Elastic-plastic Crack-tip Fields Research Articles

Elastic-plastic crack-tip field in hydride forming metals under hydrogen chemical equilibrium

Elastic-plastic crack-tip field in hydride forming metals under hydrogen chemical equilibrium

Mixed-mode crack growth analysis using a cyclic plasticity model

The effects of two singularity fields, namely, the HRR solution and modified form of the nonlinear kinematic hardening (NKH) solution, on the pure mode I and mixed mode I/II fatigue crack propagation were investigated. Experiments and computations were performed on compact tension shear (CTS) specimens of P2M steel, aluminium 7050, and Ti-6Al-4V with different monotonic and cyclic elastic–plastic properties. The analysis of the experimental data on the crack growth rate for all tested materials is given in terms of plastic stress intensity factors (SIFs) according to the HRR and NKH models. The elastic–plastic crack tip fields were obtained by FE computation as a function of the position along the curvilinear crack path for the monotonic condition and the first several cycles of fatigue deformation considering the Bauschinger effect. As a complement to the FE modelling, digital image correlation (DIC) measurements of the displacement and strain fields around the crack tip in the CTS specimens were performed. This study aimed to obtain more detailed information on the local cyclic strain ahead of the mixed mode crack tip by using DIC. From the sensitivity analysis, it was established that the DIC value depended significantly on the location along the crack direction of the pair of points selected before and behind the crack tip. In this study, acceleration or retardation of the crack growth rate in terms of the NKH plastic SIF with respect to the HRR model was observed because of the coupling effect of the mode mixity and cyclic plastic properties of the tested P2M steel, aluminium 7050, and Ti-6Al-4V.

On [formula omitted] characterization of elastic–plastic crack-tip fields under fatigue loading conditions

In the present paper, the applicability of the ΔJ-integral is computationally investigated under the Masing postulate. It confirms that the path-dependency of ΔJ is more severe than the J-integral under incremental plasticity. ΔJ from loading is up to 20% larger than ΔJ from unloading, which is independent of loading amplitudes but due to non-proportional stress variations. The stress range field can be characterized by ΔJ and shows an analogous mathematical structure as the known HRR solution, but not for the strain field under fatigue loading. The ΔJ dominant zone is much smaller than the HRR zone and further decreases with applied loading.

Crack-tip-opening-displacement-based description of three-dimensional elastic-plastic crack border fields

Characterization of elastic-plastic crack-tip fields has been great developed in the framework of the classical dominating parameter J-integral, which has a limit of the basic assumptions of small deformation and simple proportional loading. A stable and effective parameter is still a challenge for three-dimensional (3D) elastic-plastic fracture problems. Based on the crack-tip-opening-displacement (CTOD) conception and out-of-plane stress constraint factor Tz, a new elastic-plastic stress intensity factor Kδ-Tz is proposed to dominate the 3D elastic-plastic crack border fields. Detailed 3D finite element simulations are performed for four typical testing specimens, which are the centre-cracked tension specimens, compact specimens, single-edge cracked tension specimens and single-edge-notched bending specimens under three-point bending. It is shown that for specimens with different geometries and thicknesses, Kδ-Tz is proven to be more stable than the classical J-integral via the experiment data and simulation results, in which the maximum change in J-integral can be over 340% while the change in Kδ-Tz is within 7.78%. Good agreements are obtained between the CTOD-based Kδ-Tz description and simulation results for the 3D elastic-plastic crack border stress fields under all the simulated conditions.

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.

Asymptotic methods and their applications in nonlinear fracture mechanics: a review

Asymptotic methods, perturbation theory techniques and their applications in non-linear fracture mechanics and continuum damage mechanics are reviewed. I analysed asymptotic stress, strain and damage fields near the crack tip for power-law materials and the influence of the damage accumulation processes on the stress-strain state in the vicinity of the crack tip. I reviewed fundamental results obtained in nonlinear fracture mechanics by means of asymptotic methods and perturbation theory approaches. Particular attention is focused on power-law materials and asymptotic stress and strain fields in the vicinity of the crack in both non-damaged and damaged materials under mixed mode loadings. I discussed asymptotic elastic-plastic crack-tip fields derived by Hutchinson, Rice and Rosengren as a singular dominant term of the asymptotic expansion for the stress field in a power-law hardening material. I presented and discussed new solutions of nonlinear fracture mechanics and continuum damage mechanics obtained by perturbation methods.

Nonlinear Fracture Resistance Parameters for Elements of Aviation Structures under Biaxial Loading

This study is concerned with practical application of nonlinear deformation and fracture resistance parameters for residual durability estimation of the fuselage elements with operation damage. By means of numerical calculations for the fragment of fuselage skin with central crack the governing parameters of the elastic-plastic crack-tip stress field are determined as a function of biaxial loading. Two variants of modeling of the crack-tip stress field were performed for the panel of homogeneous isotropic material and a set of the special cohesive elements. As a result the assessment of the biaxial loading influence of fracture damage zone state on crack tip stress field in fuselage panel is given.

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.

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.

Characterisation of crack tip stresses in elastic-perfectly plastic material under mode-I loading

Asymptotic crack tip stress fields are developed for a stationary plane strain crack in incompressible elastic-perfectly plastic material under mode-I loading. Detailed investigations have revealed that in between the two extreme conditions of crack tip constraint, that is, between the fully plastic Prandtl [1] field and the uniform stress field the most general elastic–plastic crack tip fields can be completely described by the 5-sector stress solution proposed in this article. The 3-sector stress field proposed by Li and Hancock [2] and the 4-sector field proposed by Zhu and Chao [3] are subsets of the general elastic–plastic field proposed in this work. This study has revealed that cases arise where the severe loss of crack tip constraint can lead to compressive yielding of crack flank. This particular situation leads to 5-sector stress field. Detailed studies have revealed that, in the most general case of elastic–plastic crack tip fields, the T π parameter proposed by Zhu and Chao [3] cannot be used as a constraint parameter to represent a unique state of stress at the crack tip. A new constraint-indexing parameter T CS-2 is proposed, which along with T p is capable of representing the entire elastic–plastic crack tip stress fields over all angles around a crack tip. Excellent agreement is obtained between the proposed asymptotic crack tip stress field and the full-field finite element results for constraint levels ranging from high to low. It is demonstrated that the proposed constraint parameters are adequate to represent the crack tip constraint arising due to specimen geometry and loading conditions as well as the additional constraint that arises due to weld strength mismatch.

Asymptotic Crack-Tip Fields for Mixed Stationary Crack under Plane Stress Conditions in Pressure Sensitive and Elastic-Plastic Materials

Based on the parabolic yield criterion reflected by pressure sensitive and the SD effect, the material constitutive equation is given by orthogonal flow rule. In the plane stress condition, a basic solution in elastic-plastic crack tip field is given. The different structures of solutions with different plastic mixity are analyzed. Different parameters in crack-tip stress field and the singularity in the strain field are discussed. These results provide a theoretical reference for engineering applications.

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.

Elastic T-Stress Solutions of Embedded Elliptical Cracks Subjected to Uniaxial and Biaxial Loadings

Abstract The elastic T-stress has been found to be an important parameter in characterizing the near crack tip elastic-plastic stress states of 2-D and 3-D crack problems. In addition to the J-integral, the T-stress provides an effective two-parameter characterization of 2-D and 3-D elastic-plastic crack tip fields in a variety of crack configurations under small scale yielding conditions. Embedded elliptical crack in an infinite solid subjected to uniaxial or biaxial tension conditions represents an excellent model for many engineering components with embedded cracks. Although extensive analyses are available for stress intensity factors (SIFs) for embedded elliptical cracks, to date no analytical T-stress solutions are available. This paper presents the exact close form solutions for elastic T-stress of an embedded elliptical crack in an infinite body under remote uniaxial or biaxial tension. The foundation of the derivation is based on the potential method for three-dimensional elasticity. Solutions of stress components parallel to and perpendicular to the cracked plane are derived first. T-stress is then obtained from the asymptotic stress field near the crack front. Complete solutions of T-stress along the crack front are established. Further, three-dimensional finite element analyses are conducted to verify the derived solutions; excellent agreements are achieved. When combined with the corresponding K or J solutions, these T-stress solutions are suitable for the analysis of constraint effects for embedded elliptical shaped cracks in engineering components.

Characterization of crack-tip field and constraint for bending specimens under large-scale yielding

Elastic-plastic crack-tip fields and constraint levels in bending specimens under large-scale yielding (LSY) are examined. The J−A2 three-term solution is modified by introducing an additional term caused by the global bending. Three different methods, i.e. two-point matching, constant A2 and elastic stress estimation method, are proposed to determine the fourth term. It is shown that the elastic stress estimation method is the simplest, yet effective, in that the fourth term can be derived from the strength theory of materials and the concept of plastic hinge, and effectively quantifies the contribution of the global bending moment on the crack-tip field. Consequently, the modified J−A2 solution, with the inclusion of the correction for global bending, does not introduce any new parameter. The two parameters remain as the loading (J and M) and the constraint level (A2). To validate the present solution, detailed finite element analyses (FEA) were conducted for a Three Point Bend (TPB) specimen with a/W=0.59 in A285 steel, and Single Edge Notched Bend (SENB) specimen subjected to pure bending with a/W=0.5 in A533B steel at different deformation levels ranging from small-scale yielding (SSY) to LSY. Results show that the modified J−A2 solution matches fairly well with the FEA results for both TPB and SENB specimens at all deformation levels considered. In addition, the fourth stress term is (a) proportional to the global bending moment and inversely proportional to the ligament length; (b) negligibly small under SSY; and (c) significantly large under LSY or fully plastic deformation. Accordingly, the present model effectively characterizes the crack-tip constraint for bending dominated specimens with or without the large influence from the global bending stress on the crack-tip field.

J-integral and fatigue life computations in the incremental plasticity analysis of large scale yielding by p-version of F.E.M.

Since the linear elastic fracture analysis has been proved to be insufficient in predicting the failure of strain hardening materials, a number of fracture concepts have been studied which remain applicable in the presence of plasticity near a crack tip. This work thereby presents a new finite element model to predict the elastic-plastic crack-tip field and fatigue life of center-cracked panels(CCP) with ductile fracture under large-scale yielding conditions. Also, this study has been carried out to investigate the path-dependence of J-integral within the plastic zone for elastic-perfectly plastic, bilinear elastic-plastic, and nonlinear elastic-plastic materials. Based on the incremental theory of plasticity, the p-version finite element is employed to account for the accurate values of J-integral, the most dominant fracture parameter, and the shape of plastic zone near a crack tip by using the J-integral method. To predict the fatigue life, the conventional Paris law has been modified by substituting the range of J-value denoted by DJ for DK. The experimental fatigue test is conducted with five CCP specimens to validate the accuracy of the proposed model. It is noted that the relationship between the crack length a and DK in LEFM analysis shows a strong linearity, on the other hand, the nonlinear relationship between a and DJ is detected in EPFM analysis. Therefore, this trend will be depended especially in the case of large scale yielding. The numerical results by the proposed model are compared with the theoretical solutions in literatures, experimental results, and the numerical solutions by the conventional h-version of the finite element method.

완전소성하 변형경화 이종접합재의 계면균열선단 구속상태 및 J-적분

This paper investigates the effects of T-stress and plastic hardening mismatch on the interfacial crack-tip stress field via finite element analyses. Plane strain elastic-plastic crack-tip fields are modeled with both MBL formulation and a full SEC specimen under pure bending. Modified Prandl slip line fields illustrate the effects of T-stress on crack-tip constraint in homogeneous material. Compressive T-stress substantially reduces the interfacial crack-tip constraint, but increases the J-contribution by lower hardening material, J_L. For bimaterials with two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and J_L. The fracture toughness for bimaterial joints would consequently be much lower than that of lower hardening homogeneous material. The implication of unbalanced J-integral in bimaterials is also discussed.

항복강도 불일치 반타원 계면균열 선단에서의 응력장

Many research works have been performed on the J-T approach for elastic-plastic crack-tip stress fields in a variety of plane strain specimens. To generalize the validity of J-T method, further investigations are however needed for more practical 3D structures than the idealized plane strain specimens. The present study deals mainly with 3D finite element (FE) modeling of welded plate and straight pipe, and accompanying elastic, elastic-plastic FE analyses. Manual 3D modeling is almost prohibitive, since the models contain semi-elliptical interfacial cracks which require singular elements. To overcome this kind of barrier, we develop a program generating the meshes for semi-elliptical interfacial cracks. We then compare the detailed 3D FE stress fields to those predicted with J-T two parameters. The validity of J- T approach is thereby extended to 3D yield-strength-mismatched weld joints, and useful information is inferred for the design or assessment of pipe welds.

J-T에 의한 3차원 반타원 계면균열선단 응력장의 기술

Many research works have validated the J-T approach to elastic-plastic crack-tip stress fields in a variety of plane strain specimens. To generalize the validity of J-T method, further investigations are however needed for more practical 3D structures than the idealized plane strain specimens. In this work, we perform 3D finite element (FE) modeling of welded plate and straight pipe, and accompanying elastic, elastic-plastic FE analyses. Manual 3D modeling is almost prohibitive, since the models contain semi-elliptical interfacial cracks which require singular elements. To overcome this kind of barrier, we develop a program generating the meshes for semi-elliptical interfacial cracks. We then compare the detailed 3D FE stress fields to those predicted with J-T two parameters. Thereby we extend the validity of J-T application to 3D structures and infer some useful informations for the design or assessment of pipe welds.

Interfacial crack-tip constraints and J-integrals in plastically mismatched bi-materials

The effect of the T-stress and plastic mismatch on interfacial crack-tip stress under small scale yielding condition is investigated in this paper via detailed finite element analyses. The mismatch in plastic strength/hardening is focused, and plane strain elastic–plastic crack-tip fields have been modeled with modified boundary layer formulation. Plastic mismatches as well as compressive T-stresses in bi-material are shown to affect the interfacial crack-tip constraint substantially. Simple patching of slip-line fields is presented to characterize the interfacial crack-tip stress fields. Effects of T-stress and plastic hardening mismatch on asymmetry of the J-integral for bi-materials, and its implication on interfacial toughness are also discussed.

Constraint effects on crack-tip fields in elastic-perfectly plastic materials

Asymptotic crack-tip stress fields accounting for constraint effects are developed for a stationary plane strain crack under mode-I, mode-II or mixed-mode I/II loading. The mixed-mode loading is considered only within small-scale yielding. Materials are taken into account in incompressible, elastic-perfectly plastic materials, and plastic deformation of materials obeys von Mises yield criterion. This investigation is an extension of the solution obtained by Li and Hancock [Li, J., Hancock, J.W., 1999. Mode I and mixed mode fields with incomplete crack tip plasticity. International Journal of Solids and Structures 36 (5), 711–725] with special attention on what constraint parameters existed in the elastic-plastic crack-tip fields. Results indicate that the asymptotic crack-tip field is a 4-sector solution for mode-I cracks and a 6-sector solution for mixed-mode cracks, and is comprised of plastic sectors and elastic sector(s), and contain two undetermined parameters T p and T π which are hydrostatic stresses ahead of the crack tip and on the crack flank, respectively. When T p and T π vanish, the present elastic-plastic crack-tip field reduces to the fully plastic Prandtl slip-line field. Comparison shows that the asymptotic crack-tip stress fields can precisely match with elastic-plastic finite element results over all angles around a crack tip for various fracture specimens with constraint levels from high to low. The magnitudes of T p and T π determine the level of crack-tip constraint in plastic sectors and in elastic sector, respectively, due to geometric and loading configurations or mode mixity. Thus the parameters T p and T π can be used as constraint parameters to effectively characterize the entire crack-tip field in elastic-perfectly plastic materials under the plane strain conditions.