Modified Boundary Layer Model Research Articles

A comparative study of fatigue crack driving force considering in-plane constraint effect

Nonlinear fracture mechanics parameters, ΔCTODp and ΔJ are implicitly used to characterize the crack driving force (CDF) in estimating Fatigue Crack Growth Rate (FCGR) in ductile material. However, in the presence of structure geometry effect, evaluation of effective CDF remains to be clarified. In this regard, a comparative study was conducted and the in-plane-constraint parameter was introduced to quantify geometry effect based on the constraint theory. Using modified boundary layer model (MBL) with multi-stage node releasing embedded, the effect of in-plane-constraint on ΔCTODp and ΔJ was investigated. Results indicate that ΔCTODp can demonstrate geometry effect on CDF, as it correlates well with plastic deformation. Compartively, the effective cyclic J-integral (ΔJop) was found to have a significant dependency only on the crack closure level, thus incapable of capturing the effect of geometry on CDF. A humpbacked curve was found between in-plane constraint level and ΔCTODp. Further investigation reveals that the gain of in-plane constraint on FCGR can be attributed to the interaction between in-plane constraint and crack closure, as well as the interplay between in-plane and out-of-plane constraint. This work not only reveals the mechanism of the in-plane constraint effect on CDF, but also provides a basis for establishing FCG model across different structure geometries.

Fracture of four semi-regular lattices regulated by T-stress in modified boundary layer models

The development of additive manufacturing technology has promoted the rise of various lattice metamaterials, and evaluating their fracture properties is critical for both application and safety. Here, by conducting finite element simulations on four types of elastic-brittle semi-regular lattices (kagome, snub-trihexagonal, elongated-triangular, and snub-square), we indicate that the fracture toughness of lattice metamaterials is not solely determined by the crack-tip singular term, i.e., the critical stress intensity factor, but is also influenced by the non-singular T-stress term. T-stress regulates fracture toughness by modulating geometrical distortions and force-carrying modes of lattice bars around the crack front. This regulation effect depends on the lattice topology and becomes weaker under mixed-mode loading. As the relative density of lattices decreases, T-stress can force elastic buckling of bars around the crack front, which may deteriorate the fracture toughness. Additionally, positive T-stress tends to promote the crack deflection and negative T-stress tends to inhibit. Our results offer new insights into the structure-property relationships in the presence of T-stress, which may contribute to design the geometry of micro/nano-lattice metamaterials.

Influence of heterogeneity due to toughness variations on weakest-link modeling for brittle failure

The effect of heterogeneous microstructures on the macroscopic probability of failure is studied by use of weakest-link modeling. Heterogeneity is here associated with a local variation of toughness, where a size scale characteristic of this variation defines a length parameter. The ratio between this length parameter and the size of the active fracture process zone, defined as the heterogeneity ratio, is key to evaluating the impact of a heterogeneous microstructure. Two extremes are identified; small-scale heterogeneity (SSH) and large-scale heterogeneity (LSH). For these cases, it is possible to formulate analytical expressions based on the weakest-link concept, and references are made to existing models in the literature. Typically, heterogeneity along the crack front, where gradients of the mechanical fields are small, falls under the category of SSH. On the other hand, the effect of heterogeneity in a plane perpendicular to the crack front depends strongly on the heterogeneity ratio. Cases that can neither be identified with SSH nor LSH must be addressed with care. How this can be done is discussed, and examples are given for four different microstructure configurations of interest. The investigation is carried out by numerical analysis of a modified boundary layer model. The cumulative probability of failure by cleavage fracture is evaluated in a post-processing step, where two different statistical models are examined; the Beremin model and the Kroon–Faleskog model. Both models render the same conclusion about the alteration of the overall failure probability distributions caused by heterogeneity.

The Numerical Analysis of the In-Plane Constraint Influence on the Behavior of the Crack Subjected to Cyclic Loading.

The paper presents the influence of in-plane constraints defined by T-stress on the behavior of a crack subjected to cyclic loading. In the analysis, a modified boundary layer model approach was used in which the cohesive model was introduced. In the simulations, the constant maximum value of the stress intensity factor and four levels of T-stress were defined. The model was subjected to ten repeated stress cycles. Based on the results obtained, an analysis of the effect of the in-plane constraint on selected aspects of crack behavior was made. The strong influence of in-plane constraint applied in the model on the crack closure and the fatigue crack growth rate was proven. Since the in-plane constraint described the influence of geometry on the stress field surrounding the fatigue crack tip in real geometry, the results suggested that it is possible to create precise formulae connecting the level of the in-plane constraint with the effective stress intensity factor range and to incorporate the T-stress or Q-stress level in the Paris law.

Weibull stress solutions and applications under mixed mode I-II loading for 2-D cracks in elastic and elastic–plastic materials

Since the fracture toughness of materials in the ductile and brittle transition region presents obvious scatter, Weibull stress based on the local approach to fracture is widely used to clarify the randomness. A large number of detailed finite element analyses are necessary to be carried out for the calculation of Weibull stress, especially for the complex structures and loads. In order to simplify the calculation of Weibull stress, this paper derives Weibull stress solutions as functions of stress intensity factor or J integral for elastic and elastic–plastic materials under mode I-II loading. These solutions are verified by using modified boundary layer model and applied in the analysis of fracture toughness test results by four-point bending specimens. The relationship between failure probability Pf and SIF or J integral under mode I-II loading is derived and verified with experimental data. The result shows that the failure probability can predict the experimental data well.

The Effects of Nonproportional Biaxial Loading Paths on Ductile Fracture Initiation: A Void Growth Analysis

Abstract The effects of nonproportional biaxial loading paths on ductile fracture initiation toughness are studied in this article. To this end, the growth of a cylindrical void (hole) located in front of a mode I plane strain crack has been studied using large-deformation finite element analysis. A specific microstructural feature of a steel alloy was thoroughly studied by having a single void positioned at a fixed distance from the crack tip and void that was equal to 10 times the diameter of the void. In particular, the nonproportional biaxial loading path effects on the crack tip blunting, void growth, ligament reduction, and near-tip stress fields are investigated computationally. Under small-scale yielding conditions, one proportional loading and two nonproportional loading paths are applied to the modified boundary layer models, covering both low-constraint (negative T-stress) and high-constraint (positive T-stress) crack tip conditions. We have observed that the nonproportional load paths have a marked effect on the void growth, crack tip blunting, and the resulting ligament reduction. By applying the simple criteria for the coalescence of the crack tip and void, the ductile fracture initiation toughness is estimated. It is shown that the ductile fracture toughness is dependent on loading paths and constraint conditions (the T-stress ratios). Compared with the proportional biaxial loading case, different load sequences of the biaxial loads will result in either an increase or a decrease the fracture initiation toughness. This effect is particularly significant for specimen geometry with low-constraint conditions (i.e., negative T-stress ratios). Results from this study are of relevance to ductile fracture assessment of components or structures, such as pressure vessels that operate under nonproportional biaxial loading conditions.

Weibull stress solutions for 2D cracks under mode II loading

Weibull stress is widely used to predict the probability of cleavage fracture with local approaches. However, the calculation of Weibull stress requires detailed elastic plastic finite element analyses, which limit the application of local approaches in engineering practice. In this paper, analytic and semi-analytic solutions of Weibull stress for 2D crack/notch models with elastic and elastic–plastic materials under mode II loading are obtained, which allow the Weibull stress to be estimated according to the macroscopic fracture mechanics parameters like stress intensity factor K or J integral together with material properties. The obtained Weibull stress solutions are verified by using results of finite element analyses with modified boundary layer models and a single-edge-notched specimen. The results show that the Weibull stress solutions of this paper can predict the results calculated by direct finite element analyses very well.

Two-parameter J-A estimation for weld centerline cracks of welded SE(T) specimen under tensile loading

This work examines the J–A two-parameter characterization of elastic–plastic crack front fields for weld centerline cracks under tensile loading. Extensive finite element analyses (FEA) have been conducted to obtain solutions of the A-term for modified boundary layer (MBL) model and welded SE(T) fracture specimen. Solutions of the constraint parameter A were obtained for the material following the Ramberg-Osgood power law with variation of strain hardening exponent between 5 and 20. The crack geometries analyzed include shallow to deep cracks and remote tension loading levels cover from small-scale to large-scale yielding conditions for welded SE(T) specimen. The effects of material mismatch and weldment geometries on crack tip constraint were considered in the FEA based on an idealized tri-material weldment. A constraint parameter AMis, only caused by material mismatch, is defined and its parametric equation was obtained from MBL model. The total constraint in the tri-material weld structure for welded SE(T) specimen can be predicted by adding together AMis and A of homogeneous SE(T) specimen. Overall, Good agreements were achieved for welded SE(T) specimen with different crack sizes and weldment geometries from small-scale to large-scale yielding conditions. This methodology would be useful for performing constraint-based elastic-plastic fracture analyses of other welded test specimens.

Weibull stress analysis for the corner crack in reactor pressure vessel nozzle

The nozzle region of a reactor pressure vessel experiences higher and more complex stresses than the remaining part of the reactor pressure vessel. If a corner crack is postulated in the nozzle region, it is necessary to consider the potential strong influence of constraint on fracture behavior due to inelastic deformations of crack tip. In accordance with the requirement of probabilistic fracture mechanics, Weibull stress in local approach to fracture is analyzed with the consideration of constraint effect. Conventional fracture analysis is also carried out using a three-dimensional crack model, and the fracture driving forces ( KIand J) and T-stress are obtained. Weibull stress along the crack tip is also computed by three-dimensional models. The modified boundary layer model with plastic correction is developed for a corner crack in the reactor pressure vessel nozzle. Under the J- T stress field from three-dimensional models, Weibull stress values obtained using the modified boundary layer model are compared and discussed with that by three-dimensional models. It is found that the modified boundary layer model can effectively predict the Weibull stress under the J- T stress field, which simplified the Weibull stress calculation process for complex structures.

Weibull stress analysis in local approach to fracture

As a measure of the probability of cleavage fracture, the Weibull stress within the framework of local approach has the potential capability to predict constraint effects on fracture of structural steels. This paper mainly analyzes Weibull stress considering constraint effect using constraint parameters T-stress and Q. Weibull stress is solved with constraint effects characterized by T-stress for elastic material and Q for elastic plastic material. These solutions are verified with existing solutions for T = 0 and finite element solutions including modified boundary layer models, contact tension models and single-edge bend models. Good agreement has been obtained in all cases. The Weibull stress solutions can be further adopted to predict scale fracture toughness.

Numerical Investigations on the Effects of T-Stress in Mode I Creep Crack

The effects of [Formula: see text]-stress on the stress field, creep zone and constraint effect of the mode I crack tip in power-law creeping solids are presented based on finite element (FE) analysis in the paper. The characteristics of the crack tip field in power-law creep solids by considering low negative [Formula: see text]-stress and high positive [Formula: see text]-stress are studied in the paper. The differences of [Formula: see text]-stress effect on the crack tip field between power-law creeping solids and elastoplastic materials are also clarified. A modified parameter is proposed to characterize the influence of [Formula: see text]-stress on creep zone. The constraint parameter [Formula: see text] under both small-scale creep and large-scale creep with various [Formula: see text]-stresses for the modified boundary layer (MBL) model and various specimens with different crack depths are given. The applicability and the limitation of the MBL model for creep crack are also investigated. The inherent connection between [Formula: see text]-stress and [Formula: see text]-parameter is discussed. The investigations given in this paper can further promote the understanding of [Formula: see text]-stress effect and constraint effect on the mode I creep crack.

Effect of the Lüders plateau on ductile fracture with MBL model

In this study, the effect of Lüders plateau on the ductile crack growth resistance has been investigated with the Gurson damage model and the modified boundary layer (MBL) model, under mode I plane strain condition. The Lüders plateau is modeled as horizontal by keeping the plateau stress equaling to the yield stress. A family of Lüders elongations ranging from 0 to 5% has been considered. The remote boundary condition of the MBL model is governed by the elastic K-field and T-stress. Numerical results show that the existence of the Lüders plateau on the stress-strain curve reduces the ductile crack growth resistance. The degree of reduction depends on the scale of the Lüders elongation. The crack tip stress field analysis indicates that the existence of the Lüders plateau varies the crack tip stress striaxiality distribution and the magnitude. It is also found that the size of plastic zone ahead of the crack tip is reduced, compared with the reference case for material without Lüders plateau. It is demonstrated that the effect of Lüders plateau on ductile crack growth is more significant at lower T-stress or for materials with higher toughness. The dependence of the initial void volume fraction and the T-stress on the ductile crack growth resistance are alleviated when the Lüders elongation is large.

Modeling of coupled motion and growth interaction of equiaxed dendritic crystals in a binary alloy during solidification

Motion of growing dendrites is a common phenomenon during solidification but often neglected in numerical simulations because of the complicate underlying multiphysics. Here a phase-field model incorporating dendrite-melt two-phase flow is proposed for simulating the dynamically interacted process. The proposed model circumvents complexity to resolve dendritic growth, natural convection and solid motion simultaneously. Simulations are performed for single and multiple dendritic growth of an Al-based alloy in a gravity environment. Computing results of an isolated dendrite settling down in the convective supersaturated melt shows that solid motion is able to overwhelm solutal convection and causes a rather different growth morphology from the stationary dendrite that considers natural convection alone. The simulated tip growth dynamics are correlated with a modified boundary layer model in the presence of melt flow, which well accounts for the variation of tip velocity with flow direction. Polycrystalline simulations reveal that the motion of dendrites accelerates the occurrence of growth impingement which causes the behaviors of multiple dendrites are distinct from that of single dendrite, including growth dynamics, morphology evolution and movement path. These polycrystalline simulations provide a primary understanding of the sedimentation of crystals and resulting chemical homogeneity in industrial ingots.

The effects of nonproportional loading on the elastic-plastic crack-tip fields

In this paper, the biaxial loading path effects on the mode I plane strain elastic-plastic crack-tip stress fields are investigated computationally. First, three different loading sequences including one proportional loading and two non-proportional loading paths are applied to the modified boundary layer (MBL) model under small-scale yielding conditions. For the same external displacement field applied at the outer boundary of the MBL model, the mode I K field and T-stress field combined as the different loading paths are applied to investigate the influence of the nonproportional loading. The results show that for either the compressive or tensile T-stress, the loading path which applied K field first followed by T-stress field generates the lower crack-tip constraint comparing to proportional loading. There is only minor difference between the results from proportional loading path and that with the T-stress field applied first following by K field. Next, two finite width specimens under non-proportional biaxial loading conditions that generate the same three loading paths are analyzed. Similar crack tip characteristics are observed in these specimens as these obtained from the MBL model, and it is demonstrated that the near-tip behavior in specimens can be predicted accurately using the results from MBL models. The present results show that it is very important to include the load sequence effects in elastic-plastic fracture analysis when dealing with nonproportional loading conditions.

Stationary and propagating cracks in a strain gradient visco-plastic solid

A first order visco-plastic strain gradient constitutive model and a cohesive zone formulation are embedded within a modified boundary layer (MBL) model for the analysis of crack tip fields and crack growth. The MBL model is loaded using a mode I asymptotic crack tip solution, with the boundary displacement calculated from the stress intensity factor. The influence of the rate-dependent constitutive parameters and the intrinsic material length on fracture relevant quantities are investigated in parametric study. Two scenarios are considered: (1) a stationary crack under constant loading and (2) a crack advance under monotonic loading. Finite element model analyzes are performed. For stationary cracks it was found that the effects of the intrinsic lengthscale of the strain gradient plasticity model are more prominent for large visco-plastic power exponents, and increase with hold time. The results of computations of crack advances under monotonic loading suggest that plastic strain gradients reduce crack growth resistance and crack initiation toughness, especially for a large visco-plastic power exponent. For small values of intrinsic material length the dependence of the initiation toughness and tearing modulus on the intrinsic length is strong, but then saturates for large values of intrinsic material length. Loading rate effects are found to be more pronounced for cases with a small value of the intrinsic lengthscale.

The Effects of Nonproportional Loading on the Elastic-Plastic Crack-Tip Fields

In this paper, the loading path effects on the plane strain elastic-plastic crack-tip stress field are investigated computationally. Three different loading sequences include one proportional loading and two non-proportional loading paths are applied to the modified boundary layer (MBL) model under small-scale yielding conditions. For the same external displacement field applied at the outer boundary of the MBL model, the mode I K field and T-stress field combined as the different loading paths are applied to investigate the influence of the nonproportional loading. The results show that for either the compressive or tensional T-stress, the loading path which applied K field followed by T field generates the lower crack-tip constraint. There is only slightly difference between the proportional loading path and that with the T-stress field following by K field. The results show that it is very important to include the load sequence effects in fracture analysis when dealing with nonproportional loading conditions.

Development and validation of a high constraint modified boundary layer finite element model

When a notched structure is loaded, its behaviour is not only affected by the material propertiesbut also by the geometry (of both the structure and the defect) and loading condition, alternatively termedas constraint condition. Therefore, the relation between the failure behaviour of a small scale fracturemechanics test and a full scale structure needs to be elucidated.In an attempt to understand and describe such relationships, the crack tip stress fields are analysed bymeans of finite element simulations and compared for several test specimen geometries. A reference forcomparison is the crack tip stress field obtained from a high constraint reference geometry, further called amodified boundary layer model.First, this article provides some theoretical background on the modified boundary layer model. Second, thedevelopment of a 2D model is outlined in detail, focussing on the mesh design in the vicinity of the crack tipand the applied boundary conditions. Afterwards, an analytical and numerical validation is provided, basedon the level of the applied load and, on the other hand, on the magnitude of the crack tip stress fields.Finally, this validated model is used for the comparison of several constraint parameters. This comparisonindicates a weak influence of the T-stress on the Q-parameter for positive T-stresses. In contrast, negativeT-stresses result in more pronounced negative Q-values.

Effect of residual stresses on ductile crack growth resistance

Abstract 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.

Effect of residual stresses on the crack-tip constraint in a modified boundary layer model

Residual stresses play a crucial role in structural integrity assessment. In this study, a large cracked cylinder with a weld in the center is applied to investigate the effect of residual stresses on the crack-tip constraint. A modified boundary layer model with a remote displacement-controlled elastic K-field and T-stress under small scale yielding has been used to simulate the problem. A two-dimensional tensile residual stress field due to the weld is introduced into the model by the so-called eigenstrain method. It has been shown that the residual stresses can significantly elevate the crack-tip constraint and thus increase the probability for cleavage fracture. The constraint parameter R introduced by the authors can be used to rank the crack-tip constraint induced by the bi-axial residual stresses. The R value decreases with the increase in the applied J-integral. The residual stress-induced constraint is also coupled with the T-stress. The R value becomes smaller with larger T-stress.

Cleavage fracture modeling of pressure vessels under transient thermo-mechanical loading

The next generation of fracture assessment procedures for nuclear reactor pressure vessels (RPVs) will combine non-linear analyses of crack front response with stochastic treatments of crack size, shape, orientation, location, material properties and thermal-pressure transients. The projected computational demands needed to support stochastic approaches with detailed 3-D, non-linear stress analyses of vessels containing defects appear well beyond current and near-term capabilities. In the interim, 2-D models become appealing to approximate certain classes of critical flaws in RPVs, and have computational demands within reach for stochastic frameworks. The present work focuses on the capability of 2-D models to provide values for the Weibull stress fracture parameter with accuracy comparable to those from very detailed 3-D models. Weibull stress approaches provide one route to connect non-linear vessel response with fracture toughness values measured using small laboratory specimens. The embedded axial flaw located in the RPV wall near the cladding-vessel interface emerges from current linear-elastic, stochastic investigations as a critical contributor to the conditional probability of initiation. Three different types of 2-D models reflecting this configuration are subjected to a thermal-pressure transient characteristic of a critical pressurized thermal shock event. The plane-strain, 2-D models include: the modified boundary layer (MBL) model, the middle tension (M(T)) model, and the 2-D RPV model. The 2-D MBL model provides a high quality estimate for the Weibull stress but only in crack front regions with a positive T-stress. For crack front locations with low constraint ( T-stress < 0), the M(T) specimen provides very accurate Weibull stress values but only for pressure load acting alone on the RPV. For RPVs under a combined thermal-pressure transient, Weibull stresses computed from the 2-D RPV model demonstrate close agreement with those computed from the corresponding crack front locations in the 3-D RPV model having large negative T-stresses. Applications of this family of 2-D models provide Weibull stress values in excellent agreement with very detailed 3-D models while retaining practical levels of computational effort.