Cohesive zone laws for fatigue crack growth: Numerical field projection of the micromechanical damage process in an elasto-plastic medium

Cohesive zone failure models are widely used to simulate fatigue crack propagation under cyclic loading, but the model parameters are phenomenological and are not closely tied to the underlying micromechanics of the problem. In this paper, we will inversely extract the cohesive zone laws for fatigue crack growth in an elasto-plastic ductile solid using a field projection method (FPM), which projects the equivalent tractions and separations at the cohesive crack-tip from field information outside the process zone. In our small-scale yielding model, a single row of discrete voids is deployed directly ahead of a crack in an elasto-plastic medium subjected to cyclic mode I K-field loading. Damage accumulation under cyclic loading is captured by the growth of voids within the micro-voiding zone ahead of the crack, while the evolution of the cohesive zone law representing the micro-voiding zone is inversely extracted via the FPM. We show that the field-projected cohesive zone law captures the essential micromechanisms of fatigue crack growth in the ductile medium: from loading and unloading hysteresis caused by void growth and plastic hardening, to the softening damage locus associated with crack propagation via a void by void growth mechanism. The results demonstrate the effectiveness of the FPM in obtaining a micromechanics-based cohesive zone law in-place of phenomenological models, which opens the way for a unified treatment of fatigue crack problems.     

Fatigue damage modeling in solder interconnects using a cohesive zone approach

The objective of this work is to model the fatigue damage process in a solder bump subjected to cyclic loading conditions. Fatigue damage is simulated using the cohesive zone methodology. Damage is assumed to occur at interfaces modeled through cohesive zones in the material, while the bulk material is assumed to be linear elastic. The state of damage at a cohesive zone is incorporated into the cohesive zone constitutive law by a elasticity-based damage variable. The gradual degradation of the solder material and the corresponding damage accumulation throughout the cycling process is accounted for by a damage evolution law which captures the main damage characteristics. The model prediction of the solder bump life-time is shown to be in good agreement with one of the commonly used empirical life-time prediction laws.     

Integrated numerical–experimental analysis of interfacial fatigue fracture in SnAgCu solder joints

In ball grid array (BGA) packages, solder balls are exposed to cyclic thermo-mechanical strains arising from the thermal mismatch between package components. Thermo-mechanical fatigue crack propagation in solder balls is almost always observed at the chip side of the bump/pad junction. The objective of the experimental part of this study is to characterize the bump/pad interface under fatigue loading. Fatigue specimens are prepared by reflowing Sn3.8Ag0.5Cu lead-free solder alloy on Ni/Au substrates. Obtained results show that fatigue damage evolution strongly depends on the microstructure. Applied strain and solder volume both have an influence on the fatigue damage mechanism. In the numerical part of the study, fatigue experiments are modeled using the finite element technique. A cohesive zone approach is used to predict the fatigue damage evolution in soldered connections. Crack propagation is simulated by an irreversible linear traction–separation cohesive zone law accompanied by a non-linear damage parameter. Cohesive zone elements are placed where failure is experimentally observed. Damage evolution parameters for normal and tangential interaction are scrutinized through dedicated fatigue tests in pure tensile and shear directions. The proposed cohesive zone model is quantitatively capable of describing fatigue failure in soldered joints, which can be further extended to a numerical life-time prediction tool in microelectronic packages.     

An analysis of the delamination of an environmental protection coating under cyclic heat loads

A computational analysis of a coated high temperature composite material under cyclic heat flux loading is conducted. The material system considered is a carbon–carbon (CC) composite laminate with a SiC environmental protection coating. Interface crack growth between the CC laminate and the SiC coating as well as the concurrent changes in the effective conductivity of the material system are modeled by the use of an irreversible thermo-mechanical cohesive zone model. In this model, the degradation of the cohesive strength due to cyclic loading is accounted for through a damage variable for which an evolution equation is given in terms of cohesive zone tractions and displacement jumps. Furthermore, interface failure progressively degrades the heat transfer across the interface. In the model the crack tip singularities for both stress analysis and heat transfer are removed. While the crack growth rates predicted by the simulation results can in principle be described by a Paris law approach, the strength of the proposed model is that the use of global loading parameters is in fact not necessary. Instead, material failure is described through local changes to the coupled mechanical and thermal fields.     

Cohesive zone laws for void growth — II. Numerical field projection of elasto-plastic fracture processes with vapor pressure

Modeling ductile fracture processes using Gurson-type cell elements has achieved considerable success in recent years. However, incorporating the full mechanisms of void growth and coalescence in cohesive zone laws for ductile fracture still remains an open challenge. In this work, a planar field projection method, combined with equilibrium field regularization, is used to extract crack-tip cohesive zone laws of void growth in an elastic-plastic solid. To this end, a single row of void-containing cell elements is deployed directly ahead of a crack in an elastic-plastic medium subjected to a remote K-field loading; the macroscopic behavior of each cell element is governed by the Gurson porous material relation, extended to incorporate vapor pressure effects. A thin elastic strip surrounding this fracture process zone is introduced, from which the cohesive zone variables can be extracted via the planar field projection method. We show that the material's initial porosity induces a highly convex traction-separation relationship — the cohesive traction reaches the peak almost instantaneously and decreases gradually with void growth, before succumbing to rapid softening during coalescence. The profile of this numerically extracted cohesive zone law is consistent with experimentally determined cohesive zone law in Part I for multiple micro-crazing in HIPS. In the presence of vapor pressure, both the cohesive traction and energy are dramatically lowered; the shape of the cohesive zone law, however, remains highly convex, which suggests that diffusive damage is still the governing failure mechanism.     

混凝土黏聚开裂模型若干进展

黏聚模型是用来描述混凝土断裂行为的基本模型, 首先介绍了混凝土的黏聚开裂模型的基本概念,总结了确定黏聚区的本构方程的各种方法,即直接单轴拉伸测试、J积分方法、R曲线法、柔度法和逆推法.然后介绍了黏聚模型在I型和复合型裂纹问题、疲劳断裂问题中的应用以及黏聚模型与混凝土尺寸效应的关系.最后对黏聚开裂模型与桥联模型、带状裂缝模型进行了比较和总结, 指出了该模型存在的问题, 并对其以后的发展方向提出了建议.      

A cohesive zone model for fatigue crack growth allowing for crack retardation

A damage-based cohesive model is developed for simulating crack growth due to fatigue loading. The cohesive model follows a linear damage-dependent traction–separation relation coupled with a damage evolution equation. The rate of damage evolution is characterized by three material parameters corresponding to common features of fatigue behavior captured by the model, namely, damage accumulation, crack retardation and stress threshold. Good agreement is obtained between finite element solutions using the model and fatigue test results for an aluminum alloy under different load ratios and for the overload effect on ductile 316 L steel.     

A thermodynamic method for the construction of a cohesive law from a nonlocal damage model

Several published papers deal with the possibility of replacing a damage finite element model by a combination of cohesive zones and finite elements. The focus of the paper is to show under which conditions this change of model can be done in an energy-wise manner.The objective is to build a cohesive model based on a known damage model, without making any assumption on the shape of the cohesive law. The method is characterized, on the one hand, by the use of a well-defined thermodynamic framework for the cohesive model and, on the other hand, by the idea that the main quantity which must be maintained through the change of model is the energy dissipated by the structure. An analysis of the stability criteria enables us to determine the domains of validity of the different models. Thus, we show that it is consistent to derive the cohesive law from a given nonlocal damage model because the occurrence of a discontinuity can be viewed as an alternative way to limit localization. The method is illustrated on one-dimensional examples and a numerical resolution method for the problem with a cohesive zone is presented.     

Real-time Detection and Analysis of Damage in High-performance Concrete under Cyclic Bending

In order to quantitatively evaluate the damage level in high-performance concrete (HPC) with pozzolanic minerals under constant amplitude cyclic loads, three methods for real-time damage detection are employed in the present work, i.e., dynamic modulus instrument, real-time strain collector, and digital speckle correlative method (DSCM). Six mechanical parameters at different numbers of loading cycles are real-time captured by these three methods. For a maximum applied fatigue stress equal to 70% of the static flexural strength, a cohesive crack is detected on the specimen surface by the DSCM system from 10% of concrete fatigue life. The nucleation and propagation of the cohesive crack is reflected by the change of the strain concentration zone in 2-dimensional strain fields. The experimental results show that the admixtures of Class F Fly Ash (FA) and S95 Ground Granulated Blast-furnace Slag (GGBS) in high proportions increase the strain and cohesive-crack opening displacement as well as remarkably improve the fatigue performance of HPC.     

一个高周疲劳损伤演化修正模型

将连续损伤力学应用到疲劳问题中, 得到合理的疲劳损伤演化方程,被认为是预测疲劳寿命最有效的方法之一. 在研究Lemaitre最新疲劳损伤演化方程基础上, 根据试验数据提出了一个修正的高周疲劳损伤演化方程, 该方程可以考虑应力幅、平均应力等影响因素. 以2A12-T4铝合金为例, 得到了修正模型的材料常数. 将该修正模型以UMAT子程序形式嵌入ABAQUS主程序, 计算了两种构件14种不同受载情况下的疲劳寿命, 所得计算寿命与试验结果误差均值约15%, 说明修正的疲劳损伤演化方程可以很好地计算金属构件的高周疲劳寿命.      

基于损伤-虚拟张拉裂纹模型的地下爆炸围岩破坏规律研究

地下硐室作为爆炸危险物的隐蔽贮藏空间,有潜在的内爆炸风险。为研究内爆炸作用下硐室围岩的动态响应机制,提出了一种基于岩石HJC (Holmquist-Johnson-Cook)模型和节理内聚力单元的损伤-虚拟裂纹模型。分析了模拟方法的可靠性,并在此基础上,通过多物质ALE算法对球形硐室内爆炸过程进行数值模拟,分析了围岩损伤范围和分区破坏规律。研究表明:插入内聚力单元弥补了HJC模型无法模拟低静水压力下张拉破坏的不足,且尺寸效应易于处理。模拟方法同时考虑了岩体内张拉裂纹的扩展和岩石材料的塑性损伤,能够真实地反映岩石破坏的全过程。以红砂岩为例,根据数值模拟结果,填实（耦合装药）爆炸时围岩分区破坏规律明显,破碎区比例半径为0.26 m/kg1/3、裂隙区比例半径为0.47 m/kg1/3。随着硐室尺寸的增大,空气的间隔作用可以减小爆炸荷载对围岩的损伤作用,比例半径达到0.52 m/kg1/3时,可以实现爆炸荷载的完全解耦。     

Theoretical development and experimental validation of a thermally dissipative cohesive zone model for dynamic fracture of amorphous polymers

A thermally dissipative cohesive zone model is developed for predicting the temperature increase at the tip of a crack propagating dynamically in a nominally brittle material exhibiting a cohesive-type failure such as crazing. The model assumes that fracture energy supplied to the crack tip region that is in excess of that needed for the creation of new free surfaces during crack advance is converted to heat within the cohesive zone. Bulk dissipation mechanisms, such as plasticity, are not accounted for. Several cohesive traction laws are examined, and the model is then used to make predictions of crack tip heating at various crack propagation speeds in the nominally brittle amorphous polymer PMMA, observed to fail by a crazing-type mechanism. The heating predictions are compared to experimental data where the temperature field surrounding a high speed crack in PMMA was measured. Measurements are made in real time using a multi-point high speed HgCdTe infrared radiation detector array. At the same time as temperature, simultaneous measurement of fracture energy is made by a strain gauge technique, and crack tip speed is monitored through a resistance ladder method. Material strength can be estimated through uniaxial tension tests, thus minimizing the need for parameter fitting in the stress-opening traction law. Excellent agreement between experiments and theory is found for two of the cohesive traction law temperature predictions, but only for the case where a single craze is active during the dynamic fracture of PMMA, i.e. crack tip speed up to approximately 0.2c_R. For higher speed fracture where subsurface damage becomes prominent, the line dissipation model of a cohesive zone is inadequate, and a distributed damage model is needed.     

A cohesive plastic and damage zone model for dynamic crack growth in rate-dependent materials

Mode I steady-state dynamic crack growth in rate-dependent viscoplastic solids containing damage, under small scale yielding conditions, is analyzed based on a modified cohesive zone model. A multi-scale approach is used to describe the entire non-linear zone consisting of a plastic region and a damage region, each of which has its own constitutive law. Traction in the damage region is characterized by a softening power-law, in terms of the ultimate strength, a softening index and a rate sensitivity factor. In the plastic region, the cohesive law is assumed to be both strain hardening and rate dependent. The critical crack opening displacement at the physical crack-tip controls crack growth. The governing integral equations are derived and solved by a collocation method combined with associated boundary conditions. Numerical results are presented for the traction and opening profiles along the cohesive zone, the fracture energy and lengths of the damage and non-linear zones at different crack speeds and for different material parameters. The importance of factors, such as material softening, plastic deformation, crack speed and viscosity, is identified by parametric studies. In addition, the competition of plastic flow and material damage, and its effect on crack growth, are discussed.     

Determination of effective stress range and its application on fatigue stress assessment of existing bridges

This paper presents a unified approach on determination of the effective stress range based on equivalent law of strain energy and fatigue damage model, so as to provide an efficient approach for accurately assessing effective fatigue stress of existing bridge under traffic loading. A new theoretical framework to relate variable- and constant-amplitude fatigue is proposed in this paper. Different formulation for calculating effective stress range can be derived by the proposed theory, which include the effective stress range by the root mean square, by Miner's law and a new effective stress range based on the nonlinear fatigue damage model. Comparison of the theoretical results of fatigue damage under the effective stress range of the variable-amplitude stress spectrum and experimental data of fatigue damage under realistic traffic loading has confirmed the validity of the proposed theory. As a way to relate variable-amplitude fatigue data with constant-amplitude data, the effective stress range provides the most convenient way for evaluating fatigue damage under variable-amplitude loading. The proposed theory is then applied to provide an efficient approach for accurately assessing fatigue damage of existing bridges under traffic loading, in which online strain history data measured from bridge structural health monitoring system is available. The proposed approach is applied to evaluate the effective stress range for the purpose of the fatigue analysis of a deck section of a long-span steel bridge––the Tsing Ma Bridge in Hong Kong.     

Intersonic crack propagation in homogeneous media under shear-dominated loading: theoretical analysis

The mechanics of cohesive failure under mixed-mode loading is investigated for the case of a steadily propagating subsonic and intersonic dynamic crack subjected to a follower tensile and shear distributed load. The cohesive failure model chosen in this study is rate independent but accounts for the coupling between normal and tangential damage. Special emphasis is placed here on mixed-mode cases with predominantly shear loading. The analysis shows that the size of the mixed-mode cohesive zone is smaller than that obtained in the pure shear case. The relative extent of the shear and tensile cohesive damage zones depends on the crack speed and the mode mixity. In the intersonic regime, the failure process takes place exclusively in shear, even under remote mixed-mode loading conditions.     

粘结单元在模拟FRP层合板低速冲击响应中的应用

纤维增强树脂基复合材料层合板（fibre reinforced plastic composites，FRP）在航空、航天、交通、造船等诸多工程中得到了日益广泛的应用，而其在冲击载荷下的响应和破坏特别是分层一直为学术界所关注。本文中对FRP层合板在冲击载荷下的响应和破坏进行数值模拟，并通过引入粘结层重点研究其分层破坏。首先，介绍一种基于改进的粘结区域方法的粘结层损伤模型；其次，详细介绍了有限元模型建模过程和建模细节；最后，对有限元模型进行验证，并分析分层损伤发生的原因。模拟结果表明，该模型不仅能准确预测FRP层合板在低速冲击载荷下的载荷-时间曲线和载荷-位移曲线，还能成功地预测其分层破坏。     

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.     

Fatigue life predictions by integrating EVICD fatigue damage model and an advanced cyclic plasticity theory

An event independent cumulative damage (EVICD) fatigue prediction model was previously developed for the fatigue damage prediction under general multiaxial stress state and loading conditions. The model takes the plastic strain energy as the major contributor to the fatigue damage. The application of the EVICD model does not require a cycle counting method for general random loading. In the current effort, derivations were made to explicitly and directly relate the material constants in the fatigue model to the parameters in the Manson–Coffin equations and the cyclic stress–strain curve of the material. In addition, an advanced cyclic plasticity theory was implemented for the determination of the detailed stress–strain response that was required as the input for the EVICD fatigue model. Three metallic materials were used to demonstrate the capability of the modified fatigue model for the predictions of fatigue lives under different loading conditions. The results show that the fatigue model can provide fatigue life predictions in close agreement with the experimental observations.