Elastic–plastic near field solution of an eccentric crack under shear in a finite width plate

The near crack line analysis method is used to investigate an eccentric crack loaded by shear forces in a finite width plate, and the analytical solution is obtained in this paper. The solution includes: the unit normal vector of the elastic–plastic boundary near the crack line, the elastic–plastic stress fields near crack line, variations of the length of the plastic zone along the crack line with an external loads, and the bearing capacity of a finite plate with a centric crack loaded by shear stress in the far field. The results obtained in this paper are sufficiently precise near the crack line because the assumptions of small scale yielding theory have not been made and no other assumptions have been taken. Subsequently, the present results are compared with the traditional line elastic fracture mechanical solutions and elastoplastic near field solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

THE EXACT SOLUTIONS OF ELASTIC－PLASTIC CRACK LINE FIELD FOR MODE ⅡPLANE STRESS CRACK

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EXACT SOLUTIONS OF NEAR CRACK LINE FIELDS FOR MODE I CRACK UNDER PLANE STRESS CONDITION IN AN ELASTIC－PERFECTLY PLASTIC SOLID

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THE PRECISE AND NEW ANALYSIS FOR A MODE Ⅲ GROWING CRACK IN AN ELASTIC-PERFECTLY PLASTIC SOLID

The near crack line field analysis method has been used to investigate into ModeⅢ quasistatically propagating crack in an elastic-perfectly plastic material.Thesignificance of this paper is that the usual small scale yielding theory has been brokenthrough.By obtaining the general solutions of the stresses and the displacement rate ofthe near crack line plastic region,and by matching the general solutions with theprecise elastic fields(not the usual elastic K-dominant fields)at the elastic-plasticboundary,the precise and new solutions of the stress and deformation fields,the sizeof the plastic region and the unit normal vector of the elastic-plastic boundary havebeen obtained near the crack line.The solutions of this paper are sufficiently precisenear the crack line region because the roughly qualitative assumptions of the smallscale yielding theory have not been used and no other roughly qualitative assumptionshave been taken,either.The analysis of this paper shows that the assumingly“steady-state cas     

静止裂纹顶端塑性区内应力场

本文对不可压缩的理想塑性材料裂纹顶端塑性区内的应力场进行了数学分析,证明了当塑性区包围着裂纹顶端而应力函数可用分离变匱型的级数展开且该级数展开的首项与第一类渐近解相同时,第一类渐近解即是塑性区内应力场的精确解。本文又提出了第二类渐近解,说明应力场的渐近解不是唯一的。     

Near-tip fields for a crack growing along an elastic perfectly plastic interface

The stress and deformation fields near the tip of an anti-plane crack growing quasi-statically along an interface of elastic perfectly plastic materials are given in this paper. A family of solutions for the growing crack fields is found covering all admissible crack line shear stress ratios. The project supported by the National Natural Science Foundation of China     

铁中Ⅰ型裂纹裂尖场与裂尖形变机制分析

应用有限元及连续介质力学计算了平面应交条件下体心立方铁中不同取向裂纹的裂尖应力场和弹性应变能密度分布,从宏微观相结合角度分析了裂尖场分布与裂尖微结构演化的相互关联.指出裂尖塑性变形的具体形式与裂尖滑移面上分切应力的大小密切相关,并从能量角度解释了裂尖相变产生的原因,最后结合相关分子动力学研究成果探讨了裂尖奇异性区域.     

Torsion of an elastic solid cylinder with a radial variation in the shear modulus

A Green's function approach is used to formulate and obtain the stress field, under torsional loads in a radially finite solid cylinder with radially variable elastic modulus. With this approach a certain dual static-geometric analogy in the solution is readily proved and applied to generate the solution with stress boundary conditions from that with displacement boundary conditions and vice-versa.The problem is solved using both boundary conditions and for an exponentially varying shear modulus. In particular, under displacement boundary conditions, the stress field in the solid with a generalised Reissner-Sagoci boundary condition is easily deduced. With stress boundary conditions, the criteria for crack propagation in such elastic models are also obtained using the Griffith-Irwin condition of rupture.     

Crack growth under plane stress conditions in an elastic perfectly-plastic material

Mode-I crack growth under conditions of generalized plane stress has been investigated. It has been assumed that near the plane of the crack in the loading zone, the simple stress components corresponding to a central fan field maintain validity up to the elastic-plastic boundary. By the use of expansions of the particle velocities in the coordinate y, and by matching of the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary, a complete solution has been obtained for ε_y in the plane of the crack. The solution applies from the propagating crack tip up to the moving elastic-plastic boundary. The strain fields for a self-similar crack nucleating at a point and for steady-state propagation of a crack have been considered as special cases.     

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.     

ASYMPTOTIC ANALYSIS OF MODE Ⅱ STATIONARY GROWTH CRACK ON ELASTIC-ELASTIC POWER LAW CREEPING BIMATERIAL INTERFACE

A mechanical model was established for mode Ⅱ interfacial crack static growing along an elastic-elastic power law creeping bimaterial interface. For two kinds of boundary conditions on crack faces, traction free and frictional contact, asymptotic solutions of the stress and strain near tip-crack were given. Results derived indicate that the stress and strain have the same singularity, there is not the oscillatory singularity in the field; the creep power-hardening index n and the ratio of Young' s module notably influence the cracktip field in region of elastic power law creeping material and n only influences distribution of stresses and strains in region of elastic material. When n is bigger, the creeping deformation is dominant and stress fields become steady, which does not change with n.Poisson ' s ratio does not affect the distributing of the crack- tip field.     

弹性波与单侧界裂纹相互作用问题的边界元法

措助边界元法设计了一种迭代修正方法来求解单侧界面裂纹模型与弹性波的相互作用问题,作为对算法的检验,用这种方法我们具体地分析了平面简谐弹性波对一个则界面裂纹的入射,给出了裂纹面的接触形态及应力场。     

A continuum approach to the analysis of the stress field in a fiber reinforced composite with a transverse crack

The stress field in a periodically layered composite with an embedded crack oriented in the normal direction to the layering and subjected to a tensile far-field loading is obtained based on the continuum equations of elasticity. This geometry models the 2D problem of fiber reinforced materials with a transverse crack. The analysis is based on the combination of the representative cell method and the higher-order theory. The representative cell method is employed for the construction of Green’s functions for the displacements jumps along the crack line. The problem of the infinite domain is reduced, in conjunction with the discrete Fourier transform, to a finite domain (representative cell) on which the Born–von Karman type boundary conditions are applied. In the framework of the higher-order theory, the transformed elastic field is determined by a second-order expansion of the displacement vector in terms of local coordinates, in conjunction with the equilibrium equations and these boundary conditions. The accuracy of the proposed approach is verified by a comparison with the analytical solution for a crack embedded in a homogeneous plane.Results show the effects of crack lengths, fiber volume fractions, ratios of fiber to matrix Young’s moduli and matrix Poisson’s ratio on the resulting elastic field at various locations of interest. Comparisons with the predictions obtained from the shear lag theory are presented.     

Finite element analysis for steady-state hydride-induced fracture in metals by composite model

Delayed hydride cracking, which is observed in hydride-forming metals, due to the precipitation of hydrides near the crack tip, is investigated under conditions of constant temperature and crack velocity, plane strain and small-scale hydride-precipitation. The coupling of the operating physical processes of hydrogen-diffusion, hydride precipitation and material deformation is taken into account. The material is assumed to be an elastic composite made of hydrides and solid solution, with properties depending locally on the volume fraction of the hydrides. In the present analysis, the composite elastic properties have been derived by a generalized self consistent model for particulate composites. With respect to hydride-precipitation, two cases have been considered: (i) precipitation in a homogeneous medium with elastic properties, equal to the effective properties of the composite and (ii) precipitation in an inhomogeneous medium, where the expanding hydride has different elastic properties than those of the surrounding solid solution. The differences between the near-tip field distributions, produced by the two precipitation models, are relatively small. The effect of the hydrogen concentration far from the crack tip, on the near-tip field is also studied. It is shown that for small crack growth velocities, near the threshold stress intensity factor, the remote hydrogen concentration weakly affects the normalized stress distribution in the hydride-precipitation zone, which is controlled by the thermodynamically required hydrostatic stress, under hydrogen chemical equilibrium. However, for values of the applied stress intensity factor and the crack tip velocity, away from the threshold stress intensity factor and crack arrest, the effect of remote hydrogen concentration on the normalized near-tip stress field is strong. Reduction of the remote hydrogen concentration generally leads to reduction of the hydride-precipitation zone and increase of the near-tip stresses. Also reduction of the remote hydrogen concentration leads to distributions closer to those under hydrogen chemical equilibrium.     

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)     

载流薄板中裂纹形成瞬间尖端附近的应力场

利用电磁场的热效应对带有裂纹的载流导体进行裂纹止裂,是为了达到延长其工作寿命,提高安全性、可靠性的一种行之有效的方法。本文在文献[1]的基础上,以导电弹性体的麦克斯威尔方程为出发点,借助于边界条件和初始条件,推得了载流无限大薄板在形成裂纹的瞬间,裂纹尖端附近电流密度、温度和应力的具体表达式。通过算例分析证实了：在给定参数的情况下,通入适当强度的电流时,在电流所产生的焦耳热源的作用下,裂尖区域处的温度将瞬时升高,同时伴有压应力的产生,从而可达到阻止裂纹扩展的目的。     

Perturbed magnetic field of an infinite plate with a centered crack

Deforming a cracked magnetoelastic body in a magnetic field induces a perturbed magnetic field around the crack. The quantitative relationship between this perturbed field and the stress around the crack is crucial in developing a new generation of magnetism-based nondestructive testing technologies. In this paper, an analytical expression of the perturbed magnetic field induced by structural deforma- tion of an infinite ferromagnetic elastic plate containing a centered crack in a weak external magnetic field is obtained by using the linearized magnetoelastic theory and Fourier transform methods. The main finding is that the perturbed magnetic field intensity is proportional to the applied tensile stress, and is dominated by the displacement gradient on the boundary of the magnetoelastic solid. The tangential component of the perturbed magnetic-field intensity near the crack exhibits an antisymmetric distribution along the crack that reverses its direction sharply across its two faces, while the normal component shows a symmetric distribution along the crack with singular points at the crack tips.     

Comparative study of an interface crack for different wedge-interface models

Summary  An interface crack problem is investigated under various assumptions on an interface between two elastic materials. The interface is modeled by an additional third structure (thin elastic wedge of differing elastic properties) matching the bonded materials, or by introducing special boundary conditions on the crack line ahead. The main emphasis of the paper is placed on a comparison of the asymptotic expansion of the elastic solutions near the crack tip obtained for the different models. In particular, the behaviour of the stress singularity exponent and the generalized SIF are discussed. Numerical examples are presented. Received 16 August 2000; accepted for publication 26 May 2001     

Finite deformation analysis of crack tip fields in plastically compressible hardening–softening–hardening solids

Crack tip fields are calculated under plane strain small scale yielding conditions. The material is characterized by a finite strain elastic–viscoplastic constitutive relation with various hardening–softening–hardening hardness functions. Both plastically compressible and plastically incompressible solids are considered. Displacements corresponding to the isotropic linear elastic mode I crack field are prescribed on a remote boundary. The initial crack is taken to be a semi-circular notch and symmetry about the crack plane is imposed. Plastic compressibility is found to give an increased crack opening displacement for a given value of the applied loading. The plastic zone size and shape are found to depend on the plastic compressibility, but not much on whether material softening occurs near the crack tip.On the other hand, the near crack tip stress and deformation fields depend sensitively on whether or not material softening occurs. The combination of plastic compressibility and softening(or softening–hardening) has a particularly strong effect on the near crack tip stress and deformation fields.     

Dynamic crack-tip fields in rate-sensitive solids

The asymptotic stress and strain fields near the tip of a crack which propagates dynamically in a rate-sensitive solid are obtained under anti-plane shear and plane strain conditions. The problem is formulated within the context of a small-strain theory for a solid whose mechanical behavior under high strain rates is described by an elastic-viscoplastic constitutive relation. It is shown that, if the stresses are singular at the crack-tip, the viscoplastic relation is equivalent asymptotically to an elastic-non-linear viscous relation. Furthermore, for a certain range of the material parameter which characterizes the rate-sensitivity of the material, the elastic strain-rates near the propagating crack tip are shown to have the same asymptotic radial dependence near the propagating crack-tip as the inelastic strain-rates. This determines the order of the stress singularity uniquely. The governing equations for anti-plane shear and plane strain are then derived. The numerical results for the stress and strain fields are presented for anti-plane shear and plane strain. For the present model, the results suggest that under small-scale yielding conditions, there exists a minimum velocity for stable steady crack propagation. The implication that a terminal velocity for a running crack may exist is also discussed.