Thermodynamic-based cohesive zone healing model for self-healing materials

This work presents a thermodynamic-based cohesive zone framework to model healing in materials that tend to self-heal. The nominal, healing and effective configurations of continuum damage-healing mechanics are extended to represent cohesive zone configurations. To incorporate healing in a cohesive zone model, the principle of virtual power is used to derive the local static/dynamic macroforce balance and the boundary traction as well as the damage and healing microforce balances. A thermodynamic framework for constitutive modeling of damage and healing mechanisms of cracks is used to derive the evolution equations for the damage and healing internal state variables. The effects of temperature, resting time, crack closure, history of healing and damage, and level of damage on the healing behavior of the cohesive zone are incorporated. The proposed model promises solid basis for understanding the self-healing phenomena in self-healing materials.     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

平面碰撞与强激光加载下金属铝的层裂行为

在轻气炮和神光Ⅱ强激光装置上开展了金属铝的层裂实验。针对激光打靶层裂实验中样品自由面速度剖面后期振荡容易丢失问题,改进靶设计,获得很好效果。利用轻气炮加载和强激光加载层裂实验应变率的显著差异,并通过数值模拟,讨论了在建立具有预测能力的理论建模中需要关注的损伤成核、演化与汇合问题中的材料特性与应变率相关特性因素。结果表明,对于我们以前建立的动态损伤与断裂模型,微孔洞成核的平均半径、阈值压力、成核速率相关参数以及微孔洞长大的阈值压力等具有材料特性属性,但微孔洞的表面能以及决定材料发生完全层裂的临界损伤度等具有明显的应变率效应。另外,分析还发现,虽然层裂强度具有明显的应变率效应,但是在样品层裂当地,样品由持续拉伸向收缩转变的临界行为,取决于一个很小的临界损伤,这个临界值很可能是材料常数,与应变率无关。     

A thermodynamic consistent damage and healing model for self healing materials

Thermodynamics of the damage and the healing processes for viscoplastic materials is discussed in detail and constitutive equations for coupled inelastic-damage-healing processes are proposed in a thermodynamic consistent framework. Small deformation state is utilized and the kinematic and the isotropic hardening effects for the damage and healing processes are introduced into the governing equations. Two new yield surfaces for the damage and healing processes are proposed that take into account the isotropic hardening effect. The computational aspect for solving the coupled elasto-plastic-damage-healing problem is investigated, and the mechanical behavior of the proposed polymeric based self healing system is obtained. Uniaxial compression tests are implemented on a shape memory polymer based self healing system and the damage and the healing are captured by measurement of the changes in the modulus of elasticity. It is concluded that the proposed constitutive equations can model the damage and healing effectively and the mechanical behavior of a shape memory polymer based self healing system can be precisely modeled using this formulation.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

基于介观力学信息的颗粒材料损伤——愈合与塑性宏观表征

本文在二阶计算均匀化框架下提出了颗粒材料损伤--愈合与塑性的多尺度表征方法. 颗粒材料结构在宏观尺度模型化为梯度Cosserat连续体,在其有限元网格的每个积分点处定义具有离散颗粒介观结构的表征元. 建立了表征元离散颗粒系统的非线性增量本构关系. 表征元周边介质作用于表征元边界颗粒的增量力与增量力偶矩以表征元边界颗粒的增量线位移与增量转动角位移、当前变形状态下表征元离散介观结构弹性刚度、以及凝聚到表征元边界颗粒的增量耗散摩擦力表示. 基于平均场理论与Hill定理,导出了基于介观力学信息的梯度Cosserat连续体增量非线性本构关系. 在等温热动力学框架下定义了表征颗粒材料各向异性损伤--愈合和塑性的损伤、愈合张量因子与综合损伤、愈合效应的净损伤张量因子和塑性应变. 此外,定义了损伤和塑性耗散能密度与愈合能密度,以定量比较材料损伤、愈合、塑性对材料失效的效应. 应变局部化数值例题结果显示了所建议的颗粒材料损伤--愈合--塑性表征方法的有效性.      

Micromechanical modeling of low cycle fatigue under complex loadings — Part I. Theoretical formulation

A micromechanical model of the early fatigue damage initiation is proposed based on the slip theory. For each slip system, a local micro-damage variable is introduced to describe globally all phenomena related to the level lower than the crystallographic slip system, such as dislocations, atoms, molecules, lattice defects, etc., of FCC polycrystalline materials. This transgranular damage variable is fully coupled with micro inelastic constitutive equations. It is supposed that the local damage appears when the dislocation density reaches some critical values. The obtained model is devoted to describing the cyclic behavior of metallic materials under proportional and non-proportional loading paths neglecting the quasi-unilateral effect as well as the localization of the fatigue damage on the free surface of the specimen.     

Elastoplastic modeling of circular fiber-reinforced ductile matrix composites considering a finite RVE

A micromechanical elastoplastic damage model considering a finite RVE is proposed to predict the overall elastoplastic damage behavior of circular fiber-reinforced ductile (matrix) composites. The constitutive damage model proposed in our preceding work (Kim and Lee, 2009) considering a finite Eshelby’s tensor (Li et al., 2005, Wang et al., 2005) is extended to accommodate the elastoplastic behavior of the composites. On the basis of the exterior-point Eshelby’s tensor for circular inclusions and the ensemble-averaged effective yield criterion, a micromechanical framework for predicting the effective elastoplastic damage behavior of ductile composites is derived. A series of numerical simulations are carried out to illustrate stress–strain response of the proposed micromechanical framework and to examine the influence of a Weibull parameter on the elastoplastic behavior of the composites. Furthermore, comparisons between the present predictions and experimental data available in the literature are made to further assess the predictive capability of the proposed model.     

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.     

A thermo-viscoelastic–viscoplastic–viscodamage constitutive model for asphaltic materials

A temperature-dependent viscodamage model is proposed and coupled to the temperature-dependent Schapery’s nonlinear viscoelasticity and the temperature-dependent Perzyna’s viscoplasticity constitutive model presented in Abu Al-Rub et al., 2009, Huang et al., in press in order to model the nonlinear constitutive behavior of asphalt mixes. The thermo-viscodamage model is formulated to be a function of temperature, total effective strain, and the damage driving force which is expressed in terms of the stress invariants of the effective stress in the undamaged configuration. This expression for the damage force allows for the distinction between the influence of compression and extension loading conditions on damage nucleation and growth. A systematic procedure for obtaining the thermo-viscodamage model parameters using creep test data at different stress levels and different temperatures is presented. The recursive-iterative and radial return algorithms are used for the numerical implementation of the nonlinear viscoelasticity and viscoplasticity models, respectively, whereas the viscodamage model is implemented using the effective (undamaged) configuration concept. Numerical algorithms are implemented in the well-known finite element code Abaqus via the user material subroutine UMAT. The model is then calibrated and verified by comparing the model predictions with experimental data that include creep-recovery, creep, and uniaxial constant strain rate tests over a range of temperatures, stress levels, and strain rates. It is shown that the presented constitutive model is capable of predicting the nonlinear behavior of asphaltic mixes under different loading conditions.     

Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces

A micromechanical framework is proposed to predict the effective elastic behavior and weakened interface evolution of particulate composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu, J., 1993a. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. Journal of Applied Mechanics 60 (4), 1048–1050; Qu, J., 1993b. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mechanics of Materials 14, 269–281] is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to assess the potential of the present framework.     

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.     

Micromechanical analysis of damage in saturated quasi brittle materials

In this paper, we propose a micromechanical analysis of damage and related inelastic deformation in saturated porous quasi brittle materials. The materials are weakened by randomly distributed microcracks and saturated by interstitial fluid with drained and undrained conditions. The emphasis is put on the closed cracks under compression-dominated stresses. The material damage is related to the frictional sliding on crack surface and described by a local scalar variable. The effective properties of the materials are determined using a linear homogenization approach, based on the extension of Eshelby’s inclusion solution to penny shaped cracks. The inelastic behavior induced by microcracks is described in the framework of the irreversible thermodynamics. As an original contribution, the potential energy of the saturated materials weakened by closed frictional microcracks is determined and formulated as a sum of an elastic part and a plastic part, the latter entirely induced by frictional sliding of microcracks. The influence of fluid pressure is accounted for in the friction criterion through the concept of local effective stress at microcracks. We show that the Biot’s effective stress controls the evolution of total strain while the local Terzaghi’s effective stress controls the evolution of plastic strain. Further, the frictional sliding between crack lips generates volumetric dilatancy and reduction in fluid pressure. Applications of the proposed model to typical brittle rocks are presented with comparisons between numerical results and experimental data in both drained and undrained triaxial tests.     

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.      

Damage and healing effects in rubber-like balloons

Inflation experiments on thin rubber-like balloons show a complex, history-dependent hysteretic behavior, important for many technological applications. Typically, this is ascribed to the occurrence of damage processes at the micro-scale level. The experimental pressure–strain and stress–strain responses [Johnson, M.A., Beatty, F.M., 1995. The Mullins effect in equibiaxial extension and its influence on the inflation of a balloon. Int. J. Eng. Sci. 33(2), 223–245], suggest that for successive cyclic experiments also the occurrence of healing for previously damaged material may play a crucial role (see [Diani, J., Fayolle, B., Gilormini, P., 2009. A review on the Mullins effect, Eur. Polym. J. 45, 601–612] and references therein). In this work we apply a recently proposed, micro-structure-based model for damage and healing effects in rubber-like materials to the inflation problem of a thin spherical balloon. The model, while keeping a computational efficiency, is shown to be in a significant qualitative agreement with the available experimental results.     

A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids

In order to simulate quasi-brittle failure in porous elastic solids, a continuum damage model has been developed within the framework of strain gradient elasticity. An essential ingredient of the continuum damage model is the local strain energy density for pure elastic response as a function of the void volume fraction, the local strains and the strain gradients, respectively. The model adopts Griffith’s approach, widely used in linear elastic fracture mechanics, for predicting the onset and the evolution of damage due to evolving micro-cracks. The effect of those micro-cracks on the local material stiffness is taken into account by defining an effective void volume fraction. Thermodynamic considerations are used to specify the evolution of the latter. The principal features of the model are demonstrated by means of a one-dimensional example. Key aspects are discussed using analytical results and numerical simulations. Contrary to other continuum damage models with similar objectives, the model proposed here includes the effect of the internal length parameter on the onset of damage evolution. Furthermore, it is able to account for boundary layer effects.     

Microstructure damage related to stress–strain curve for grain composites

Mathematical model of micro-heterogeneous medium with random deformation and strength properties of microstructure is developed assuming that the tensor of macroscopic deformations is known for the structure. Green–Somigliana tensor is used to obtain the formulas for random stress distribution in microstructure elements. The probability of the stress exceeding the ultimate strength in an element determines the probability of fracture in this element and the relative micro-damage. The correlation functions of stochastic microstructure ultimate strength condition are calculated for various types of stress. Normal distribution is used to calculate the damage. The distribution density can be adjusted through the stress moments to the fourth order.Micro-fractions change the composite’s macro modules of elasticity. Therefore, changes the relationship between stress and strain. Setting an increment step on the macro-strain axis, the stress–strain curve is plotted taking into account changes in composite properties. Stress–strain curves are obtained for different types of load.The increase of the factor of safety corresponds with the reduction of microstructure damage permitted in the design. Critical microstructure damage also depends on the dispersion of the microstructure properties. It is shown that the microstructure properties of composite significantly influence the behavior of materials under load and the shape of stress–strain curve. Findings are compared with experiment data.     

含热冲击预损伤的陶瓷基复合材料损伤本构模型

陶瓷基复合材料结构在服役过程中不可避免地经受热冲击(较高的热应力梯度)而产生热机械损伤, 因此, 建立含循环热冲击预损伤材料的损伤本构模型, 以描述材料在热机械载荷作用下的力学行为, 对材料结构损伤容限设计与结构完整性评估非常重要. 本文首先对经历了循环热冲击的材料进行单调拉伸损伤实验, 发现对于含循环热冲击预损伤的材料, 其弹性模量的下降与所施加的应变直接相关. 然后在连续介质损伤力学的框架下, 基于平面应力假设, 建立了含循环热冲击预损伤材料的损伤演化模型, 该模型所涉及的参数可通过一个偏轴(45$^\circ$)以及两个正轴(平行于两个主方向)的单调拉伸试验获得. 最后, 采用经典塑性理论对由基体损伤引起的非弹性应变进行了描述. 本文所提出的应变损伤宏观模型可以描述陶瓷基复合材料在热机械载荷作用下的损伤演化, 同时弥补了含预损伤的陶瓷基复合材料在机械载荷下损伤本构模型在理论及实验研究方面的不足.