Steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modeling

This work employs high resolution, finite element computations to investigate key features of the elastic–plastic fields near a steadily advancing crack at quasi-static rates under three-dimensional, small-scale yielding conditions. The model represents a structurally thin component constructed of a material (e.g., Al and Ti alloys) with flow stress and fracture toughness properties that together limit the size of the in-plane plastic zone during steady-growth to no more than several multiples of the plate thickness. The computational approach generalizes the streamline integration procedure used previously for two-dimensional studies into three dimensions to represent steady-state growth on a fixed mesh in a boundary-layer framework. The plate thickness provides the only geometrical length scale. Crack extension occurs at the remotely applied, fixed loading without the need for a local growth criterion. In the first computations of this type, the present work considers a straight crack front advancing under local and global mode I loading with zero T-stress in a moderately hardening material. Applied remote loads at steady growth generate plastic zone sizes ahead of the advancing crack front ranging from 0.25 to 6.4 times the thickness. Key results include: (1) the crack-front fields exhibit a self-similar scaling characterized by a non-dimensional loading parameter; (2) three-dimensional effects extend to distances of approximately 1.5–2.5 times the thickness ahead of the advancing crack front for key values of this loading parameter, beyond which the fields (elastic–plastic then linear-elastic at greater distances) become uniform over the thickness; and (3) crack opening profiles on the outside surface reveal a “wedge-like”, opening shape which simplifies the definition of a crack-tip opening angle.     

Mode I fatigue crack growth under biaxial loading

This paper is centred on the role of the T-stress during mode I fatigue crack growth. The effect of a T-stress is studied through its effect on plastic blunting at crack tip. As a matter of fact, fatigue crack growth is characterized by the presence of striations on the fracture surface, which implies that the crack grows by a mechanism of plastic blunting and re-sharpening (Laird C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue crack propagation, STP 415. Philadelphia: ASTM; 1967. p. 131–68 [8]). In the present study, plastic blunting at crack tip is a global variable ρ, which is calculated using the finite element method. ρ is defined as the average value of the permanent displacement of the crack faces over the whole K-dominance area. The presence of a T-stress modifies significantly the evolution of plastic deformation within the crack tip plastic zone as a consequence of plastic blunting at crack tip. A yield stress intensity factor K_Y is defined for the cracked structure, as the stress intensity factor for which plastic blunting at crack tip exceeds a given value. The variation of the yield stress intensity factor was studied as a function of the T-stress. It is found that the T-stress modifies significantly the yield point of the cracked structure and that the yield surface in a (T, K_I) plane is independent of the crack length. Finally, a yield criterion is proposed for the cracked structure. This criterion is an extent of the Von-Mises yield criterion to the problem of the cracked structure. The proposed criterion matches almost perfectly the results obtained from the FEM. The evolution of the yield surface of the cracked structure in a (T, K_I) plane was also studied for a few loading schemes. These results should develop a plasticity model for the cracked structure taking into account the effect of the T-stress.     

Mode I cracks subjected to large T-stresses

There are several criteria for predicting brittle fracture in mode I and mixed mode loading. In this paper, the modified maximum tangential stress criterion originally proposed for mixed mode loading, is employed to study theoretically brittle fracture for mode I cracks. In particular, the effect of the non-singular term of stress, often known as the T-stress, on the angle of initiation of fracture and the onset of crack growth is explored. The T-stress component of the tangential stress vanishes along the crack line. Therefore, it is often postulated for linear elastic materials that the effect of T-stress on mode I brittle fracture can be ignored. However, it is shown here that the maximum tangential stress is no longer along the line of initial crack when the T-stress exceeds a critical value. Thus, a deviation in the angle of initiation of fracture can be expected for specimens having a large T-stress. It is shown that the deviation angle increases for larger values of T-stress. Theoretical results show that the apparent fracture toughness decreases significantly when a deviation in angle occurs. Earlier experimental results are used to corroborate the findings. The effect of large T-stresses is also explored for a crack specimen undergoing moderate scale yielding. The elastic-plastic investigation is conducted using finite element analysis. The finite element results reveal a similar deviation in the angle of maximum tangential stress for small to moderate scale yielding.     

Numerical simulation of constraint effects in fatigue crack growth

The computational analysis of constraint effects on fatigue crack growth is discussed. An irreversible cohesive zone model is used in the computations to describe the processes of material separation under cyclic loading. This approach is promising for the investigation of fatigue crack growth under constraint as the energy dissipation due to the formation of new crack surface and cyclic plastic deformation is accounted for independently. Fatigue crack growth in multi-layer structures under consideration of different levels of T-stress are conducted with a modified boundary layer model. Fatigue crack growth is computed as a function of layer thickness and T-stress for constant and variable amplitude loading cases.     

In-plane and out-of-plane crack-tip constraint effects under biaxial nonlinear deformation

In-plane and out-of-plane constraint effects on crack-front stress fields under both elastic–plastic and creep conditions are studied by means of three-dimensional numerical analyses of finite thickness boundary layer models and plane strain reference solutions. This investigation is an extension of the plane strain solution obtained by Shlyannikov and Boychenko in 2008, with special attention on what constraint parameters existed in the nonlinear crack-tip fields in a finite thickness solid. Characterization of constraint effects is given by using the non-singular T-stress, the local triaxiality parameter, the factor of the stress-state in 3D cracked body and the second order term amplitude factor. The influence of nominal stress load biaxiality and creep time on the behavior of constraint factors is considered. Stresses and constraint factors from FEA at the crack-front on different planes in the thickness direction of the plate are compared with plane strain reference solutions. The results show that 3D-stress fields can be characterized in common with the local triaxiality parameter and factor of the stress-state in 3D solid by the three-term solution throughout the thickness even in the region near the free surface. It is found that there is a distinct relationship between the in-plane and the crack-front out-of-plane constraint factors which can be well captured using the relation between the second order term amplitude factor and remote boundary layer stress.     

Stress intensity factor K and the elastic T-stress for corner cracks

The stress intensity factor K and the elastic T-stress for corner cracks have been determined using domain integral and interaction integral techniques. Both quarter-circular and tunnelled corner cracks have been considered. The results show that the stress intensity factor K maintains a minimum value at the mid-plane where the T-stress reaches its maximum, though negative, value in all cases. For quarter-circular corner cracks, the K solution agrees very well with Pickard's (1986) solution. Rapid loss of crack-front constraint near the free surfaces seems to be more evident as the crack grows deeper, although variation of the T-stress at the mid-plane remains small. Both K and T solutions are very sensitive to the crack front shape and crack tunnelling can substantially modify the K and T solutions. Values of the stress intensity factor K are raised along the crack front due to crack tunnelling, particularly for deep cracks. On the other hand, the difference in the T-stress near the free surfaces and at the mid-plane increases significantly with the increase of crack tunnelling. These results seem to be able to explain the well-observed experimental phenomena, such as the discrepancies of fatigue crack growth rate between CN (corner notch) and CT (compact tension) test pieces, and crack tunnelling in CN specimens under predominantly sustained load.     

Quasi-static crack tip fields in rate-sensitive FCC single crystals

In this work, the effects of loading rate, material rate sensitivity and constraint level on quasi-static crack tip fields in a FCC single crystal are studied. Finite element simulations are performed within a mode I, plane strain modified boundary layer framework by prescribing the two term (K − T) elastic crack tip field as remote boundary conditions. The material is assumed to obey a rate-dependent crystal plasticity theory. The orientation of the single crystal is chosen so that the crack surface coincides with the crystallographic (010) plane and the crack front lies along [10[`1]][10\overline 1] direction. Solutions corresponding to different stress intensity rates [(K)\dot]\dot{{K}}, T-stress values and strain rate exponents m are obtained. The results show that the stress levels ahead of the crack tip increase with [(K)\dot]\dot{{K}} which is accompanied by gradual shrinking of the plastic zone size. However, the nature of the shear band patterns around the crack tip is not affected by the loading rate. Further, it is found that while positive T-stress enhances the opening and hydrostatic stress levels ahead of crack tip, they are considerably reduced with imposition of negative T-stress. Also, negative T-stress promotes formation of shear bands in the forward sector ahead of the crack tip and suppresses them behind the tip.     

Cracked asphalt pavement under traffic loading – A 3D finite element analysis

An asphalt pavement containing a transverse top-down crack is investigated under traffic loading using 3D finite element analysis. The stress intensity factors (SIFs) and the T-stress are calculated for different distances between the crack and the vehicle wheels. It is found that all the three Modes (I, II and III) are present in the crack deformation. The signs and magnitudes of K_I, K_II, K_III and T are significantly dependent on the location of the vehicle wheels with respect to the crack plane. The magnitude of T-stress is considerable, if compared to the stress intensity factors, when one of the wheels is very close to the crack plane.     

Crack growth characteristics of carbon nanotube-based polymer composites subjected to cyclic loading

This paper presents the characterization of crack growth in carbon nanotube (CNT)-based polymer composites under fatigue loading. Fatigue crack growth tests were performed on single-edge cracked plate specimens of CNT/polycarbonate composites at room temperature and liquid nitrogen temperature (77 K). An elastic–plastic finite element analysis was also conducted to determine the J-integral range. The crack growth rate data were expressed in terms of the J-integral range, and the effect of nanotube addition on the fatigue crack growth behavior was examined. In addition, possible mechanisms of the crack growth in the nanocomposites are discussed based on microscopic observations of the specimen fracture surfaces.     

T-Stress Effects on Isochromatic Fringe Patterns in Mode II

The theory of photoelasticity is used to study analytically the effects of T-stress on the fringe patterns around the crack tip in mode II crack specimens. The locus of an isochromatic fringe determined by taking into account the T-stress is compared with the locus of a fringe with no T-stress. It is shown for mode II cracks that in the presence of T-stress, the fringe loops are neither symmetric nor continuous. Asymmetric and discontinuous fringe patterns predicted in this paper are consistent with the experimental results observed previously in photoelasticity tests.     

J resistance behavior in functionally graded materials using cohesive zone and modified boundary layer models

This paper describes elastic–plastic crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) under mode I loading conditions using cohesive zone and modified boundary layer (MBL) models. For this purpose, we first explore the applicability of two existing, phenomenological cohesive zone models for FGMs. Based on these investigations, we propose a new cohesive zone model. Then, we perform crack growth simulations for TiB/Ti FGM SE(B) and SE(T) specimens using the three cohesive zone models mentioned above. The crack growth resistance of the FGM is characterized by the J-integral. These results show that the two existing cohesive zone models overestimate the actual J value, whereas the model proposed in the present study closely captures the actual fracture and crack growth behaviors of the FGM. Finally, the cohesive zone models are employed in conjunction with the MBL model. The two existing cohesive zone models fail to produce the desired K–T stress field for the MBL model. On the other hand, the proposed cohesive zone model yields the desired K–T stress field for the MBL model, and thus yields J _R curves that match the ones obtained from the SE(B) and SE(T) specimens. These results verify the application of the MBL model to simulate crack growth resistance in FGMs.     

Three‐dimensional mixed‐mode (I and II) crack‐front fields in ductile thin plates — effects of T‐stress

It has been well‐established that the non‐singular T‐stress provides a first‐order estimate of geometry and loading mode (e.g. tension versus bending) effects on elastic–plastic crack‐front field under mode I loading conditions. The objective of this paper is to exam the T‐stress effect on three‐dimensional (3D) crack‐front fields under mixed‐mode (modes I and II) loading. To this end, detailed 3D small strain, elastic–plastic simulations are carried out using a 3D boundary layer (small‐scale yielding) formulation. Characteristics of near crack‐front fields are investigated for a wide range of T‐stresses (T/σ₀ = ?0.8, ?0.4, 0.0, 0.4, 0.8). The plastic zones and thickness and angular and radial variations of the stresses are studied, corresponding to two values of the remote elastic mixity parameters M_e = 0.3 and 0.7, under both low and high levels of applied loads. It is found that different T‐stresses have a significant effect on the plastic zones size and shapes, regardless of the mode mixity and load level. The thickness, angular and radial distributions of stresses are also affected markedly by T‐stress. It is important to include these effects when investigating the mixed‐mode ductile fracture failure process in thin‐walled structural components.     

Variation of elastic T-stresses along slightly wavy 3D crack fronts

The non-singular terms in the series expansion of the elastic crack-tip stress field, commonly referred to as the elastic T-stresses, play an important role in fracture mechanics in areas such as the stability of a crack path and the two-parameter characterization of elastic-plastic crack-tip deformation. In this paper, a first order perturbation analysis is performed to study some basic properties of the T-stress variation along a slightly wavy 3D crack front. The analysis employs important properties of Bueckner-Rice 3D weight function fields. Using the Boussinesq-Papkovitch potential representation for the mode I weight function field, it is shown that, for coplanar cracks in an infinite isotropic and homogeneous linear elastic body, the mean T-stress along an arbitrary crack front is independent of the shape and size of the crack. Further, a universal relation is discovered between the mean T-stress and the stress field at the same crack front location under the same loading but in the absence of a crack. First-order-accurate solutions are given for the T-stress variation along a slightly wavy crack front with nearly circular or straight confifurations. Specifically, cosine wave functions are adopted to describe smooth polygonal and slightly undulating planar crack shapes. The results indicate that T ₁₁, the 2D T-stress component acting normal to the crack front, increases with the curvature of the crack front as it bows out but T ₃₃, acting parallel to the crack front, decreases with the crack front curvature.     

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.     

J-Dominance in Mixed Mode Ductile Fracture Specimens

The asymptotic mixed mode crack tip fields in elastic-plastic solids are scaled by the J-integral and parameterized by a near-tip mixity parameter, M _{_p} . In this paper, the validity and range of dominance of these fields are investigated. To this end, small strain elastic-plastic finite element analyses of mixed mode fracture are first performed using a modified boundary layer formulation. Here, a two term expansion of the elastic crack tip field involving the stress intensity factor |K| the elastic mixity parameter M _{_e} as well as the T-stress is prescribed as remote boundary conditions. The analyses are conducted for different values of M _{_e} and the T-stress. Next, several commonly used mixed mode fracture specimens such as Compact Tension Shear (CTS), Four Point Bend (4PB), and modified Compact Tension specimen are considered. Here, the complete range of loading from contained yielding to large scale yielding is analyzed. Further, different crack to width ratios and strain hardening exponents are considered. The results obtained establish that the mixed mode asymptotic fields dominate over physically relevant length scales in the above geometries, except for predominantly mode I loading and under large scale yielding conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

J–Q characterized stress fields of surface-cracked metallic liners – II. Composite overwrapped pressure vessels

Two-parameter J–Q elastic–plastic crack front fields are developed for surface-cracked metallic liners of composite overwrapped pressure vessels (COPV). Uniaxial tensile data from 6061-T6 aluminum coupon specimens for the metallic liner and anisotropic elastic material properties for the filament wound carbon fiber epoxy are used in three-dimensional finite element models. Modified boundary layer (MBL) finite element solutions are used to evaluate near tip dominance and parameterization limits. Semicircular surface cracks of varying depths inserted in the inner liner surface are investigated. J and Q crack front distributions, and the corresponding parameterization limits, and near tip triaxiality trends are obtained and the effects of elastic–plastic material discontinuity of the heterogeneous joint and the biaxiality of stresses are evaluated. J–Q predicted fields maintain accuracy for higher far-field loads and lower near tip deformations compared to the liner only models for angles near the free surface. However, for the critical crack growth region, Q does not maintain a radially independent measure of constraint for loads seen in a typical COPV; therefore, these fracture predictors may not be applicable. In the COPV, large-scale yielding marks a transition where triaxiality is higher as a function of constraint compared to the linear relationship common to homogeneous structures. Results from this study will facilitate the implementation of proof test logic and accurate fracture prediction of COPV liners with emphasis on geometric limits and fracture specimen applicability.     

A New Fracture Toughness Test Covering Mixed-Mode Conditions and Positive and Negative <Emphasis Type="Italic">T</Emphasis>-Stresses

It is known that sign of T-stress in cracked specimens affects fracture toughness under mixed mode conditions. We suggest a new test involving an inclined edge cracked semi circular specimen subjected to asymmetric three-point bend loading (IASCB specimen) that covers a broad range of modes I and II SIFs and T-stress values. It can provide both positive and negative T-stresses. This is illustrated by FEM computations.     

History effect in fatigue crack growth under mixed-mode loading conditions

Plastic deformation within the crack tip region introduces internal stresses that modify subsequent behaviour of the crack and are at the origin of history effects in fatigue crack growth. Consequently, fatigue crack growth models should include plasticity-induced history effects. A model was developed and validated for mode I fatigue crack growth under variable amplitude loading conditions. The purpose of this study was to extend this model to mixed-mode loading conditions. Finite element analyses are commonly employed to model crack tip plasticity and were shown to give very satisfactory results. However, if millions of cycles need to be modelled to predict the fatigue behaviour of an industrial component, the finite element method becomes computationally too expensive. By employing a multiscale approach, the local results of FE computations can be brought to the global scale. This approach consists of partitioning the velocity field at the crack tip into plastic and elastic parts. Each part is partitioned into mode I and mode II components, and finally each component is the product of a reference spatial field and an intensity factor. The intensity factor of the mode I and mode II plastic parts of the velocity fields, denoted by dρ_I/dt and dρ_II/dt, allow measuring mixed-mode plasticity in the crack tip region at the global scale. Evolutions of dρ_I/dt and dρ_II/dt, generated using the FE method for various loading histories, enable the identification of an empirical cyclic elastic–plastic constitutive model for the crack tip region at the global scale. Once identified, this empirical model can be employed, with no need of additional FE computations, resulting in faster computations. With the additional hypothesis that the fatigue crack growth rate and direction can be determined from mixed-mode crack tip plasticity (dρ_I/dt and dρ_II/dt), it becomes possible to predict fatigue crack growth under I/II mixed-mode and variable amplitude loading conditions. To compare the predictions of this model with experiments, an asymmetric four point bend test system was setup. It allows applying any mixed-mode loading case from a pure mode I condition to a pure mode II. Initial experimental results showed an increase of the mode I fatigue crack growth rate after the application of a set of mode II overload cycles.     

Constraint effects on plastic crack-tip fields for plane strain mode-I, II and III cracks in non-hardening materials

To explore constraint effects on fully plastic crakc-tip fields, analytical solutions are examined for mode-I, II and III loading in non-hardening materials under plane strain conditions. The results reveal that under mode-II and III loading the crack-tip stress fields are unique, and thus can be characterized by a `single parameter'. Under mode-I loading, however, the crack-tip stress field is non-unique but can be characterized by two sets of solutions or `two parameters'. One set of the solutions is the well-known Prandtl field and the other is a plastic T-stress field. This conclusion corroborates the observation of McClintock (1971) that the slip-line field is non-unique for plane strain tensile cracks. A two-term plastic solution which combines the Prandtl field and the plastic T-stress field with two parameters B ₁ and B ₂ can then characterize the crack-tip stress field of plane strain mode-I crack over the plastic region and quantify the magnitude of crack-tip constraints. These characters are similar to those for hardening materials. Analyses and examples show that the two-term plastic solution can match well with the slip-line field or finite element results over plastic region. Thus the parameters B ₁ and B ₂ can be used to characterize the constraint level for mode-I finite-sized crack specimens in non-hardening materials under plane strain conditions.     

Elastic–plastic J and COD estimation schemes for 90° elbow with throughwall circumferential crack at intrados under in-plane opening moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic–plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic–plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, such schemes for elbow (or pipe bend), which is one of the important components for LBB analyses, is very meager. Recently, elastic–plastic J and COD estimation scheme has been developed for throughwall circumferentially cracked elbow subjected to closing bending moment. However, it is well known that the elbow deformation characteristics are distinctly different for closing and opening bending modes because the ovalisation patterns of elbow cross section are different under these two modes. Development of elastic–plastic J and COD estimation scheme for an elbow with throughwall circumferential crack at intrados subjected to opening bending moment forms the objective of the present paper. Experimental validation of proposed J-estimation scheme has been provided by comparing the crack initiation, unstable ductile tearing loads and crack extension at instability with the test data. The COD estimation scheme has been validated by comparing the COD of test data with the predictions of the proposed scheme.