Analytic solutions to crack tip plastic zone under various loading conditions

Based on stress field equations and Hill yield criterion, the crack tip plastic zone is determined for orthotropic materials and isotropic materials under small-scale yielding condition. An analytical solution to calculating the crack tip plastic zone in plane stress states is presented. The shape and size of the plastic zone are analyzed under different loading conditions. The obtained results show that the crack tip plastic zones present “butterfly-like” shapes, and the elastic–plastic boundary is smooth. The size of the plastic zone for orthotropic composites is less at the crack tip for various loading conditions, compared with the case of isotropic materials. Crack inclination angle and loading conditions affect greatly the size and shape of crack tip plastic zone. The mode I crack has a crucial effect on the plastic zone for mixed mode case in plane stress state. The plastic zone for pure mode I crack and pure mode II crack have a symmetrical distribution to the initial crack plane.     

考虑应力松驰复合型裂纹顶端塑性区分析

Ｓｈｉｈ［１］应用奇异单元，获得了不考虑应力松驰小范围屈服条件下复合型裂纹尖端塑性区形状。Ｚ．Ｚ．Ｚｕ等［２］采用Ｒｉｃｅ［５］给出的裂纹尖端应力关系式，利用有限元分析获得了不考虑应力松驰下复合型裂纹尖端塑性区，本文基于静力学中内力与外力平衡条件，用线弹性的全场解代替局部解，给出了考虑应力松驰下复合型裂纹尖端塑性区边界方程，获得了考虑应力松驰下的任意方向的塑性区尺寸及塑性区形状     

Limiting state of an elastoplastic orthotropic plate with a periodic system of collinear cracks

A modified Dugdale model is used to study the fracture of an orthotropic elastoplastic plate with a periodic system of rectilinear cracks. The material of the plate obeys a general yield criterion. The general form of solution is obtained in terms of Kolosov-Muskhelishvili potentials. The size of the plastic zone is expressed in terms of the external load and geometrical parameters. The equations for the determination of the stresses in the plastic zone and the crack opening displacement are derived. The effect of anisotropy on the formation of the plastic zones at the crack tip is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 80–88, May 2007.     

Simulation of edge dislocation emission from a crack under in-plane shear

Considered is the tandem emission of dislocations and dislocation dipoles from a crack under in-plane shear in one slip system as well as multiple slip systems. Effective stress intensity factors are determined by considering zones of local distortion similar to that in macro-plasticity. The dislocation free zone (DFZ) is also obtained which is analogous to the core region in fracture mechanics. Studied are effects of dislocation emission or development of plastic zone in front of the crack tip on the potential crack propagation based on the strain energy density factor criterion.     

Analysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone models

Crack initiation and crack growth resistance in elastic plastic materials, dominated by crack-tip plasticity are analyzed with the crack modeled as a cohesive zone. Two different types (exponential and bilinear) of cohesive zone models (CZMs) have been used to represent the mechanical behavior of the cohesive zones. In this work, it is suggested that different forms of CZMs (e.g., exponential, bilinear) are the manifestations of different micromechanisms-based inelastic processes that participate in dissipating energy during the fracture process and each form is specific to each material system. It is postulated that the total energy release rate comprises the plastic dissipation rate in the bounding material and the separation energy rate within the fracture process zone, the latter is determined by CZMs. The total energy release rate then becomes a function of the material properties (e.g., yield strength, strain hardening exponent) and cohesive properties of the fracture process zone (e.g., cohesive strength and cohesive energy), and the form of cohesive zone model (CZM) that determines the rate of energy dissipation in the forward and wake regions of the crack. The effects of material parameters, cohesive zone parameters as well as the form/shape of CZMs in predicting the crack growth resistance and the size of plastic zone (SPZ) surrounding the crack tip are systematically examined. It is found that in addition to the cohesive strength and cohesive energy, the form (shape) of the traction–separation law of CZM plays a very critical role in determining the crack growth resistance (R-curve) of a given material. It is further observed that the shape of the CZM corresponds to inelastic processes active in the forward and wake regions of the crack, and has a profound influence on the R-curve and SPZ.     

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.     

Initial kinking of an interface crack between two elastic media

The initial kinking of a thin fracture process zone near the crack tip under plane strain is studied using the Wiener-Hopf method. The crack is located at the interface between dissimilar elastic media. The fracture process zone is modeled by a straight line of normal displacement discontinuity emerging from the crack tip at an angle to the interface. The angle between the process zone and the interface is determined from the condition of strain energy maximum in the process zone. The dependences of the length and angle of the process zone on the external load and other parameters of the problem are studied. The results are compared with theoretical and experimental data obtained by other researchers __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 28–41, October 2007.     

Mode II dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

An asymptotic solution is given for Mode II dynamic fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic—perfectly-plastic solid (plane strain). It is shown that, like for Modes I and III (Gao and Nemat-Nasser, 1983), the complete dynamic solution for Mode II predicts a logarithmic singularity for the strain field, but unlike for those modes which involve no elastic unloading, the pure Mode II solution includes two elastic sectors next to the stress-free crack surfaces. This is in contradiction to the quasi-static solution which predicts a small central plastic zone, followed by two large elastic zones, and then two very small plastic zones adjacent to the stress-free crack faces. The stress field for the complete dynamic solution varies throughout the entire crack tip neighborhood, admitting finite jumps at two shock fronts within the central plastic sector. This dynamic stress field is consistent with that of the stationary crack solution, and indeed reduces to it as the crack growth speed becomes zero.     

Mode I Fracture at Spot Welds in Dual-Phase Steel: An Application of Reverse Digital Image Correlation

Strain fields in 600 grade dual-phase steel V-notch tensile specimens, both with and without a spot weld, were measured after mode I fracture initiation. Starting with the final image of a fully developed crack, a novel reverse digital image correlation (DIC) analysis was used to determine the path that the crack followed at the onset of fracture as well as the crack tip deformation field. This gave the pixel coordinates of grid points on both sides (i.e. fracture surfaces) of the crack path in the undeformed image. Strain fields that develop in the base material regions surrounding the two fracture surfaces were subsequently measured with forward DIC analysis. Steady state values of the crack tip opening displacement (CTOD) and crack tip opening angle (CTOA), which are important fracture parameters, were measured for the base DP600 metal. Notch tip opening displacement (NTOD) and notch tip opening angle were also measured. It was found that steady state values of the CTOD and CTOA are reached within 2 mm or so of crack growth following completion of the flat-to-slant transition of the fracture surface and stabilization of the crack tunneling effect.     

Fatigue fracture model for thin isotropic plates with cracks in axial loading

Conclusion We have constructed a model of the growth of a fatigue crack in a thin, isotropic plate, taking the two-stage evolution of the fracture process into account. The model is based on concepts of the mechanics of a continuous defective state and on a schematic representation of the neighborhood of the tip of a fatigue crack as a plastic zone moving together with the crack. The model takes into account the influence of the cumulative defective state (damage level) along the crack propagation front on the speed of propagation.We have formulated solutions for the cases when the length of the plastic zone is constant and when it varies during the growth of fatigue cracks. We have established the fact that the plastic zone at the crack tip tends to disrupt the stability of the motion immediately at the time of inception or opening of the crack. The speed of crack propagation decreases as the plastic zone grows in size.We have shown that the problem of estimating the kinetics of fatigue cracks in thin plates can be reduced to calculating the growth rate as a function of the peak-to-peak amplitude of the stress intensity factor while preserving the structure of the governing equations of the model. We have also shown that the concept of a plastic zone of constant length induces a power-law dependence of the crack rate on K, the power exponent varying from 2 to 10–12. The Dugdale model gives a square-law dependence of the crack rate on K, which for the most part is applicable to plastic materials.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 7, pp. 53–63, July, 1994.     

Study of the plastic zone near a crack in an anisotropic body

The plastic zone at a crack tip in a finite anisotropic body is studied. A boundary-value problem is formulated in terms of the components of the covariant displacement vector for small strains. Particular attention is given to the case of plain strain. In this case, a numerical solution is found for a long rectangular body with a central crack under tension. As a result, conditions for the occurrence and development of a plastic zone at the crack tip are established. A plastic zone on the lateral surface of the body is discovered. How both zones extend and coalesce is elucidated. The effect of anisotropy on the occurrence of a plastic zone is evaluated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 29–44, July 2006.     

Two-parameter failure criterion for elastoplastic bodies with mode I cracks

The generalized Dugdale crack model is used to formulate two-parameter failure criteria for the cases of quasibrittle state and developed plastic zones at a mode I crack tip. The failure criteria relate the fracture strength characteristics and the stress mode at the crack tip through the plastic constraint factor. The critical state of bodies with cracks under uni-and biaxial loading is analyzed in the cases of plane stress and plane strain using the Tresca and von Mises yield criteria. A small-scale yield criterion, which is an analytic relation between the critical stress intensity factor and T-stresses, is established __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 47–57, July 2007.     

裂尖曲率对裂纹前缘塑性区的影响

考虑尖端为圆弧形的钝头裂纹模型，在外围取线弹性无裂纹体的解，应用线场分析方法。形成一套估计钝头裂纹前缘塑性区尺寸的方法。对含径向裂纹和圆弧形裂尖的圆盘受均匀张力作用情况，给出了塑性区的裂纹前缘尺寸与裂纹尖端曲率的关系。得到的结论是，塑性区的裂纹前缘尺寸与裂纹尖端曲率有关；对于给定的塑性区的裂纹前缘尺寸，载荷反比于外缘尺寸的平方。前一结论说明了塑性区的裂前尺寸作为裂纹失稳扩展判断的局限性；后一结论说明了裂纹体强度失效的尺寸效应规律：抗断强度与总体线尺寸的平方成反比。     

煤层倾角与覆岩变形破裂分带

地下采煤中三带(冒落带、裂隙带、弯曲带)高度的判断是预测采空引起地表变形破坏程度的依据。在诸多的影响因素中, 煤层倾角是控制“三带”高度的最主要因素。建立在实测统计基础上的经验范围值用于“三带”高度的判断带有一定的人为性。急倾斜煤层中冒落带和裂隙带高度随煤层倾角变化的规律已被褐示[7].通过弹塑性岩石材料的非线性有限元模拟, 本文提出了利用应力重分布图判断中、缓倾角煤层采空区覆岩“三带”高度的方法, 并应用于实际工程。     

On the direction of development of a thin fracture process zone at the tip of an interfacial crack between dissimilar media

The Wiener-Hopf method is used to study, under the conditions of plane strain, the direction of development of a thin fracture process zone at the tip of an interfacial crack in a piecewise homogeneous isotropic elastoplastic body. The zone is modeled by a straight line of tangential displacement discontinuity that emerges from the crack tip at an angle to the interface. The dependences of the zone length and the angle on the load and other parameters of the problem are investigated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 14–23, February 2006.     

Initial kinking of an interface crack between two elastic media under tension and shear

The initial kinking of a thin fracture process zone near the tip of an interface crack between two elastic media under plane strain is studied using the Wiener–Hopf method. The zone is modeled by the discontinuity plane of the normal displacement. This plane is assumed to emerge from the crack tip at an angle to the interface. The angle between the fracture process zone and the interface is determined from the condition that the potential energy is maximum in the zone. The dependence of the length of the zone and its angle on the external load and other parameters is analyzed in the cases of biaxial tension and pure shear. The results obtained are compared with theoretical and experimental data reported by other researchers     

复合型裂纹断裂的新准则

以Ⅰ-Ⅱ复合型裂纹为研究对象,对裂纹尖端的塑性区分布规律进行了理论分析。引入两组评价裂纹尖端应力场对裂纹扩展影响的参数,考虑裂纹尖端存在的局部塑性变形,并采纳如下两个假设,(1)裂纹沿最短路径穿过塑性区向弹性区扩展,(2)当在扩展方向上的弹塑性边界极半径r大于其临界极半径rC时,裂纹开始扩展。在此基础上,导出了新的复合型裂纹断裂准则,并与现有部分断裂准则及实验结果进行了对比。结果表明：新建立的复合裂纹断裂准则与实验结果吻合程度非常高。最后,阐明了以单一KIC或KIIC建立的复合裂纹扩展准则的局限性以及考虑裂纹尖端当前应力场的必要性。     

Refined Dugdale plastic zones of an external circular crack

The refined Dugdale-type plastic zones ahead of an external circular crack, subjected to a uniform displacement at infinity, are evaluated both analytically and numerically. The analytical method utilizes potential theory in classical linear elasticity with emphasis on the contrast from the internal crack problem. A closed-form solution to the mixed boundary problem is obtained to predict the length of the plastic zone for a Tresca yield condition. The analytical solution is also used to benchmark the results obtained from the numerical method, which show good agreement. Through an iterative scheme, the numerical technique is able to estimate the size of crack tip plasticity zone, which is governed by the non-linear von Mises criterion. The relationships between the applied displacement and the length of the plastic zone are compared for the different yielding conditions. Computational modeling has demonstrated that the plastic constraint effect based on the true yield condition can significantly influence the load-bearing capacity. It is also discovered from the comparative study that the stress components predicted by the three different yield conditions may differ notably; however, the stress triaxiality in the ligament region has only small deviations. The proposed study may find applications in predicting the plastic flow in a circumferentially notched round bars under tension.     

过载作用下聚乙烯疲劳裂纹扩展行为研究

论文针对中密度聚乙烯材料(MDPE)，采用平板试样进行了I型疲劳裂纹扩展和单次过载下裂纹扩展试验．发现与金属材料类似，单次拉伸过载对聚乙烯(PE)的疲劳裂纹扩展有明显的迟滞作用，降低了裂纹扩展速率．试验还通过变载荷刻线法获取疲劳裂纹扩展前缘的实际形貌和变化规律，对常规变载荷刻线方法进行了调整和验证，其修正方法对高分子材料的疲劳裂纹扩展前缘刻线具有较好的效果．通过观察发现含楔形塑性区的裂尖钝化是裂纹迟滞的主要原因．过载引入的塑性区内残余应力对裂纹迟滞也起了重要作用．论文利用Dugdale模型计算了塑性区尺寸，使用基于残余应力的Wheeler模型对过载迟滞进行了很好的拟合．