Three-dimensional Crack Front Research Articles

Fracture toughness assessment of surface cracks in slender ultra-high-strength steel plates

Safe design against unstable fractures in load-bearing structures is crucial at sub-zero temperatures where the brittle fracture toughness can be unfavourable, especially for high-stress designs incorporating ultra-high-strength steels. The brittle fracture toughness of surface cracks in structural steel with a minimum yield strength of 1300 MPa is, for this reason, tested in the present study at sub-zero temperatures. The realistic flaws are compared with single-edge notched specimens (SEN(B)) from thicker plates with the same chemical composition, using a representative fracture toughness for a three-dimensional crack front according to the Master Curve method. A novel approach determines the latter without considering the local temperature and constraint variation through empirical relations. The experimental result shows a difference in the reference temperature between the two specimen types, which likely is the natural variation of the manufactured materials and/or a loss of constraint due to the difference in the scaled specimen deformation level.

Analysis of a method to compute mixed-mode stress intensity factors for non-planar cracks in three-dimensions

In this work, we present and prove results underlying a method which uses functionals derived from the interaction integral to approximate the stress intensity factors along a three-dimensional crack front. We first prove that the functionals possess a pair of important properties. The functionals are well-defined and continuous for square-integrable tensor fields, such as the gradient of a finite element solution. Furthermore, the stress intensity factors are representatives of such functionals in a space of functions over the crack front. Our second result is an error estimate for the numerical stress intensity factors computed via our method. The latter property of the functionals provides a recipe for numerical stress intensity factors; we apply the functionals to the gradient of a finite element approximation for a specific set of crack front variations, and we calculate the stress intensity factors by inverting the mass matrix for those variations.

Analysis of 3D crack patterns in a free plate caused by thermal shock using FEM-bifurcation

Damage to components made of brittle material due to thermal shock represents a high safety risk. Predicting the degree of damage is therefore very important to avoid catastrophic failure. An energy-based linear elastic fracture mechanics bifurcation analysis using a three-dimensional finite element model is presented here, which allows the determination of crack length and crack spacing for a defined thermal load in a free plate. It is assumed that a hierarchical crack pattern is formed due to cooling penetration. The constant growth of the ideal regular pattern of hexagons can change into a pattern with a different symmetry by slightly changing the cooling conditions. This bifurcation point is determined by the second derivative of the mechanical potential with respect to the geometry of the crack front. The very high computational effort for the second derivative is reduced by describing the three-dimensional crack front with a limited number of Fourier coefficients. A one-dimensional transient temperature field at a sufficient distance from the plate edge is assumed. For alumina, the crack length and crack spacing curves are computed for different quenching temperatures and heat transfer coefficients. The corresponding final crack lengths are also calculated as a measure of damage. Comparison with a two-dimensional model confirms the expected 1/2 difference in crack spacing. Data from thermal shock experiments are also presented. However, due to the cracks caused by the strong cooling at the edge, these correspond to the results of the two-dimensional model.

Edge stress intensity functions along elliptic and part-elliptic 3D cracks

Three-dimensional part-elliptic crack fronts are common in engineering practice. Herein we formulate a computational method, known as the quasidual function method (QDFM) for the functional representation of edge stress intensity functions (ESIFs) along such cracks in 3D domains. It is an efficient post-processing algorithm that may use the finite element solution away from the singular edge. Numerical examples including an internal elliptic crack, a surface half-elliptic crack and a quarter elliptic crack from a hole are provided, demonstrating the efficiency and accuracy of the proposed method.

On the effect of an out-of-plane constraint on the three-dimensional crack front fields in a thin elastic plate

Two-dimensional theories of fracture are still applied widely today and provide theoretical foundations for solutions to many practical problems. These two-dimensional theories are based on the plane strain or plane stress assumption. However, strictly speaking, for a thin elastic plate with a through-thickness crack under tension, plane strain conditions can be met only at the crack front (except the corn point) and plane stress conditions exist at a distance of about one half of the plate thickness from the crack front in the mid-plane. What are the stress fields in the region where both plane strain and plane stress conditions cannot be met? In this paper, further investigations into the problem are carried out. Three-dimensional Maxwell stress functions, the principle of minimum complementary potential energy and three-dimensional J-integrals are employed to obtain an analytical solution to depict the relationship among out-of-plane constraints, three-dimensional J-integrals and stress intensity factors. Three-dimensional finite element simulations with fine meshes are carried out to verify the analytical results. Compared with the corresponding plane strain solution, the solution proposed is valid in a larger region.

Elastic-plasticJ-Tzdominance in bending and tension loadings

PurposeCrack tip stresses are used to relate the ability of structures to perform under the influence of cracks and defects. One of the methods to determine three-dimensional crack tip stresses is through theJ-Tzmethod. TheJ-Tzmethod has been used extensively to characterize the stresses of cracked geometries that demonstrate positiveT-stress but limited in characterizing negativeT-stresses. The purpose of this paper is to apply theJ-Tzmethod to characterize a three-dimensional crack tip stress field in a changing crack length from positive to negativeT-stress geometries.Design/methodology/approachElastic-plastic crack border fields of deep and shallow cracks in tension and bending loads were investigated through a series of three-dimensional finite element (FE) and analyticalJ-Tzsolutions for a range of crack lengths ranging from 0.1⩽a/W⩽0.5 for two thickness extremes ofB/(W−a)=1 and 0.05.FindingsBoth the FE and theJ-Tzapproaches showed that the combined in-plane and the out-of-plane constraint loss were differently affected by theT-stress and the out-of-plane size effects when the crack length changed from deep to shallow cracks. The conditions of theJ-Tzdominance on the three-dimensional crack front tip were shown to be limited to positiveT-stress geometries, and theJ-Tz-Q2Dapproach can extend the crack border dominance of the three-dimensional deep and shallow bend models along the crack front tip until perturbed by an elastic-plastic corner field.Practical implicationsThe paper reports the limitation of theJ-Tzapproach, which is used to calculate the state of three-dimensional crack tip stresses in power law hardening materials. The results from this paper suggest that the characterization of the three-dimensional crack tip stress in power law hardening materials is still an open issue and requires other suitable solutions to solve the problem.Originality/valueThis paper demonstrates a thorough analysis of a three-dimensional elastic-plastic crack tip fields for geometries that are initially either fully constrained (positiveT-stress) or unconstrained (negativeT-stress) crack tip fields but, subsequently, theT-stress sign changes due to crack length reduction and specimen thickness increase. TheJ-Tzstress-based method has been tested and its dominance over the crack tip field is shown to be affected by the combined in-plane and the out-of-plane constraints and the corner field effects.

FEM-bifurcation analysis for 3D crack patterns

During the shrinkage of volcanic rock, columnar joints evolve with an almost regular hexagonal pattern formation and a very distinct column diameter defining the crack spacing. This phenomenon is caused by an inhomogeneous, constantly moving temperature field. The constant column growth is presumably always close to a bifurcation point at which the ideal regular pattern of hexagons could branch off to a pattern with a different symmetry due to slightly changing cooling conditions. This paper presents an energy-based fracture mechanical bifurcation analysis to determine this point and its stability by evaluating first and second order derivatives of the free Helmholtz energy with respect to a limited number of coefficients of a truncated Fourier series expansion of the three-dimensional crack front contour. The energy is obtained by linear elastic simulations on three-dimensional Finite Element models of the periodically repeatable patterns before and after bifurcation. While providing an understanding of the column growth procedure of volcanic rock, the characteristics of the approach suggest applications for various complex three-dimensional crack growth systems in engineering.

Theoretical approach of characterizing the crack-tip constraint effects associated with material’s fracture toughness

The conversion problem of plane strain fracture toughness ( $$K_\mathrm{IC}$$ ) which is necessarily measured according to ASTM standards, to lower constraint applications generally existing in engineering structures, has led to extensive material and labor costs. The present paper explores and quantifies the crack-tip constraint effects by the crack-tip plastic zone to serve the prediction of material’s fracture toughness. Firstly, the approximate three-dimensional crack front displacement fields are obtained by using the variable separation method. The three-dimensional stress field is then used to predict the shape and size of the crack-tip plastic zone. Secondly, the two-dimensional in-plane and the three-dimensional out-of-plane geometric constraint effects are quantified separately, and two constraint factors, i.e., $$\alpha _\mathrm{in}$$ and $$\alpha _\mathrm{out}$$ are proposed. A series of configurations of the two-dimensional and three-dimensional crack-tip plastic zones that vary with the specimen’s geometric size (plate width, thickness, etc.) are presented, which will facilitate a better understanding of the crack-tip constraint effect. Finally, the present method is applied to elucidate the significant “thickness effect” of the X70 pipeline steel’s fracture toughness by using the out-of-plane constraint factor $$\alpha _\mathrm{out}$$ . The corresponding results are compared with the experimentally measured and FEM-predicted fracture toughness $$K_\mathrm{C}$$ . It is concluded that the parameters $$\alpha _\mathrm{out}$$ and the fracture toughness $$K_\mathrm{C}$$ can be correlated with each other, which will be beneficial to the prediction of material’s fracture toughness and the avoidance of experimental costs.

Analysis of fatigue crack configuration influence on fatigue life

When a crack initiates and grows in a plain specimen under constant cyclic load amplitude, fatigue crack growth behavior is not reproducible. The fatigue crack length l0 when the scatter of the fatigue crack growth rate converges is reported by observing the crack growth behavior on the specimen surface. The l0 is reported that it is approximately six times as long as the grain size in carbon steels. However, the crack shape of the inside is not observed and we considered that the three-dimensional irregular fatigue crack front shape affects the fatigue crack growth behavior on the specimen surface. Furthermore, the physical meaning and controlling factors of the l0 is still uncertain. Therefore, in this study, we propose two factors that affect the local fatigue crack growth rate: local microstructural and mechanical factors. The former causes a variation of the three-dimensional fatigue crack front shape, and the fatigue crack front shape synergistically affects the mechanical condition at the crack tip. Then we investigated the stress intensity factor values along the tip of the crack including a part of the locally grown crack front. And we propose a concept of force caused by a stable growth part which prevents local growth parts from growing.

Determination of Stress Intensity Factor K I for a Three-Dimensional Crack Front by the Caustic Method

In this paper, an experimental technique to determine the stress intensity factor (SIF) KI for a three-dimensional crack front using the caustic method in combination with the stress-freezing and stress-releasing techniques is proposed. A comparison between the experimental results determined by the proposed method and the traditional photoelastic method is made; in addition, the experimental results of the present work are compared with the theoretical solution of Smith-Alavi. The experimental results obtained by the caustic method coincide favorably with the results obtained by the photoelastic method. The caustic method in combination with the stress-freezing and stress-releasing techniques proposed in the present work makes it possible to consecutively determine the distribution of the SIF KI for a three-dimensional crack front.

In-plane and out-of-plane unified constraint-dependent creep crack growth rate of 316H steel

The unified characterization parameter Ac of in-plane and out-of-plane creep constraints along three-dimensional crack front in different specimen geometries has been analyzed. Based on the parameter Ac and the experimental creep crack growth rate data of different specimen geometries in the literature, the in-plane and out-of-plane unified constraint-dependent creep crack growth rate equation of 316H steel has been obtained. The predicted creep crack growth rate by using the equation for each specimen agrees well with the experimental data. Based on the two-parameter C∗−Ac concept, the equation may be used in creep crack growth life assessments incorporating in-plane and out-of-plane constraint effects.

On the Quantification of the Constraint Effect Along a Three-Dimensional Crack Front

Elbows with a shallow surface cracks in nuclear pressure pipes have been recognized as a major origin of potential catastrophic failures. Crack assessment is normally performed by using the J-integral approach. Although this one-parameter-based approach is useful to predict the ductile crack onset, it depends strongly on specimen geometry or constraint level. When a shallow crack exists (depth crack-to-thickness wall ratio less than 0.2) and/or a fully plastic condition develops around the crack, the J-integral alone does not describe completely the crack tip stress field. In this paper, we report on the use of a three-term asymptotic expansion, referred to as the J-A 2 methodology, for modeling the elastic-plastic stress field around a three-dimensional shallow surface crack in an elbow subject to internal pressure and out-of-plane bending. The material, an A 516 Gr. 70 steel, used in the nuclear industry, was modeled with a Ramberg-Osgood power law and flow theory of plasticity. A finite deformation theory was included to account for the highly nonlinear behavior around the crack tip. Numerical finite element results were used to calculate a second fracture parameter A 2 for the J-A 2 methodology. We found that the used three-term asymptotic expansion accurately describes the stress field around the considered three-dimensional shallow surface crack.

X-FEM Applied to Three-Dimensional Curvilinear Crack Front

A three-dimensional extended finite element algorithm coupled with the level set method is developed for curvilinear crack front problems. This method does not need to force the discontinuities to be in conformity with the mesh. In order to compute the pointwise stress intensity factors along the curvilinear crack front, an orthogonal curvilinear coordinate system is constructed at the intersection point between the crack front and a surface of the crack front element. As the crack surface is expressed by the level set method, no explicit meshing of the crack geometry emerges on the crack front. This makes the domain integral more difficult. An independent grid of hexahedral cells around the crack front needs to be built as the virtual extension domain for accomplishing the domain integral. To assess the accuracy of the approach, an example with semi-elliptic crack surface is considered. The results show excellent agreement with the results calculated by the standard finite element method and those obtained from the literature.

Three-dimensional Analysis of Fatigue Crack Propagation using X-Ray Tomography, Digital Volume Correlation and Extended Finite Element Simulations

Fatigue crack propagation is usually analyzed as a one-dimensional problem and the identification of Paris’ type propagation law is often performed using standardized samples. The present paper is devoted to the analysis of three dimensional fatigue crack propagation using advanced experimental, imaging, measurement and numerical simulation techniques. Fatigue experiments are performed in situ in a tomograph. The images are analyzed using digital volume correlation to extract displacement fields and crack geometry. Stress intensity factors are evaluated along three-dimensional crack fronts and compared to numerical simulations using measured boundary conditions and detected crack shape. Last, using a single experiment, local da/dN vs. ΔK laws are estimated within a large range of ΔK values.

Fracture mechanics of the three-dimensional crack front: vertex singularity versus out of plain constraint descriptions

The effect of a free surface on stress distribution along the three-dimensional crack front is discussed. Generalized stress intensity factor methodology and out-of-plane constraint based procedure are used for a reliable estimation of the stress state in front of the crack tip. The two present methodologies are discussed and compared. The out-of plane, in-plane constraint parameters and generalized stress intensity factor were estimated. It is shown significant changes of the singularity exponents and fracture parameters values close to the free surface. Therefore, conventional methodologies can lead to inaccurate description of the stress state and cannot be used for reliable description of fatigue crack propagation in this area.

Finite element crack propagation of adhesively bonded repaired panels in general mixed-mode conditions

In order to simplify the analyses, in the previously developed mixed-mode fracture analyses of the single-side repaired panels with composite patches, the effect of mode-III was neglected. In those analyses, it was also assumed that the crack-front remains perpendicular to the panel's surfaces during the crack propagation which is in defeat with the experimental observations. In this paper, finite element analyses are performed to investigate the crack propagation of single-side repaired panels using composite patches containing an initial central inclined crack. In these analyses the general mixed-mode fracture conditions of mode I, II and III and the real three-dimensional crack-front are considered in the finite element crack growth modeling. For this purpose, Modified Virtual Crack Closure Technique (MVCCT) is used to calculate the three strain energy release rate components corresponding to various fracture modes at the crack-front. The obtained crack-front shapes at various crack growth steps, crack trajectories at both patched and un-patched sides and life of the repaired panels with various patch lay-ups are discussed.

On the quantification of the constraint effect along a three-dimensional crack front

Abstract In this paper we examined the characterization of constraint effects for surface cracked plates under uniaxial and biaxial tension loadings. First, three-dimensional (3D) modified boundary layer analyses were conducted using the finite element method to study the constraint effect at a typical 3D crack front. The analyses were carried out using small geometry change formulation and deformation plasticity material model. Elastic-plastic crack front stress fields at a constant J and various T-stress levels were obtained. Three-dimensional elastic-plastic analyses were performed for semi-circular surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. Then, J-Q and J-T two-parameter approaches are used in characterizing the elastic-plastic crack-tip stress fields along the 3D crack front. It is found that the J-Q characterization provides good estimate for the constraint effect for crack-tip stress fields. Reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis.

A domain-independent integral for computation of stress intensity factors along three-dimensional crack fronts and edges by BEM

The present work deals with an evaluation of stress intensity factors (SIFs) along straight crack fronts and edges in three-dimensional isotropic elastic solids. A new numerical approach is developed for extraction, from a solution obtained by the boundary element method (BEM), of those SIFs, which are relevant for a failure assessment of mechanical components. In particular, the generalized SIFs associated to eigensolutions characterized by unbounded stresses at a neighbourhood of the crack front or a reentrant edge and also that associated to T-stress at the crack front can be extracted. The method introduced is based on a conservation integral, called H-integral, which leads to a new domain-independent integral represented by a scalar product of the SIF times some element shape function defined along the crack front or edge. For sufficiently small element lengths these weighted averages of SIFs give reasonable pointwise estimation of the SIFs. A proof of the domain integral independency, based on the bi-orthogonality of the classical two-dimensional eigensolutions associated to a corner problem, is presented. Numerical solutions of two three-dimensional problems, a crack problem and a reentrant edge problem, are presented, the accuracy and convergence of the new approach for SIF extraction being analysed.

Boundary element method for J-integral and stress intensity factor computations in three-dimensional interface cracks

A general numerical tool for the analysis of three–dimensional bimaterial interface cracks is presented in this paper. The proposed tool is based on a multidomain formulation of the Boundary Element Method (BEM), with the crack located at the interface of the domain. Mixed mode stress intensity factors are computed along the three-dimensional crack fronts using the Energy Domain Integral (EDI) methodology and decoupled via the Interaction Integral. The capability of the procedure is demonstrated by solving a number of examples. The last of these examples consists in a thick centre cracked panel for which the behaviour of the J-integral and the mixed-mode stress intensity factors along the crack front is studied as a function of the material mismatch.

Design of a new fracture specimen by 3D singularity analyses

AbstractA new design of a fracture mechanics specimen is presented. It ensures a valid square‐root singularity along the whole threedimensional crack front under mode‐I conditions. Within the proposed concept it is possible to adjust the angle between the crack front and the free surface in advance. Therefore, the shape of the whole crack front can be influenced in a desired manner. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)