Three-dimensional stress and displacement fields near an elliptical crack front

Local stress and deformation fields for an elliptical crack embedded in an infinite elastic body subjected to normal, shear and mixed loads are considered. Particular emphasis is placed on the direction of propagation of points along the crack border. A confocal curvilinear coordinate system related to a fundamental ellipsoid, and a local spherical coordinate system attached to the crack border are adopted. Using asymptotic analysis, this paper obtains the stress and displacement fields in a plane inclined to the 3D crack front. Results show that the present solutions are independent of the curvature of the ellipse, and different from those given by Sih (1991). Based on two different fracture criteria, crack growth analysis shows that a 3D crack would propagate in the direction of the normal plane along the crack front. As a result, the fracture initiation and propagation of a 3D flat crack can be analyzed in the plane normal to the crack front, and the local fields in the normal plane are the linear superposition of the plane strain mode-I, mode-II, and mode-III crack-tip fields.     

Coupling of the BEM with the MFS for the numerical simulation of frequency domain 2-D elastic wave propagation in the presence of elastic inclusions and cracks

This paper proposes a coupling formulation between the boundary element method (BEM displacement and TBEM traction formulations) and the method of fundamental solutions (MFS) for the transient analysis of elastic wave propagation in the presence of multiple elastic inclusions to overcome the specific limitations of each of these methods. The full domain of the original problem is divided into sub-domains, which are handled separately by the BEM or the MFS. The coupling is enforced by imposing the required boundary conditions.The accuracy, efficiency and stability of the proposed algorithms, using different combinations of BEM and MFS, are verified by comparing the solutions against reference solutions. The computational efficiency of the proposed coupling formulation is illustrated by computing the CPU time and the error at high frequencies.The potential of the proposed procedures is illustrated by simulating the propagation of elastic waves in the vicinity of an empty crack, with null thickness placed close to an elastic inclusion.     

A novel boundary-domain element method of initial stress, finite deformation and discrete cracks in multilayered anisotropic elastic solids

We introduce a novel boundary-domain element method of initial stress, finite deformation (due to large rotation) and discrete cracks in multilayered anisotropic elastic solids. Because the special Green’s function that satisfies the interfacial continuity and surface boundary conditions is employed, the numerical discretization is reduced to be along one side of the cracks and over the subdomains of finite deformation. Two examples are presented. First, the process of interfacial delamination is simulated around a growing through-thickness crack in a pre-stretched film bonded to a flexible substrate. It is shown that the progression of delamination damage is stable but the initiation of delamination crack can be a snap-back instability. This simultaneous damage and fracture process is approached by the cohesive zone model. Second, the postbuckling of a circular delaminated and pre-compressed film is simulated on a flexible substrate. It is shown that the compliance of substrate can play a significant role on the critical behavior of buckling. If the substrate is more compliant or stiffer than the film, the instability initiates as a subcritical hard or a supercritical soft bifurcation. The critical magnitude of pre-strain for the initiation of buckling increases with substrate stiffness. Also, the transition of buckling from the first to the second mode is captured in the simulation.     

On improved crack tip plastic zone estimates based on T‐stress and on complete stress fields

Cracked ductile structures yield locally to form a plastic zone (pz) around their crack tips, which size and shape controls their structural behaviour. Classical pz estimates are based solely on stress intensity factors (SIF), but their precision is limited to very low σ_n/S_Y nominal stress to yield strength ratios. T‐stresses are frequently used to correct SIF‐based pz estimates, but both SIF and SIF plus T‐stress pz estimates are based on truncated linear elastic (LE) stress fields that do not satisfy boundary conditions. Using Griffith's plate complete LE stress field to avoid such truncated pz estimates, the influence of its Williams’ series terms on pz estimation is evaluated, showing that T‐stress improvements are limited to medium σ_n/S_Y values. Then, corrections are proposed to introduce equilibrium requirements into LE pz estimates. Finally, these improved estimates are compared with pz calculated numerically by an elastic–plastic finite element analysis.     

Non‐oscillatory and non‐singular asymptotic solutions to stress fields at interface cracks

This work concerns the complex oscillatory singularities revealed in Williams's asymptotic solutions to stress fields around arbitrary interface cracks, which are the foundation of phenomenological interface fracture mechanics. First, we highlight the fatal discrepancy between the asymptotic stress fields for cracks in a homogeneous material obtained by assigning an identical material on both regions embracing an interface crack, and the solutions directly derived from cracks in a single material. Next, following a brief introduction to Williams's formulation process, we adopt the method of repeatedly eliminating variables instead of solving the determinant equation for the coefficient matrix to reformulate the asymptotic analysis of stress fields at arbitrary interface cracks. The resultant stresses get rid of oscillatory character. Further, under two specific loading conditions, namely, remotely uniaxial tension or shear, non‐oscillatory and non‐singular asymptotic solutions to stress fields around interface cracks are obtained.     

The elastic displacement and stress in a material to promote reflection and transmission of an incoming wave

The conditions on elastic displacement and stress in a material that will promote reflection and transmission of an incoming wave are calculated. It is found, for example, that to optimize reflection and transmission, scalar potentials of the displacement in the wave and in the material will be related to rotations in planes perpendicular and parallel to the direction of propagation to the wave. When a pulse is constructed and its path analyzed through short distances, it is shown that abrupt transitions in tension and compression in a material will maximize reflection of the pulse. When strain energy is minimized where reflection and refraction are to occur, differences in tension and compression become prominent again. Finally, an approximate volume of material is calculated for an electron to harness the restoring forces in a material to balance the energy lost in inelastic scattering.     

Acoustic measurements of stress fields and microstructure

A very precise system for measuring two-dimensional velocity fields in solid samples has been used for nondestructive measurements of both externally applied and residual inhomogeneous stresses in solids,J integrals, stress intensity factors of cracks, and hardness of quenched steel. The longitudinal velocity measurement is based on precise determination of the propagation transit time through the stressed solid specimen using a small diameter, water-coupled acoutic transducer, which is scanned mechanically over the sample. Changes in velocity are then related to changes of stress in the sample by the theory of acoustoelasticity. Similar measurements show a high degree of correlation between longitudinal velocity changes and changes in microstructure in steel samples. Applications to problems of solid mechanics and material science illustrate the utility of this nondestructive measuring technique.     

Linear elastic fracture mechanics (LEFM) analysis of the effect of residual stress on fatigue crack propagation rate

In this research, both residual and applied stresses are converted to stress intensity factors independently and combined using the superposition principle. The fatigue crack propagation rates are predicted. Experiments using two different loading modes, constant applied stress intensity factor (SIF) range, and constant applied load modes are done for samples with and without initial tensile residual stresses. The samples with initial tensile residual stresses exhibit accelerations of the crack propagation rates. The results show that the weight function method combined with the three-component model provides a good prediction of fatigue crack propagation rates in tensile residual stress fields.     

The numerical solution of boundary-value problems for an elastic body with an elliptic hole and linear cracks

Using the boundary-element method which is a combination of a fictitious load and a displacement discontinuity, numerical solutions are obtained for two-dimensional (plane deformation) boundary-value problems for the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length. Using the method of separation of variables, the boundary-value problem is solved in the case of an infinite domain containing an elliptic hole with a linear cut on whose contour the symmetry conditions are fulfilled.     

The optimal control of fracture and stress parameters in a piezoceramic halfspace with cracks

An approach to the determination of the optimal control of fracture and strength parameters in a piezoceramic halfspace with cracks under antiplane deformation conditions is proposed. The distribution over certain part of the halfspace boundary, harmonically changing with time under the influence of axial forces or electric charges is analyzed as a control interaction. The solution of the inverse problem in fracture mechanics is obtained from the solution of the corresponding direct boundary value problem; in this case, the optimization problem is reduced to the momentum problem. The solution of the direct electroelastic problem using the method of the boundary integral equations is obtained. Various control functions permitting to realize the optimal process of control, i.e. the minimal energetic expenses, are given.     

Numerical simulation of fatigue crack propagation in compressive residual stress fields of notched round bars

In this paper, the influence of the residual compressive stresses induced by roller burnishing on fatigue crack propagation in the fillet of notched round bar is investigated. A 3D finite element simulation model of rolling has allowed to introduce a residual stress profile as an initial condition. After the rolling process, fatigue loading has been applied to three‐point bending specimens in which an initial crack has been introduced. A numerical predictive method of crack propagation in roller burnished specimens has also been implemented. It is based on a step‐by‐step process of stress intensity factor calculations by elastic finite element analyses. These stress intensity factor results are combined with the Paris law to estimate the fatigue crack growth rate. In the case of roller burnished specimens, a numerical modification concerning experimental crack closure has to be considered. This method is applied to three specimens: without roller burnishing, and with two levels of roller burnishing (type A and type B). In all these cases, the computational finite element predictions of fatigue crack growth rate agree well with the experimental measurements. The developed model can be easily extended to crankshafts in real operating conditions.     

Analysis of the stress intensity factor dependence with the crack velocity using a lattice model

In this work, the influence of crack propagation velocity in the stress intensity factor has been studied. The analysis is performed with a lattice method and a linear elastic constitutive model. Numerous researchers determined the relationship between the dynamic stress intensity factor and crack propagation velocity with experimental and analytical results. They showed that toughness increases asymptotically when the crack tip velocity is near to a critical. However, these methods are very complex and computationally expensive; furthermore, the model requires the use of several parameters that are not easily obtained. Moreover, its practical implementation is not always feasible. Hence, it is usually omitted. This paper aims to capture the physics of this complex problem with a simple fracture criterion. The selected criterion is based on the maximum principal strain implemented in a lattice model. The method used to calculate the stress intensity factor is validated with other numerical methods. The selected example is a finite 2D notched under mode I fracture and different loads rates. Results show that the proposed model captures the asymptotic behaviour of the SIF in function of crack speed, as reported in the aforementioned models.     

The finite element absolute nodal coordinate formulation incorporated with surface stress effect to model elastic bending nanowires in large deformation

Surface stress was incorporated into the finite element absolute nodal coordinate formulation in order to model elastic bending of nanowires in large deformation. The absolute nodal coordinate formulation is a numerical method to model bending structures in large deformation. The generalized Young-Laplace equation was employed to model the surface stress effect on bending nanowires. Effects from surface stress and large deformation on static bending nanowires are presented and discussed. The results calculated with the absolute nodal coordinate formulation incorporated with surface stress show that the surface stress effect makes the bending nanowires behave like softer or stiffer materials depending on the boundary condition. The surface stress effect diminishes as the dimensions of the bending structures increase beyond the nanoscale. The developed algorithm is consistent with the classical absolute nodal coordinate formulation at the macroscale.     

Relaxation element method in calculations of stress state of elastic plane with the plastic deformation band

On the basis of relaxation element method an analytical representation of the band of localized plastic deformation, as the defect with its own internal stress field in the plane under tensile loading, is given. The influence of orientation of the band with respect to tensile axis on the stress concentration in the plane is analyzed.     

Minimizing tooth bending stress in spur gears with simplified shapes of fillet and tool shape determination

The strength of a gear is typically defined relative to durability (pitting) and load capacity (tooth-breakage). Tooth-breakage is controlled by the root shape and this gear part can be designed because there is no contact between gear pairs here. The shape of gears is generally defined by different standards, with the ISO standard probably being the most common one. Gears are manufactured using two principally different tools: rack tools and gear tools. In this work, the bending stress of involute teeth is minimized by shape optimization made directly on the final gear. This optimized shape is then used to find the cutting tool (the gear envelope) that can create this optimized gear shape. A simple but sufficiently flexible root parameterization is applied and emphasis is put on the importance of separating the shape parameterization from the finite element analysis of stresses. Large improvements in the stress level are found.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.     

The influence of higher order terms of Williams series on a more accurate description of stress fields around the crack tip

In this work, the crack tip stress field is investigated in two different configurations loaded in mode I as well as in mode I + II. The detailed two‐dimensional solution includes a conventional finite elements analysis that is compared to results obtained analytically from what is referred to as Williams expansion. The influence of consideration of various numbers of terms of the series expansion on the stress distribution is discussed, and the significance of the multi‐parameter fracture mechanics approach is emphasized for some engineering applications, for example, failure of quasi‐brittle materials and estimation of plastic zone extent.     

The behavior of statically-indeterminate structural members and frames with cracks present

Crack stability is discussed as affected by their presence in statically-indeterminate beams, frames, rings, etc. loaded into the plastic range. The stability of a crack in a section, which has become plastic, is analyzed with the remainder of the structure elastic and with subsequent additional plastic hinges occurring. The reduction of energy absorption characteristics for large deformations is also discussed. The methods of elastic-plastic tearing instability are incorporated to show that in many cases the fully plastic collapse mechanism must occur for complete failure.     

Three‐dimensional mixed‐mode (I and II) crack‐front fields in ductile thin plates — effects of T‐stress

It has been well‐established that the non‐singular T‐stress provides a first‐order estimate of geometry and loading mode (e.g. tension versus bending) effects on elastic–plastic crack‐front field under mode I loading conditions. The objective of this paper is to exam the T‐stress effect on three‐dimensional (3D) crack‐front fields under mixed‐mode (modes I and II) loading. To this end, detailed 3D small strain, elastic–plastic simulations are carried out using a 3D boundary layer (small‐scale yielding) formulation. Characteristics of near crack‐front fields are investigated for a wide range of T‐stresses (T/σ₀ = ?0.8, ?0.4, 0.0, 0.4, 0.8). The plastic zones and thickness and angular and radial variations of the stresses are studied, corresponding to two values of the remote elastic mixity parameters M_e = 0.3 and 0.7, under both low and high levels of applied loads. It is found that different T‐stresses have a significant effect on the plastic zones size and shapes, regardless of the mode mixity and load level. The thickness, angular and radial distributions of stresses are also affected markedly by T‐stress. It is important to include these effects when investigating the mixed‐mode ductile fracture failure process in thin‐walled structural components.