The configuration of moving crack tip zones

The dynamic elasticity solution of a steady state crack is used for determination of the geometrical characteristics and of the displacement rates within the discrete crack zones which are formed as a result of the selective propagation of cleavage microcracks ahead of the tip of a running brittle crack in a mild steel plate. The zone length and the stress distribution in the zone are found to be strongly dependent on the assumed form of the stress-displacement relation for the progressively fracturing metal. On the other hand, the crack opening displacement rate is much less sensitive to the assumed form. The magnitudes of the nominal plastic strain rates, found by an approximate procedure, are of the order of 10⁵ to 10⁶ sec^?1. A comparison with the results of dynamic tension tests on similar steels indicates that the flow stress at these strain rates would exceed the twinning stress, and therefore, it is suggested in agreement with experimental observations, that twinning is the principal deformation mode in the crack zone. Accordingly, the crack tip boundary conditions considered in the solution of crack propagation problems can be assumed as independent of crack velocity. Also, their possible dependence on temperature would not be related to the deformation mode in the crack zone.     

The flow of energy into the tip of a moving crack

A modification of Craggs' method for calculating the flow of energy into the tip of a moving crack is proposed. For a plane crack extending uniformly at both ends the flow falls to zero at the Rayleigh velocity, contrary to Craggs' result, but in agreement with that of Broberg.

---

Zusammenfassung Eine Modifikation des Craggs'schen Verfahrens zur Berechnung des Energieflusses in die Spitze eines sich bewegenden Risses wind vorgeschlagen. Im Fall eines zweidimensionalen Risses, dessen beide Enden gleichmässig auswärts laufen, findet an; days der Fluss bei der Geschwindigkeit der Rayleighwellen Null wird. Dieses Resultat stimmt mit der Behauptung von Broberg überein, aber mit dem Schluss von Craggs nicht.

---

Résumé On propose une modification de la méthode de Craggs pour calculer le flux d'énergie qui entre dans l'extrémité dúne fissure en mouvement. Pour une fissure à deux dimensions dont les extrémités se propagent uniformément en directions contraires le flux se réduit à zéro à la vitesse d'ondes de Rayleigh. Cc résultat est en accord avec celui de Broberg, mais non avec celui de Craggs.

     

The flow of energy into the tip of a moving crack

A modification of Craggs' method for calculating the flow of energy into the tilt of a moving crack is proposed. For a plane crack extending uniformly at both ends the flow falls to zero at the Rayleigh velocity contrary to Craggs' result, but in agreement with that of Broberg.

---

Zusammenfassung Eine Modifikation des Craggs'schen Verfahrens zur Berechnung des Energieflusses ih die Spitze eines sich bewegenden Risses wird vorgeschlagen. Im Fall eines zweidimensionalen Risses, dessen beide Enden gleichmässig auswärts laufen, findet man, dass der Fluss bei der Geschwindigkeit der Rayleighwellen Null wird. Dieses Resultat stimmt mit der Behauptung von Broberg überein, aber mit dem Schluss von Craggs nicht.

---

Résumé On propose une modification de la méthode de Craggs pour calculer le flux d'énergie qui entre dans l'extrémité d'une fissure en mouvement. Pour une fissure à deux dimensions dont les extrémités se propagent uniformément en directions contraires le flux se réduit à zéro à la vitesse d'ondes de Rayleigh. Ce résultat est en accord avec celui de Broberg, mais non avec celui de Craggs.

     

The effects of hyperbolic heat conduction around a dynamically propagating crack tip

Using infrared detectors, Zehnder and Rosakis (1991, J. Mech. Phys. Solids 39(3), 385), Zehnder and Kallivayalil (1991, SPIE ISS4A, 48) and Mason and Rosakis (1992, SN Report 92-2), have recorded the temperature field around a dynamically propagating crack tip travelling at constant velocity in several metals. At the same time, Tzou (1990a, J. Heat Transfer 112, 21, 1990b, Inst. Heat Mass Transfer 33(5), 877) has suggested that the temperature field around a propagating crack tip might exhibit some of the characteristics of hyperbolic heat conduction. In this paper a corrected solution of the hyperbolic heat conduction equation for a travelling point source is derived. Then an experimental estimate of the active plastic zone (heat generating zone) at a crack tip is used for various experimental conditions to examine the possible effects of hyperbolic heat conduction around a propagating crack tip. Finally, using the actual experimental conditions of Zehnder and Rosakis (1991) Zehnder and Kallivayalil (1991) and Mason and Rosakis (1992) it is shown that no effects of hyperbolic heat conduction are observed around a propagating crack tip. It is seen that, due to adiabatic conditions at the crack tip during these experiments, the solution of the hyperbolic heat equation is indistinguishable from the solution of the parabolic heat conduction equation for crack propagation in steel.     

Higher order asymptotic field at the moving crack tip

The higher order asymptotic field at a smoothly curved dynamic crack tip under mixed loading conditions is investigated by the use of a moving coordinate system and the mathematical framework of the complex potential theory. On the basis of the general representation for stress functions in a moving coordinate system, the recurrence formulae for determining the higher order solutions from lower order ones are derived. The calculation in this paper shows that the higher order asymptotic field can be separated into two parts: the steady state asymptotic field which depends on the crack velocity; and the non-steady state asymptotic field which is determined by the time rate of change of the intensity factor, the crack tip acceleration and rotation speed. The second order solution giving the sress distributions at the moving crack tip are presented. The important result that the experimentally observed crack branching velocities are estimated to be much smaller than Yoffe's prediction might be explained by this second order asymptotic solution.     

Experimental evaluation of stress field around crack tip by caustic method

The caustic method is an optical technique which is useful to determine stress intensity factor values. In this paper, the caustics method was applied to specimens which have an oblique crack, various thicknesses and an open notch to investigate the stress field around the crack tip. The results are summarized as follows:

1. The caustic method is a useful technique to determine the stress intensity factor values of the specimens which have an oblique crack or various thickness and an open notch.

2. The conventional theory of measurement concerning this method is effective when the initial curve r₀ is larger than the minimum initial curve r₀^min which was obtained in this study. It is observed that the values of r₀^min decrease as the ratio of K_II to K_I increases under mixed-mode loading, the one increases with an increase of thickness and notch opening angle.

3. The 3D stress field exists in the vicinity of crack tip; however, the stress state is nearly plane strain deformation in the case of mode I loading. In the case of mixed-mode loading, the stress state approximates to plane stress deformation as the ratio of K_II to K_I increases.

4. A method based on the distribution of the three-dimensional (3D) stress field is proposed to expediently yield the values of K_I using the caustic method in the case of r₀<r₀^min.

The crack tip opening angle (CTOA) of the plane stress moving crack

Recently, Crack Tip Opening Angle (CTOA) was proposed by C.F. Shih et al. to describe the instability criterion of ductile crack propagation during plane strain (flat crack) conditions, and was derived by J. R. Rice analytically by means of the slip line field theory and the incremental theory of plasticity. CTOA appears to be applicable in (some or most) cases, but does not accurately describe the plane stress growing crack (slant crack).Unstable ductile crack propagation of the plane stress crack is widely studied for the safe design of highly pressurized gas pipelines. The impact absorption energy of the Charpy test is well correlated to the fracture arresting properties of the structures, but the mechanics of the fracture are not yet well established.In this paper, CTOA of the plane stress growing crack is derived from the plane stress plasticity of perfectly plastic materials by Sokolovsky's approach. Our proposed modification of CTOA expressed as follows: $\text{CTOA}\text{= (α/δ}{}_0\text{)(}\text{d}\text{J/}\text{d}\text{l) + β(δ}{}_0\text{/E)}\text{ln}\text{(eR/r)}$ where $\text{β = 1.40}$ under the plane stress conditions.CTOA in the Dugdale model is also defined and compared with the results of laboratory test. The results show that $\text{α = 0.5}$, and $\text{β = 1.27}$ for plane stress crack growth. These analyses give similar results to those obtained by Rice et al. for CTOA under plane strain conditions, that is, $\text{α = 0.65}$ from the experimental results and $\text{β = 5.08}$ from the slip line theory.The CTOA obtained for plane stress ductile crack growth is applied to the wide plate tensile crack growth test. The results of the present analysis coincide well with those of the plane stress finite element method (FEM) computed by T. Kanazawa et al. The phenomena of plane stress ductile crack propagation are also explained by the CTOA criterion under plane stress conditions.     

An approximate solution of the annular crack problem

A method is presented for the approximate solution of the axisymmetric problem of an annular crack Embedded in an infinite elastic solid and subjected to a normal internal pressure. The numerical results Obtained for the stress intensity factors are in good agreement with those obtained by Smetanin².     

Thickness effect on crack tip deformation at fracture

The crack tip deformation at the onset of surface crack growth for four single edge cracked specimens having four different thicknesses were studied. The specimens were made of the ductile and tough HY-80 steel. The thickness contraction in the crack tip region and the tensile strain in the direction of load were measured. The crack tip necking-in acts like a notch. The depths, the root radii, and the angle changes of the necking-in notches were measured. The results indicated that the near tip strain can be used as a fracture criterion of ductile and tough materials. It was also found that the fracture strengths of the cracked specimens can be correlated with the tensile ductilities measured with Clausing specimens.     

An analysis of the crack tip stress field in DCB adhesive fracture specimens

The problem of a cracked adhesive bonded DCB-type fracture specimen has been analyzed using a hybrid stress model finite element analysis which incorporates an advanced crack tip element. Stresses in the near and far fields have been studied as a function of adherend/adhesive modulus ratio and adhesive thickness. The results are compared to monolithic systems with regard to the stress intensity factor and the localization of the singular stress domain associated with the crack tip.

---

Résumé Le problème d'une éprouvette de rupture Double Poutre Cantilever comportant une liaison par adhésif fissurée a été analysée en utilisant un modèle de contrainte hybride et une analyse par élément fini qui incorpore un élément particulier à la pointe de la fissure. Les contraintes dans le champ proche et dans le champ éloigné ont été étudiées en fonction du rapport des modules de l'adhésif et de l'adhérent, et de l'épaisseur de l'adhésif. Les résultats sont comparés à un système monolitique en ce qui regarde le facteur d'intensité des contraintes et la localisation du domaine singulier des contraintes associé à l'extrémité de la fissure.

     

XFEM fracture analysis of shells: The effect of crack tip enrichments

In this study the effect of crack tip enrichment functions in the extended finite element analysis of shells is investigated. Utilization of crack tip enrichments leads to reduction of the required number of elements, mesh independency and increased accuracy in computation of fracture mechanics parameters such as the stress intensity factor, the crack tip opening displacement and the crack tip opening angle. The procedure is verified by modeling various shell and plate problems and available benchmark tests. Also, effects of enrichments of in-plane, out-of-plane and rotational degrees of freedom and high order out-of-plane enrichments on different fracture modes are studied. Moreover, reduction of the dependency of crack tip opening angle on the element size in crack propagation problems is discussed.     

Transient effect on ideally plastic stationary crack tip fields

This paper analyses the transient effect on ideally plastic stationary crack tip fields under mode I plane strain conditions, when the inertial forces are not negligible. It is shown that the governing equation for such a problem can be expressed in formal simplicity when referred to a system of moving curvilinear coordinates, which is a generalization of the system defined by the slip-line field in quasi-static plasticity. A perturbation method of solving the equations is described and illustrated by application to problems of ideally plastic stationary crack tip fields when the inertial forces are not negligible.     

The effect of tip plastic energy on mixed-mode crack initiation

An approach to the fracture initiation prediction of a ductile crack with mixed-mode loading (mode I and II) conditions is presented. The tip plastic energy around the crack tip is applied for evaluating the crack initiation load and the plastic zone shape. It is proposed that a mixed-mode crack will initiate as the tip plastic energy reaches a critical value. Numerical results for various loading conditions are illustrated. These results indicate that the predicted crack initiation loads correlate well with the experimental data available. This revised version was published online in August 2006 with corrections to the Cover Date.     

An analysis of the crack tip fields in a ductile three-point bend specimen subjected to impact loading

Analyses of an impact fracture test of a precracked, three-point beam of HY100 steel were performed to determine the dynamic fracture toughness. During impact, the crack tip opening displacement (CTOD) 100 μm behind the crack tip was measured using an optical measuring device called the interferometric strain/displacement gage. Since fracture initiates when stress wave effects dominate, a numerical simulation of the fracture event was conducted to obtain relevant near crack tip field parameters. The specimen was modeled by a plane stress finite element simulation using a rate sensitive elastoplastic material law. Since the simulated CTOD was to be compared with the measured CTOD in a region of residual strains due to crack closure, this effect was included in the model. The simulation produces a CTOD versus time response within 10% of the observed response, indicating that the other field quantities (such as the J-integral) should also be reliable. The loading rate /.K₁ was approximately 8 × 10⁶MPa√m/sec. If the fracture initiation time is assumed to coincide with the time at which the simulated and observed CTOD curves diverge, then the impact fracture toughness is 56% higher than the static fracture toughness.     

An elastic-plastic finite element analysis of crack tip fields under biaxial loading conditions

A square plate containing a central crack and subjected to biaxial stresses has been studied by a finite element analysis. An elastic analysis shows that the crack opening displacement and stress of separation ahead of the crack tip are not affected by the mode of biaxial loading and therefore the stress intensity factor adequately describes the crack tip states in an elastic continuum.

An elastic-plastic analysis involving more than localized yielding at the crack tip provides different solutions of crack tip stress fields and crack face displacements for the different modes of biaxial loading.

The equi-biaxial loading mode causes the greatest separation stress but the smallest plastic shear ear and crack displacement. The shear loading system induces the maximum size of shear ear and crack displacement but the smallest value of crack tip separation stress.

     

An integral associated with the state of a crack tip in a non-elastic material

An integral has been proposed for a non-elastic material whose value is determined by the magnitude of the singularities at the tip of a crack but which may be evaluated mainly far away from the crack, where the state of deformation may be determined numerically with higher accuracy than near the crack. The integral is intended for situations when plasticity is too great for linear elastic fracture mechanics to be appropriate, but may be related to the stress intensity factors in the linear elastic case. Its value has been calculated for a central or edge crack in a uniformly loaded and unloaded plate with a non-work-hardening elastic plastic material when the loading is either tension or longitudinal shear. It has also been calculated for a non-work-hardening material for a central sloping crack under tension and for a central crack under a quadratic temperature gradient for which previously suggested contour integrals are no longer path independent even in the linear elastic case.

---

Résumé On propose une intégrale, applicable à un matériau non élastique, dont la valeur est déterminée par la grandeur des singularités à la pointe d'une fissure, mais dont l'évaluation peut être faite en grande partie assez loin de la fissure, c'est-à-dire en une zône où l'état de déformation peut être établi par voie numérique avec une précision meilleure qu'au voisinage de la fissure. Cette intégrale est d'application lorsqu'il se trouve que la plasticité est trop importante pour que soit applicable la mécanique de la rupture linéaire et élastique, et où elle peut cependant être associée aux facteurs d'intensité des contraintes correspondant au cas linéaire et élastique.On a calculé sa valeur pour une fissure centrale ou latérale située dans une plaque soumise à chargement ou à déchargement uniforme-il s'agit de tension ou de cisaillement longitudinal-constituée d'un matériau élastique plastique non sujet à l'écrouissage.On a également considéré le calcul d'un tel matériau comportant une fissure inclinée par rapport à la direction des efforts, ou comportant une fissure centrale soumise à un gradient quadratique de températures, cas pour lesquels les intégrales de contour qui avaient été précédemment suggérées s'avérent ne plus être indépendantes du chemin parcouru, même en conditions élastiques.