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.     

Analyses on micro-fracture of crack tip under large deformation conditions

In the present paper, Gurson's constitutive equation, which takes into account the development of voids, is used to study the behaviour of the material in the region near crack tip. Furthermore, the effect of void development on Young's modulus, which was not considered by Gurson, is taken into consideration. The analyses on void development, on stress distribution near crack tip, and on the variance of COD for the plane strain mode I problem are carried out with the large elastic-plastic deformation finite element method. The results are compared with those estimated from the Prandtl-Reuss constitutive equation.     

复合型韧性断裂实验和控制参数

通过对不同韧性材料在各种平面复合载荷形式下裂纹启裂阶段裂端变形、启裂位置和扩展方向的系统的宏微观实验验证及计算分析,考察了韧性断裂参数空穴扩张比的分布特征和裂纹启裂及扩展方向的关系．得到：对于不同韧性的材料,在裂端的钝化变形区域,空穴扩张比的极大值区对应干裂纹的启裂位置,裂纹启裂时钝化裂端前缘空穴扩张比的临界值不敏感于复合比的变化．而对于裂纹启裂后的扩展方向,则需根据具体材料在相应的特定区域中比较空穴扩张比参数极大值的分布特征,需经进一步的分析,从而确定裂纹的扩展方向．实验及计算结果表明,尽管复合型断裂时裂纹启裂及扩展的机理极其复杂,用于韧性材料复合型断裂的空穴扩张比参数仍能很好地预测裂端的启裂及扩展方向,可作为复合型韧性断裂过程的控制参数．     

Effect of residual surface stress on the fracture of nanoscale materials

The problem of a mode I crack in nanomaterials under a remote mechanical load is investigated. The effect of the residual surface stress on the crack surface is considered and the solutions to the crack opening displacement (COD) and the stress intensity factor (K_I) are obtained. The results show that the surface effect on the crack deformation and crack tip field are prominent at nanoscale. Moreover, COD and K_I are influenced by the residual surface stress not only on the surface near the crack tip region but also on the entire crack surface.     

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

裂尖大应变细观断裂研究

本文用反映空穴形核成长的Gurson本构方程来描述裂尖区域材料在大应变情形下的力学特性，并进一步考虑了空穴演变对材料杨氏模量的影响。文中用上述本构方程分别结合弹塑性大应变有限元方法对平面应变I型裂纹问题作了计算，分析了裂尖应力分布、裂尖形状变化和裂尖空穴演变过程，并与用Prandtl-Reuss本构方程教育处的结果作了比较。     

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.     

Mixed mode (I and II) crack tip fields in bulk metallic glasses

Stationary crack tip fields in bulk metallic glasses under mixed mode (I and II) loading are studied through detailed finite element simulations assuming plane strain, small scale yielding conditions. The influence of internal friction or pressure sensitivity on the plastic zones, notch deformation, stress and plastic strain fields is examined for different mode mixities. Under mixed mode loading, the notch deforms into a shape such that one part of its surface sharpens while the other part blunts. Increase in mode II component of loading dramatically enhances the normalized plastic zone size, lowers the stresses but significantly elevates the plastic strain levels near the notch tip. Higher internal friction reduces the peak tangential stress but increases the plastic strain and stretching near the blunted part of the notch. The simulated shear bands are straight and extend over a long distance ahead of the notch tip under mode II dominant loading. The possible variations of fracture toughness with mode mixity corresponding to failure by brittle micro-cracking and ductile shear banding are predicted employing two simple fracture criteria. The salient results from finite element simulations are validated by comparison with those from mixed mode (I and II) fracture experiments on a Zr-based bulk metallic glass.     

Mode I and mode II crack tip asymptotic fields with strain gradient effects

The strain gradient effect becomes significant when the size of fracture process zone around a crack tip is comparable to the intrinsic material lengthl, typically of the order of microns. Using the new strain gradient deformation theory given by Chen and Wang, the asymptotic fields near a crack tip in an elastic-plastic material with strain gradient effects are investigated. It is established that the dominant strain field is irrotational. For mode I plane stress crack tip asymptotic field, the stress asymptotic field and the couple stress asymptotic field can not exist simultaneously. In the stress dominated asymptotic field, the angular distributions of stresses are consistent with the classical plane stress HRR field; In the couple stress dominated asymptotic field, the angular distributions of couple stresses are consistent with that obtained by Huang et al. For mode II plane stress and plane strain crack tip asymptotic fields, only the stress-dominated asymptotic fields exist. The couple stress asymptotic field is less singular than the stress asymptotic fields. The stress asymptotic fields are the same as mode II plane stress and plane strain HRR fields, respectively. The increase in stresses is not observed in strain gradient plasticity for mode I and mode II, because the present theory is based only on the rotational gradient of deformation and the crack tip asymptotic fields are irrotational and dominated by the stretching gradient. The project supported by the National Natural Science Foundation of China (19704100), National Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20), CAS K.C. Wong Post-doctoral Research Award Fund and Post-doctoral Science Fund of China     

Intergranular strain evolution near fatigue crack tips in polycrystalline metals

The deformation field near a steady fatigue crack includes a plastic zone in front of the crack tip and a plastic wake behind it, and the magnitude, distribution, and history of the residual strain along the crack path depend on the stress multiaxiality, material properties, and history of stress intensity factor and crack growth rate. An in situ, full-field, non-destructive measurement of lattice strain (which relies on the intergranular interactions of the inhomogeneous deformation fields in neighboring grains) by neutron diffraction techniques has been performed for the fatigue test of a Ni-based superalloy compact tension specimen. These microscopic grain level measurements provided unprecedented information on the fatigue growth mechanisms. A two-scale model is developed to predict the lattice strain evolution near fatigue crack tips in polycrystalline materials. An irreversible, hysteretic cohesive interface model is adopted to simulate a steady fatigue crack, which allows us to generate the stress/strain distribution and history near the fatigue crack tip. The continuum deformation history is used as inputs for the micromechanical analysis of lattice strain evolution using the slip-based crystal plasticity model, thus making a mechanistic connection between macro- and micro-strains. Predictions from perfect grain-boundary simulations exhibit the same lattice strain distributions as in neutron diffraction measurements, except for discrepancies near the crack tip within about one-tenth of the plastic zone size. By considering the intergranular damage, which leads to vanishing intergranular strains as damage proceeds, we find a significantly improved agreement between predicted and measured lattice strains inside the fatigue process zone. Consequently, the intergranular damage near fatigue crack tip is concluded to be responsible for fatigue crack growth.     

Asymptotic character of mixed mode in plane deformation of crack in rubber-like material

The asymptotic stress and strain distribution near a crack tip in rubber-like materials is determined by finite element for in-plane mixed mode loading. For large strain, the crack tip field is always in a state of uniaxial tension. The shear load affects only the orientation of the deformed near tip field in the space. A good agreement is obtained between the theoretical and numerical results.     

Study on fracture behavior of ferroelectric ceramics under combined electromechanical loading by using a moiré interferometry technique

This paper discusses an in situ observation of fracture behavior around a crack tip in ferroelectric ceramics under combined electromechanical loading by use of a moiré interferometry technique. The deformation field induced by the electric field and the stress concentration near the crack tip in three-points bending experiments was measured. By analysis of the moiré images it is found that under a constant mechanical load, the electric field almost has no effect on the crack extension in the case that the directions of the poling, electric field and crack extension are perpendicular to each other. When the poling direction is parallel to the crack extension direction and perpendicular to the electric field, the strain decreases faster than that calculated by FEM with and without electrical loading as one goes away from the crack tip. In addition, as the electric field intensity increases, the strain near the crack tip increases, and the strain concentration becomes more significant. The project supported by the National Natural Science Foundation of China (10132010, 10025209, 10232023)     

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)     

LC4—M材料复合型韧断主要影响因素的分析

通过分析LC4－M铝合金材料在不同复合比载荷下的断裂试验结果,参照常规断裂现象,修正了一般断裂试验中认定裂纹启裂方向的方法,结合不同复合比下裂尖附近应力三维、主应力方向的计算分析,得到：在裂端的钝化变形区域、应力三维度的极大值处,对应于裂纹的启裂位置,即使在高韧性材料中发生剪切断裂的情况下也是如此;裂纹的启裂方向在拉伸断裂时与启裂点最大拉应力方向有关,在剪切断裂时启裂点最大剪应力方向有关,引起两种形式断裂的主要因数和破坏机理有很大不同。     

Stress and energy density fields near a blunted crack front in power hardening material

Using Jaumann and Dienes rates of Euler stress in elastic-plastic constitutive equations of finite deformation, plane strain finite element analysis for a compact tension specimen with a blunted crack front is made. The Euler stress, Kirchhoff stress and volume strain energy density near a blunted crack tip are computed. Constitutive relations with different deformation rates affect the the near crack tip solution in a region within an order of magnitude of the crack opening displacement. The results differed from the corresponding solution of deformation plasticity (or nonlinear elasticity) with increasing deformation. They are smaller in a local region of about 2 to 10 times of the crack opening distance.The volume energy density near the crack tip is computed, the stationary values of which determine the locations of extensive yielding and possible sites of crack initiation. It remained nearly constant with increasing deformation. Such a character tends to support the volume energy density criterion as a means for quantifying the ductile fracture behavior of metals.     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.     

Micro-crack tip fracture of commercial grade aluminum under mixed mode loading

In situ tensile tests were made in a scanning electron microscope (SEM) to investigate the deformation and micro-fracture in the immediate vicinity of a micro-crack tip in commercial pure aluminum with large-size crystal. Examined are the slip line field, stress intensity factor, strain energy density factor and crack tip opening displacement (CTOD) for mixed mode loading. Blunting and sharpening effects are observed. The latter is controlled by localized slip while the former by uniformed slip of the operating slip system with the highest crack tip Schmid factor. The operating slip system depends on the crystallographic orientation of crystal containing micro-cracks.The damage and fracture take place in the blunted region and depend on the coarsening and spacing of uniformed slip lines. The mixed mode micro-crack propagates along the direction where the voids grow and coalesce into the micro-crack. The direction also depends on the orientation of the applied loading. This suggests that the formation of macro-fracture mechanics could be applied. In particular, the minimum strain energy density criterion is suitable for determining the direction of micro-crack instability in the mixed mode. The in situ data were used to yield a nearly constant critical, minimum strain energy density factor for onset of micro-cracking.     

分子动力学模拟含不同初始缺陷单晶镍的断裂行为与应力演化

断裂是一个跨尺度复杂的物理过程,对宏观尺度的断裂行为已有深入的研究和发展,然而对微观尺度的断裂行为及断裂过程中应力场的变化缺乏深入的理解。本文通过分子动力学模拟,研究了具有不同初始缺陷(尖锐裂纹、钝裂纹和孔洞)的单晶镍的断裂行为和应力分布特征。结果表明,不同的初始缺陷导致了不同的断裂机制、断裂强度和抗断裂性能。含初始孔洞的单晶镍样品有最高的断裂强度和最强的抗断裂性能,这与孔洞扩展过程中堆积层错的形成密切相关。其次是含初始钝裂纹的样品,在裂纹扩展过程中出现由[100]超位错发射引起的裂尖钝化;含尖锐裂纹的样品表现为脆性断裂,裂尖原子没有出现微结构的变化,其强度和抗断裂性能最低。此外,不同的初始缺陷也会导致断裂过程中应力分布的变化,对含有尖锐裂纹的脆性断裂试样,高应力(拉伸应力、平均应力和米塞斯应力)总是出现在扩展裂纹的裂尖。而对于含有钝裂纹或孔洞的韧性断裂试样,高应力不仅分布在裂尖,也分布在位错发射和堆积层错形成的区域,在裂纹/孔洞扩展之前,应力随着加载时间的增加而迅速增加,而一旦裂纹或孔洞开始扩展,应力增加非常缓慢或几乎不增加,但拉伸应力值始终大于平均应力和米塞斯应力值。这表明,在I型...     

The singular elastostatic field due to a crack in rubberlike materials

Within the framework of finite-strain elastostatics an asymptotic analysis is carried out in order to calculate the singular field near the crack tip in a slab under conditions of plane deformation. A class of Ogden-Ball hyperelastic rubberlike materials and general loading conditions ensuring vanishing tractions on the crack faces near the crack tip are considered. It is shown that the singular deformation field near the crack tip can be specified by applying a rigid body rotation with a subsequent parallel translation to a so-called canonical field. The adjective canonical is adopted here to denote the field with symmetrically opening crack faces, just resembling the displacement field of the symmetric mode in linear elastic fracture mechanics. No analogy with the antisymmetric mode is possible, and the crack equilibrium criterion requires only one stress intensity factor to be determined.     

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).