Plane-strain crack tip field and a dual-parameter fracture-criterion for non-linear fracture mechanics

Hancock and Cowling measured the critical crack tip opening displacements, δ_f, at fracture initiation in HY-80 steel specimens of six different configurations. δ_f varied from 90 μm in a deeply double-edge-cracked tensile panel to 900 μm in a single-edge-cracked tensile panel.McMeeking and Parks, and Shih and German have shown by their finite element calculations that the characteristics of the plane strain crack tip fields in both large scale yielding and general yielding are strongly dependent on specimen geometry and load level.In this study, the plane strain crack tip fields in the specimens tested by Hancock and Cowling were calculated using the finite element method. The crack tip triaxial tensile stress field is strongly affected by specimen geometric constraint, and the state of the triaxial tensile stress in a crack tip region is monitored by the ratio between the local tensile stress and the effective stress, i.e., ( ), at a distance x=2δ from the crack tip. The values of ( ) vary from 3.1 for the double-edge-cracked tensile panel to 1.7 for the single-edge-cracked tensile panel. The δ_f measured by Hancock and Cowling correlates very well with the ratio ( ). δ_f is a measure of the fracture ductility of the material ahead of the crack tip, and the ductility decreases with an increase in the triaxial tensile stress, i.e., the ratio ( ).     

Characterization of crack tip stresses in plane-strain fracture specimens having weld center crack

Fracture toughness of metals depends strongly on the state of stress near the crack tip. The existing standards (like R-6, SINTAP) are being modified to account for the influence of stress triaxiality in the flaw assessment procedures. These modifications are based on the ability of so-called ‘constraint parameters’ to describe near tip stresses. Crack tip stresses in homogeneous fracture specimens are successfully described in terms of two parameters like J–Q or J–T. For fracture specimens having a weld center crack, strength mismatch ratio between base and weld material and weld width are the additional variables, along with the magnitude of applied loading, type of loading, and geometry of specimen that affect the crack tip stresses. In this work, a novel three-parameter scheme was proposed to estimate the crack tip opening stress accounting for the above-mentioned variables. The first and second parameters represent the crack tip opening stress in a homogeneous fracture specimen under small-scale yielding and are well known. The third parameter accounts for the effect of constraint developed due to weld strength mismatch. It comprises of weld strength mismatch ratio (M, i.e. ratio of yield strength of weld material to that of base material), and a plastic interaction factor (I_p) that scales the size of the plastic zone with the width of the weld material. The plastic interaction factor represents the degree of influence of weld strength mismatch on crack tip constraint for a given mismatch ratio. The proposed scheme was validated with detailed FE analysis using the Modified Boundary Layer formulation.     

Analysis of deformation-induced crack tip toughening in ductile single crystals by nano-indentation

This paper reports on the experimental examination of the deformation characteristics near a crack tip in a cyclically work-hardened copper single crystal using a 2D surface scans with nano-indentation. The experimental methodology enables the characterization of the primary deformation field near a crack tip via the modulation of the imposed secondary deformation field by a nano-indentation. In a heavily deformed 4-point bend specimen, the measurements showed an existence of an asymptotic field around the crack tip at a distance of R ⩾ 2.5J/σ₀. The measurements also showed the qualitative details of toughness evolution within the high-gradient deformation field around the crack tip. The nature of dislocation distribution (i.e. statistically distributed vs. distributions necessitated by geometry) around the crack tip is quantified. The measurements indicate the dominance of the geometrically necessary dislocation within the finite deformation zone ahead of the tip up to a distance of R ≈ 3J/σ₀. Thereafter, it is confined in radial rays coinciding with the sector boundaries around the crack tip. These measurements elucidate the origin of the inhomogeneous hardening and the size dependent macroscopic response close to crack tip.     

Rate sensitivity of mixed mode interface toughness of dissimilar metallic materials: Studied at steady state

Crack propagation in metallic materials produces plastic dissipation when material in front for the crack tip enters the active plastic zone traveling with the tip, and later ends up being part of the residual plastic strain wake. Thus, the macroscopic work required to advance the crack is typically much larger than the work needed in the near tip fracture process. For rate sensitive materials, the amount of plastic dissipation typically depends on the rate at which the material is deformed. A dependency on the crack velocity should therefore be expected. The objective of this paper is to study the macroscopic toughness of crack advance along an interface joining two dissimilar rate dependent materials, characterized by an elastic-viscoplastic material model that approaches the response of a J₂-flow material in the rate independent limit. The emphasis here is on the rate sensitivity of the macroscopic fracture toughness under mixed Mode I/II loading. Moreover, special cases of joined similar rate dependent materials, as well as dissimilar materials where one substrate remains either elastic or approaches the rate independent limit is also included. The numerical analysis is carried out using the SSV model [Suo, Z., Shih, C., Varias, A., 1993. A theory for cleavage cracking in the presence of plastic flow. Acta Metall. Mater. 41, 1551–1557] embedded in a steady state finite element formulation, here assuming plane strain conditions and small-scale yielding. Results are presented for a wide range of material parameters, including noteworthy observations of a characteristic crack velocity at which the macroscopic toughness becomes independent of the material rate sensitivity. The potential of this phenomenon is elaborated on from a modeling point of view.     

Dynamic fracture analysis by moiré interferometry

Dynamic moiré interferometry was used to measure separately theu- andv-displacement fields surrouding a rapidly propagating crack tip in Homalite-100 and 7075-T6 aluminum-alloy plates. These transient crack-tip-displacement data were then used to compute the dynamic stress-intensity factor and the remote stress component.J-integral values were also estimated using the static approximate procedure of Kanget al. This static analysis provided the correctJ when the contour integral was taken within 3 mm of the crack tip.Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WI on June 9–13.     

A centrally cracked thin circular disk, Part I: 3D Elastic–plastic finite element analysis

A full field solution, based on small deformation, three-dimensional elastic–plastic finite element analysis of the centrally cracked thin disk under mode I loading has been performed. The solution for the stresses under small-scale yielding and lo!cally fully plastic state has been compared with the HRR plane stress solution. At the outside of the 3D zone, within a distance of rσ_o/J=18, HRR dominance is maintained in the presence of a significant amount of compressive stress along the crack flanks. Ahead of this region, the HRR field overestimate the stresses. These results demonstrate a completely reversed state of stress in the near crack front compared to that in the plane strain case. The combined effect of geometry and finite thickness of the specimen on elastic–plastic crack tip stress field has been explored. To the best of our knowledge, such an attempt in the published literature has not been made yet. For the qualitative assessment of the results some of the field parameters have been compared to the available experimental results of K, gives a fair estimate of the crack opening stress near the crack front at a distance of order 10⁻² in. On the basis of this analysis, the Linear Elastic Fracture Mechanics approach has been adopted in analyzing the fatigue crack extension experiments performed in the disk (Part II).     

Energy dissipation and contour integral characterizing fracture behavior of incremental plasticity

Jep -integral is derived for characterizing the frac- ture behavior of elastic-plastic materials. The J ep -integral differs from Rice’s J-integral in that the free energy density rather than the stress working density is employed to define energy-momentum tensor. The J ep -integral is proved to be path-dependent regardless of incremental plasticity and deformation plasticity. The J epintegral possesses clearly clear physical meaning: (1) the value J ep tip evaluated on the infinitely small contour surrounding the crack tip represents the crack tip energy dissipation; (2) when the global steadystate crack growth condition is approached, the value of J ep farss calculated along the boundary contour equals to the sum of crack tip dissipation and bulk dissipation of plastic zone. The theoretical results are verified by simulating mode I crack problems.     

Crack tip field andJ-integral with strain gradient effect

The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani. The theory retains the essential structure of the incremental version of the conventionalJ ₂ deformation theory. No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required. The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus. The strain gradient measures are included into the tangent modulus as internal parameters. Therefore the boundary value problem is the same as that in the conventional theory. Two typical crack problems are studied: (a) the crack tip field under the small scale yielding condition induced by a linear elastic mode-IK-field and (b) the complete field for a compact tension specimen. The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory. The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it. Consequently, theJ-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases. The project supported by the National Natural Science Foundation of China (19704100 and 10202023) and the Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20)     

J-integral and crack driving force in elastic–plastic materials

This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

Limitations to the small scale yielding approximation for crack tip plasticity

Recent finite-element results by S.G. Larsson and A.J. Carlsson suggest a limited range of validity to the ‘small scale yielding approximation’, whereby small crack tip plastic zones are correlated in terms of the elastic stress intensity factor. It is shown with the help of a model for plane strain yielding that their results may be explained by considering the non-singular stress, acting parallel to the crack at its tip, which accompanies the inverse square-root elastic singularity. Further implications of the non-singular stress term for crack tip deformations and fracturing are examined. It is suggested that its effect on crack tip parameters, such as the opening displacement and J-integral, is less pronounced than its effect on the yield zone size.     

Calculation for path-domain independentJ integral with elasto-viscoplastic consistent tangent operator concept-based boundary element methods

This paper presents an elasto-viscoplastic consistent tangent operator (CTO) based boundary element formulation, and application for calculation of path-domain independentJ integrals (extension of the classicalJ integrals) in nonlinear crack analysis. When viscoplastic deformation happens, the effective stresses around the crack tip in the nonlinear region is allowed to exceed the loading surface, and the pure plastic theory is not suitable for this situation. The concept of consistency employed in the solution of increment viscoplastic problem, plays a crucial role in preserving the quadratic rate asymptotic convergence of iteractive schemes based on Newton's method. Therefore, this paper investigates the viscoplastic crack problem, and presents an implicit viscoplastic algorithm using the CTO concept in a boundary element framework for path-domain independentJ integrals. Applications are presented with two numerical examples for viscoplastic crack problems andJ integrals. The project supported by National Natural Science Foundation of China (9713008) and Zhejiang Natural Science Foundation Special Funds No. RC.9601     

Finite deformation analysis of COD,J-integral and crack tip intense strain region in plane strain large-scale yielding

A finite element analysis was performed to simulate crack tip blunting and the development of the intense strain region in a small compact tension specimen (0.4 T CT) of SA533B-1 under plane strain large-scale yielding, with the condition of large-geometry change around the crack tip taken into consideration. The region where the equivalent plastic strain $\text{}\text{?}$g3_p is greater than 0.15 was defined as the intense strain region, which corresponded to the recrystallized-etched zone delineated experimentally around the blunting crack tip. The development of the intense strain region was discussed as a function of the J-integral and the crack opening displacement. A linear relationship was obtained between the plastic work W_p dissipated within the intense strain region and (J/σ_y)² or b², where b is the crack opening displacement, defined as the separation of the two points at which the boundary of the intense strain region surrounding the crack tip intersects with the free surfaces of the crack.     

Two general integrals of singular crack tip deformation fields

The Eshelby tensor E has vanishing divergence in a homogeneous elastic material, whereas the invariance of the crack tip J integral suggests, in accord with known solutions, that the product rE will have a finite limit at the tip. Here r is distance from the tip. These considerations are shown to lead to two general integrals of the equations governing singular crack tip deformation fields. Some of their consequences are discussed for analysis of crack tip fields in linear and nonlinear materials.     

The kinked crack solved by Mellin transform

The problem of the evaluation of the stress intensity factors (SIF) at a kinked crack-tip is treated by the use of the most appropriate tool for this problem, the Mellin transform technique. The advantage of this technique lies on the validity of the solution everywhere in the stress field and for every length of the branched crack. Since in the limiting case of the kinked crack, numerical difficulties appear in the evaluation of the kernel functions of the corresponding Fredholm integral-equation system, an asymptotic development, with respect to the length of the kinked crack, is constructed for these functions and therefore for the SIFs at the kinked tip. This analysis yields the interesting result that the SIFs at the crack tip tend to vanish as the kinked crack reduces in length, a result which is similar to the already established behavior of the SIFs at an interfacial tip.     

High-order asymptotic analysis for the crack in nonlinear material

Accurate high-order asymptotic analyses were carried out for Mode II plane strain crack in power hardening materials. The second-order crack tip fields have been obtained. It is found that the amplitude coefficientk ₂ of the second term of the asymptotic field is correlated to the first order field as the hardening exponentn<n ^* (n ^*≈5), but asn≥n ^*,k ₂ turns to become an independent parameter. Our results also indicated that, the second term of the asymptotic field has little influence on the near-crack-tip field and can be neglected whenn<n ^*. In fact,k ₂ directly reflects the effects of triaxiality near the crack tip, the crack geometry and the loading mode, so that besidesJ-integral it can be used as another characteristic parameter in the two-parameter criterion. The project supported by National Natural Science Foundation of China     

Fatigue properties of age-hardened Al alloy 2017-T4 under ultrasonic loading

Fatigue properties of age-hardened Al alloy 2017-T4 under ultrasonic loading frequency (20 kHz) were investigated and compared with the results under conventional loading of rotating bending (50 Hz). The growth of a crack retarded at about 500 µm in surface length under ultrasonic loading, while at about 20 µm under rotating bending. Although striations being a typical fracture mechanism were observed under conventional loading, most of fracture surface was covered with many facets under ultrasonic loading. These facets were also observed under rotating bending in nitrogen gas. The difference in growth mechanism depending on the loading frequency and the retardation of a crack growth under ultrasonic loading may be caused by the environment at the crack tip due to high crack growth rate under ultrasonic loading.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

Plastic flow at a crack tip. Energy failure criterion and its relationship to the J integral

Plastic flow at the tip of a crack moving in an elastic body is considered using the theory of an ideally rigid-plastic body. The material at the crack tip is treated as a body consisting of an elastic outer region and a rigid-plastic inner region. It is shown that this representation is energetically justified for small plastic regions. The distribution of the specific dissipation of the work of internal forces and deformations along the particle trajectory at the crack tip is obtained. The relationship between the specific dissipation of the work of internal forces and the J integral under plane-strain conditions is established.