Numerical simulations of fracture initiation in ductile solids under mode I, dynamic loading

In this paper, an overview of some recent computational studies by the authors on ductile crack initiation under mode I, dynamic loading is presented. In these studies, a large deformation finite element procedure is employed along with the viscoplastic version of the Gurson constitutive model that accounts for the micro-mechanical processes of void nucleation, growth and coalescence. A three-point bend fracture specimen subjected to impact, and a single edge notched specimen loaded by a tensile stress pulse are analysed. Several loading rates are simulated by varying the impact speed or the rise time and magnitude of the stress pulse. A simple model involving a semi-circular notch with a pre-nucleated circular hole situated ahead of it is considered. The growth of the hole and its interaction with the notch tip, which leads to plastic strain and porosity localization in the ligament connecting them, is simulated. The role of strain-rate dependence on ductile crack initiation at high loading rates, and the specimen geometry effect on the variation of dynamic fracture toughness with loading rate are investigated.     

Overload analysis and Je-based fatigue life prediction of spot-welded auto seat belt anchors

We demonstrate the validity of J_e, which comprehensively describes the effects of specimen geometry and loading type, in predicting the fatigue life of auto seat belt anchor panel. We first simplify the heat‐affected zone model to reduce the number of finite elements. We then establish finite‐element models reflecting the actual overload behaviour of three types of seat belt anchor specimens. Using elaborate finite‐element models, we obtain the effective crack‐driving parameter J_e composed of its ductility‐dependent modal components. It is confirmed that the J_e concept successfully predicts the fatigue life of multi‐spot‐welded panel structures represented by auto seat belt anchors here.     

Constraint effects on ductile fracture processes near a notch tip under mixed-mode loading

In this work, the effect of constraint on ductile fracture process of microvoid growth and coalescence near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, two sets of finite element simulations are carried out under two-dimensional plane strain conditions. In the first set, a modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. Next, ductile four point bend specimens subjected to mode I and mixed mode loading are considered. In both sets of simulations, the interaction between a notch tip and a pre-nucleated hole ahead of it is modelled. The background material is represented by the Gurson constitutive model and micro-void nucleation at uniformly distributed small scale particles is also taken into account. The accumulation of matrix plastic strain and porosity in the ligament between the notch tip and the hole as well as the growth of the hole are studied. Finally, the effect of crack tip constraint on the relationship between the fracture toughness and mode mixity is examined.     

A method for evaluating the thermal fatigue resistance of ceramics and its application to a fibre reinforced refractory material

An original method of characterizing thermal fatigue of ceramic materials has been proposed. This method is based on after-shock measurements of the degree of damage through a compliance calibration using compact tension (CT) test pieces. This method has been applied to a fibre-reinforced refractory material subjected to repeated thermal shock between 20 and 800° C. It has been demonstrated from both experiments and finite element analysis that the CT specimen is a convenient shape for the evaluation of thermal fatigue behaviour. In these specimens it has been established that the damage primarily affects the notch tip. The thermal fatigue behaviour of the CT specimens depends on notch length: when the notch length is greater than 30 mm, catastrophic failure occurs after a few cycles. When the notch length is less than 30 mm, the crack formed at the notch tip during the first cycle grows slowly during subsequent cycles. This behaviour has been explained by the variation of the stress intensity factor K _I.     

Constraint Loss under Dynamic Loading in Rate Independent Plastic Solids

The objectives of this paper are to examine the loss of crack tip constraint in dynamically loaded fracture specimens and to assess whether it can lead to enhancement in the fracture toughness at high loading rates which has been observed in several experimental studies. To this end, 2-D plane strain finite element analyses of single edge notched (tension) specimen and three point bend specimen subjected to time varying loads are performed. The material is assumed to obey the small strain J ₂ flow theory of plasticity with rate independent behaviour. The results demonstrate that a valid J–Q field exists under dynamic loading irrespective of the crack length and specimen geometry. Further, the constraint parameter Q becomes strongly negative at high loading rates, particularly in deeply cracked specimens. The variation of dynamic fracture toughness K _dc with stress intensity rate K for cleavage cracking is predicted using a simple critical stress criterion. It is found that inertia-driven constraint loss can substantially enhance K _dc for .     

The behaviour of cracks in elastic-plastic materials under plane normal and shear loadings

Cracks in structures are often subjected to complex loading conditions. The direction of the crack extension depends on the normal and the shear components of the load. This paper is based on the kinking behaviour of cracks taking elastic-plastic behaviour of materials into account. The J-integral and the mixed-mode components J _I and J _II were determined after having performed several finite element analyses for different loading conditions. The path independence of J, J _I and J II is investigated for both, the line integral proposed by Rice and the volume integral proposed by deLorenzi. For correctly determined crack deflection angles the J _II-component vanishes when a FE-model with a kinked crack is considered. Hence, cracks propagate perpendicularly to the local mode I load.     

A finite element analysis of dynamic fracture initiation by ductile failure mechanisms in a 4340 steel

In some recent dropweight impact experiments [5] with pre-notched bend specimens of 4340 steel, it was observed that considerable crack tunneling occurred in the interior of the specimen prior to gross fracture initiation on the free surfaces. The final failure of the side ligaments happened because of shear lip formation. The tunneled region is characterized by a flat, fibrous fracture surface. In this paper, the experiments of [5] (corresponding to 5 m/s impact speed) are analyzed using a plane strain, dynamic finite element procedure. The Gurson constitutive model that accounts for the ductile failure mechanisms of micro-void nucleation, growth and coalescence is employed. The time at which incipient failure was observed near the notch tip in this computation, and the value of the dynamic J-integral, _J d, at this time, compare reasonably well with experiments. This investigation shows that J-controlled stress and deformation fields are established near the notch tip whenever J _d , increases with time. Also, it is found that the evolution of micro-mechanical quantities near the notch root can be correlated with the time variation of J _d .The strain rate and the adiabatic temperature rise experienced at the notch root are examined. Finally, spatial variations of stresses and deformations are analyzed in detail.     

Numerical simulations of hole growth and ductile fracture initiation under mixed-mode loading

Numerical simulations of ductile fracture initiation caused by the interaction between a notch tip and a nearby hole under mixed-mode loading involving modes I and II are performed. Attention is restricted to plane strain, small-scale yielding conditions. The Gurson constitutive model that accounts for the ductile failure mechanisms of micro-void nucleation, growth and coalescence is employed within the framework of a finite deformation plasticity theory. The failure of the ligament connecting the notch tip and the hole by either microvoid coalescence or by intense plastic strain localization is modelled. The effect of mode-mixity on the notch tip deformation, hole growth and the critical value of J at fracture initiation is examined. The dominant failure mechanism which is operative near the notch tip for various extents of mixity of modes I and II is identified.     

Effect of thermomechanical loads on the propagation of crack near the interface brittle/ductile

The purpose of this paper is to understand the combined effect of thermal and mechanical loading on the initiation and behaviour of sub-interface crack in the ceramic. In this study a 2D finite element model has been used to simulated mixed mode crack propagation near the bimaterial interface. The assembly ceramometalic is subjected simultaneously to thermomechanical stress field. The extent of a plastic zone deformation in the vicinity of the crack-tip has a significant influence on the rate of its propagation. The crack growth at the joint specimen under four-point bending (4PB) loading and the influence of residual stresses was also evaluated by the maximum tensile stress criterion. The J-integral at the crack tip is generally expressed by the thermomechanical local stresses. The results obtained show the effect of the temperature gradient ΔT, the size of the crack and the applied stresses on the J-integral.     

Comparison of mode I and mode II elastic-plastic fracture toughness for two low alloyed high strength steels

In the present work, mode I and mode II tests were carried out on two low alloyed high strength steels. An asymmetrical four point bend specimen and J _II-integral vs. crack growth resistance curve technique were used for determining the mode II elastic-plastic fracture toughness, J _IIc · J _II-integral expression of the specimen was calibrated by finite element method. The results indicate that the present procedure for determining the J _IIc values is easy to use. Moreover, the mode I fracture toughness J _Ic is very sensitive to the rolling direction of the test steels, but the mode II fracture toughness J _IIc is completely insensitive to the rolling direction of the steels, and the J _IIc /J _Ic ratio is not a constant for the two steels, including the same steel with different orientations. Finally, the difference of the fracture toughness between the mode I and mode II is discussed with consideration of the different fracture mechanisms.     

Fracture behaviour of UPVC thin tubes at high loading rates

The use of fracture mechanics concepts in describing the failure behaviour of unplasticized poly(vinyl chloride) at high loading rates is studied. An impact test is employed which uses a gas pulse generated by a shock tube to load thin cylindrical specimens. These controlled pulses are used to internally pressurize specimens containing machined notches of different lengths or unnotched specimens. The specimen and the apparatus are both instrumented such that the pressure pulse and the resulting strain in the specimen are monitored throughout the tests to the point of fracture. Linear Elastic Fracture Mechanics (LEFM) is successfully applied to describe the fracture behaviour of the specimens at room temperature over a wide range of loading rates. A criterion is proposed by which the values ofK _c andG _c can independently be obtained over the range of the loading rates applied.     

Modelling the softening within the fracture process zone associated with the fissuring mode of crack growth in Zr-2.5Nb CANDU pressure tube material

The fissuring mode of fracture in CANDU pressure tube material, and in particular Stage 1 crack growth (essentially flat J _R curve) as observed in some irradiated compact toughness specimens has been investigated. Models are presented of the fracture process zone associated with a crack that tunnels at the specimen mid-section, which extends preliminary work reported earlier. Various types of process zone behaviour are analysed, and based on an appropriate value for J _c, the J value associated with the cumulative mode of crack propagation in irradiated material, together with an estimate of the tensile stress at the leading edge of the process zone, the known failure mechanism (formation, growth and coalescence of voids) of the ligaments between the fissures is shown to be reasonably consistent with the experimental measurements of the fissure spacing and fissure length.     

Critical tearing energy in a circumferentially-cracked circular cylinder of rubber under finite deformation

In this paper the circumferentially-cracked cylindrical (CCC) specimen is subjected to both asymptotic and finite element analyses to determine its suitability for measuring the critical tearing energy, T _c of rubber-like materials. When deformed axially, these specimens produce uniform radial crack deformation and extension along the circular crack front. The local/pointwise value of tearing energy along the crack front is also uniform and therefore identical to the tearing energy magnitude obtained by global measurement techniques such as the compliance method.Application of the CCC specimen is illustrated on a non-crystallizing rubber (SBR). Experimental results and non-linear finite element analysis are used to show that the CCC specimen yields a constant value of T _c which is statistically independent of crack depth and the two sizes of the specimen used in the experiments. The CCC specimen represents an improvement over pre-cracked thin sheet specimens used for measuring T _c where it has been demonstrated (see Mazich, et al. [2]), that crack extension along the crack front is pointwise non-uniform resulting in a globally measured value for critical tearing energy that is very dependent on specimen thickness and therefore different from the purportedly invariant material parameter T _c. The asymptotic analysis yields an expression relating tearing energy, T, deformed crack-tip radius, ₀, and material shear modulus, G of the form: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2% da9maalaaabaGaaG4maaqaaiaaiIdaaaGaeqiWdaNaam4raiabeg8a% YnaaBaaaleaacaaIWaaabeaaaaa!409C!\[T = \frac{3}{8}\pi G\rho _0 \]. This expression yields numerical results which are in excellent agreement with the finite element predictions and test data.     

Mixed-mode fracture toughness testing of concrete beams in three-point bending

Results obtained for mixed-mode fracture toughness parameters K _c , G _c , J _c , G _F (plane strain mixed-mode stress-intensity factor, energy release rate, J-integral and fracture energy, respectively) for small notched concrete beams in bending indicate that all these parameters decrease with x/S (x is the distance from support, S is the span) in general to values near midspan consistent with Mode I results. All the parameters except J _c vary with notch depth in a similar manner for each notch location.     

A numerical method for evaluation of J‐integral in plates made of functionally graded materials with sharp and blunt V‐notches

The main purpose of the paper is to propose a numerical method for evaluation of J‐integral in plates made of functionally graded materials (FGM) with sharp and blunt V‐notches under Mode I loading. The material properties have been assumed to be varied exponentially along the specimen width (notch direction). Using the proposed method, the effect of material gradient on the J‐integral for two cases of sharp and blunt V‐notches has been studied. The results have shown that in FGMs with sharp V‐notches, the J‐integral is not proportional to . So, the parameter J_L is path dependent. It has been observed that the material gradient has larger effect on the J‐integral in sharp V‐notch compared with that in blunt V‐notch.     

Fracture mechanics of short glass fibre-reinforced nylon composite

The fracture behaviour of injection-moulded short glass fibre-reinforced, thermoplastic nylon 6.6 plaques has been studied under static loading using compact tension specimens and under impact loading using single-edge notched charpy specimens. The influences of specimen position as taken from the plaque mouldings, notch direction, notch sharpness and the rate of testing on the fracture toughness of this composite system were investigated. Results indicated that the fracture toughness is highest for the cracks perpendicular to the mould fill direction and is lowest for cracks parallel to the mould fill direction. A single fracture parameter, K _c, seems to be inadequate for fracture toughness characterization. Evaluation of the fracture toughness as a function of notch sharpness indicated that for notches perpendicular to the mould fill direction the fracture toughness is not affected by the sharpness of the initial notch. However, for cracks in the mould fill direction, sharpness of the initial notch had a significant effect upon the measured value of the fracture toughness. Results also indicated, that the fracture toughness is rate insensitive over the crosshead speed ranging from 0.5–50 mm min^–1. Finally, the specimen position, as taken from plaque mouldings, had no significant effect on the measured value of the fracture toughness.     

J II fracture testing of a plastically deforming material

A compact model II fracture specimen was previously analyzed and employed to determine the mode II fracture toughness K _IIc , of perspex. In employing this specimen for a more ductile material such as aluminium, it was observed that the load vs. crack sliding displacement record becomes nonlinear for small loads. Thus, concepts of linear elastic fracture mechanics cannot be employed. To this end, the specimen was calibrated for J-integral testing, so that J _IIc mesurements can be performed.In this study, mode I and II tests are carried out on an aircraft aluminium alloy, AI 7075-T7351. First, standard K _Ic tests are performed leading to a value of 27.9 15-1 which would be equivalent to a J _Ic of 10.7 kN/m. Then standard J _Ic tests are carried out on this material with specimen thicknesses, of 5, 7.5 and 9.9 mm, leading to an average J _Ic value of 10.5 kN/m. Methods for J _II testing are proposed; a series of specimens of six thicknesses between 5 and 16 mm are employed for testing. An average J _IIc value was found to be 40.2 kN/m which yields a K _IIc value of 54.1 15-2. Thus, K _IIc is seen to be approximately twice that of K _Ic for this material.     

J-INTEGRAL SOLUTIONS FOR SHALLOW CRACK GEOMETRIES UNDER COMBINED TENSION AND BENDING

This paper presents J-integral solutions for geometries with shallow edge cracks (0.05 ≤a/W≤ 0.1) subjected to combined tension and bending. Tension and bending is applied simultaneously and sequentially. Curves of J versus structural strain are obtained from two-dimensional plane strain finite element analyses using medium to low work hardening power laws to represent material behaviour. Empirical equations are derived from the finite element results to estimate the J-integrals, using previously developed J-estimation schemes for SENT and SENB specimens with only minor modifications. The predictions presented are invariably safe, but due to the complexity of the problems studied and the variability of the loadings considered, the degree of conservatism is found to be high in some cases.     

A STUDY OF SUPERIMPOSED FRACTURE MODES I, II AND III ON PMMA

A test apparatus has been developed to study the fracture behaviour of engineering materials subjected to superimposed tensile and shear (Mode I and II) loads using a single edge notch specimen. Stress intensity factors were calculated using finite element analysis. Test results for PMMA are reported. Results from circumferentially notched round bar specimens, subjected to combined tension and torsion (Mode I and III) loading are also reported. The Mode I/II results are consistent with the mixed mode fracture response of a wide range of brittle materials, although there is some evidence of non-linear behaviour. The fracture behaviour for superimposed Modes I and III indicates the strong influence of non-linear deformation which causes the mixed mode toughness to be dependent on the sequence and type of loading.     

Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading

In the present work the J-integral (indicated here as J_Vρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of J_Vρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, J_Vρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of J_Vρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.