Dugdale-type model for a mixed-mode crack: plane stress state

Amixed-mode (I + II) crack model with a plastic strip on its continuation is proposed. The three unknown stress components within the strip are determined from the yield conditions, stress limitation, and relationship between the normal stress components defined via the principal stress state. The crack parameters are analyzed for the Mises yield condition     

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

Dynamic Crack Growth Past a Stiff Inclusion: Optical Investigation of Inclusion Eccentricity and Inclusion-matrix Adhesion Strength

Interactions between a dynamically growing matrix crack and a stationary stiff cylindrical inclusion are studied optically. Test specimens with two different bond strengths (weak and strong) and three crack-inclusion eccentricities (e = 0, d/2 and 3d/4, d being inclusion diameter) are studied using reflection mode Coherent Gradient Sensing (CGS) and high-speed photography. These variants produce distinct dynamic crack trajectories and failure behaviors. A weaker inclusion-matrix interface attracts a propagating crack while a stronger one deflects the crack away. The former results in a propagating crack lodging (‘key-hole’) into the inclusion-matrix interface whereas in the latter the crack tends to circumvent the inclusion. When the inclusion is in the prospective crack path, the maximum attained crack speed is much higher in the weakly bonded inclusion cases relative to the strongly bonded counterparts. For a crack propagating towards a weakly bonded inclusion, the effective stress intensity factor (K _e) value remains constant for each inclusion eccentricity considered. But these constant K _e values increase with increasing eccentricity. A distinct drop in K _e occurs when the crack is near the inclusion. In strongly bonded inclusion cases, on the other hand, monotonically increasing K _e before the crack reaches the inclusion is observed. A drop in K _e is seen just before the crack reaches the inclusion. The mode-mixity estimates are of opposite signs for weakly and strongly bonded inclusions in case of the largest eccentricity studied, confirming the observed crack attraction and deflection mechanisms.

A Crack with an Electric Displacement Saturation Zone in an Electrostrictive Material

A crack with an electric displacement saturation zone in an electrostrictive material under purely electric loading is analyzed. A strip saturation model is here employed to investigate the effect of the electrical polarization saturation on electric fields and elastic fields. A closed form solution of electric fields and elastic fields for the crack with the strip saturation zone is obtained by using the complex function theory. It is found that the K _I -dominant region is very small compared to the strip saturation zone. The generalized Dugdale zone model is also employed in order to investigate the effect of the saturation zone shape on the stress intensity factor. Using the body force analogy, the stress intensity factor for the asymptotic problem of a crack with an elliptical saturation zone is evaluated numerically.     

Effects of specimen thickness, hardening and crack closure for the plastic strip model

The triaxial stress constraint T_z and the effective yield stress distribution in the plastic zone for strain hardening materials are considered for the strip where the thickness effect is investigated by introducing a plastic constraint factor α. This factor depends on the specimen thickness, crack length, load level and hardening exponent. A simple expression of the plastic zone length and an expression involving α are obtained. Application of the strip model to crack closure shows that the specimen thickness has an influence and the results are compared with those found finite element.     

Contribution of point defects to the temperature anomaly of the yield stress in single crystals of the Ni<Subscript>3</Subscript>Ge alloy

The temperature dependence of the yield stress τ_* Ni ₃ Ge single crystals is studied. The temperature dependence τ_*(T) in the high-temperature region (above 420 K) is found to be conditioned by thermally activated accumulation of the density of non-screw components of superdislocations. Interaction of point defects with edge dislocations and its effect on the temperature anomaly of the yield stress in Ni ₃ Ge single crystals are analyzed. The calculated results are found to agree with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 154–161, July–August, 2007.     

Thermoelastic investigations for fatigue life assessment

An investigation is presented on the suitability and accuracy of a thermoelastic technique for the analysis of fatigue cracks. The stress intensity factor ranges ΔK _I and ΔK _II are determined from thermoelastic data recorded from around the tip of a sharp slot in a steel specimen under biaxial load, in order to assess the accuracy of the technique. ΔK _I and ΔK _II are determined to within 4% and 9% of a theoretical prediction, respectively. The results from a similar test on a fatigue crack under biaxial load are also presented. These show that thermoelastic stress analysis is a rapid and accurate way of analyzing mixed-mode fatigue cracks. A discussion is given on the potential of thermoelastic stress analysis of propagating cracks.     

Analysis of dynamic mixed-mode isochromatics

The mixed-mode, elastodynamic state of stress in the neighborhood of a constant-velocity crack tip is used to generate numerically unsymmetric isochromatics. Unsymmetry associated with the third-order terms of a mixed-mode stress field, with and without the Mode II singular stress term, is also investigated. In extractingK _I from an unsymmetric isochromatic pattern, errors in the Mode I fracture parameters due to the assumed presence ofK _II in aK _I stress field were found to be significant when data are taken more than 4 mm from the crack tip. Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

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.     

Unsteady Dynamic Crack Propagation in a Brittle Polymer

Dynamic crack propagation in a brittle polymer, poly(methyl-methacrylate) (PMMA), was studied using the method of caustics in combination with a Cranz–Schardin high-speed camera. Four different types of specimen geometry and loading method were employed to achieve the crack acceleration, deceleration, and/or reacceleration processes in one fracture event. The dynamic stress intensity factor K _ID and crack velocity were obtained in the course of the crack propagation and the corresponding relationship was determined. The effect of the crack acceleration and deceleration on the K _ID-velocity relationships was as follows: (1) the variations of K _ID and the velocity were strongly influenced by the specimen geometry and loading method; (2) the velocity change was qualitatively in accord with K _ID; (3) K _ID for a constant crack velocity was larger when the crack decelerated than it was when the crack accelerated or reaccelerated; (4) K _ID for an acceleration-free crack was uniquely related to the velocity; and (5) K _ID could be expressed as two parametric functions of the velocity and acceleration.     

An overdeterministic approach for measuringK I using strain gages

An overdeterministic method for determining the opening-mode stress-intensity factor,K _I , from many measurements of the radial strain,∈ _rr , is described. The method was verified with an experimental study of a compact-tension specimen where strains along the 0, 45 and 90-degree lines were measured using strip gages with ten strain sensors per strip gage. The results indicated errors in the range of one to three percent with three or four parameter models of the strain field in the region near the crack tip. Paper was presented at the 1986 SEM Spring Conference on Experimental Mechanics which was held in New Orleans, LA on June 8–13.     

Stress state in the vicinity of an inclined elliptical defect and stress intensity factors for biaxial loading of a plate

The problem of determining the stress state of a plate with an inclined elliptical notch under biaxial loading is considered. The Kolosov-Muskhelishvili method is used to obtain an expression for the stress near the vertex of an inclined ellipse, whose particular case are expressions for the stress in the case of an inclined crack. The stress intensity factors K _I and K _II were determined experimentally by holographic interferometry in the case of extension of a plate with an inclined crack-like defect. The calculation results are compared with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 118–127, January–February, 2009.     

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.     

A photoelastic study of crack repair

Birefringent coatings have been employed to study the effectiveness of an adhesively bonded repair of a center-cracked tension panel. The repair was one sided, with photoelastic coatings applied to the opposite side. Photoelastic coatings were also applied over the patch. Analysis methods are presented to permit the stress intensity factor to be determined from the isochromatic fringe patterns recorded from both continuous andX- andY-edged coatings. The results showed that the one-sided adhesively bonded patch reduced the stress intensity factor; however, the repair did not markedly change the character of stress distributions. Fringe loops formed near the crack tips for both the cracked and repaired tension panels. The primary difference was in the size of the loops. The reduction inK _I due to repair was smaller than anticipated, but even small improvements in ΔK _I markedly enhance the life of a repaired panel. The Paris power law is used to show the relation between the reduction in ΔK _I and the improvement in the crack growth rateda/dN. Fringe patterns from the birefringent coatings applied to the patch provided a means not only to investigate the stresses in the patch but also to detect the initiation of the local debonding of the adhesive in the neighborhood of the crack. The birefringent coating on the patch is an approach for producing an optically “smart” repair.     

Bentonite slurries—zeta potential,yield stress,adsorbed additive and time-dependent behaviour

Bentonite clay is a vital ingredient of drilling mud. The time-dependent and high shear thinning yield stress behaviour of drilling mud is essential for maintaining wellbore stability and to remove cuttings, cool and clean the drill bit of debris. As-prepared 3, 5 and 7 wt.% bentonite clay slurries displayed time-dependent behaviour where the yield stress (measured after quick stirring) decreased with time of rest. An equilibrium value is reached after 24 h. Despite the low solids concentration, the yield stress is already relatively high and is displayed at all pH level. The yield stress is maximum at pH 2 and minimum at pH ∼ 7. This yield stress is due to the formation of gel structure by the swelling clay particles. However the addition of phosphate additives such as (PO₃)^19 −, (P₃O₁₀)^5 − and (P₂O₇)^4 − completely dispersed the clay slurries at pH above 6. At pH below 6, yield stress is still present but is 3-folds smaller than that of the pure bentonite slurry. With phosphate additives, the magnitude of the critical zeta potential at the complete dispersion pH is ca 48 mV. However for the pure bentonite, the slurry remained flocculated at zeta potential of >50 mV in magnitude. Interestingly, (P₂O₇)^4 − anions is more effective than the other two phosphate additives in reducing the yield stress at low pH, ∼ 2.0.     

Axial crack propagation and arrest in a pressurized cylinder: An experimental-numerical analysis

The feasibility of using a previously developed crack-kinking criterion to predict crack arrest at a tear strap in a pressurized fuselage was studied with instrumented axial rupture tests of 21 models of an idealized fuselage. A rapidly propagating axial crack, which was initiated from a precrack, kinked immediately upon extension and propagated diagonally until it turned circumferentially and propagated along the tear straps. An elastodynamic finite element analysis of the rupturing model fuselage yielded the mixed-mode stress intensity factors,K _I andK _II , and the remote stress component, σ _OX . This numerical procedure was also used to predict the crack trajectories in full-scale fuselage rupture tests. All numerical results agreed well with their measured counterparts regardless of size.     

Fatigue crack growth rate in ferrite-martensite dual-phase steel

An empirical study is made on the fatigue crack growth rate in ferrite-martensite dual-phase (FMDP) steel. Particular attention is given to the effect of ferrite content in the range of 24.2% to 41.5% where good fatigue resistance was found at 33.8%. Variations in ferrite content did not affect the crack growth rate da/dN when plotted against the effective stress intensity factor range ΔK_eff which was assumed to follow a linear relation with the crack tip stress intensity factor range ΔK. A high ΔK_eff corresponds to uniformly distributed small size ferrite and martensite. No other appreciable correlation could be ralated to the microstructure morphology of the FMDP steel. The closure stress intensity factor K_cl, however, is affected by the ferrite content with K_cl/Kmax reaching a maximum value of 0.7. In general, crack growth followed the interphase between the martensite and ferrite.Dividing the fatigue crack growth process into Stage I and II where the former would be highly sensitive to changes in ΔK and the latter would increase with ΔK depending on the R = σ_min/σ_max ratio. The same data when correlated with the strain energy density factor range ΔS showed negligible dependence on mean stress or R ratio for Stage I crack growth. A parameter α involving the ratio of ultimate stress to yield stress, percent reduction of area and R is introduced for Stage II crack growth so that the da/dN data for different R would collapse onto a single curve with a narrow scatter band when plotted against αΔS.     

Determining the stress intensity factor by using the principle of minimum potential energy

Expressing the total potential energy of the system of a cracked body П by Williams’ infinite series solution of stress and displacement components containing coefficients A_n(n = 1,2,...), we obtain a set of simultaneous linear equations of unknown coefficients A_n by using the principle of minimum potential energy. When the set of equations is solved, the stress intensity factor K₁ can be easily determined. It is equal to √2πaA₁ Take a sample plate as an example. A single-edgc-cracked plate under tension, with the ratio of crack length to the width of the plate being 0.5 and the ratio of half plate height to the width of the plate being 2.0 and 2. 5, has been calculated. Only 20 - 30 coefficients are taken, and the errors in stress intensity factors are within 5%.     

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 local elastic softening on crack extension

Elastic softening materials are brittle materials such that crack extension is associated with a softening zone behind the crack tip, with the material elements within this zone exerting a restraining effect on the crack tip. Crack extension is sometimes characterised in terms of the stress intensity K_F, due to the applied loadings, at the front of the softening zone, i.e. the actual crack tip. This paper is concerned with the determination of the maximum load K_F value for a general positive geometrical configuration, for the case where the softening zone size is small compared with a solid's characteristic dimension. The resulting expression for K_F is compared with the maximum load stress intensity value K_T measured with regard to the initial crack position, i.e. the trailing edge of the softening zone.