresses may serve as a measure of the critical stress level when fracture and failure may occur. Thus, both the stress and elastic strain singularities are removed in such a simple gradient theory. In addition, we give the relation to the nonlocal stresses in Eringen’s nonlocal elasticity for the nonsingular stresses.