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 ( ).     

A special crack tip displacement discontinuity element

Based on the analytical solution to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid and the note of the crack tip element by Crouch, in the present paper, the special crack tip displacement discontinuity element is developed. Further the analytical formulas for the stress intensity factors of crack problems in general plane elasticity are given. In the boundary element implementation the special crack tip displacement discontinuity element is placed locally at each crack tip on top of the non-singular constant displacement discontinuity elements that cover the entire crack surface. Numerical results show that the displacement discontinuity modeling technique of a crack presented in this paper is very effective.     

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.     

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

A numerical study of crack tip constraint in ductile single crystals

In this work, the effect of crack tip constraint on near-tip stress and deformation fields in a ductile FCC single crystal is studied under mode I, plane strain conditions. To this end, modified boundary layer simulations within crystal plasticity framework are performed, neglecting elastic anisotropy. The first and second terms of the isotropic elastic crack tip field, which are governed by the stress intensity factor K and T-stress, are prescribed as remote boundary conditions and solutions pertaining to different levels of T-stress are generated. It is found that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the crack tip are also strongly influenced by the level of T-stress, with progressive loss of crack tip constraint occurring as T-stress becomes more negative. A family of near-tip fields is obtained which are characterized by two terms (such as K and T or J and a constraint parameter Q) as in isotropic plastic solids.     

Problems of dynamic fracture mechanics without contact of the crack faces

Conclusion In this, the first part of the survey we have discussed only certain aspects of dynamic fracture mechanics. The surveyed material has been selected with a preference for the most highly developed parts of the theory, specifically those elements which have direct bearing on the second part of the survey. We have also included information on the dynamic fracture mechanics of initially stressed materials, in the development of which one of the authors has been a major contributor.Since many problems of dynamic fracture mechanics have been overlooked in the survey, we have added supplementary references to the literature. Various aspects of the strength and fracture of materials under dynamic loading are set forth in [11, 12, 40, 57, 60, 73, 80, 83]. Criteria of the initiation, motion, branching, and arrest of cracks are discussed in [7, 9, 60, 102, 111, 113, 124]. Among the most interesting elements of dynamic fracture mechanics are the problems of crack propagation. Certain analytical results pertinent to this topic have been obtained in [43–45, 47, 67–72, 78, 87, 92, 96, 97].S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Highway Engineering University, Kharkov. Translated from Prikladnaya Mekhanika, Vol. 30, No. 10, pp. 3–28, October, 1994.     

On the determination of elastodynamic crack tip stress fields

A linear elastic body in plane strain which contains a stationary crack and which is initially at rest and stress free is considered. It is shown that if the elastodynamic displacement field and stress intensity factor are known, as functions of crack length, for any symmetrical distribution of time-varying forces which acts on the body, subsequent to t=0, then the stress intensity factor due to any other symmetrical load system whatsoever which acts on the same body may be directly determined. The other load system may be of arbitrary spatial distribution and time variation. Further, that part of the elastodynamic displacement field due to the other load system, which arises from the presence of the crack, may also be directly determined. The results are obtained by extension of Rice's mode of derivation of the corresponding Bueckner-Rice elastostatic results to Laplace-transformed elastodynamic variables. Likewise, the existence of a universal elastodynamic “weight function” for any given cracked body is demonstrated. As an application, Freund's recent result for the stress intensity factor due to suddenly applied concentrated forces on the crack surfaces is derived directly by our method, from de Hoop's earlier solution for suddenly applied uniform pressures.     

On the applicability of theI-integral in fracture mechanics

Summary The use of conservation integrals as fracture mechanics parameters has been extended since Rice's work on theJ-integral originally restricted to hyperelastic materials, to other classes of constitutive laws in the past few years. The so-calledI-integral being valid for any constitutive law seemeed to be suitable as a generalization of theJ-integral in the elastic-plastic regime.In this paper a proof is given that theI-integral vanishes under standard boundary conditions. Therefore, theI-integral has not the quality of a fracture mechanics parameter. A comparison betweenI andJ leads to a new representation of theJ-integral.

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Über die Anwendbarkeit desI-Integrals in der Bruchmechanik

Übersicht Die Verwendung von Erhaltungsintegralen als Bruchmechanikparameter wurde seit der Arbeit von Rice über dasJ-integral, das ursprünglich auf hyperelastisches Material beschränkt war, in den letzten Jahren auf andere Klassen von Materialgesetzen erweitert. Das sogenannteI-Integral, das für ein beliebiges Materialgesetz gültig ist, schien als Verallgemeinerung desJ-Integrals im elastisch-plastischen, Bereich in Frage zu kommen.In dieser Arbeit wird bewiesen, daß dasI-Integral unter Standardrandbedingungen stets verschwindet. Es ist daher kein Bruchmechanik-Parameter. Ein Vergleich zwischenI undJ führt zu einer neuen darstellung desJ-Integrals.

     

数值岩石试样破裂过程多重分形特征

采用数值分析的方法和FLAC3D软件,建立服从应变软化关系的非均质岩石试样模型。引入多重分形理论,通过盒计数法计算了单轴加载过程中峰前某一阶段单元体积应变的多重分形谱f(α)。分析了f(α)-α曲线特征参数#f和#α随加载过程的阶段特征,不同均匀程度的岩石试样在临近峰值强度前#α和f(α)max的变化规律。研究结果表明,采用多重分形理论对单元体积应变的空间分布特征进行描述,可以在较大的尺度内满足标度不变性。通过不同的参量进行概率计算得出的f(α)略有不同,但动态变化规律相近,f(α)曲线呈左钩形状,随着载荷的增加#f变大,而#α变小。当不是所有的盒子内都存在可测度的参量时,f(α)max随载荷的增加而增长。对于均匀程度不同的试样,峰前f(α)max与#α随均匀性的增加而减小,但在均匀性较高以后,逼近于某一常数。     

On discrete models in fracture mechanics

In two papers published in 1969, where the foundations of the theory of brittle fracture were laid, Valentin Valentinovich Novozhilov introduced a necessary and sufficient criterion for estimating the strength of an elastic body weakened by a cut and formulated the concept of brittle fracture. The criterion is based on taking account of the body discrete structure, and the fracture process is treated as loss of stability of elastic equilibrium. These two facts, fracture discreteness and loss of stability in the deformation of the breaking bond, underlie numerous phenomena, which could not be discovered in the framework of continuum mechanics. In the present paper, the following effects of taking these facts into account are briefly discussed: an unstable leg of the stress-strain diagram and the capture of part of the fracture energy by the lattice, the role of the lattice dimension, radiation in quasi-static and dynamic crack propagation including the radiation structure, the role of the dynamic factor, irregularities in the crack growth, crack propagation under the action of high-frequency waves including steady-state modes with periodically varying speed, and cracks in continuum discrete models.     

压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向同性压电材料中受拉伸和电载荷作用的平片裂纹I型断裂力学问题. 使用有限部积分概念，从三维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意.     

Modeling the fracture process zone near a crack tip in a nonlinear elastic body

The model of a fracture process zone near the tip of a mode I crack in a nonlinear elastic body is proposed. Using the numerical solution of an appropriate boundary-value problem, the effect of the fracture process zone on a crack opening displacement is examined     

Combination of fracture and damage mechanics for numerical failure analysis

A new approach for the analysis of crack propagation in brittle materials is proposed, which is based on a combination of fracture mechanics and continuum damage mechanics within the context of the finite element method. The approach combines the accuracy of singular crack-tip elements from fracture mechanics theories with the flexibility of crack representation by softening zones in damage mechanics formulations. A super element is constructed in which the typical elements are joined together. The crack propagation is decided on either of two fracture criteria; one criterion is based on the energy release rate or the J-integral, the other on the largest principal stress in the crack-tip region. Contrary to many damage mechanics methods, the combined fracture⧹damage approach is not sensitive to variations in the finite element division. Applications to situations of mixed-mode crack propagation in both two- and three-dimensional problems reveal that the calculated crack paths are independent of the element size and the element orientation and are accurate within one element from the theoretical (curvilinear) crack paths.     

On physically incorrect results in fracture mechanics

The problem of physically incorrect results due to neglecting the interaction of the crack edges in fracture mechanics is discussed. The cases of dynamic loading, interface cracks, and thermal loading in fracture mechanics are analyzed. The main conclusion is that the majority of results obtained in these areas of fracture mechanics are physically incorrect     

On the correctness of the energy criterion in fracture mechanics

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 31, No. 10, pp. 28–34, October, 1995.     

Problems of dynamic fracture mechanics with allowance for contact of the crack edges

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine. Kharkov Institute of Highway Engineering. Translated from Prikladnaya Mekhanika, Vol. 31, No. 1, pp. 3–36, January, 1995.     

On use of the Galerkin method to solve the fracture mechanics problem for a linear crack under normal loading

The problem of contact interaction of the opposite faces of a linear crack under a normally incident harmonic tension-compression wave is numerically solved by the Galerkin method with piecewise-linear continuous elements. The dependence of the stress intensity factor (opening mode) on the wave number is investigated Published in Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 137–142, November 2005.     

Special approach for the determination of fracture mechanical parameters at an interface crack tip

An interface crack of finite length is considered between two semi-infinite planes with an artificial contact zone at one of the two crack tips. A transcendental equation and certain simple asymptotic formulas are established for the real contact zone (in the Comninou-Dundurs sense) in terms of the stress intensity factors (SIFs) of the considered model. In these terms analytical expressions are also provided for the energy release rate and for the SIF of the classical interface crack model with an oscillating singularity at the crack tip. The appropriate length of the artifical contact zone is shown to be attainable on the basis of the analysis of the stresses at the crack tip. The use of the proposed model is suggested for integrity assessment of inhomogeneous structural elements of composites containing interface cracks. Received 26 March 1997; accepted for publication 12 September 1997