Effect of residual stresses on ductile crack growth resistance

It is well known that residual stresses influence the ductile fracture behaviour. In this paper, a numerical study was performed to assess the effect of residual stresses on ductile crack growth resistance of a typical pipeline steel. A modified boundary layer model was employed for the analysis under plane strain, Mode I loading condition. The residual stress fields were introduced into the finite element model by the eigenstrain method. A sharp crack was embedded in the center of the weld region. The complete Gurson model has been applied to simulate the ductile fracture by microvoid nucleation, growth and coalescence. Results show that tensile residual stresses can significantly reduce the crack growth resistance when the crack growth is small compared with the length scale of the tensile residual stress field. With the crack growth, the effect of residual stresses on the crack growth resistance tends to diminish. The effect of residual stress on ductile crack growth resistance seems independent of the size of geometrically similar welds. When normalized by the weld zone size, the ductile crack growth resistance collapses into one curve, which can be used to assess the structural integrity and evaluate the effect of residual stresses. It has also been found that the effect of residual stresses on crack growth resistance depends on the initial void volume fraction f₀, hardening exponent n and T-stress.     

A cohesive XFEM model for simulating fatigue crack growth under mixed-mode loading and overloading

Structures are subjected to cyclic loads that can vary in direction and magnitude, causing constant amplitude mode I simulations to be too simplistic. This study presents a new approach for fatigue crack propagation in ductile materials that can capture mixed-mode loading and overloading. The extended finite element method is used to deal with arbitrary crack paths. Furthermore, adaptive meshing is applied to minimize computation time. A fracture process zone ahead of the physical crack tip is represented by means of cohesive tractions from which the energy release rate, and thus the stress intensity factor can be extracted for an elastic-plastic material. The approach is therefore compatible with the Paris equation, which is an empirical relation to compute the fatigue crack growth rate. Two different models to compute the cohesive tractions are compared. First, a cohesive zone model with a static cohesive law is used. The second model is based on the interfacial thick level set method in which tractions follow from a given damage profile. Both models show good agreement with a mode I analytical relation and a mixed-mode experiment. Furthermore, it is shown that the presented models can capture crack growth retardation as a result of an overload.     

Experimentelle und numerische Ermittlung dynamischer Risswiderstandskurven im Kerbschlagbiegeversuch

Experimental and numerical determination of crack resistance curves in the notched‐bar impact test The assessment of the reliability of components requires the knowledge of crack resistance curves, which are often not available due to lack of specimen material. More likely is the availability of typical material parameters such as the yield strength, tensile strength, uniform elongation, elongation at rupture as well as upper shelf impact energy and the lateral elongation of notched‐bar impact test specimens. The material model of Gurson describes ductile crack growth due to the nucleation, growth and coalescence of voids in the material. Although dependent on the material and temperature, the material parameters of the Gurson model are independent of the specimen geometry and rate of loading. This latter fact allows one to use the values of these parameters determined on statically‐loaded fracture mechanics specimens to model specimens with other geometries and subjected to different loading conditions, in particular to model impact loaded Charpy‐V specimens. A method is proposed to construct crack resistance curves based on available data of tension tests and on quasi‐static yield curves. Dynamic yield curves are determined using proven procedures as based on the analysis of the dislocation activation energy. The ductile damage parameters are then obtained via simulation of tests on notched tensile specimens and notched‐bar impact tests as well as the fitting to the upper shelf impact energy. In this way, the ductile damage parameters are determined, which in turn enable the determination of the required J‐resistance curves via simulation of ductile crack growth in fracture mechanics specimens. Thus, the application of the classical J‐integral concept gets possible. Furthermore, the independence of the identified material parameters from the geometry of the specimen then allows the use of the Gurson model to analyse the safety of structural components with cracks directly.     

A modified cohesive zone model for a high‐speed expanding crack

The present work studies a self‐similar high‐speed expanding crack of mode‐I in a ductile material with a modified cohesive zone model. Compared with existing Dugdale models for moving crack, the new features of the present model are that the normal stress parallel to crack faces is included in the yielding condition in the cohesive zone and traction force in the cohesive zone can be non‐uniform. For a ductile material defined by von Mises criterion without hardening, the present model confirms that the normal stress parallel to crack face increases with increasing crack speed and can be even larger than the normal traction in the cohesive zone, which justifies the necessity of including the normal stress parallel to the crack faces in the yielding condition at high crack speed. In addition, strain hardening effect is examined based on a non‐uniform traction distribution in the cohesive zone.     

ANALYSIS AND MODELISATION OF SHORT CRACK GROWTH BY DUCTILE FRACTURE MICROMECHANISMS

Abstract— A ductile medium strength steel has been modelled by means of the Gurson model, and been used to investigate the effect of crack tip constraint in several fracture mechanics specimens. Both numerical and experimental results have been obtained, in the course of the crack extension process, for single edge notch bending specimens with different crack length-to-width ratios. The geometries with the shorter cracks always exhibited higher J values at initiation and steeper J crack growth resistance curves, and these results have been explained in terms of the stress and strain fields and damage development in the region ahead of the crack tip.     

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.     

Comparison of cohesive zone model and linear elastic fracture mechanics for a mode I crack near a compliant/stiff interface

Cohesive zone model has been widely applied to simulate crack growth along interfaces, but its application to crack growth perpendicularly across the interface is rare. In this paper, the cohesive zone model is applied to a crack perpendicularly approaching a compliant/stiff interface in a layered material model. One aim is to understand the differences between the cohesive zone model and linear elastic fracture mechanics in simulating mode I crack growth near a compliant/stiff interface. Another aim is to understand the effects of elastic modulus mismatch and cohesive strength of the stiff layer on the crack behavior near the interface. To simulate crack growth approaching an interface, the cohesive zone model which incorporates both the energy criterion and the strength criterion is an effective method.     

Crack growth resistance behavior of a functionally graded material: computational studies

This paper describes crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) using a cohesive zone ahead of the crack front. The plasticity in the background (bulk) material follows J₂ flow theory with the flow properties determined by a volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model). A phenomenological, cohesive zone model with six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) describes the constitutive response of the cohesive zone. Crack growth occurs when the complete separation of the cohesive surfaces takes place. The crack growth resistance of the FGM is characterized by a rising J-integral with crack extension (averaged over the specimen thickness) computed using a domain integral (DI) formulation. The 3-D analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. The paper describes applications of the cohesive zone model and the DI method to compute the J resistance curves for both single-edge notch bend, SE(B), and single-edge notch tension, SE(T), specimens having properties of a TiB/Ti FGM. The numerical results show that the TiB/Ti FGM exhibits significant crack growth resistance behavior when the crack grows from the ceramic-rich region into the metal-rich region. Under these conditions, the J-integral is generally higher than the cohesive energy density at the crack tip even when the background material response remains linearly elastic, which contrasts with the case for homogeneous materials wherein the J-integral equals the cohesive energy density for a quasi-statically growing crack.     

Crack Growth Across a Strength Mismatched Bimaterial Interface

Crack growth across an interface between materials with different strength is examined by a cohesive zone model. The two materials have identical elastic properties but different fracture process properties, or different yield stresses, which is modeled by different cohesive stresses. The fracture criteria is a critical crack opening displacement. Load is represented by a stress intensity factor defining a remote square root singular stress field. The results show that the ratio between the cohesive stresses of the two materials primarily determines the behavior of the critical stress intensity factor. When the crack approaches a material with a higher cohesive stress the crack tip is shielded, but if the crack approaches a material with smaller critical crack opening displacement the maximum level of shielding is determined by the ratio between the critical crack opening displacements. When a crack approaches a material with a lower cohesive stress it is exposed to an amplified load. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture toughness of layered structures: Embrittlement due to confinement of plasticity

The fracture toughness of a layered composite material is analyzed employing a combined two dimensional dislocation dynamics (DD)-cohesive zone (CZ) model. The fracture mechanism of an elastic-plastic (ductile) material sandwiched within purely elastic layers approaches ideally brittle behaviour with decreasing layer thickness. We investigate the influence of different constitutive parameters concerning dislocation plasticity as well as the effect of cohesive strength of the ductile material on the scaling of fracture toughness with layer thickness. For a constant layer thickness, the results of the numerical model are consistent with the expectation that fracture toughness decreases with increasing yield strength, but increases with the cohesive strength of the material. The scaling behaviour of the fracture toughness with layer thickness depends on these material parameters, but also on the dislocation microstructure in the vicinity of the crack tip. Strain localization due to easy dislocation generation right at the crack tip improves toughness in thin layers and leads to a jump-like increase of fracture toughness with layer thickness. However, the fracture toughness for films that are thick enough to exhibit bulk behaviour proves to be higher when the distribution of dislocations is more homogeneous, because in this case the crack grows in a stable fashion over some distance.     

Crack growth predictions in polyethylene using measured traction–separation curves

The accuracy of predicting the crack growth in any cohesive zone model calculation depends critically on the choice of cohesive law. A novel experimental method was used to measure directly such a cohesive law or ‘traction–separation curve’ in polyethylene. Deep notched tensile specimens were tested under constant displacement rate conditions, which facilitated a localisation of the damage mechanisms thought to precede crack growth and allowed a quantification of these processes independent of bulk deformation. Results showed that both the fracture energy and cohesive strength measured in this manner are a function of the applied rate and specimen geometry.Here we present a cohesive zone model within the finite-volume method to predict crack initiation and propagation history in three grades of polyethylene of different toughness, using the experimental measurements described above. The choice of cohesive law is crucial as it has a fundamental bearing on the predicted crack growth rates, particularly in tough polymers, where changes in the prevailing rate, constraint and temperature may affect the magnitude of the holding tractions within the damage zone. Initially a single experimentally measured, fixed rate traction–separation curve was used in the model as the fracture criterion but was unable to provide satisfactory crack growth predictions. By contrast, use of a more physically realistic family of curves measured at different rates provided better agreement of the prediction with experiment for the tough polyethylenes and very good agreement for the more brittle polyethylene. It was concluded that along with a rate dependent cohesive law, an accurate prediction of the crack growth history of tough polyethylenes would also require an incorporation of the effects of variations in constraint and perhaps also temperature. The ultimate goal may therefore be the development of a physical material model, sufficiently calibrated by experimental data, which would be able to accurately describe the local fracture process via a rate, constraint and temperature dependent traction–separation law.     

Comparison of cohesive zone parameters and crack tip stress states between two different specimen types

The cohesive zone parameters (separation energy and cohesive strength) and the crack tip triaxialities are compared between a compact tension (CT) and a double edge notched tension (DENT) specimen with smooth side-surfaces. The material is a pressure vessel steel 20MnMoNi55. The cohesive zone parameters are determined by fitting the simulated crack extensions near the midsection to the experimental data. The purpose of the study is to understand the relationship between the cohesive zone parameters and the crack tip stress triaxiality. The results show that for the same cohesive zone parameters the crack tip triaxiality near the midsection is lower in DENT specimens than in CT specimens. When the separation energy is set constant for CT and DENT specimens, the cohesive strength for the DENT specimens should be significantly lower than that for the CT specimens in order to make the simulated crack extensions near the midsection fit to the experimental data. Near the midsection, the cohesive strength and crack tip triaxiality influence each other: the specimen with a higher stress triaxiality has a higher cohesive strength; an increase of cohesive strength results in an increase of the crack tip triaxiality.     

A discrete cohesive model for fractal cracks

The fractal crack model described here incorporates the essential features of the fractal view of fracture, the basic concepts of the LEFM model, the concepts contained within the Barenblatt-Dugdale cohesive crack model and the quantized (discrete or finite) fracture mechanics assumptions proposed by Pugno and Ruoff [Pugno N, Ruoff RS. Quantized fracture mechanics. Philos Mag 2004;84(27):2829-45] and extended by Wnuk and Yavari [Wnuk MP, Yavari A. Discrete fractal fracture mechanics. Engng Fract Mech 2008;75(5):1127-42]. The well-known entities such as the stress intensity factor and the Barenblatt cohesion modulus, which is a measure of material toughness, have been re-defined to accommodate the fractal view of fracture.For very small cracks or as the degree of fractality increases, the characteristic length constant, related to the size of the cohesive zone is shown to substantially increase compared to the conventional solutions obtained from the cohesive crack model. In order to understand fracture occurring in real materials, whether brittle or ductile, it seems necessary to account for the enhancement of fracture energy, and therefore of material toughness, due to fractal and discrete nature of crack growth. These two features of any real material appear to be inherent defense mechanisms provided by Nature.     

Mode I fracture of adhesive joints using tailored cohesive zone models

Cohesive zone models are explored in order to study cleavage fracture in adhesive bonded joints. A mode I cohesive model is defined which correlates the tensile traction and the displacement jump (crack faces opening) along the fracture process zone. In order to determine the traction-separation relation, the main fracture parameters, namely the cohesive strength and the fracture energy, as well as its shape, must be specified. However, owing to the difficulties associated to the direct measurement of the fracture parameters, very often they are obtained by comparing a measured fracture property with numerical predictions based on an idealized traction separation relation. Likewise in this paper the cohesive strength of an adhesive layer sandwiched between elastic substrates is adjusted to achieve a match between simulations and experiments. To this aim, the fracture energy and the load-displacement curve are adopted as input in the simulations. In order to assess whether or not the shape of the cohesive model may have an influence on the results, three representative cohesive zone models have been investigated, i.e. exponential, bilinear and trapezoidal. A good agreement between experiments and simulations has been generally observed. However, a slight difference in predicting the loads for damage onset has been found using the different cohesive models.     

Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials

This work investigates elastic-plastic crack growth in ceramic/metal functionally graded materials (FGMs). The study employs a phenomenological, cohesive zone model proposed by the authors and simulates crack growth by the gradual degradation of cohesive surfaces ahead of the crack front. The cohesive zone model uses six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) to describe the constitutive response of the material in the cohesive zone. A volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model) describes the elastic-plastic response of the bulk background material. The numerical analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. Numerical values of volume fractions for the constituents specified at nodes of the finite element model set the spatial gradation of material properties with isoparametric interpolations inside interface elements and background solid elements to define pointwise material property values. The paper describes applications of the cohesive zone model and the computational scheme to analyze crack growth in a single-edge notch bend, SE(B), specimen made of a TiB/Ti FGM. Cohesive parameters are calibrated using the experimentally measured load versus average crack extension (across the thickness) responses of both Ti metal and TiB/Ti FGM SE(B) specimens. The numerical results show that with the calibrated cohesive gradation parameters for the TiB/Ti system, the load to cause crack extension in the FGM is much smaller than that for the metal. However, the crack initiation load for the TiB/Ti FGM with reduced cohesive gradation parameters (which may be achieved under different manufacturing conditions) could compare to that for the metal. Crack growth responses vary strongly with values of the exponent describing the volume fraction profile for the metal. The investigation also shows significant crack tunneling in the Ti metal SE(B) specimen. For the TiB/Ti FGM system, however, crack tunneling is pronounced only for a metal-rich specimen with relatively smaller cohesive gradation parameter for the metal.     

A combined dislocation—cohesive zone model for fracture in a confined ductile layer

The collective dislocation behavior near a crack tip in a ductile layer sandwiched between two brittle solids is analyzed via two-dimensional dislocation dynamics (DD) simulations that incorporate a cohesive zone (CZ) model. The cohesive crack tip is treated as part of a much larger finite crack confined in the ductile layer. The underlying boundary value problem is formulated with a set of boundary integral equations and numerically evaluated with a collocation method. The fracture energy of the layered composite material is shown to be strongly correlated with the layer thickness and is directly influenced by the cohesive strength of the ductile layer (Hsia KJ et al. (1994) J Mech Phys Solids 6 877–896).     

A finite element analysis of fracture initiation in ductile/brittle periodically layered composites

A near-tip plane strain finite element analysis of a crack terminating at and normal to the interface in a laminate consisting of alternate brittle and ductile layers is conducted under mode-I loading. The studies are carried out for a system representing steel/alumina composite laminate. The Gurson constitutive model, which accounts for the ductile failure mechanisms of microvoid nucleation, growth and coalescence, is employed within the framework of small deformation plasticity theory. Evolution of plastic zone and damage in the ductile layer is monitored with increasing load. High plastic strain localization and microvoid damage accumulation are found to occur along the brittle/ductile interface at the crack-tip. Fracture initiation in the ductile phase is predicted and the conditions for crack renucleation in the brittle layer ahead of the crack are established for the system under consideration. Ductile fracture initiation has been found to occur before plasticity spreads in multiple ductile layers. Effects of material mismatch and yield strength on the plastic zone evolution are briefly discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Constraint effect on the ductile crack growth resistance of circumferentially cracked pipes

A systematic study has been carried out by using 2D axisymmetric models to understand the ductile fracture behavior of pipes with internal and external circumferential cracks. Crack growth resistance curves have been computed using the complete Gurson model. Pipes with various diameter-to-thickness ratios, internal pressure, crack depths and material properties are analyzed. The results have been compared with those of corresponding SENT and standard SENB specimens. It clearly indicates that the SENT specimen is a good representation of circumferentially flawed pipes and an alternative to the conventional standard SENB specimen for the fracture mechanics testing in engineering critical assessment of pipes.     

Three-dimensional modeling of ductile crack growth: Cohesive zone parameters and crack tip triaxiality

For 10 mm thick smooth-sided compact tension specimens made of a pressure vessel steel 20MnMoNi55, the interrelations between the cohesive zone parameters (the cohesive strength, T_max, and the separation energy, Γ) and the crack tip triaxiality are investigated. The slant shear-lip fracture near the side-surfaces is modeled as a normal fracture along the symmetry plane of the specimen. The cohesive zone parameters are determined by fitting the simulated crack extensions to the experimental data of a multi-specimen test. It is found that for constant cohesive zone parameters, the simulated crack extension curves show a strong tunneling effect. For a good fit between simulated and experimental crack growth, both the cohesive strength and the separation energy near the side-surface should be considerably lower than near the midsection. When the same cohesive zone parameters are applied to the 3D model and a plane strain model, the stress triaxiality in the midsection of the 3D model is much lower, the von-Mises equivalent stress is distinctly higher, and the crack growth rate is significantly lower than in the plane strain model. Therefore, the specimen must be considered as a thin specimen. The stress triaxiality varies dramatically during the initial stages of crack growth, but varies only smoothly during the subsequent stable crack growth. In the midsection region, the decrease of the cohesive strength results in a decrease of the stress triaxiality, while the decrease of the separation energy results in an increase of the triaxiality.