T-stress Component Research Articles

Investigation on in-plane and out-of-plane constraint effects for plates with I-II mixed mode cracks under biaxial compressive loading

Crack sizes, geometrical configurations of specimens and structures, as well as loading conditions have significant effect on in-plane and out-of-plane constraint effects. Safety assessment with constraint effects can effectively improve the accuracy. There are prominent discrepancies between compressive fracture and tensile fracture. Most of engineering structures are usually under biaxial loading conditions, so that the influence of biaxial compressive loading on fracture analysis with constraint effects is worth paying attention. The size effect factor (relative crack length a/W, thickness to width ratio t/W and crack inclination angle β) and load factor (lateral pressure coefficient λ) of central cracked plate (CCP) specimens with I-II mixed mode crack on constraint effects under biaxial compressive loading have been investigated through theoretical analysis and finite element analysis. In-plane non-singular T-stress (T11) and its components (Tx, Ty, and Txy), out-of-plane non-singular T-stress (T33), out-of-plane constraint parameter Tz and stress triaxiality factor h have been adopted to quantify constraint effects. It demonstrates that λ has remarkable influence on SIF, T11 and T33 under biaxial compressive loading. The effect of higher a/W (long cracks) on B11 is more noticeable when I-II mixed mode crack is closer to pure mode II crack or pure mode I crack. The proportion of shear fracture increases under biaxial compression with λ. With the increase of β, B33 decreases sharply and then increases rapidly with higher λ. Tz can directly reflect stress states and adequately describe the loss of out-of-plane constraint. The stress state of crack tip is more likely to asymptotic plane-strain condition ahead of crack tip under compression. There are crucial differences in variation range and tendency of Tz subjected to biaxial tensile and biaxial compressive loading. Moreover, Tz is insensitive to λ. Stress triaxiality factor h can illustrate in-plane and out-of-plane constraint effects including size effect and load factor (sensitive to a/W, t/W, β and λ), especially can be adopted as a unified constraint parameter for long cracks under biaxial compressive loading. The in-plane T-stress components can be directly calculated by T11 under biaxial compression. The empirical equations of KII and T11 containing size effect and load factor simultaneously are effective and convenient for applications.

Stress field analysis of an infinite plate with a central closed inclined crack under uniaxial compression

In view of the shortcomings of existing studies, the mechanical behavior of the closed crack under uniaxial compression is studied. Based on the far-field and the crack surface stress boundary conditions, Kolossoff-Muskhelishvili stress function of an infinite plate with a central closed inclined crack is derived with Muskhelishvili complex theory. Then the calculation methods of the stress components, stress intensity factors and T-stress at the crack tip are proposed. The mainly findings are as follows. First of all, the crack surface stress boundary conditions vary with variation of the far-field stress condition from tension to compression, so it’s false to directly introduce the theory for the crack under tension to that for the closed crack under compression. Next, it is found that there is no KI singular term at the crack tip and meanwhile KII and T-stress are influenced by the effective shear stress on the crack surface. When the effective shear stress is zero, there are two T-stress components (i.e. Tx and Ty, which are parallel and perpendicular to the crack surface respectively). While the effective shear stress is greater than zero, the third T-stress component (namely Txy which is along the xy direction) also exists. Finally, the theoretical predictions with the proposed method are verified with the isochromatic fringe patterns of photoelasticity experiment near the crack tip under uniaxial compression. Meanwhile the distribution of each stress component around the crack is effected obviously by the friction coefficient of the crack surface and the dip angle of crack, but the distribution tendency of these stress components is similar.

T-stresses solution and out-of-plane constraint for central cracked plate (CCP) with I-II mixed mode crack under uniaxial compression

The crack tip constraint effect should be considered in the fracture assessment of structures with cracks, and the in-plane constraint and out-of-plane constraint should be illustrated simultaneously for the three-dimensional cracks in practical engineering structures. The constraint effect of central cracked plate (CCP) specimens with I-II mixed mode crack under uniaxial compression has been investigated quantitatively in present work. The relationship between T-stress components (Tx, Ty, Txy) and in-plane constraint parameter T11 has been established. The effects of a wide range of in-plane and out-of-plane geometrical configurations (including relative crack length a/W and thickness to width ratio t/W), material parameters (including elastic modulus E and Poisson's ratio ν), crack inclination angle β and friction coefficient μ corresponding to stress intensity factors (K), in-plane constraint parameter T11 and out-of-plane constraint parameters T33 and Tz at the crack tip have been calculated and discussed theoretically and numerically. It demonstrates that the constraint parameters can effectively express the stress-strain field at the crack tip. The relative crack length a/W has an influence on the in-plane and out-of-plane constraint parameters, and the thickness to width ratio t/W mainly affects the out-of-plane constraint parameters T33 and Tz. The out-of-plane constraint parameters should be considered for long cracks. There is a loss of out-of-plane constraint under compression when t/W less than or equal to 0.9, which can be much more directly and evidently investigated through the analysis of Tz than T33. The empirical equations for KII, T11 and Tz have been obtained according to theoretical analysis and simulation results, which are convenient and effective for application.

Analytical solution of T-stresses for an inclined crack in compression

It is widely accepted that T-stress near crack tip has a considerably effect on crack behavior. In the classic fracture mechanics, only Tx (T-stress component in x-direction) was addressed, and recently one researcher pointed out that both Tx and Ty exist for a crack in compression. In order to theoretically study the T-stress for an inclined crack under compression, analytical solution of stresses near crack tip was derived from the complex stress functions which were obtained according to the crack surfaces stress conditions and boundary conditions. The analytical solution of stresses consists of two parts: one is singular term which is the same as the classical stress expressed in terms of stress intensity factor, and the other is non-singularity term which can be expressed in terms of three components of T-stress, i.e. Tx, Ty and Txy, which indicates that the T-stress should have three components. In order to validate the result of T-stress, the crack initiation angle was calculated by the MTS criterion for four cases: (1) without T-stress, (2) Tx only, (3) Tx and Ty, and (4) Tx, Ty and Txy. The prediction results of the crack initiation angles by MTS criterion were compared to the previous test results of PMMA and the test results conducted in this study by using sandstone specimens with an inclined crack under uniaxial compression. The results show that the prediction results by considering the three components Tx, Ty and Txy can better fit both test results of PMMA and sandstone, which indicates that the T-stress near crack tip should have three components Tx, Ty and Txy, and all of them have effect on crack initiation behavior.

The effect of crack-tip constraint in some problems of fracture mechanics

The effect of in-plane and out-of-plane crack-tip constraint parameters, namely, the non-singular T-stress components and the parameter A, in some problems of elastic and elastic-plastic fracture mechanics is discussed. The three-dimensional crack tip stresses including the non-singular T-stress components are employed to describe the crack tip plastic zone, the failure assessment diagrams and the constraint-dependent fracture toughness for a solid with a finite notch/crack under mode I loading and different angles of surface crack growth under conditions of rolling-sliding. Theoretical and numerical aspects of the two-parameter J-A approach in elastic-plastic fracture mechanics are presented in connection with crack-tip constraint. It is shown that the parameter A can be introduced in a fracture criterion as a constraint parameter.

The Effect of the Non-singular T-stress Components on Crack Tip Plastic Zone under Mode I Loading

Theoretical and numerical analysis of the joint influence of the non-singular components of the T-stresses on the crack tip plastic zone estimated by the von Mises yield criterion is carried out under mode I loading in connection with specimen thickness. Calculations are performed for three thicknesses of the CT specimen.The amplitudes of the non-singular terms in the three-dimensional series expansion of the crack front stress field are the terms Txx and Tzz which describe in-plane and out-of-plane constraint, respectively. The plastic zone is analyzed for various combinations of the terms Txx to Tzz ratios.The plastic zone size distribution is mapped in the specimen thickness direction. It is shown that the plastic zone is affected by the Tzz-stress, i.e. there is strong effect of out-of-plane constraint on crack tip plastic zones. Crack tip plastic zone in the middle plane of the specimen decreases with the increase of specimen thickness.

Effect of nonsingular T-stress components on the plastic-deformation zones near the tip of a mode I crack

The effect of nonsingular T-stress components on the plastic-deformation zone size near the tip of a mode I crack as a function of the sample thickness is analyzed theoretically and numerically. The calculations are performed for compact samples of three thicknesses. The plastic-deformation zone size in the middle plane of the sample is found to decrease with increasing sample thickness. To determine the shape of the plastic-deformation zone more correctly, it is necessary to take into account the angular distribution of the nonsingular T stresses near the crack tip.

Multiparametric full-field representations of the in-plane stress fields ahead of cracked components under mixed mode loading

In some cases of engineering interest, the contribution of the higher order stress terms (besides the stress intensity factors and the T-stress) is not negligible in the crack tip stress field. The main aim of the present investigation is to present a set of equations for accurately describing the crack tip stress components particularly for those cases where the modes I and II stress intensity factors used in combination with the T-stress component, are unable to capture with satisfying precision the complete stress field ahead the crack tip. The case of a plate with a central crack under mixed mode (I+II) loading is discussed to show the different contributions of the higher order terms in the overall stress field. The second example deals with a thin welded lap joint characterized by a jointing face width to thickness ratio d/t ranging from 0.5 to 5. The third example investigates the case of an inclined crack departing from the slit tip of a welded lap joint.The strain energy averaged over a control volume can be analytically evaluated with high precision by taking into account only four additional terms besides KI, KII and T. Finally, the averaged SED is used to provide a demarcation line between the elastic and the elastic–plastic behaviors as well as to create a possible bridging with the CJP (Christopher, James and Patterson) model.

Fatigue crack propagation law measured from integrated digital image correlation: the example of Ti35 thin sheets

In this study, a crack in commercially-pure Titanium (CP-Ti) is observed with optical means during its propagation in fatigue. Digital image correlation is then used to extract directly, with no post-processing step, the crack tip location and the corresponding stress intensity factor amplitude to evaluate the parameters of Paris’ law, the T-stress component and an estimate of the plastic zone shape.

On crack path evolution rules

The models of crack growth in mixed mode conditions are reviewed for the plane and three-dimensional (3D) states of stress. Both critical load value and crack path or surface growth are predicted by different criteria in terms of elastic singular stress states and T-stress component. Monotonic and cyclic loading induced crack growth is considered. The energy and critical plane criteria based on local or non-local measure of stress and strain are most useful in developing predictive crack growth simulation. The finite critical distance from the crack tip should be specified to provide averaged or local stress and strain states. The application of MK-criterion of crack growth expressed in terms of volumetric and deviatoric strain energies is presented for several specific cases of monotonic and cyclic loading. The concepts of smooth and rough crack surface are discussed with application to 3D crack surface growth.

Comportamento a fatica dei giunti saldati in funzione della densità di energia di deformazione locale: influenza dei campi di tensione singolari e non singolari

In the Strain Energy Density (SED) approach for fatigue strength assessments of welded joints a well-defined control volume is considered. This volume surrounds the weld root or weld toe, both modelled like sharp (zero radius) V-notches with different opening angles. The volume becomes a circular sector under plane strain conditions, with the radius being about 0.3 mm for welded joints made of structural steel. The mean value of the SED mainly depends on the singular stress fields when the main plate thickness is large enough, whereas the influence of the T-stress component cannot be neglected in the case of thin-walled welded joints. Both contributions are directly accounted for by using finite element models, also when the relevant meshes are quite coarse. This fact makes the application of the SED approach easier than any stress-based approach in the case of complex structures.Due to three-dimensional effects, a non conventional out-of-plane singular mode can be present, in addition with respect to modes I and II of the Williams’ solution. This out-of-plane mode, analogous to the Mode III, is discussed here with reference to welded (seam) lap joints under tensile-shear loads.

On all components of the elastic <i>T</i>-stress for embedded elliptical cracks under linearly varying loading

Explicit analytical expressions of all components of the elastic T-stress for an embedded elliptical crack in an infinite elastic body under linearly varying loadings (pressure and shift) on the crack surfaces are obtained in this paper. Previously, many interesting results for embedded elliptical cracks have been obtained, but mainly they have been devoted to determination of the stress intensity factors. In the current paper, formulae for three components of the elastic T-stress are derived by means of the potential method and a special set of harmonic functions. The expressions for the elastic T-stress components for a penny-shaped crack follow from the present results as special cases. It is shown that all components of the elastic T-stress components strongly depend on Poisson’s ratio of the material, a parametric angle, the ratio of the semiaxes of the ellipse, and the type of loading.

Elastic stresses in bodies with elliptic, parabolic, and tunnel cracks

The paper gives explicit expressions of the elastic T-stress components TI, TII, and TIII for an elliptic crack in an unbounded body under uniform pressure and bending and expressions of all the T-stress components for parabolic and tunnel cracks under uniform loading. These formulas are derived by analyzing the asymptotic behavior of the stress components near the crack front using special harmonic functions. The dependence of the T-stresses on Poisson’s ratio, semiaxes and parametric angle of the elliptic crack is studied. The expressions of TI, TII, and TIII for a penny-shaped crack under arbitrary uniform pressure and bending follow as a special case from the respective expressions for an elliptic crack

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.

Effect of residual stresses on interface crack growth by void expansion mechanism

Crack growth along an interface between two adjacent elastic–plastic materials in a layered solid is analysed, using special interface elements to represent the fracture process ahead of the crack-tip. These interface elements account for ductile failure by the nucleation and growth of voids to coalescence. In these elements the stress components normal to the interface and the shear stresses are given by equilibrium with the surrounding material, and the stress component tangential to the interface is determined by the requirement of compatibility with the surrounding material in the tangential direction. It is assumed that the layers are sufficiently thick, so that the plastic regions around the crack-tip are much smaller than the thickness of the nearest layers. The analyses focus on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The results show that the value of the T-stress component in the softer material adjacent to the interface crack plays the dominant role, such that a negative value of this stress component gives a significant increase of the interface fracture toughness.

Influence of plasticity on interface toughness in a layered solid with residual stresses

For crack growth along an interface between two adjacent elastic–plastic materials in a layered solid, the resistance curve behaviour is analysed by approximating the behaviour in terms of a bi-material interface under small scale yielding conditions. Thus, it is assumed that the layers are thick enough so that the extent of the plastic regions around the crack tip are much smaller than the thickness of the nearest layers. The focus is on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The fracture process is represented in terms of a cohesive zone model. It is found that the value of the T-stress component in the softer material adjacent to the interface crack plays a dominant role, such that a negative value of this T-stress gives a significant increase of the interface fracture toughness, while a positive value gives a reduction of the fracture toughness.

Mode I cracks subjected to large T-stresses

There are several criteria for predicting brittle fracture in mode I and mixed mode loading. In this paper, the modified maximum tangential stress criterion originally proposed for mixed mode loading, is employed to study theoretically brittle fracture for mode I cracks. In particular, the effect of the non-singular term of stress, often known as the T-stress, on the angle of initiation of fracture and the onset of crack growth is explored. The T-stress component of the tangential stress vanishes along the crack line. Therefore, it is often postulated for linear elastic materials that the effect of T-stress on mode I brittle fracture can be ignored. However, it is shown here that the maximum tangential stress is no longer along the line of initial crack when the T-stress exceeds a critical value. Thus, a deviation in the angle of initiation of fracture can be expected for specimens having a large T-stress. It is shown that the deviation angle increases for larger values of T-stress. Theoretical results show that the apparent fracture toughness decreases significantly when a deviation in angle occurs. Earlier experimental results are used to corroborate the findings. The effect of large T-stresses is also explored for a crack specimen undergoing moderate scale yielding. The elastic-plastic investigation is conducted using finite element analysis. The finite element results reveal a similar deviation in the angle of maximum tangential stress for small to moderate scale yielding.

Variation of elastic T-stresses along slightly wavy 3D crack fronts

The non-singular terms in the series expansion of the elastic crack-tip stress field, commonly referred to as the elastic T-stresses, play an important role in fracture mechanics in areas such as the stability of a crack path and the two-parameter characterization of elastic-plastic crack-tip deformation. In this paper, a first order perturbation analysis is performed to study some basic properties of the T-stress variation along a slightly wavy 3D crack front. The analysis employs important properties of Bueckner-Rice 3D weight function fields. Using the Boussinesq-Papkovitch potential representation for the mode I weight function field, it is shown that, for coplanar cracks in an infinite isotropic and homogeneous linear elastic body, the mean T-stress along an arbitrary crack front is independent of the shape and size of the crack. Further, a universal relation is discovered between the mean T-stress and the stress field at the same crack front location under the same loading but in the absence of a crack. First-order-accurate solutions are given for the T-stress variation along a slightly wavy crack front with nearly circular or straight confifurations. Specifically, cosine wave functions are adopted to describe smooth polygonal and slightly undulating planar crack shapes. The results indicate that T11, the 2D T-stress component acting normal to the crack front, increases with the curvature of the crack front as it bows out but T33, acting parallel to the crack front, decreases with the crack front curvature.

Three-Dimensional Slightly Nonplanar Cracks

Abstract Three-dimensional slightly nonplanar cracks are studied via a perturbation method valid to the first-order accuracy in the deviation of the crack shape from a perfectly planar reference crack. The Bueckner-Rice crack-face weight functions are used in the perturbation analysis to establish a relationship, within first-order accuracy, between the apparent and local stress intensity factors for the nonplanar crack. Perturbation solutions for a cosine wavy crack with arbitrary wavelengths are used to examine the effects of three T-stress components, Txx, TXZ, TZZ, on the stability of a mode 1 planar crack in the x-z plane with front lying along the z-axis. A condition for the mode 1 crack to be stable against three-dimensional wavy perturbations of wavelengths λx and λz is determined as Txx + Tzzg < 0 where g is negative, with a very small magnitude, for 0<λx/λz<1/3 and positive for 1/3<λx/λz<∞; this suggests that when Txx = 0, a compressive stress Tzz may cause crack deflection with large wavelengths parallel to the crack front and a tensile stress Tzz may cause deflection with small wavelengths parallel to the front. For comparable T-stress values, it is shown that a negative Txx always enhances the stability of a mode 1 planar crack and a negative Tzz ensures the stability of a mode 1 crack against perturbations parallel to the crack front. The shear component Txz, while not affecting the mode 1 path stability, induces a mode 3 stress intensity factor once crack deflection occurs, and thus promotes the formation of en echelon-type cracking patterns.