Phenomenological formulation of viscoplastic constitutive equation for polyethylene by taking into account strain recovery during unloading

A viscoplastic constitutive equation for polyethylene that properly describes significant strain recovery during unloading was proposed. The constitutive equation was formulated by combining the kinematic hardening creep theory of Malinin and Khadjinsky with the nonlinear kinematic hardening rule of Armstrong and Frederick. In order to describe the strain recovery, the nonlinear kinematic hardening rule was modified. First, a loading surface was defined in a viscoplastic strain space. A loading–unloading criterion was then introduced using the loading surface. Moreover, a new parameter was defined by the relationship between the loading surface and the current state of the viscoplastic strain, and the evolution equation of back stress was modified using this parameter, which has some value only during unloading. Experimental results for polyethylene were simulated by using the modified constitutive equations, and cyclic inelastic deformation in both uniaxial and biaxial states of stress was predicted. Finally, the validity of the above-described modification was verified, and the features of the constitutive equation and the deformation were discussed.     

An improved unified viscoplastic constitutive model for strain-rate sensitivity in high temperature fatigue

An improved unified cyclic viscoplastic material model for high temperature fatigue of P91 steel is presented. The primary enhancement over existing models is in relation to strain-rate independence of parameters, for accurate interpolation and extrapolation across a range of strain-rates and stress regimes, as relevant to flexible operation of high temperature power generation plant. The model combines a hyperbolic sine constitutive equation with anisothermal cyclic evolution of isotropic and kinematic hardening variables. The material model is developed from a thermodynamic framework and is implemented in multi-axial form within a user material subroutine. The user material subroutine is calibrated and validated for P91 steel across a range of cyclic (isothermal fatigue and thermo-mechanical fatigue) and non-cyclic high temperature loading conditions. A novel method for the identification of the cyclic viscoplastic material parameters is also presented.     

Constitutive modeling of creep-fatigue interaction for normal strength concrete under compression

Conventional approaches to model fatigue failure are based on a characterization of the lifetime as a function of the loading amplitude. The Wöhler diagram in combination with a linear damage accumulation assumption predicts the lifetime for different loading regimes. Using this phenomenological approach, the evolution of damage and inelastic strains and a redistribution of stresses cannot be modeled. The gradual degration of the material is assumed to not alter the stress state. Using the Palmgren–Miner rule for damage accumulation, order effects resulting from the non-linear response are generally neglected.In this work, a constitutive model for concrete using continuum damage mechanics is developed. The model includes rate-dependent effects and realistically reproduces gradual performance degradation of normal strength concrete under compressive static, creep and cyclic loading in a unified framework. The damage evolution is driven by inelastic deformations and captures strain rate effects observed experimentally. Implementation details are discussed. Finally, the model is validated by comparing simulation and experimental data for creep, fatigue and triaxial compression.     

Experimental study on dynamic damage evolution of concrete under multi-axial stresses

The effect of stress state on the dynamic compressive strength and the dynamic damage evolution process of concretes are investigated by use of a Spilt Hopkinson Pressure Bar (SHPB) and the ultrasonic technique. The columned concrete specimen is encircled by a steel sleeve. The multi-axial loading includes the axial and the radial loadings. The axial loading is supplied by the incidence bar, and the radial ones are produced by the steel sleeve. Analysis of the dynamic damage evolution of the samples is based on the measurement of the changes of ultrasonic wave velocities before and after the impact tests. The waveforms in the test bars, the stress strain curves, the confining pressure of the specimen, the dynamic compressive strength and other information about the samples are obtained during the SHPB experiments. The results of the tests show that the loading rate and stress states of the specimen apparently influence the damage evolution process in concretes. The dynamic damage evolutions are accelerated with the increase of the strain rate and are delayed significantly under the confined pressure.     

Finite element analysis of time-dependent inelastic deformation in the presence of transient thermal stresses

A finite element formulation for the solution of time-dependent inelastic deformation problems for metallic structures, in the presence of transient thermal stresses, is presented in this paper. A rate formulation of the equations is used and any of a number of recently proposed combined creep-plasticity constitutive models with state variables can be adopted to describe material behaviour. The computer program developed can solve planar (plane strain and stress) and axisymmetric problems. Using one of the above-mentioned constitutive models, numerical results are presented for several illustrative problems, and comparisons of results, using either the quasi-steady or the unsteady diffusion equation for the determination of the temperature field, are carried out.     

Large strain constitutive modelling and computation for isotropic,creep elastoplastic damage solids

A three-dimensional fully coupled creep elastoplastic damage model at finite strain for isotropic non-linear material is developed. The model is based on the thermodynamics of an irreversible process and the internal state variable theory. A hyperelastic form of stress–strain constitutive relation in conjunction with the multiplicative decomposition of the deformation gradient into elastic and inelastic parts is employed. The pressure-dependent plasticity with strain hardening and the damage model with two damage internal variables are particularly considered. The rounding of stress–strain curves appearing in cycling loading is reproduced by introduction of the creep mechanism into the model. A numerical integration procedure for the coupled constitutive equations with three hierarchical phases is proposed. A consistent tangent matrix with consideration of the fully coupled effects at finite strain is derived. Numerical examples are tested to demonstrate the capability and performance of the present model at large strain.     

复合固体推进剂黏弹性微裂纹损伤本构模型

为建立复合固体推进剂的损伤本构模型,在介观尺度上视其为微裂纹损伤,选取微裂纹密度为损伤内变量。在Abdel-Tawab本构方程的基础上,基于微裂纹均匀化理论,推导了损伤映射张量的一般形式。该张量通常具有非完全对称性,其物理意义是将真实应力空间中各向异性材料的多轴加载映射为等效应力空间中各向同性材料的更为复杂的多轴加载。其次,基于黏弹性动态裂纹扩展模型和裂纹扩展阻力曲线的概念,建立了损伤内变量的演化方程。该演化方程仅含4个物理意义明确的细观参数,并且参数的取值规律与宏观应力曲线的变化规律相一致。数值结果表明,建立的模型能够有效反映材料损伤的应变率、温度依赖性及各向异性特征,并且具有一定的蠕变损伤预测能力。     

On the Effect of Triaxial State of Stress on Ductility Using Nonlinear CDM Model

Ductility takes into account the material capability to plastically deform. This parameter is not only modified by temperature but it is strongly affected by the stress triaxiality that, in the case of positive hydrostatic stress, reduces the material strain to failure. Due to the importance of this parameter in engineering design many attempts to predict the evolution of ductility with stress triaxiality have been done. Here, a nonlinear continuum damage model, as proposed by the author, is used to obtain the evolution of material ductility with stress triaxiality. The expression found relates the strain to failure in multi-axial state of stress regime only to the uniaxial strain to failure, to the damage strain threshold, to the material Poisson's ratio, and, of course, to stress triaxiality. The proposed model was successfully verified comparing the predicted evolution of material ductility with the experimental data relative to several metals. The procedure for the damage parameters identification is also discussed in details.     

Implicit integration and consistent tangent modulus of a time‐dependent non‐unified constitutive model

This paper describes the implicit integration and consistent tangent modulus of an inelastic constitutive model with transient and steady strain rates, both of which are time‐ and temperature‐dependent; the transient rate is influenced by the evolution of back stress decomposed into parts, while the steady rate depends only on applied stress and temperature. Such a non‐unified model is useful for high‐temperature structural analysis and is practical owing to the ease in determining material constants. The implicit integration is shown to result in two scalar‐valued coupled equations, and the consistent tangent modulus is derived in a quite versatile form by introducing a set of fourth‐rank constitutive parameters into the discretized evolution rule of back stress. The constitutive model is, then, implemented in a finite element program and applied to a lead‐free solder joint analysis. It is demonstrated that the implicit integration is very accurate if the multilinear kinematic hardening model of Ohno and Wang is employed, and that the consistent tangent modulus certainly affords quadratic convergence to the Newton–Raphson iteration in solving nodal force equilibrium equations. Copyright © 2003 John Wiley & Sons, Ltd.     

Damage tolerance reliability analysis of automotive spot-welded joints

This paper develops a damage tolerance reliability analysis methodology for automotive spot-welded joints under multi-axial and variable amplitude loading history. The total fatigue life of a spot weld is divided into two parts, crack initiation and crack propagation. The multi-axial loading history is obtained from transient response finite element analysis of a vehicle model. A three-dimensional finite element model of a simplified joint with four spot welds is developed for static stress/strain analysis. A probabilistic Miner's rule is combined with a randomized strain-life curve family and the stress/strain analysis result to develop a strain-based probabilistic fatigue crack initiation life prediction for spot welds. Afterwards, the fatigue crack inside the base material sheet is modeled as a surface crack. Then a probabilistic crack growth model is combined with the stress analysis result to develop a probabilistic fatigue crack growth life prediction for spot welds. Both methods are implemented with MSC/NASTRAN and MSC/FATIGUE software, and are useful for reliability assessment of automotive spot-welded joints against fatigue and fracture.     

一种新的用于各向同性损伤模型的四参数等效应变

基于Hsieh-Ting-Chen强度破坏准则的思想,提出由于求解混凝土各向同性损伤模量的等效应变。首先通过混凝土试件材料特性以及强度试验资料来确定四参数,然后求解出由多轴应力状态下的应变退化得到的等效应变以及相应的损伤模量。对一混凝土立方体试件进行有限元分析,计算结果表明:采用该等效应变计算所得的应力应变曲线与试验曲线吻合性较好,从而说明了该等效应变的合理性,并且可说明利用该等效应变的损伤模型不但可以描述单轴应力状态下混凝土破坏的全过程曲线,而且能描述多轴应力状态下混凝土破坏的全过程曲线。该等效应变形式单一,计算简单。     

An efficient complex variable technique for the boundary element solution of stress concentration problems arising in notched creeping metallic structural components

The present paper presents, for the first time, a direct formulation of the boundary element method using complex variable theory techniques for the solution of the inelastic stress concentration problem arising in V-notched creeping metallic structural components in the presence of high temperature gradients. The application of this method is illustrated by obtaining stress and strain components distribution fo specimens with a single edge notch subjected to tension and bending loading conditions. The stress-strain constitutive relations are assumed to be described by an internal state variable viscoplastic model. Numerical examples presented and solved by the proposed methodology are further compared with other numerical solutions obtained for different loading rates.     

Crack growth in a new nickel-based superalloy at elevated temperature

Crack growth at elevated temperature has been simulated using the finite element method for sustained and cyclic loading conditions, representative of time-dependent and time-independent crack growth. Elastic-creep (EC) and elastic-plastic-creep (EPC) models have been used to simulate the crack growth under sustained loads at 650 and 725C. Crack mouth opening displacements as well as the evolution of the inelastic zones due to creep and plasticity have been obtained. Elastic-plastic finite element analysis has been carried out to simulate the crack growth under cyclic load using a constitutive model. Fatigue crack growth was simulated for plane stress, plane strain and generalized plane strain loading conditions. The influence of plasticity on the effective crack driving force was also examined.Creep damage was found to be very limited at both temperatures for this alloy. Plasticity-induced crack closure was found to be absent in plane strain or generalized plane strain conditions, overestimated in plane stress loading conditions by the conventional compliance method.     

循环荷载作用下粉砂岩疲劳流变损伤模型研究

针对现有流变模型难以有效描述循环荷载作用下岩石变形及疲劳损伤演化特征等问题,开展了粉砂岩循环加卸载试验,分析了不同上限荷载下岩石的流变规律与疲劳特性。基于Kachanov蠕变损伤理论建立损伤变量,引入一个带应变触发和应力阈值的黏塑性元件,与Burgers模型串联构建循环荷载作用下岩石疲劳流变损伤模型;将正弦波应力函数替换流变微分本构方程中的恒定应力,推导岩石在循环荷载下的一维、三维微分型损伤本构方程,再根据叠加原理得到模型的黏弹塑性流变损伤方程。适用性验证表明,新建模型不仅可以精确地反映循环加卸载过程中粉砂岩的衰减、稳态流变阶段,还可以有效地描述上限荷载高于疲劳强度时的加速流变阶段。通过粉砂岩疲劳损伤流变全过程定量化分析,提出加速流变阶段的临界损伤阈值和破坏失稳判据,并给出加速流变阶段的启始时间、持续时间及疲劳寿命预测方法,模型对岩体工程长期稳定性评价具有一定的理论指导意义。     

THERMAL FATIGUE UNDER MULTIAXIAL STRESSES

Thermal fatigue under multiaxial stresses has been investigated. Circular cylindrical specimens were tested under thermal fatigue which was in-phase with axial mechanical fatigue loading. The axial forces on the specimens were varied throughout the test programme, but all the temperature cycles were identical, so various biaxial stress and strain ratios were obtained. Straight thermal fatigue cracks occurred in different directions and also net-like crazes in various patterns were observed on the surfaces of the specimens. The transient temperature, stress and strain fields have been calculated with a thermal elasto-plastic Finite Element Method. Comparing test results with calculations, it appears that the patterns of thermal fatigue cracks are dependent on the stress state and the plastic strain state, not on the strain state. The direction of cracks is perpendicular to the maximum principal stress and the maximum plastic strain. Net-like thermal fatigue crazes will occur when one principal stress is about the same as the other one and one plastic strain component is approximately equal to another.     

Viscoplastic response and creep failure time prediction of polymers based on the transient network model

The nonlinear viscoelastic/viscoplastic response of polymeric materials is described by a new model based on previous works in terms of monotonic loading, stress–relaxation, and creep. In the proposed analysis, following a constitutive equation of viscoelasticity, based on the transient network theory, essential modifications are introduced, which account for the nonlinearity and viscoplasticity at small elastic and finite plastic strain regime. In addition, viscoplastic response is successfully analyzed by a proper kinematic formulation, which is combined with a functional form of the rate of plastic deformation. A three-dimensional constitutive equation is then derived for an isotropic incompressible medium. This analysis is capable of capturing the main aspects of inelastic response and the instability stage taking place at the tertiary creep, related to the creep failure. Model simulations described successfully the experimental data of polypropylene, which were performed elsewhere.     

A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete

An existing viscoelastic constitutive model which accounts for the effects of rate-dependent damage growth is described and applied successfully to characterize the uniaxial stress, constant strain rate behavior of asphalt concrete. The special case of an elastic continuum damage model with multiaxial loading, which is based upon thermodynamics of irreversible processes with internal state variables, is first reviewed and then it is shown how this model has been extended to a corresponding viscoelastic damage model through the use of an elastic-viscoelastic correspondence principle. The general mathematical model is next specialized to uniaxial loading. A rate-type evolution law, similar in form to a crack growth law for a viscoelastic medium, is adopted for describing the damage growth within the body. Results from laboratory tests of uniaxial specimens under axial tension at different strain rates are then shown to be consistent with the theory. The discussion of data analysis describes the specific procedure used here to obtain the material parameters in the constitutive model for uniaxial loading and how the method may be generalized for multiaxial loading.     

5182-O铝合金塑性成形性能表征

目的 结合复杂加载状态试验、塑性和损伤断裂本构模型及有限元应用,实现AA5182-O铝合金在复杂加载状态下塑性变形和损伤断裂行为的精确表征。方法 通过拉伸、剪切等试验,研究5182-O在剪切、单向拉伸、平面应变拉伸等复杂应力状态下的力学性能,应用pDrucker方程来表征其复杂加载状态下的塑性变形和损伤断裂特性。采用逆向工程方法实现pDrucker屈服方程和pDrucker断裂准则的精确标定。将标定后的塑性本构模型和损伤断裂准则应用到ABAQUS/Explicit中,预测不同试件的塑性变形和损伤断裂情况。结果 通过有限元模拟与试验结果的对比,发现有限元仿真准确预测了5182-O在复杂加载状态下的力-位移曲线和损伤断裂情况。结论 有限元模拟与试验结果的对比表明,pDrucker方程可以实现5182-O铝合金在复杂加载状态下塑性成形性能的精确表征。标定的pDrucker方程可应用于5182-O冲压成形过程的有限元分析、模具设计和工艺优化中。     

中低应变率加载下弱胶结软岩动力学特性

我国西部侏罗系煤层上覆巨厚白垩系富水软岩,为了解此类软岩在冲击荷载作用下的力学本构关系及损伤演化规律,利用Hopkinson压杆装置对干燥、饱和红砂软岩进行中低应变率下的冲击试验,结果表明:红砂软岩峰值应力、峰值应变均表现出明显的应变率效应,其中峰值应力与应变率呈指数关系;相同应变率下,干燥红砂软岩的强度大于饱和状态,对冲击荷载表现出更强的抵抗能力,但饱和红砂软岩的宏观破坏强度大于干燥状态;低应变率加载下,干燥红砂软岩出现负损伤;结合微观机理分析,低应变率下,水对红砂软岩的弱化作用占据主导地位,随着应变率的增大,在惯性效应和水的Stefan效应共同作用下,饱和红砂软岩的动态强度得到强化;基于Z-W-T模型和应变等效原理,建立了服从Weibull分布的损伤本构方程,经验证能很好的反映红砂软岩的动态本构关系,具有一定的工程实际意义。