Framework using functional forms of hardening internal state variables in modeling elasto-plastic-damage behavior

This work gives the thermodynamically consistent theoretical formulations and the numerical implementation of a plasticity model fully coupled with damage. The formulation of the elasto-plastic-damage behavior of materials is introduced here within a framework that uses functional forms of hardening internal state variables in both damage and plasticity. The damage is introduced through a damage mechanics framework and utilizes an anisotropic damage measure to quantify the reduction of the material stiffness. In deriving the constitutive model, a local yield surface is used to determine the occurrence of plasticity and a local damage surface is used to determine the occurrence of damage. Isotropic hardening and kinematic hardening are incorporated as state variables to describe the change of the yield surface. Additionally, a damage isotropic hardening is incorporated as a state variable to describe the change of the damage surface. The hardening conjugate forces (stress-like terms) are general nonlinear functions of their corresponding hardening state variables (strain-like terms) and can be defined based on the desired material behavior. Various exponential and power law functional forms are studied in this formulation. The paper discusses the general concept of using such functional forms. however, it does not address the relevant appropriateness of certain forms to solve different problems. The proposed work introduces a strong coupling between damage and plasticity by utilizing damage and plasticity flow rules that are dependent on both the plastic and damage potentials. However, in addition to that the coupling is further enhanced through the use of the functional forms of the hardening variables introduced in this formulation.The use of this formulation in solving boundary value problems will be presented in future work. The fully implicit backward Euler scheme is developed for this model to be solved in a Newton–Raphson solution procedure.     

基于Rayleigh-Ritz算法的梁裂缝损伤识别及试验研究

结构动态特性的变化则预示结构出现损伤,基于Ritz线性近似,文中提出一种裂缝损伤识别的方法,用以识别结构裂缝损伤位置及损伤程度。该方法分两步：首先用模态子空间近似关系．消去模态应变能表达式中的整体刚度矩阵和整体质量矩阵,避免模型误差对识别结果的影响。再采用计算向量空间夹角的方法分离损伤位置与损伤程度的耦合影响,进而识别出单元裂缝损伤位置;其次,识别损伤程度采用二次线性规划方法,不再计算特征值灵敏度。线性约束条件保证了二次规划问题的解是唯一的。模拟筒支梁几种裂缝损伤情况进行数值计算与模态试验,利用所得模态参数对该算法程序进行了验证,识别出了裂缝损伤的确切位置及损伤程度,并进行了误差对比。结果表明,该算法由于不用整体结构的数值模型,从而避免了边界条件、连接条件及材料特性参数等因素对识别结果的干扰,识别精度得到提高,将其用于结构损伤识别是可行的。     

On the extension of the Gurson-type porous plasticity models for prediction of ductile fracture under shear-dominated conditions

One of the major drawbacks of the Gurson-type of porous plasticity models is the inability of these models to predict material failure under low stress triaxiality, shear dominated conditions. This study addresses this issue by combining the damage mechanics concept with the porous plasticity model that accounts for void nucleation, growth and coalescence. In particular, the widely adopted Gurson–Tvergaard–Needleman (GTN) model is extended by coupling two damage parameters, representing the volumetric damage (void volume fraction) and the shear damage, respectively, into the yield function and flow potential. The effectiveness of the new model is illustrated through a series of numerical tests comparing its performance with existing models. The current model not only is capable of predicting damage and fracture under low (even negative) triaxiality conditions but also suppresses spurious damage that has been shown to develop in earlier modifications of the GTN model for moderate to high triaxiality regimes. Finally the modified GTN model is applied to predict the ductile fracture behavior of a beta-treated Zircaloy-4 by coupling the proposed damage modeling framework with a recently developed J₂–J₃ plasticity model for the matrix material. Model parameters are calibrated using experimental data, and the calibrated model predicts failure initiation and propagation in various specimens experiencing a wide range of triaxiality and Lode parameter combinations.     

A generalized coupled viscoplastic-viscodamage-viscohealing theory for glassy polymers

A thermodynamic consistent, small-strain, non-unified model is developed to capture the irregular rate dependency included in the strain controlled inelastic responses of polymers at the glassy state. The model is considered as a generalized Frederick-Armstrong-Philips-Chaboche (FAPC) theory proposed by [Voyiadjis and Basuroychowdhury, 1998] and [Voyiadjis and Abu Al-Rub, 2003] which is based on a von Mises and Chaboche isotropic hardening type viscoplasticity formulation. Using the proposed model, different experimental results are simulated and the range of viscoplastic related material constants are obtained through a parametric study. The thermodynamic framework is used to incorporate the effect of coupling between viscodamage and viscohealing phenomena into the inelastic deformation of glassy polymers. This coupling effect is crucial for polymeric based self healing systems in which different damage mechanisms are active and the efficiency of the healing processes are highly dependent on the damage. The computational aspect for general coupled inelastic-damage-healing processes together with the required solution algorithms are elaborated and the inelastic-damage-healing response of a polymeric based self-healing system is simulated. The proposed viscoplasticity theory constitutes a physically consistent approach to model the irregular mechanical responses of glassy polymers and the viscodamage model provides an exquisite predicting tool to evaluate the ductile damage associated with the large inelastic deformation and low cycle fatigue in polymeric based material systems. In conclusion, a well structured viscohealing theory is formulated for polymeric based self healing systems.     

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.      

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.     

多尺度材料模拟的自适应有限元方法研究

研究了材料模拟中一类新型耦合多尺度的自适应有限元方法. 采用微观分子动力学耦合宏观有限元的桥尺度方法来模拟材料破坏的前期行为,其中宏观有限元计算推广到了一般非结构三角形网格. 材料破坏形成后,停止微观尺度的计算,它的进一步发展和演化通过一个宏观模型来描述,采用自适应有限元方法来求解这一宏观模型. 其中,后验误差估计的基础是变分多尺度理论,即自适应网格加密是基于粗尺度上残差分布和细尺度上单元Green's函数. 计算中采用了破坏准则来模拟材料的断裂. 数值实验表明了方法的有效性.      

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.     

On the coupling of anisotropic damage and plasticity models for ductile materials

In this contribution various aspects of an anisotropic damage model coupled to plasticity are considered. The model is formulated within the thermodynamic framework and implements a strong coupling between plasticity and damage. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. The damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The model considers different interaction mechanisms between damage and plasticity defects in such a way that two-isotropic and two-kinematic hardening evolution equations are derived, one of each for the plasticity and the other for the damage. An additive decomposition of the total strain into elastic and inelastic parts is adopted in this work. The elastic part is further decomposed into two portions, one is due to the elastic distortion of the material grains and the other is due to the crack closure and void contraction. The inelastic part is also decomposed into two portions, one is due to nucleation and propagation of dislocations and the other is due to the lack of crack closure and void contraction. Uniaxial tension tests with unloadings have been used to investigate the damage growth in high strength steel. A good agreement between the experimental results and the model is obtained.     

A multiaxial constitutive model for concrete in the fire situation: Theoretical formulation

This paper aims to develop a multiaxial concrete model for implementation in finite element software dedicated to the analysis of structures in fire. The need for proper concrete model remains a challenging task in structural fire engineering because of the complexity of the concrete mechanical behavior characterization and the severe requirements for the material models raised by the development of performance-based design. A fully three-dimensional model is developed based on the combination of elastoplasticity and damage theories. The state of damage in concrete, assumed isotropic, is modeled by means of a fourth order damage tensor to capture the unilateral effect. The concrete model comprises a limited number of parameters that can be identified by three simple tests at ambient temperature. At high temperatures, a generic transient creep model is included to take into account explicitly the effect of transient creep strain. The numerical implementation of the concrete model in a finite element software is presented and a series of numerical simulations are conducted for validation. The concrete behavior is accurately captured in a large range of temperature and stress states. A limitation appears when modeling the concrete post-peak behavior in highly confined stress states, due to the coupling assumption between damage and plasticity, but the considered levels of triaxial confinement are unusual stress states in structural concrete.     

Crack shielding and material deterioration in damaged materials: an antiplane shear fracture problem

Summary A simple damage evolution model is proposed for a quasibrittle material in the framework of continuum damage mechanics. The model is used to obtain a closed form solution for a mode-III stationary crack under small scale damage conditions. It is found that the crack tip stress intensity factor is reduced, i.e., the crack is shielded by the damage. However, this shielding effect is completely offset by the material deterioration caused by the damage. It also holds for steady state crack growth. When the most effective shielding is reached for the stationary crack, the zone dominated by the stress intensity factor shrinks to the crack tip. The results for the antiplane shear problem should shed some light on the in- plane fracture problem. Received 4 August 1997; accepted for publication 7 October 1997     

弹体贯穿混凝土数值模拟的改进材料模型

研究混凝土结构在冲击载荷下的力学特性对武器以及防护结构的设计和评估具有重要意义,而合适的材料模型可以更准确地预测混凝土结构的力学行为和破坏模式。因此,本文中提出了一种改进的混凝土塑性损伤材料模型来描述其在冲击载荷下的力学响应。该改进模型考虑了压力-体积应变关系、应变率效应、洛德角效应和塑性损伤累积对混凝土材料力学特性的影响,并引入了一个与损伤相关的硬化/软化函数来描述压缩状态下的应变硬化和软化行为。随后,通过对3个独立的强度面进行线性插值得到了该改进模型的破坏强度面,并采用部分关联流动法则考虑了混凝土材料的体积膨胀特性。最后,开展了单个单元在不同加载条件下和弹体贯穿钢筋混凝土靶的数值模拟,验证了该改进模型的可行性、准确性以及预测性能提升。     

Modeling stress state dependent damage evolution in a cast Al–Si–Mg aluminum alloy

Internal state variable rate equations are cast in a continuum framework to model void nucleation, growth, and coalescence in a cast Al–Si–Mg aluminum alloy. The kinematics and constitutive relations for damage resulting from void nucleation, growth, and coalescence are discussed. Because damage evolution is intimately coupled with the stress state, internal state variable hardening rate equations are developed to distinguish between compression, tension, and torsion straining conditions. The scalar isotropic hardening equation and second rank tensorial kinematic hardening equation from the Bammann–Chiesa–Johnson (BCJ) Plasticity model are modified to account for hardening rate differences under tension, compression, and torsion. A method for determining the material constants for the plasticity and damage equations is presented. Parameter determination for the proposed phenomenological nucleation rate equation, motivated from fracture mechanics and microscale physical observations, involves counting nucleation sites as a function of strain from optical micrographs. Although different void growth models can be included, the McClintock void growth model is used in this study. A coalescence model is also introduced. The damage framework is then evaluated with respect to experimental tensile data of notched Al–Si–Mg cast aluminum alloy specimens. Finite element results employing the damage framework are shown to illustrate its usefulness.     

脆性岩石破裂过程损伤与渗流耦合数值模型研究

大量的实验结果表明，脆性岩石的渗透性不是一个常量，而是应力和应力诱发损伤破裂的函数．建立了一个描述非均匀岩石渗流-应力-损伤耦合数学模型(FSD Model)，开发出岩石破裂过程渗流-应力-损伤耦合分析计算系统(F-RFPA^2D)．在该系统中，单元的力学、水力学性质根据统计分布而变化，以体现材料的随机不均质性，材料在开裂破坏过程中流体压力传递通过单元渗流-损伤耦合迭代来实现．该系统能够对岩石试件在孔隙水压力和双轴荷载作用下裂纹的萌生、扩展过程中渗透率演化规律及其渗流-应力耦合机制进行模拟分析．最后，给出两个算例：算例1模拟载荷作用下岩石应力应变-渗透率全过程．模拟结果表明，非均匀性对岩石的应力峰值强度、峰值前后其渗透性演化规律及其破裂机制影响十分明显，模拟结果和实验结果较为一致；算例2模拟孔隙水压力作用下岩石拉伸断裂过程，通过和物理实验对比验证，验证了模型计算结果的可靠性。     

A thermodynamics-based cohesive model for interface debonding and friction

A constitutive model for interface debonding is proposed which is able to account for mixed-mode coupled debonding and plasticity, as well as further coupling between debonding and friction including post-delamination friction. The work is an extension of a previous model which focuses on the coupling between mixed-mode delamination and plasticity. By distinguishing the interface into two parts, a cracked one where friction can occur and an integral one where further damage takes place, the coupling between frictional dissipation and energy loss through damage is seamlessly achieved. A simple framework for coupled dissipative processes is utilised to derive a single yield function which accurately captures the evolution of interface strength with increasing damage, for both tensile and compressive regimes. The new material model is implemented as a user-defined interface element in the commercial package ABAQUS and is used to predict delamination under compressive loads in several test cases.     

Fractal effect and anisotropic constitutive model for concrete

An elasto-anisotropic damage constitutive model for concrete is developed in this work. Disregarding the coupling between the isotropic and the anisotropic damage, the isotropic damage variables are defined as functions of the microcrack fractal dimension, and the anisotropic parts are expressed by the lengths of cracks in concrete which various in different directions. The Helmholtz free energy is decomposed into the elastic deforming, damage and irreversible deforming components, with the last component used to replace the plastic deformation. Therefore the damage constitutive formulas for concrete are derived based on continuum damage mechanics. Evolution laws for both isotropic and anisotropic damage variables are derived, in which the anisotropic parts are obtained by modifying an empirical model. The critical fracture stress and the fracture toughness are investigated for materials with a single fractal crack based on the fractal geometry and the Griffith fracture criterion. Numerical computation is conducted for concrete under the uniaxial and the biaxial compression. The results indicate that the material stiffness degradation can be well addressed when the anisotropic damage is incorporated; the irreversible deformation is greatly related to the behavior of the descending branch beyond the peak load. The validation of the presented model is proofed by comparing results with the experimental data. This model provides an approach to link the macro properties of a material with its micro-structure change.     

The Three-Stage Model Based on Strain Strength Distribution for the Tensile Failure Process of Rock and Concrete Materials

A three-stage model is introduced to describe the tensile failure process of rock and concrete materials.Failure of the material is defined to contain three stages in the model,which include elastic deformation stage,body damage stage and localization damage stage.The failure mode change from uniform body damage to localization damage is expressed.The heterogeneity of material is described with strain strength distribution.The fracture factor and intact factor,defined as the distribution function of strain strength,are used to express the fracture state in the failure process.And the distributive parameters can be determined through the experimental stress-strain curve.     

Calibration,validation, and verification including uncertainty of a physically motivated internal state variable plasticity and damage model

In this paper, we illustrate a formal calibration, validation, and verification process that includes uncertainty in an internal state variable plasticity-damage model that is implemented in a finite element code. The physically motivated continuum model characterizes damage evolution by incorporating material uncertainty due to microstructural spatial clustering. The uncertainty analysis is performed by introducing material variation through model validation and verification. The effect of variability in microstructural clustering and boundary conditions on the sensitivities and uncertainty of the plasticity-damage evolution for the 7075 aluminum alloy is characterized. The results show the potential of this methodology in the evaluation of material response uncertainty due to microstructure spatial clustering and its effect on damage evolution. For damage evolution, we have shown that the initial isotropic damage evolved into an anisotropic form as the deformation increased which is consistent with experimentally observed behavior for 7075 aluminum alloy in literature. Also, the sensitivities were found to be consistent with the physics of damage progression for this particular type of material. Through the sensitivity analysis, the initial defect size and number density of cracked particles are important at the beginning of deformation. As damage evolves, more voids are nucleated and grow and the sensitivity analysis illustrates this as well. Then, voids combine with each other and coalescence becomes the main driver, which is also confirmed by the sensitivity analysis. This work also shows that the microstructurally based damage evolution equations provide an accurate representation of the damage progression due to large intermetallic particles. Finally, we illustrate that the initial variation in the microstructure clustering can lead to about ±7.0%, ±8.1%, and ±9.75% variation in the elongation to failure strain for torsion, tensile, and compressive loading, respectively.     

高周疲劳的损伤-硬化模型

两级循环加载条件下,材料的剩余寿命强烈地依赖于加载历史。究其原因,不同加载历史将引起材料的微结构发生不同的变化,使得材料的硬化效果和变形行为表现出明显的差异,从而影响了损伤的演化过程。本文引入硬化状态变量来表征加载历史对疲劳损伤演化过程的影响。通过对两级循环加载下损伤演化规律和剩余寿命的研究,认为在两级（或多级）加载条件下,材料的损伤演化和剩余寿命强烈地依赖于加载历史造成的损伤和硬 化状态。