An Experimental Study of Mixed Mode Crack Initiation and Growth in Functionally Graded Materials

Quasi-static mixed mode crack initiation and growth in functionally graded materials (FGMs) was studied through fracture experiments on polymer-based FGMs manufactured by selective ultraviolet irradiation poly(ethylene carbon monoxide)—a photo-sensitive copolymer that becomes more brittle and stiffer under ultraviolet irradiation. The objective of the study was to determine whether crack kinking criteria for homogeneous materials, e.g., maximum hoop stress criterion, also hold for FGMs. Single edge notched tension specimens with different spatial variations of Young's modulus, failure stress and failure strain, were tested. Near tip mode mixity was introduced either by inclining the crack to the remote loading direction, as in the case of homogeneous materials, or to the direction of material gradient, or both. A full-field digital image correlation technique was used to measure in real-time the displacement field around the crack tip while it propagated through the graded material, and to extract the fracture parameters of stress intensity factor K _I and K _II , and the T-stress. It was found that the nonsingular T-stress term in the asymptotic expansion for stresses plays a very important role in accurately measuring fracture parameters. It was also found that the maximum tangential stress criterion can be applied to the case of FGMs to predict crack kinking provided that the effect of the T-stress is accounted for and the process zone size is small compared to the intrinsic material gradient length scale. However, for accurate crack path prediction at a length scale comparable to the material gradient, detailed material property information is required. In general, the crack will propagate towards a region that exhibits less fracture toughness, but, unlike the case of homogeneous materials, along a path where K _II is not necessarily equal to zero.     

Mode III effects on interface delamination

For crack growth along an interface between dissimilar materials the effect of combined modes I, II and III at the crack-tip is investigated. First, in order to highlight situations where crack growth is affected by a mode III contribution, examples of material configurations are discussed where mode III has an effect. Subsequently, the focus is on crack growth along an interface between an elastic-plastic solid and an elastic substrate. The analyses are carried out for conditions of small-scale yielding, with the fracture process at the interface represented by a cohesive zone model. Due to the mismatch of elastic properties across the interface the corresponding elastic solution has an oscillating stress singularity, and this solution is applied as boundary conditions on the outer edge of the region analyzed. For several combinations of modes I, II and III crack growth resistance curves are calculated numerically in order to determine the steady-state fracture toughness. For given values of K_I and K_II the minimum fracture toughness corresponds to K_III=0 in most of the range analyzed, but there is a range where the minimum occurs for a nonzero value of K_III.     

A method for measuring mode I crack tip constraint under static and dynamic loading conditions

A novel experimental technique for measuring crack tipT-stress, and hence in-plane crack tip constraint, in elastic materials has been developed. The method exploits optimal positioning of stacked strain gage rosette near a mode I crack tip such that the influence of dominant singular strains is negated in order to determineT-stress accurately. The method is demonstrated for quasi-static and low-velocity impact loading conditions and two values of crack length to plate width ratios (a/W). By coupling this new method with the Dally-Sanford single strain gage method for measuring the mode I stress intensity factorK _I , the crack tip biaxiality parameter is also measured experimentally. Complementary small strain, static and dynamic finite element simulations are carried out under plane stress conditions. Time histories ofK _I andT-stress are computed by regression analysis of the displacement and stress fields, respectively. The experimental results are in good agreement with those obtained from numerical simulations. Preliminary data for critical values ofK _I and β for dynamic experiments involving epoxy specimens are reported. Dynamic crack initiation toughness shows an increasing trend as β becomes more negative at higher impact velocities.     

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

The paper gives explicit expressions of the elastic T-stress components T _I, T _II, and T _III 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 T _I, T _II, and T _III for a penny-shaped crack under arbitrary uniform pressure and bending follow as a special case from the respective expressions for an elliptic crack __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 57–70, August 2007.     

Effect of T-stress on the mode-I fracture toughness of concrete

T-stress expressions are provided for three-point bending (TPB) beams and compact tension (CT) specimens and then its influence on mode I fracture toughness of concrete is investigated. The study shows that T-stress is dependent on the specimen's geometry and the material's property as well, and for TPB and CT specimens of regular size, T-stress is so small that its consequences can be neglected. The study also indicates that concrete specimen size should be carefully chosen to make sure the existence of K-dominance ahead of the crack tip, thus fracture toughness extracted from these specimen configurations can be reliable.     

<Emphasis Type="Italic">T</Emphasis>-stress analysis for a Griffith crack in a magnetoelectroelastic solid

T-stress as an important parameter characterizing the stress field around a cracked tip has attracted much attention. This paper concerns the T-stress near a cracked tip in a magnetoelectroelastic solid. By applying the Fourier transform, we solve the associated mixed boundary-value problem. Adopting crack-faces electromagnetic boundary conditions nonlinearly dependent on the crack opening displacement, coupled dual integral equations are derived. Then, the closed-form solution for the T-stress is obtained. A comparison of the T stresses for a cracked magnetoelectroelastic solid and for a cracked purely elastic material is made. Obtained results reveal that in addition to applied mechanical loading, the T-stress is dependent on electric and magnetic loadings for a vacuum crack.     

Fracture toughness for CNT specimens from numerically obtained critical CTOD values

Many approaches for estimating mode I fracture toughness (K_IC) using circumferentially notched tensile (CNT) specimen have been demonstrated in the literature. In this paper, an effective approach for estimating fracture toughness from the numerical solution of critical crack tip opening displacement (CTOD) is proposed. An elasto-plastic finite element analysis is used to estimate critical CTOD values for CNT specimens. A number of materials are analysed, and the relationship between K_IC and critical CTOD for CNT specimens is obtained. The proposed relationship is validated by comparing the fracture toughness values obtained from the relationship with those obtained experimentally using CNT specimens. The fracture toughness (K_IC) calculated according to this relationship from numerically obtained critical CTOD is found to be in close agreement with the experimentally obtained fracture toughness for the respective materials.     

Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions

The effect of constraint on brittle fracture of solids under predominantly elastic deformation and mode I loading conditions is studied. Using different cracked specimen geometry, the variation of constraint is achieved in this work. Fracture tests of polymethyl methacrylate were performed using single edge notch, compact tension and double cantilever beam specimens to cover a bread range of constraint. The test data demonstrate that the apparent fracture toughness of the material varies with the specimen geometry or the constraint level. Theory is developed using the critical stress (strain) as the fracture criterion to show that this variation can be interpreted using the critical stress intensity factorK _Cand a second parameterT orA ₃,whereT andA ₃are the amplitudes of the second and the third term in the Williams series solution, respectively. The implication of this constraint effect to the ASTM fracture toughness value, crack tip opening displacement fracture criterion and energy release rateG _Cis discussed. Using the same critical stress (strain) as the fracture criterion, the theory further predicts crack curving or instability under mode I loading conditions. Experimental data are presented and compared with the theory.     

Resolved shear stress intensity coefficient and fatigue crack growth in large crystals

Fatigue crack growth is caused primarily by shear decohesion due to dislocation motion in the crack tip region. The resolved shear stress, which drives dislocation in a crystal, is strongly orientation dependent, and therefore, the cyclic plastic deformation of the shear decohesion process is highly anisotropic.The crack planes are often inclined to the loading axis both in the inplane orientation and in the thickness direction. This inclination induces all three modes of the crack tip stress field, K_I, K_II, and K_III.Fatigue crack growth in large-grain Al 7029 aluminum alloy was studied. The crack tip stress fields of the test specimens are calculated with the finite element method. The values of K_I, K_II, and K_III are evaluated. The orientation of the crystal at a crack tip was determined with the Laue X-ray method. The crystal orientation and the calculated crack tip stress fields are used to compute the resolved shear stress intensity of each of the twelve slip systems of the crystal at the crack tip. The resolved shear stress field of a slip system is linearly proportional to the resolved shear stress intensity coefficient, RSSIC.The values of RSSIC thus evaluated are used to analyze the orientations of the crack plane and to correlate with the shear fatigue crack growth rate.     

Initiation and arrest of cracks on two pipeline steels at low temperature

Dynamic fracture toughness at initiationK _1d and fracture toughness at arrestK _1a were measured on two pipeline steel grades. Dynamic fracture toughness was measured at a very high loading rate with the help of split Hopkinson pressure bars. The values ofK _1d andK _1a are compared. The purpose of this work is to examine the possibilities of using dynamic fracture toughness at crack initiation as a lower bound of crack arrest toughness. This work has practical applications because crack arrest tests are difficult to perform, give scattered results and are costly and time consuming. This procedure shows that it is possible to economize and rationalize using intelligent technology.     

A bar impact tester for dynamic fracture testing of ceramics and ceramic composites

A bar impact test was developed to study the dynamic fracture responses of precracked ceramic bars, Al₂O₃ and 15/29-percent volume SiC_w/Al₂O₃. Crack-opening displacement was measured with a laser-interferometric displacement gage and was used to determine the crack velocity and the dynamic stress-intensity factorK _I ^dyn . The crack velocity andK _I ^dyn increased with increasing impact velocity while the dynamic-initiation fracture toughness,K _Id, did not vary consistently with increasing impact velocities.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas on June 8–11.     

An improved experimental method for the determination of the critical stress intensity factor KIc of brittle materials

The critical stress intensity factor K_Ic is determined by a simple and accurate method, using small test specimens and a simple procedure in this paper.Single edge V-notched tension specimens made of PMMA are subjected to a load which is slowly increased until the crack begins to move from the notch tip. During the crack propagation event shadow patterns at the tip of the crack are recorded in a video recorder. Under these loading conditions, the creating real crack propagate slowly until the crack propagation velocity take an abrupt increase and the entire fracture of the specimen takes place. The stress intensity factor which correspond to the transition from the slow to fast crack speed, is the critical stress intensity factor K_Ic and it can be the fracture toughness of the material.The results are accurate and in good agreement with those values of K_Ic which are calculated by approximate theoretical expressions.The purpose of this paper is to introduce an improved, simple and accurate experimental method for the determination of fracture toughness of brittle materials.     

Anisotropy of the fracture toughness of structurally inhomogeneous ageing alloys

The dependence of the fracture toughness K _1C of rolled ageing alloys with structural and crystallographic textures on the loading direction is established. A formula describing the anisotropy of the K _1C and including structural parameters of structurally textured alloys on planes of growth of mode I cracks is derived and validated for aluminum alloys. The influence of crystallographic planes and crack growth direction on K _1C is analyzed for titanium alloy as a rolled material with crystallographic texture     

Two-Parameter Model of a Mode I Crack in an Elastoplastic Body under Plane-Strain conditions

The Dugdale crack model is generalized to the case of plane strain. The governing equations are set up to determine the stresses in the plastic zone. Numerical results from specific problems are analyzed and compared with those for plane stress state and other cases. A relationship between the crack model and K _I-T theory is established in the case of small-scale yielding at the crack tip __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 44–55, June 2005.     

<Emphasis Type="Italic">T</Emphasis>-stress estimation by the two-parameter approach for a specimen with a V-shaped notch

In the present research, T-stress solutions are provided for a V-shaped notch in the case of surface defects in a pressurised pipeline. The V-shaped notch is analyzed with the use of the finite element method by the Castem2000 commercial software to determine the stress distribution ahead of the notch tip. The notch aspect ratio is varied. In contrast to a crack, it is found that the T-stress is not constant and depends on the distance from the notch tip. To estimate the T-stress in the case of a notch, a novel method is developed, inspired by the volumetric method approach proposed by Pluvinage. The method is based on averaging the T-stress over the effective distance ahead of the notch tip. The effective distance is determined by the point with the minimum stress gradient in the fracture process zone. This approach is successfully used to quantify the constraints of the notch-tip fields for various geometries and loading conditions. Moreover, the proposed T-stress estimation creates a basis for analyzing the crack path under mixed-mode loading from the viewpoint of the two-parameter fracture mechanics.     

Influence of indenter geometry on half-plane with edge crack subjected to fretting condition

An edge crack is analyzed to study fretting failure. A flat punch with rounded corners and a half-plane are regarded as an indenter and a substrate, respectively. Plane strain condition is considered. Contact shear traction in the case of partial slip is evaluated numerically. It is assumed that an initial crack is extended to the point of minimum strain energy density in the half-plane from the trailing edge of contact. Dislocation density function method is used to evaluate K_I and K_II. The variations of K_I and K_II during crack growth are examined in the case of indentation by a punch with different ratio of the flat region (l) to the punch width (L). Sih's minimum strain energy density theory [1] is also applied to predict the propagation direction of the initial crack. The direction evaluated is similar to that found in the experiment. Stress intensity factor ranges (ΔK_I and ΔK_II) are examined during cyclic shear on the contact. For the design of contacting bodies, a suggestible geometry of punch for alleviating cracking failure is studied.     

Mixed Mode Dynamic Fracture in Particulate Reinforced Functionally Graded Materials

A detailed analytical and experimental investigation is presented to understand the dynamic fracture behavior of functionally graded materials (FGMs) under mode I and mixed mode loading conditions. Crack-tip stress, strain and displacement fields for a mixed mode crack propagating at an angle from the direction of property gradation were obtained through an asymptotic analysis coupled with a displacement potential approach. This was followed by a comprehensive series of experiments to gain further insight into the behavior of propagating cracks in FGMs. Dynamic photoelasticity coupled with high-speed photography was used to obtain crack tip velocities and dynamic stress fields around the propagating cracks. Birefringent coatings were used to conduct the photoelastic study due to the opaqueness of the FGMs. Dynamic fracture experiments were performed using different specimen geometries to develop a dynamic constitutive fracture relationship between the mode I dynamic stress intensity factor (K _ID ) and crack-tip velocity ( ) for FGMs with the crack moving in the direction of increasing fracture toughness. A similar -K _ID relation was also obtained for matrix material (polyester) for comparison purposes. The results obtained show that crack propagation velocities in FGMs were about 80% higher than the polyester matrix. Crack arrest toughness was found to be about 10% lower than the value of local fracture toughness in FGMs.     

A numerical analysis of flat internal cracks under mixed mode loading

A numerical/analytical approach is proposed to determine the stress intensity factors K_I, K_II, and K_III of a 3D internal crack. The main point of this approach is the meshing technique that can model very sharp crack fronts. The meshing technique is based on an elliptical coordinate transformation that starts from a circular crack. It allows the obtainment of a curved crack front with elements normal to the crack front. Remarkable accuracy can be obtained for elliptical crack fronts with axes ratio smaller that 0.01. Accuracy demonstration is provided for cylindrical element with an inclined internal crack subjected to uni-axial tension. This case corresponds to crack propagation for all three modes of loading, the solution of which can checked with references’ results.     

Critique on fracture criteria in mixed-mode-loading

The paper presents an improved formulation of mixed-mode fracture criteria. Both the fracture toughness K_IC and K_IIC are taken into account. Consequently, the crack propagation direction and the fracture envelope in the and plane depends on the toughness ratio . In the plane only one envelope for each criterion is obtained.     

Stress intensity factor and crack velocity relationship for polyester/TiO2 nanocomposites

The dynamic fracture behavior of polyester/TiO₂ nanocomposites has been characterized and compared with that of the matrix material. A relationship between the dynamic stress intensity factor,K _I and the crack tip velocity,å, has been established. Dynamic photoelasticity coupled with high-speed photography has been used to obtain crack tip velocities and dynamic stress fields around the propagating cracks. Birefringent coatings were used to conduct the photoelastic study due to the opaqueness of the nanocomposites. Single-edge notch tension and modified compact tension specimens were used to obtain a broad range of crack velocities. Fractographic analysis was conducted to understand the fracture process. The results showed that crack arrest toughness in nanocomposites was 60% greater than in the matrix material. Crack propagation velocities prior to branching in nanocomposites were found to be 50% greater than those in polyester.