The range of applicability of the small-scale yielding approach in theoretical investigations of yield and fracture at a crack tip

Theoretical analysis of the processes of plastic deformation and fracture in the vicinity of a sharp crack tip becomes easier if plasticity is confined to the immediate vicinity of the crack tip, for linear elastic stress intensity factors can be used to give the surrounding elastic field, and it is possible to employ a boundary layer formulation to determine the elastic-plastic field in the immediate vicinity of the crack tip. Consideration is given to the range of applicability of the small-scale yielding approach using a simple model in which slip is confined to the crack plane. A range of stress-displacement laws is considered, the object being to assess the effect of a material's flow characteristics on the range of applicability. The conclusions are discussed with respect to plane strain crack propagation in mild steel.     

On the fracture of constrained layers

The problem of a crack embedded in a layer sandwiched between two elastic adherends is analysed accounting for the influence material property mismatch on the crack tip plastic deformation, which is contained in the layer. The cohesive crack model developed by Dugdale and Barrenblatt is adopted to model the strip yielding behaviour in a constrained layer. It is found that, due to the constraint imparted by elastic adherends with higher moduli, the near tip plastic deformation exhibits a sharp transition (plastic zone grows faster than the square of stress intensity factor) from small scale to large scale yielding. Because the region of singularity dominance for a crack embedded in a layer is generally much smaller than the layer thickness when the layer has a modulus much lower than the adherends, the prevailing failure mode of most bonded joints should be under large scale yielding conditions. A model based on energy balance is proposed to determine the fracture energy of bonded joints under such condition, taking into account of the plastic dissipation in the constrained layer. Comparison with experimental results demonstrates that the theory correctly predicts the dependence of fracture toughness on layer thickness as observed in experiments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of the optical method of caustics for dynamic crack propagation

In the interpretation of experimental data on dynamic crack propagation in solids obtained by means of the optical method of caustics, it has been customary to neglect the effect of material inertia on the stress distribution in the vicinity of the crack tip. In this paper, the elastodynamic crack tip stress field is used to establish the exact equations of the caustic envelope formed by the reflection of light rays from the surface of a planar solid near the tip of a propagating crack. These equations involve the instantaneous crack tip speed, the material parameters and the instantaneous dynamic stress intensity factor, and they can be used to determine the stress intensity factor for given material parameters and crack tip speed. The influence of inertial effects on stress intensity factor measurements for system parameters typical of experiments with PMMA specimens is considered. It is found that the stress intensity factor values inferred through a dynamic analysis may differ by as much as 30–40% from values based on a quasi-static analysis.     

An asymptotic approach applied to a longitudinal crack in an adhesive layer

The problem of a crack within an adhesive layer which is bonded to two linear elastic half-planes under tensile loading is studied. Two cases are considered. One in which the adhesive is linear elastic and, the second in which it is taken to be elasto-plastic. For the linear elastic layer, the half-planes (adherends) are assumed to be both similar and dissimilar. When the adhesive is considerably more compliant than the adherends, a method of inner and outer asymptotic expansions is employed to determine a relationship between the corresponding stress intensity factors. Expansions are determined in three regions and matched. The inner expansion relates to a region whose distance from the crack tip is much less than the adhesive thickness. The intermediate expansion relates to a region whose size is governed by the decay length of the stress in that part of the adhesive in which its compliancy is significant. The outer expansion relates to a region whose distance from the crack tip is much less than the crack length, for example, but much greater than the adhesive thickness. This method may be employed to determine all field quantities in terms of the outer stress intensity factor. For a layer which is considerably stiffer than the adherends, a similar strategy for solving the problem is sketched. In addition, for dissimilar adherends, energy considerations are employed to verify the relationship between the inner and outer stress intensity factors. It is seen that the two expressions for the stress intensity factors are identical. The problem of oscillatory stress and displacement behavior is addressed.Next, the problem of a sufficiently compliant elasto-plastic adhesive between dissimilar adherends is examined. Matched asymptotic expansions are employed to determine the plastic zone size, as well as the crack tip opening displacement. Small scale yielding is assumed. A Dugdale-Barenblatt type model is employed with the elasticity of the layer accounted for. The yield stress is taken to be constant throughout the plastic zone.     

Fracture toughness of layered structures: Embrittlement due to confinement of plasticity

The fracture toughness of a layered composite material is analyzed employing a combined two dimensional dislocation dynamics (DD)-cohesive zone (CZ) model. The fracture mechanism of an elastic-plastic (ductile) material sandwiched within purely elastic layers approaches ideally brittle behaviour with decreasing layer thickness. We investigate the influence of different constitutive parameters concerning dislocation plasticity as well as the effect of cohesive strength of the ductile material on the scaling of fracture toughness with layer thickness. For a constant layer thickness, the results of the numerical model are consistent with the expectation that fracture toughness decreases with increasing yield strength, but increases with the cohesive strength of the material. The scaling behaviour of the fracture toughness with layer thickness depends on these material parameters, but also on the dislocation microstructure in the vicinity of the crack tip. Strain localization due to easy dislocation generation right at the crack tip improves toughness in thin layers and leads to a jump-like increase of fracture toughness with layer thickness. However, the fracture toughness for films that are thick enough to exhibit bulk behaviour proves to be higher when the distribution of dislocations is more homogeneous, because in this case the crack grows in a stable fashion over some distance.     

Fracture toughness of laminated steels

Lamination occurs spontaneously in the transverse direction in many commercially available steel plates, if the transverse stresses are sufficiently high. Previous investigations have indicated that lamination is often accompanied by an improvement in the fracture toughness of the plate material. In the vicinity of the crack tip, the stress concentration is so large that the bond between adjacent layers will break before crack propagation sets in. If these layers are sufficiently thin, a state of plane stress is approached near the crack tip. In the present study, the influence of layer thickness and bond strength on the fracture toughness is investigated. It is shown that lamination does improve the toughness, if certain conditions in these variables are fulfilled. This offers a possibility to build up structures with high yield stress and high fracture toughness at the same time, since the permissible defect size to prevent unstable crack growth need not be uncomfortably small.     

Solute distribution in crack tips under mixed mode I/II conditions

Summary Concentration solutions of crack problems under steady state conditions are expressed in terms of the Westergaard stress function. The problem of a central crack in an infinite plate subjected to a biaxial stress field at any angle of inclination with respect to the crack axis is considered in detail. The concentration distribution in the vicinity of the crack tip is obtained and is expressed in terms of the opening-mode and sliding-mode stress intensity factors. Constant terms, usually omitted, are incorporated into the concentration solution. The crack growth criterion based on the maximum concentration of diffusing species in front of the crack tip is reformulated by incorporating the constant terms of the concentration solution. It is shown that the omission of the constant terms may result in a significant error in the prediction of the critical quantities for crack growth.     

铁素体管线钢的分层裂纹及其对断裂的影响

通过对针状铁素体管线钢缺口根部三维应力状态的有限元分析和不同形式的断裂实验,研究了管线钢分层裂纹产生的条件及其对断裂性能的影响.结果表明裂纹或缺口根部的三维应力状态是产生分层裂纹的必要条件,材料的强度分布影响分层裂纹的形式和方向.分层裂纹均为主裂纹扩展前材料中的弱界面在垂直该弱界面的拉应力作用下产生的,其数量和方向受裂纹端部三维应力场和材料的强度分布状态控制.分层裂纹面上的应力为零,分层裂纹有一定的间距.在断裂过程中产生的分层裂纹使裂纹或缺口根部的构形发生改变,从而对裂尖的应力状态和材料的断裂性能产生巨大的影响.穿透裂纹体的分层裂纹使其有效厚度减小,表面裂纹体的分层裂纹与裂纹扩展方向垂直.在断裂过程中产生分层裂纹需要消耗更多的能量、降低裂端三维应力约束、有效厚度降低或裂尖钝化.这些因素均使断裂扩展更加困难,而使材料韧性得到提高.     

Numerical simulations of dynamic crack growth along an interface

Dynamic crack growth is analyzed numerically for a plane strain bimaterial block with an initial central crack. The material on each side of the bond line is characterized by an isotropic hyperelastic constitutive relation. A cohesive surface constitutive relation is also specified that relates the tractions and displacement jumps across the bond line and that allows for the creation of new free surface. The resistance to crack initiation and the crack speed history are predicted without invoking any ad hoc failure criterion. Full finite strain transient analyses are carried out, with two types of loading considered; tensile loading on one side of the specimen and crack face loading. The crack speed history and the evolution of the crack tip stress state are investigated for parameters characterizing a PMMA/Al bimaterial. Additionally, the separate effects of elastic modulus mismatch and elastic wave speed mismatch on interface crack growth are explored for various PMMA-artificial material combinations. The mode mixity of the near tip fields is found to increase with increasing crack speed and in some cases large scale contact occurs in the vicinity of the crack tip. Crack speeds that exceed the smaller of the two Rayleigh wave speeds are also found.     

Stress in the vicinity of a griffith crack at the interface of a layer bonded to a half plane—an approximate method

Approximations to the stress field in the vicinity of a Griffith crack located at the interface of a layer bonded to a dissimilar half plane are determined. A systematic use of Fourier transforms reduces the problem to that of solving a set of simultaneous dual integral equations with trigonometric kernels and weighting functions. This latter problem is reduced to the solution of an uncoupled pair of singular integral equations. An approximate technique using Legendre polynomial expansions is discussed. The analysis shows that when a constant pressure is applied to the faces of the crack, the stress components have the distinctive oscillatory singularities at the crack tip. Expressions up to the order of $\text{h}{}^{\mathrm{?4}}$, where $\text{h}$ is the thickness of the layer and is much greater than 1, are derived for the stress components.     

Measurement of crack closure points during cyclic loading by the strain interference method

Effective stress intensity factor range ΔK_eff and J integral range ΔJ, which are determined using the crack tip opening or closing point, have been considered as the reasonable parameters controlling crack propagation behavior. Therefore, the development of a simple and practical measurement method of crack tip opening and closing stress is considered to be important for the evaluation of crack propagation. In this study, for a center-cracked plate in the elastic-plastic condition, it is shown that the crack tip opening and closing points can be easily determined with the relationship between the nominal cyclic stresses and the local strains measured in the vicinity of the crack center. Then, this measurement method is compared with the results obtained by the finite element method calculation.     

Electric Current-Induced Stresses at the Crack Tip in Conductors

The electromagnetic and thermal effects on the stress distribution around the crack tip in conducting materials due to electric current are investigated. Emphases are placed on quantifying the crack growth behavior affected by the interplay between these effects. A two-dimensional finite element analysis is conducted to examine the coupled problems. The results show that the compressive stress state around the crack tip plays a decisive role in preventing the crack from further growth. The resulting normal stress in front of the crack tip caused by the Joule heat generation tends to suppress the crack growth, while the stress induced by the electromagnetic forces provides a tensile normal stress with smaller magnitude in the vicinity of the crack tip, hence promotes the crack growth. Favorable agreements between numerical analysis results and existing experimental results are achieved. By utilizing these phenomena, the electric current may be used to actively control the damage propagation, hence catastrophic failure can be prevented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Crack redirection with thermal secondary loading

In the current paper crack redirection due to a strategic placing of a heat source in the vicinity of a crack tip is studied. Analysis suggests that for PMMA and considered temperature range the only factor responsible for the deviation of crack trajectory is thermal stress. The simulation of crack growth in PMMA under external tension and secondary heat loading shows that a moving heat source in the vicinity of a crack tip can serve as a pointer for the crack trajectory. In highly conductive materials, redirection can be possibly effected with low-power thermal dipoles.     

The stress distribution around a crack perpendicular to an interface between materials

The plane strain problem of a crack terminating perpendicular to a planar interface between two isotropic half spaces with different elastic constants is solved to obtain the distribution of stress in the vicinity of the crack tip. The relative elastic constants are shown to strongly affect the relative magnitudes of the various stress components as well as their radial drop off with distance from the crack tip. The implications of the results with regard to failure modes in composite materials are discussed.     

Two-parameter characterization of the near-tip stress fields for a bi-material elastic-plastic interface crack

A particular case of interface cracks is considered. The materials at each side of the interface are assumed to have different yield strength and plastic strain hardening exponent, while elastic properties are identical. The problem is considered to be a relevant idealization of a crack at the fusion line in a weldment. A systematic investigation of the mismatch effect in this bi-material plane strain mode I dominating interface crack has been performed by finite strain finite element analyses. Results for loading causing small scale yielding at the crack tip are described. It is concluded that the near-tip stress field in the forward sector can be separated, at least approximately, into two parts. The first part is characterized by the homogeneous small scale yielding field controlled by J for one of the interface materials, the reference material. The second part which influences the absolute value of stresses at the crack tip and measures the deviation of the fields from the first part can be characterized by a mismatch constraint parameter M. Results have indicated that the second part is a very weak function of distance from the crack tip in the forward sector, and the angular distribution of the second part is only a function of the plastic hardening property of the reference material.     

Analytical and numerical study of cohesive zones for a crack in an adhesive layer between identical isotropic materials

A crack in a thin adhesive elastic-perfectly plastic layer between two identical isotropic elastic half-spaces is considered. Uniformly distributed normal stress is applied to the substrates at infinity. First, stress distribution in the cohesive zones and the J-integral values are defined numerically by the finite element method (FEM). Further, a mathematical formulation of the problem is given and its analytical solution is proposed. It is assumed that, at the crack continuations, there exist cohesive zones. The interlayer thickness is neglected since it is much smaller than the crack length. The distribution of the normal stress, which was obtained by means of the FEM, is now approximated by a piecewise-constant function and assumed to be applied at the faces of the cohesive zones. The formulated problem is solved analytically and an equation for determination of the cohesive zone lengths is derived. Also, closed expressions for the crack tip opening displacement and for the J-integral are obtained in an analytical form. These parameters are found with respect to the values of the normal stress applied at infinity. Finally, a universal approximating function, which describes the stress distribution in the cohesive zones, is constructed. This function depends on the ratio between the interlayer thickness and the crack length and on the ratio between the normal stress applied at infinity and the yield limit of the interlayer’s material. Once again, the problem is solved analytically, but this time for the stress distribution prescribed by the universal approximating function. The cohesive zone lengths, the values of the crack tip opening displacement and of the J-integral are calculated. A comparative analysis of the obtained results is carried out. A good agreement of the J-integral values calculated by means of the developed analytical models and by the associated finite element analysis is demonstrated.     

Moving cracks in layered composites

The dynamic behavior of layered composites is intimately associated with inherent flaws or cracks that are induced in the material during loading. Often, one of the dominant cracks may reach a critical size and begins to spread. Whether this process of material separation would lead to global instability or not depends on the nonhomogeneous properties of the composite structure. Because of the complexity of the physical phenomenon involving interaction of stress waves with varying material properties, a three-layered composite model is assumed with a crack moving in the center layer. The material properties of the middle layer differ from those of the surrounding material. Both in-plane extensional and out-of-plane shear loading are considered. Making use of the Galilean transformation and Fourier sine and cosine transforms, the dynamic crack tip stress fields are determined analytically while the dynamic stress intensity factors are evaluated numerically from the standard Fredholm integral equations. The intensity of the local dynamic stresses are found to either increase or decrease with the crack length to layer thickness depending on the relative magnitudes of the adjoining layer material properties. The crack speed tends to amplify the effect of material nonhomogeneity. These results are discussed in terms of the dynamic stress intensity factors and displayed graphically.     

裂纹板贴补应力分析

本文采用含裂纹无限大板特殊基本解和合力边界条件,用体积力法对含裂纹金属薄板的胶贴补强问题进行应力分析。使用一满足胶贴层位移连续条件的剪切单元,把问题转化为对裂纹板和贴片的分析。由于使用的特殊基本解精确满足裂纹面自由力边界条件,避免了对裂纹尖端附近的奇异场进行离散处理,因而可以比较精确地求出裂纹尖端附近的应力分布,同时由于单位集中力引起的裂纹尖端应力强度因子可以解析得到,因而可以较准确地反映出用应力强度因子的降低来表征的贴补效果。作为贴补计算的例子,文中计算了受拉力和剪力作用时,含中心裂纹的金属裂纹板在贴补前后裂纹尖端应力强度因子的降低,给出了贴片的厚度、弹性模量和尺寸及肢贴层厚度等对贴补效果的影响。     

Two-Parameter (J-M) Description of Crack Tip Stress-Fields for an idealized weldment in small scale yielding

The crack tip stress-field in a trimaterial finite element model has been examined. The model represents an idealised steel weldment with a crack located at the fusion line. The model was loaded with a K _I displacement field to simulate small scale yielding conditions. The effect of changing the weld metal plastic properties and the HAZ layer thickness on the crack tip stress-field was studied, keeping the material properties of the HAZ and base metal constant. The results show that the calculated J-integral remains path independent in the trimaterial model. It is confirmed that the crack tip stress-fields can be normalised by the J-integral. The mismatch constraint can be characterised by a difference field, which is independent of the normalised distance from the crack tip. The results show that changes of HAZ thickness only have a small effect on the stress-fields close to the crack tip. The hardenability of the weld metal influences on the slope of the crack tip stress distribution, but for small changes in hardenability, this effect can be neglected. The results indicate that the difference fields show some radial dependence when a homogeneous reference field is used, but the radial dependence was removed by introducing an inhomogeneous reference field. The effect of changes in the weld metal yield strength has been described with a two parameter (J-M) formulation using the inhomogeneous reference field.