Meso approaches for the creep damage behavior of nickel-base directionally solidified superalloys

IntroductionNickel-basehightemperatureresistancesuperalloysarsewidelyusedingasturbinesandjetengines.DireetionallysolidificationwasintroducedtoenhancecreepsbengthbyelindnahnggrainboundariesnormaltotheappliedstresswherevoidsarelikelytOoccurundercreepload.Inpractice,however,theappliedstressmaynotbeuniaxialnordirectedparalleltOthegrainboundaries.Singlecrystalswerethusdeveloped.CongregationoflowtempefAnremeltingelementsongrainboundaries,grainboundaryoxidation)etc.,areadditionalfactorsthatwoulddeg…     

沉淀相的尺寸与形态对镍基合金蠕变行为的影响

镍基合金具有优良的高温力学性能,广泛应用于涡轮叶片等热端部件。沉淀相的尺寸和形态是影响镍基合金力学性能的重要因素。本文在考虑应变梯度的镍基合金晶体塑性本构模型的基础上,引入了各向异性损伤张量,研究了包含两种不同尺寸和三种不同长细比的沉淀相形态的镍基合金蠕变行为。结果表明,该模型能够很好地反映沉淀相的尺寸对镍基合金蠕变行为的影响,与实验结果符合较好。同时,沉淀相的形态也对镍基合金的力学性能产生重要影响,随着沉淀相长细比的增加,镍基合金的蠕变寿命延长,这体现了粗化和形态对镍基合金蠕变行为影响的一种竞争的机制。     

Evaluation of creep damage behavior of nickel-base directionally solidified superalloys with different crystallographic orientations

A crystallographic creep damage constitutive model is developed for nickel-base directionally solidified superalloys. The rates of material degradation and grain boundary void growth are considered. The governing parameters are determined from the creep test data of single crystals and directionally solidified superalloys with a crystallographic orientation. A finite element program is used to analyze the creep damage behavior of nickel-base directionally solidified superalloys for different crystallographic orientations. The results depend on the number of grains modelled and compare well with the experimental data.     

Anisotropic continuum damage modeling for single crystals at high temperatures

In single crystals, the process of creep damage is generally anisotropic. Indeed, the damage evolution does not only depend on the loading conditions, but also on the lattice orientation. And the current state of damage has an anisotropic influence on the effective stress state, so that it is represented by a tensorial damage variable. Based on the continuum damage mechanics theory, a creep damage model for F.C.C. single crystals has been developed and implemented in a three-dimensional anisotropic creep model. It is shown that the resulting material model is capable of describing the orientation dependence of the creep and damage evolution of nickel-based superalloys in the high temperature regime.     

Creep,plasticity, and fatigue of single crystal superalloy

Single crystal components in gas turbine engines are subject to such extreme temperatures and stresses that life prediction becomes highly inaccurate resulting in components that can only be shown to meet their requirements through experience. Reliable life prediction methodologies are required both for design and life management. In order to address this issue we have developed a thermo-viscoplastic constitutive model for single crystal materials. Our incremental large strain formulation additively decomposes the inelastic strain rate into components along the octahedral and cubic slip planes. We have developed a crystallographic-based creep constitutive model able to predict sigmoidal creep behavior of Ni base superalloys. Inelastic shear rate along each slip system is expressed as a sum of a time dependent creep component and a rate independent plastic component. We develop a new robust, computationally efficient rate-independent crystal plasticity approach and combined it with creep flow rule calibrated for Ni-based superalloys. The transient variation of each of the inelastic components includes a back stress for kinematic hardening and latent hardening parameters to account for the stress evolution with inelastic strain as well as the evolution for dislocation densities. The complete formulation accurately predicts both monotonic and cyclic tests at different crystallographic orientations for constant and variable temperature conditions (low cycle fatigue (LCF) and thermo-mechanical fatigue (TMF) tests). Based on the test and modeling results we formulate a new life prediction criterion suitable for both LCF and TMF conditions.     

复杂应力状态镍基单晶合金低周疲劳损伤模型

根据连续介质损伤力学理论,采用应变能释放率作为热力学广义力描述正交异性材料的疲劳损伤过程,引入取向函数考虑镍基单晶合金晶体取向对疲劳损伤的非线性影响,提出了一个各向异性疲劳损伤模型。应用多元线性回归分析方法,拟合疲劳试验数据可确定模型参数。从应变能释放率的应变空间表达式出发,导出了含有3个弹性常数的单晶合金应变三轴性因子,它既反映了材料性能的晶体取向相关性,又反映了正应力和剪应力的相互作用,并可退化为各向同性材料的应变三轴性因子。利用该模型对CMSX-2镍基单晶合金在应力控制对称循环拉-扭载荷作用下的低周疲劳寿命进行预测,预测值与试验值吻合的相当好,试验所得数据均落在2.2倍偏差的分布带内。     

镍基单晶高温合金定向粗化行为及高温蠕变力学性能研究进展

镍基单晶高温合金是一种广泛应用于航空发动机和工业燃气轮机的两相叶片材料,由软的$\gamma $ 基体相和均匀镶嵌在其中的立方状 $\gamma'$ 沉淀强化相组成.它有个显著的特征,即在高温施加应力条件下, $\gamma '$沉淀相会发生定向粗化, 形成筏状.这种筏化行为直接影响了合金的蠕变疲劳寿命,是镍基单晶高温合金强化机制研究的重点. 此外,镍基单晶高温合金无晶界, 不存在高温晶界弱化、纵向晶界裂纹等问题.因此, $\gamma$/$\gamma'$相界面的位错运动、微观结构以及在载荷和温度作用下的演化决定了其蠕变力学性能.本文从镍基单晶高温合金的微观强化机制出发对定向粗化行为及蠕变力学性能进行了综述.重点介绍了定向粗化行为发生的微观机理、驱动力、影响因素和蠕变过程中界面微结构演化、蠕变力学模型以及定向粗化对高温蠕变力学性能的影响,指出了高温蠕变力学性能研究的发展方向和仍待解决的问题.      

Stochastic damage hysteretic model for concrete based on micromechanical approach

In the present paper, a kind of stochastic damage hysteretic model is proposed to describe the damage and hysteretic behaviors of concrete material. According to the model, a parallel system made up of micro-elements, which are developed based on the micro-tensile and shear damage mechanism respectively, is adopted to obtain the overall responses of concrete. The influence of plastic strain and hysteretic energy dissipation of the material are also considered in the model. To reflect the stochastic properties of concrete, the fracture strain of the micro-element is set as a random variable. Then the monotonic, loading and unloading curves of the parallel system are derived analytically by averaging the stochastic micro-elements and two hysteretic rules are combined to the proposed model to account for complicated loading conditions. Furthermore, a nonlocal process is introduced to the model to overcome the mesh dependence issues of softening materials. Finally, several numerical examples are conducted, demonstrating that the proposed model can provide reliable results reflecting the damage, plasticity and hysteretic behaviors of concrete material.     

考虑损伤的内变量黏弹-黏塑性本构方程

基于Rice 不可逆内变量热力学框架,在约束构型空间中讨论材料的蠕变损伤问题. 通过给定具体的余能密度函数和内变量演化方程推导出考虑损伤的内变量黏弹-黏塑性本构方程. 通过模型相似材料单轴蠕变加卸载试验对一维情况下的本构方程进行参数辨识和模型验证,本构方程能很好地描述黏弹性变形和各蠕变阶段.不同的蠕变阶段具有不同的能量耗散特点. 受应力扰动后,不考虑损伤的材料系统能自发趋于热力学平衡态或稳定态. 在考虑损伤的整个蠕变过程中,材料系统先趋于平衡态再背离平衡态发展. 能量耗散率可作为材料系统热力学状态偏离平衡态的测度;能量耗散率的时间导数可用于表征系统的演化趋势;两者的域内积分值可作为结构长期稳定性的评价指标.      

Constitutive modelling of anisotropic creep deformation in single crystal blade alloys SRR99 and CMSX-4

A damage mechanics based model has been developed to model stress rupture and creep behaviour of the first and second generation single crystal superalloys SRR99 and CMSX-4. In this article the creep behaviour of CMSX-4 in several different orientations at 950°C is simulated using finite elements, these simulations are compared with the results of creep tests. In order that the effects of rotation and specimen bending can be accounted for in the analysis the entire creep specimen is modelled. The FE program ABAQUS has been used and the slip system model is written using a User MATerial subroutine (UMAT). EBSD (electron back scattered diffraction) measurements of the lattice rotations occurring during creep indicate that the active slip systems at 950°C are the <101>{111} and <112>{111} systems, our simulations show that the creep results can be explained by activating these two families of slip system. There is strong microstructural evidence that the significant components of the hardening matrix should be those causing self and latent hardening of the <101>{111} systems and latent hardening by the <101>{111} systems on the <112>{111} systems.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

含与不含晶界空穴的双晶体蠕变行为研究

基于晶体滑移理论,建立了各向异性镍基合金双晶体的蠕变本构模型和蠕变寿命预测模型,通过MARC用户子程序CRPLAW将上述本构模型进行了有限元实现,并对双晶体蠕变行为进行了计算分析,考虑了:(1)晶体取向的影响;(2)垂直、倾斜和平行于外载方向的三种位向晶界情况;(3)晶界处引进空间空穴的影响。结果表明,双晶体上特别是微空穴和晶界附近区域的蠕变应力应变呈现不同的变化规律,对此晶粒晶体取向和晶界位向有较大的影响;微空穴的存在削弱了双晶体的承载能力,显著地影响了双晶体蠕变持久寿命;相同条件下,垂直晶界对双晶体模型的蠕变损伤影响最为强烈,倾斜晶界次之,平行晶界最小;微空穴的生长与晶界位向和晶体取向有强烈的依赖关系,其中垂直晶界更有利于晶体滑移和微空穴生长。     

Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms

This work presents a model to represent ductile failure (i.e. failure controlled by nucleation, growth and coalescence) of materials whose irreversible deformation is controlled by several plastic or viscoplastic deformation mechanisms. In addition work hardening may result from both isotropic and kinematic hardening. Damage is represented by a single variable representing void volume fraction. The model uses an additive decomposition of the plastic strain rate tensor. The model is developed based on the definition of damage dependant effective scalar stresses. The model is first developed within the generalized standard material framework and expressions for Helmholtz free energy, yield potential and dissipation potential are proposed. In absence of void nucleation, the evolution of the void volume fraction is governed by mass conservation and damage does not need to be represented by state variables. The model is extended to account for void nucleation. It is implemented in a finite element software to perform structural computations. The model is applied to three case studies: (i) failure by void growth and coalescence by internal necking (pipeline steel) where plastic flow is either governed by the Gurson–Tvergaard–Needleman model or the Thomason model, (ii) creep failure (Grade 91 creep resistant steel) where viscoplastic flow is controlled by dislocation creep or diffusional creep and (iii) ductile rupture after pre-compression (aluminum alloy) where kinematic hardening plays an important role.     

A nonlocal model with strain-based damage

A thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented. The damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena. An important advantage of models with strain-based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations. A mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two-field variational formulation. Hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two-field variational formulation. Two examples concerning a one-dimensional bar in simple tension and a two-dimensional notched plate are addressed. No mesh dependence or boundary effects are apparent.     

考虑晶体滑移面分解正应力的细观损伤模型

材料内部的解理、滑移面剥离等细观损伤是引起宏观失效的根源, 从细观尺度研究损伤的发生和发展有助于深入认识材料的变形和失效过程. 本文基于晶体塑性理论, 从滑移系的受力和变形出发研究材料的细观损伤, 建立了考虑滑移面分解正应力的细观损伤模型, 为晶体材料解理断裂的分析提供了新方法. 首先, 在晶体弹塑性变形构型的基础上引入损伤变形梯度张量的概念, 从变形运动学着手建立了考虑损伤能量耗散的本构方程, 并推导了塑性流动方程与损伤演化方程; 然后, 建立了相应的数值计算方法, 给出了应力与状态变量的更新算法, 推导了Jacobian矩阵的表达式; 接着, 以$[100]$取向的单晶铜材料为例, 通过有限元计算与试验结果的对比, 并采用粒子群优化算法标定了11个材料细观参数; 最后, 将所提细观损伤模型应用于RVE单轴拉伸过程的模拟, 得到了考虑损伤影响的应力应变曲线, 并分析了材料的塑性流动与损伤演化过程. 结果表明, 本文所提模型能够计算材料在受载过程中的损伤累积效应, 合理反映晶体材料的细观损伤机理.      

Low cycle fatigue behavior of single crystal superalloy with temperature gradient

In present study, low cycle fatigue model based on rate dependent constitutive formulation with kinematic hardening and fatigue damage rule is introduced to investigate the fatigue behavior of single crystal superalloys with temperature gradient. Low cycle fatigue tests with uniform temperature and with temperature gradient are carried out to investigate the influence of temperature gradient. The microstructure reveals that the slip deformation is a principal mechanism of low cycle fatigue with temperature gradient. Complex stress experiments of low cycle fatigue are carried out to verify the life prediction rule. Bauschinger effect, ratcheting effect and fatigue damage of single crystal superalloy at different temperatures are studied. Special attention is placed on the simulation of low cycle fatigue behavior with temperature gradient. The simulation results show that the ratcheting effect at high temperature is more remarkable than that at low temperature due to the influence of temperature gradient, which results in the difference of damage evolution between high temperature zone and low temperature zone. Comparison between experiments and simulations with temperature gradient proves that the predicted model based on the damage of the second cycle is reasonable.     

An anisotropic micromechanics model for predicting the rafting direction in Ni-based single crystal superalloys

An anisotropic micromechanics model based on the equivalent inclusion method is developed to investigate the rafting direction of Ni-based single crystal superalloys. The micromechanical model considers actual cubic structure and orthogonal anisotropy properties. The von Mises stress, elastic strain energy density, and hydrostatic pressure in different inclusions of micromechanical model are calculated when applying a tensile or compressive loading along the [001] direction. The calculated results can successfully predict the rafting direction for alloys exhibiting a positive or a negative mismatch, which are in agreement with pervious experimental and theoretical studies. Moreover, the elastic constant differences and mismatch degree of the matrix and precipitate phases and their influences on the rafting direction are carefully discussed.     

基于结构化变形驱动的非局部宏-微观损伤模型的真Ⅱ型裂纹模拟

Ⅱ型载荷作用下裂纹变形模式也为Ⅱ型的破坏问题称为真Ⅱ型破坏.准确定量地把握真Ⅱ型破坏的全过程是具有挑战性的问题.本文采用结构化变形驱动的非局部宏-微观损伤模型对真Ⅱ型破坏问题进行了模拟.根据结构化变形理论将点偶的非局部应变分解为弹性应变与结构化应变两部分,进而利用Cauchy-Born准则与结构化应变计算点偶的结构化正伸长量.在本文中,结构化应变取为非局部应变的偏量部分.当点偶的结构化正伸长量超过临界伸长量时,微细观损伤开始在点偶层次发展.将微细观损伤在作用域中进行加权求和得到拓扑损伤,并通过能量退化函数将其嵌入到连续介质-损伤力学框架中进行数值求解.进一步地,本文采用Gauss-Lobatto积分格式计算点偶的非局部应变,将积分点数目降低到4个,显著降低了前处理和非线性分析的计算成本.通过对Ⅱ型加载下裂尖应变场的分析揭示了采用偏应变作为结构化应变的原因.对两个典型真Ⅱ型破坏问题的模拟结果表明,本文方法不仅可以把握Ⅱ型加载下的真Ⅱ型裂纹扩展模式,同时可以定量刻画加载过程中的载荷-变形曲线,且不具有网格敏感性.最后指出了需要进一步研究的问题.     

单晶有限变形的热力耦合计算模型

以弹性变形梯度作为基本变量,结合热力学理论构造了单晶有限变形的热、力耦合计算模型。该模型考虑了温度、变温速率以及塑性耗散等条件对单晶有限变形的影响,相对于传统的以弹性变形梯度为基本变量的晶体塑性模型,算法能够体现温度效应的影响。采用隐式的积分方法对建立的控制方程进行计算以保证求解过程的稳定。以1100Al单晶为例计算了不同升温、降温速率,以及不同应变率影响下的材料应力-应变的响应。结果表明,模型能较好地反映变温过程中,单晶各向异性性质的演化以及应力、应变之间关系的变化。