Mode II stress intensity factors for central slant cracks with frictional surfaces in uniaxially compressed plates

The effect of crack surface friction on mode II stress intensity factor (SIF) of a central slant crack in a plate uniformly loaded in uniaxial compression is quantified. A previously developed two-dimensional finite element analysis was utilised after its modification to accommodate the friction between the crack surfaces. The plane strain state was assumed. A new numerical technique was devised to avoid the iteration procedures, which had to be employed due to the existence of frictional forces.

The crack inclination angle varied between zero and 75° measured from the horizontal direction. The coefficient of friction of the crack surfaces changed from zero to 1. In case of relatively sliding crack surfaces, mode II SIF existed. As is well known, the resulting mode II SIF decreased with increasing the coefficient of friction of the crack surfaces. Further, mode II SIF increased with increasing crack line inclination angle and then decreased after reaching a maximum value. The angle corresponding to that maximum SIF increased as the coefficient of friction of the crack surfaces increased.     

Arbitrarily oriented microcracks near the tip of an interface macrocrack in bonded dissimilar anisotropic materials

It is well known that microcracking in brittle materials results in a reduction of the stress intensity factor (SIF) and energy release rate (ERR). The reduced SIF or ERR represents crack tip shielding which is of significant interest to micromechanics and material science researchers. However, the effect of microcracking on the SIF and ERR is a complicated subject even for isotropic homogeneous materials, and becomes much more formidable in case of interface cracks in bonded dissimilar solids. To unravel the micromechanics of interface crack tip shielding in bonded dissimilar anisotropic solids, an interface crack interacting with arbitrarily oriented subinterface microcracks in bonded dissimilar anisotropic materials is studied. After deducing the fundamental solutions for a subinterface crack under concentrated normal and tangential tractions, the present interaction problem is reduced to a system of integral equations which is then solved numerically. A J‐integral analysis is then performed with special attention focused on the J₂‐integral in a local coordinate system attached to the microcracks. Theoretical and numerical results reassert the conservation law of the J‐integral derived for isotropic materials 1 , 2 also to be valid for bonded dissimilar anisotropic materials. It is further concluded that there is a wastage when the remote J‐integral transmits across the microcracking zone from infinity to the interface macrocrack tip. In order to highlight the influence of microstructure on the interfacial crack tip stress field, the crack tip SIF and ERR in several typical cases are presented. It is interesting to note that the Mode I SIF at the interface crack tip is quite different from the ERR in bonded dissimilar anisotropic materials.     

Effect of vertex singularities on stress intensities near plate free surfaces

The stress intensity factor (SIF) distribution along the front of a through‐the‐thickness crack is significantly influenced by the presence of the 3D corner (vertex) singularities. All past 3D finite element studies indicated that for mode I, SIF rapidly decreases near the free surface and for mode II, it sharply increases. From the previous numerical simulations, it is unclear what the limiting values of SIF near the surface are and whether these values are infinite or bounded at the vertex point. In this paper, we conduct a careful finite element study and propose a theoretical equation, which describes the SIF behaviour near the vertex. We demonstrate that the asymptotic behaviour of SIF near the surface is governed by the difference in the strength of the corner and edge singularities. Furthermore, we validate our numerical approach and calculations by utilising the invariant properties of J‐integral.     

Stress Intensity Factor of an Edge Crack in Bonded Orthotropic Materials

The article presents the problem of an edge crack under normal point loading terminating perpendicular to the surface of an orthotropic strip of finite thickness which is bonded to another orthotropic half plane. Expressing the displacements and stresses in plane strain condition in terms of harmonic functions, the problem is reduced to a pair of simultaneous integral equations with Cauchy type singularities, which are finally been solved by the Hilbert transform technique. The analytical expression of stress intensity factor (SIF) at the crack tip for large thickness of the strip is calculated, which corresponds to the weight function of a crack under normal loading. The influences of elastic constants of two different orthotropic materials, distinct arbitrary locations of normal point loading on the crack surface and length of the crack on the dynamic SIF are depicted through graphs.     

Three-dimensional finite element analysis of fracture modes for the pull-off test of a thin film from a stiff substrate

A three-dimensional geometrically nonlinear finite element analysis model is presented to study the interfacial delamination for the pull-off test of a thin film strip debonded from a stiff substrate. The strain energy release rates of all three modes (mode I, mode II, and mode III) along the debond front are considered and calculated to investigate the mixed fracture modes for the entire deformation regime from bending plate to stretching membrane. These results indicate that the individual strain energy release rates and the total energy release rate vary along the width of the debond front and strong three-dimensional edge effects exist near the free edges of the film. Interestingly, residual stress also plays an important role in controlling mixed fracture modes and the variation of the energy release rates. Finally, the three-dimensional finite element model is compared with an analytical solution developed earlier. The three-dimensional finite element model is found to provide additional insights for interfacial delamination for the pull-off test.     

Determination of dynamic fracture toughness for PMMA

A dynamic FEM (finite element method) and a strain gage method are applied to analyze the dynamic fracture toughness and SIF (stress intensity factor) for PMMA (polymethyl methacrylate). The analyses are carried out for plates with an edge crack subjected to one-point bending in a plane of the plate. A simple procedure that the present author has proposed is applied to the problem of using a triangular element of assumed constant strain on finite element analysis. The numerical simulation by FEM provides values for the applied forces as measured with the strain gages. Also, a crack initiation time is measured with the strain gage mounted around the crack tip. The dynamic fracture toughness is determined by adapting the crack initiation time to the simulation curve of the dynamic SIF calculated by the FEM. In this study, the usefulness of the method to determine the dynamic fracture toughness is investigated by comparing predictions with the experimental results for dynamic stresses and SIFs.     

Weld root crack propagation under mixed mode I and III cyclic loading

This study examined fatigue propagation behaviour and fatigue life of weld root cracks under mixed mode I and III loading. Fatigue tests were performed on butt-welded joints with a continuous lack-of-penetration (LOP) inclined at angles of 0°, 15°, 30° or 45° to the normal direction of the uniaxial cyclic load. Branch and/or co-planar crack propagation was observed, depending on the initial mode I stress intensity factor (SIF) range. Co-planar crack propagation predominated when the SIF range was large. The fatigue crack propagation mode affected fatigue life; the life of branch crack propagation was longer than that of co-planar crack propagation. Using an initial equivalent SIF range based on a maximum strain energy release rate criterion, the results obtained from the 0°, 15°, 30°, and 45° specimens indicated almost the same fatigue lives, despite the different inclination angles.     

Multiple penny-shaped cracks interaction in a finite body and their effect on stress intensity factor

In this paper we investigate the stress intensity factors (SIFs) of multiple penny-shaped cracks in an elastic solid cylinder under mode I (axial tension) loading. The cracks are located symmetrically and in parallel to one another in the isotropic cylinder. The fractal-like finite element method (FFEM) is employed to study the interaction of multiple cracks and to demonstrate the efficiency of the FFEM for multiple crack problems. The results show that the SIF values of the inner cracks, which are denoted as crack number 1,2,3,…,(n+1)/2 of a stack of n parallel cracks, are lower than the SIF values of a single crack by between 16% and 48%. Also, the outermost crack, that is the crack closest to the boundaries of a multiple cracked body, has the highest SIF values and is, therefore, likely to fail first.     

Application of virtual crack closure integral method for interface cracks in low-k integrated circuit devices under thermal load

The virtual crack closure integral (VCCI) method is used to evaluate the stress intensity factor (SIF) and energy release rate (ERR) of an interface crack under thermal load. The VCCIs used in this work include the traditionally known “Mode I” and “Mode II” VCCIs and an additional coupling VCCI. The singularity element is used in the finite element method (FEM) implementation. The SIF and ERR calculated by the FEM are compared to the exact solution in the case of a joint dissimilar semi-infinite plates with double edge crack under thermal loading. The FEM result agrees well with the exact solution for relatively coarse meshes. The contribution of the mesh density and material mismatch to the FEM error is also explored. The VCCI method is used with the multi-scale FEM in a delamination risk assessment of a low-k integrated circuits device in flip-chip plastic ball grid array packages. The ERR is calculated for different package configurations and the prediction of the delamination risk is confirmed by reliability tests.     

Three-dimensional Correction of the Stress Intensity Factor for Plate with a Notch

The 3D effects of the 2D mode I stress intensity factor for the plate with a V-shaped straight through-thickness notch are investigated by the finite element method and three-dimensional thicknessdependent correction of SIF is suggested. The correction relies on the assumed relationship between the SIF and the constraint factor (out-of-plane degree of freedom). The 3D finite element mesh generator combining the 2D in-plane adaptive unstructured mesh with the structured through-thickness mesh Is developed and applied for the analysis purposes. Three-dimensionality was examined by using two independent indicators, namely, stress- or strain-based constraint factors. The three-point bend and tensile center-cracked plates are investigated. The results demonstrated that the developed 3D corrections may be treated as upper bound estimates of the SIF for three-point bend plate,while directly obtained numerical values are considered as lower bound estimates. Analysis of the tensile center-cracked plate demonstrated a different nature of the 3D SIF profile, which cannot be simply explained as a transitional state between plain strain and plain stress. Therefore, the suggested 3D correction concept is of a particular character.     

Effects of Stitching on Delamination of Satin Weave Carbon-Epoxy Laminates Under Mode I,Mode II and Mixed-Mode I/II Loadings

The objective of the present study is to characterize the effect of modified chain stitching on the delamination growth under mixed-mode I/II loading conditions. Delamination toughness under mode I is experimentally determined, for unstitched and stitched laminates, by using untabbed and tabbed double cantilever beam (TDCB) tests. The effect of the reinforcing tabs on mode I toughness is investigated. Stitching improves the energy release rate (ERR) up to 4 times in mode I. Mode II delamination toughness is evaluated in end-notched flexure (ENF) tests. Different geometries of stitched specimens are tested. Crack propagation occurs without any failure of stitching yarns. The final crack length attains the mid-span or it stops before and the specimen breaks in bending. The ERR is initially low and gradually increases with crack length to very high values. The mixed-mode delamination behaviour is investigated using a mixed-mode bending (MMB) test. For unstitched specimens, a simple mixed-mode criterion is identified. For stitched specimens, stitching yarns do not break during 25% of mode I ratio tests and the ERR increase is relatively small compared to unstitched values. For 70% and 50% of mode I ratios, failures of yarns are observed during crack propagation and tests are able to capture correctly the effect of the stitching: it clearly improves the ERR for these two mixed modes, as much as threefold.     

Studies of mode I delamination in monotonic and fatigue loading using FBG wavelength multiplexing and numerical analysis

Bridging by intact fibers in composite materials is one of the most important toughening mechanisms. However, a direct experimental assessment of its contribution is not easy to achieve. In this work a semi-experimental method is proposed to quantify its contribution to fracture of unidirectional carbon fiber/epoxy double cantilever beam (DCB) specimens in mode I delamination under monotonic and 1 Hz fatigue loads. In each specimen, an embedded optical fiber with an array of eight wavelength-multiplexed fiber Bragg gratings is used to measure local strains close to the crack plane. The measured strain distribution serves in an inverse identification procedure to determine the tractions in the bridging zone in monotonic and fatigue loads. These tractions are used to calculate the energy release rate (ERR) associated with bridging fibers. The results indicate that the ERR due to bridging is about 40% higher in fatigue. The bridging tractions are further included in a cohesive element model which allows to predict precisely the complete load displacement curve of monotonic DCB tests. Using the principle of superposition and the identified tractions, the total stress intensity factor (SIF) is calculated. The results show that the SIF, at initiation, is very close to the one calculated at crack propagation and bridging by intact fibers is responsible for the entire increase in toughness seen in the DCB specimens used herein.     

Examination of the fatigue crack growth equations

Most of the crack growth equations proposed so far correlate the crack growth rate (da/dN or da/dt) with crack tip parameters such as the stress intensity factor (SIF) or energy release rate (ERR). In our previous works, an experimental setup was designed to examine the applicability and the boundary of the functional relationship between da/dN and the crack tip parameters, particularly, ERR. In the present paper, the variation of the ERR along the experimentally observed curvilinear crack trajectories is obtained by means of the finite element method. The analysis shows that the Paris-Erdogan type of laws are applicable until the crack tip is located outside the strong crack-defect interaction region (SI region). A functional relationship between da/dN and ERR breaks down within this region. This suggests the existence of additional crack tip parameters that are not accounted for within conventional fracture mechanics. An approach to modeling the observed phenomenon is discussed following the concept of the Crack Layer theory.     

Mode III Interlaminar Fracture Behavior of Glass Fiber Reinforced Polymer Woven Laminates at 293 to 4 K

This paper deals with mode III delamination properties of glass fiber reinforced polymer woven laminates at room temperature (293 K), liquid nitrogen temperature (77 K), gas helium temperature (20 K), and liquid helium temperature (4 K). In order to evaluate these properties, the Split Cantilever Beam (SCB) fracture test is performed. The load is applied to a test specimen through a set of identical grips in order to reduce (in some degree) the mode II loading at the free edges. A three-dimensional finite element analysis is used to study the stress and strain state of the specimens and to interpret the experimental measurements. The strain energy release rate is calculated by using the virtual crack closure technique. It is found that the strain energy release rate is dominated by the mode III component. A non-uniform distribution of the strain energy release rate along the delamination front is obtained with mode III component having maximum at the center of the delamination front, while mode II component increases towards the free edges. The strain energy release rate is also determined using the crack closure technique. A finite element analysis is also carried out to calculate the stress intensity factors for the SCB specimens. The fracture surfaces are examined by scanning electron microscopy to identify the fracture mechanisms. The most important conclusion from the present study is that at temperature lowering from 293 to 20 K the mode III fracture toughness increases, further cooling to 4 K produces a toughness decrease.     

Fracture Analysis of Concrete Structural Components Accounting for Tension Softening Effect

This paper presents methodologies for fracture analysis of concrete structural components with and without considering tension softening effect. Stress intensity factor (SIF) is computed by using analytical approach and finite element analysis. In the analytical approach, SIF accounting for tension softening effect has been obtained as the difference of SIF obtained using linear elastic fracture mechanics (LEFM) principles and SIF due to closing pressure. Superposition principle has been used by accounting for non-linearity in incremental form. SIF due to crack closing force applied on the effective crack face inside the process zone has been computed using Green's function approach. In finite element analysis, the domain integral method has been used for computation of SIF. The domain integral method is used to calculate the strain energy release rate and SIF when a crack grows. Numerical studies have been conducted on notched 3-point bending concrete specimen with and without considering the cohesive stresses. It is observed from the studies that SIF obtained from the finite element analysis with and without considering the cohesive stresses is in good agreement with the corresponding analytical value. The effect of cohesive stress on SIF decreases with increase of crack length. Further, studies have been conducted on geometrically similar structures and observed that (i) the effect of cohesive stress on SIF is significant with increase of load for a particular crack length and (iii) SIF values decreases with increase of tensile strength for a particular crack length and load.     

Experimental and finite element studies on mode I and mixed mode (I and II) stable crack growth—II. finite element analysis

Finite element studies are presented on both mode I and mixed mode stable crack growth under static loadings through an aluminium (D16AT) alloy. A COD based criterion has been used to predict the load-displacement diagram from initiation to instability. The theoretical predictions are compared with experimental results presented in Part I. Results on computed crack profiles, stress-strain distribution ahead of the crack tip, J integrals, J resistance curves, plastic zones, etc., are included. The study indicates that the load-displacement diagram associated with a mixed mode stable crack growth in a compact tension type of specimen geometry can be predicted reasonably accurately using the criterion of a fixed crack opening displacement at a finite distance behind the crack tip provided the crack is allowed to grow in the direction of initial growth in the finite element analysis. The crack assumes a more blunted profile in a mixed mode than in the mode I at all the stages of stable extension. The distributions of normal stress and strain in the direction perpendicular to the crack extension line, ahead of the current crack tip, have similarities between the mode I and mixed mode, irrespective of loading angle. Both the stress and strain levels increase as the crack extension proceeds. In a mixed mode, the J integral at the onset of crack extension is the lowest compared with the values at the later stages of the extension. Further, the tearing modulus associated with initial kinking is very small; it becomes close to the mode I values at the later stages. The tearing modulus remained approximately constant during the whole mode I stable growth and it had a similar trend subsequent to kinking in a mixed mode. The specific work of crack extension is zero as Δa → 0 and it increases gradually with Δa irrespective of the mode of loading; the actual variation depends on the loading angle. The plastic zone size grows as the stable extension progresses; the growth is approximately the maximum along the crack extension line.     

Modified crack closure integral using six-noded isoparametric quadrilateral singular elements

Six-noded, isoparametric serendipity type quadrilateral regular/singular elements are used for the estimation of stress intensity factors (SIF) in linear elastic fracture mechanics (LEFM) problems involving cracks in two-dimensional structural components. The square root singularity is achieved in the six-noded elements by moving the in-side nodes to the quarter point position. The modified crack closure integral (MCCI) method is adopted which could generate accurate estimates of SIF for a relatively coarse mesh. The equations for strain energy release rate and SIF are derived for mixed mode situations using six-noded quadrilateral elements at the crack tip. The model is validated by numerical studies for a centre crack in a finite plate under uniaxial tension, a single edge notched specimen under uniaxial tension, an inclined crack in a finite rectangular plate and cracks emanating from a pin-loaded lug (or lug attachment). The results compare very well with reference solutions available in the literature.     

动荷载作用下含裂缝公路结构体的应力强度因子

以沈阳-大连高速公路为工程背景，基于弹性动力学理论，采用平面应变有限单元法，分析了车辆荷载对含裂缝路面体的动态作用，分析过程中，车辆荷载简化为正弦分布柔性荷载；路面结构体计算模型抽象为平面应变模型；路面结构体为弹性的连续介质，为了反映裂尖应力，位移场的奇异性和减少模型网格数，在裂尖环向设置了奇异单元。通过计算得到裂尖的位移场，由位移外插得到I-型应力强度因子随加载时间的变化规律。同时探讨了初始裂缝长度和公路结构材料阻尼比的变化对I-型应力强度因子分布规律的影响，为路面体的动态破坏研究提供了一定的理论参考。     

Analysis of the driving force for a generally oriented crack in a functionally graded strip sandwiched between two homogeneous half planes

The driving forces for a generally oriented crack problem embedded in a Functionally Graded strip sandwiched between two half plane are analyzed using singular integral equations with Cauchy kernels, and integrated using Lobatto-Chebyshev collocation. Mixed-mode Stress Intensity Factors (SIF) and Strain Energy Release Rates (SERR) are calculated. The Stress Intensity Factors are compared for accuracy with previously published results. Parametric studies are conducted for various non-homogeneity ratios, crack lengths, crack orientation and thickness of the strip. It is shown that the SERR is more complete and should be used for crack propagation analysis.     

Calculating stress intensity factors for a semi-elliptical surface crack in a heterogeneous stress field

The author proposes an equation for calculating the stress intensity factor (SIF) for a semi-elliptical surface crack for uniform, linear, and quadratic laws of variation of the load applied to its edges. The derivation of the equation is based on the well-known Newman—Raju solution for a bent plate. The distribution of the values of SIF along the crack front, obtained using the empirical equations, coincides with the results of calculations carried out using the finite element method (FEM).Translated from Problemy Prochnosti, No. 7, pp. 38–41, July, 1990.