On the mechanics of large inelastic deformations: noncoaxiality,axial effects in torsion and localization

Summary The macroscopic behavior of materials subjected to large deformations is investigated by considering the structure and mechanics of the single slip. Physically based constitutive relations for the plastic flow and material spin are then rigorously derived by resorting to the concept of non-coaxiality; yielding a class of vertex-type plasticity models incorporating features like noncoaxiality and large material rotation. It is shown that these phenomena are inherently coupled through the concept of plastic spin as it relates to the persistence of noncoaxiality. The implication of such phenomena to large deformations is investigated by examining their influence on the development of axial effects accompanying finite shear deformation, as well as on the onset of localized shear bands.     

Application of activated barrier hopping theory to viscoplastic modeling of glassy polymers

An established statistical mechanical theory of amorphous polymer deformation has been incorporated as a plastic mechanism into a constitutive model and applied to a range of polymer mechanical deformations. The temperature and rate dependence of the tensile yield of PVC, as reported in early studies, has been modeled to high levels of accuracy. Tensile experiments on PET reported here are analyzed similarly and good accuracy is also achieved. The frequently observed increase in the gradient of the plot of yield stress against logarithm of strain rate is an inherent feature of the constitutive model. The form of temperature dependence of the yield that is predicted by the model is found to give an accurate representation. The constitutive model is developed in two-dimensional form and implemented as a user-defined subroutine in the finite element package ABAQUS. This analysis is applied to the tensile experiments on PET, in some of which strain is localized in the form of shear bands and necks. These deformations are modeled with partial success, though adiabatic heating of the instability causes inaccuracies for this isothermal implementation of the model. The plastic mechanism has advantages over the Eyring process, is equally tractable, and presents no particular difficulties in implementation with finite elements.     

On the mechanics of large inelastic deformations: kinematics and constitutive modeling

Summary In this paper we examine the complexities associated with the kinematics of finite elastoplastic deformations and other issues related to the development of constitutive equations. The decomposition of the total strain and strain rate tensors into elastic and plastic constituents is investigated by considering both a multiplicative decomposition of the deformation gradient and an additive decomposition of the deformation vector field. Physically based definitions for the elastic and plastic strain rate tensors are given and compared with other values found in the literature. Constitutive equations for the plastic flow are derived by considering both a phenomenological-energy approach and a physically motivatedmesomechanical approach based on the double-slip idealization. It is shown that by resorting to the mechanics of the double slip, specific relations for the plastic stretching and plastic spin can be rigorously derived, taking into account the effect of noncoaxiality and material rotation. Finally, the implication of such effects to large deformations is examined in connection with the localization phenomenon.     

Numerical simulation of the plastic flow properties of iron single crystals

Summary The present paper deals with the numerical simulation of the plastic flow properties of iron single crystals as well as their influence on the macroscopic elastic-plastic deformation and localization behavior affected by superimposed hydrostatic pressure. Based on experimental observations the onset of plastic yielding on the microscale is described by an extended microscopic yield condition taking into account various microscopic stress components acting on the respective slip systems. In addition, to be able to compute inelastic deformations from a plastic potential, the latter is expressed in terms of workconjugate microscopic stress and strain measures which leads to a non-associated flow rule for the macroscopic plastic strain rate. On the numerical side, generalized functions for constitutive parameters will be used to be able to simulate the single crystal's microscopic deformation behavior observed in experiments. Estimates of the current microscopic stresses and strains are obtained via an efficient and remarkably stable plastic predictor-elastic corrector technique which is incorporated into a nonlinear finite element program. Numerical simulations of uniaxial tests show quantitatively the influence of hydrostatic pressure on current material data. Further numerical studies on the additional constitutive non-Schmid terms elucidate their effect on iron single crystal's macroscopic deformation and localization behavior.     

Finite element methods for the analysis of softening plastic hinges in beams and frames

This paper presents new finite element methods for the analysis of localized failures in plastic beams and frames in the form of plastic hinges. The hinges are modeled as discontinuities of the generalized displacements of the underlying Timoshenko beam/rod theory. Hinges accounting for a discontinuity in the transversal and longitudinal displacements and the rotation field are developed in this context. A multi-scale framework is considered in the incorporation of the dissipative effects of these discontinuities in the large-scale problem of a beam and a general frame. A localized softening cohesive law relating these generalized displacements with the stress resultants acting at the level of the cross section is effectively introduced in the frame response. The resulting models, referred to as localized models, are then able to capture the localized dissipation observed in the localized failures of these structural members, avoiding altogether the inconsistencies observed for classical models in the stress resultants with strain softening. The constructive approach followed in the development of these models leads naturally to the formulation of enhanced strain finite elements for their numerical approximation. In this context, we develop new finite elements incorporating the singular strains associated to the plastic hinges at the element level. A careful analysis is presented so the resulting finite elements avoid the phenomenon of stress locking, that is, an overstiff response in the softening of the hinge, not allowing for the full release of the stress. The accurate approximation of the kinematics of the hinges requires a strain enhancement linking the jumps in the deflection and the rotation fields, given the coupled definition of the transverse shear strain in these two fields. Different enhanced strain elements, involving different base finite elements and different enhancement strategies, are considered and analyzed in detail. Their performance are then compared in several representative numerical simulations. These analyses identify optimally enhanced finite elements for the accurate modeling the localized failures observed in common framed structures.Financial support for this research has been provided by the ONR under grant no. N00014-00-1-0306 with UC Berkeley. This support is gratefully acknowledged.     

Analysis of the Plastic Deformation Micrononuniformity Effect on the Strain Resistance of the Metal

A simplified model of micrononuniform deformation of the metal at a constant strain rate (uniaxial macrouniform stress state) is presented. The account of a temperature increase upon plastic strain and its localization due to the deformation micrononuniformity is shown to give a qualitative explanation of impact indentation results that point to a decrease in dynamic hardness over the range of high strain rates. Calculation results obtained with this model demonstrated that the localization effect was growing with strain rates and characteristic localized flow layer sizes. At small strains of the material, containing thin localized flow layers, the deformation micro- nonuniformity effect on strain resistance is insignificant.     

A class of variational strain‐localization finite elements

We present a class of finite elements for capturing sub‐grid localization processes such as shear bands and void sheets. The elements take the form of a double surface and deform in accordance with an arbitrary constitutive law. In particular they allow for the development of displacement and velocity jumps across volume element boundaries. The thickness of the localized zone is set by an additional field variable which is determined variationally. The localization elements are inserted, and become active, only when localized deformations become energetically favourable. The implementation presented in this work is three‐dimensional and allows for finite deformations. The versatility and predictive ability of the method are demonstrated through a simple shear test and the simulation of the dynamic impact of a pre‐notched C300 steel sample by a steel projectile. Copyright © 2004 John Wiley & Sons, Ltd.     

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。     

Hyper-elastic constitutive equations of conjugate stresses and strain tensors for the Seth-Hill strain measures

The use of hypo-elastic constitutive equations for large strains in nonlinear finite element applications usually requires special considerations. For example, the strain does not tend to zero upon unloading in some elastic loading-unloading closed cycles. Furthermore, these equations are based on objective material time rate tensors, which require incrementally objective algorithms for numerical applications and integration. Hyper-elastic constitutive equations on the other hand do not require such considerations. However, their behaviour for large elastic strains is important and may differ in tension and compression. In the present work, Hyper-elastic constitutive equations for the Seth-Hill strains and their conjugate stresses are explored as a natural generalisation of Hook’s law for finite elastic deformations. Based on the uniaxial and simple shear tests, the response of the material for different constitutive equations is examined. Together with an objective rate model, the effect of different constitutive laws on Cauchy stress components is compared. It is shown that the constitutive equation based on logarithmic strain and its conjugate stress gives results closer to that of the rate model. In addition, the use of Biot stress-strain pairs for a bar element results in an elastic spring which obeys the Hook’s law even for large deformations and has the same behaviour in both tension and compression. The effect of the constitutive equation on the volume change of the material has also been considered here.     

Elastoviscohysteresis constitutive law in convected coordinate frames: application to finite deformation shear tests

 The aim of this contribution is to present a theoretical and numerical model applied to the finite deformations of elastoviscoplastic metallic materials at room temperature. The constitutive equations are integrated numerically in the context of a finite element formulation and numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms. The model is then compared to experimental results obtained from shear tests of a mild steel for various strain rates. Model predictions show a general agreement with experimental data for monotonic and cyclic conditions on this particular material and reproduce correctly the fast plastic loadings and unloadings in the large deformation field. Received 8 June 2001     

考虑应变梯度及刚度劣化的剪切带局部变形分析

基于梯度塑性理论,研究了应变软化阶段的刚度劣化对剪切带内部的局部应变及相对剪切位移的影响。剪切带被看作一维剪切问题,本构关系为线弹性及线性应变软化。考虑刚度劣化后,剪切带的弹性应变由弹性剪切模量、损伤变量及残余剪切模量确定。剪切带的非局部总应变由双线性的本构关系确定。将非局部总应变减去弹性应变,可得剪切带的非局部塑性应变。剪切带非局部塑性应变与流动应力及损伤变量等参数有关,此关系即为在经典弹塑性理论框架之内的考虑刚度劣化的屈服函数。将二阶应变梯度项引入该函数,可得剪切带内部的局部塑性剪切应变及局部总剪切应变的分布规律。对局部塑性剪切应变积分,得到了局部塑性剪切位移。结果表明:考虑了刚度劣化后,剪切带内部的弹性剪切应变及位移增加,而局部塑性剪切应变及位移降低。若不考虑刚度劣化,理论结果可蜕化为以前的结果。理论结果与岩石局部变形的观测结果在定性是一致的。     

A study on micro-mechanical deformation behaviour of heat-treated 20MnMoNi55 steel

Several heat treatment procedures are designed considering critical temperatures of phase transformation evaluated through dilatometric testing of 20MnMoNi55 steel to transform low carbon bainitic as-received material into ferrite-martensite dual-phase steels consisting of varied martensite fractions. A thorough metallographic study correlated with the micro-hardness of constituent phases ensures morphological characteristics along with its fractional variations in as-received and dual-phase steels. The impact of fractional variation in constituent phases on the uniaxial monotonic deformation characteristics of dual-phase steels has been observed with a correlation study between experimental tensile and finite element simulated results. Therefore, a physical-based model with a 2-dimensional representative volume element has been established, addressing actual morphological characteristics obtained from metallographic studies. Moreover, the constitutive flow behaviours of ferrite and martensite are also derived from the dislocation-based hardening model to address the actual deformation phenomenon. Finally, an inhomogeneous deformation behaviour among constituent phases and localization of plastic strain in ferrite matrix has been observed with von-Mises stress, and equivalent plastic strain distribution through finite element simulated results. This phenomenon is again confirmed with kernel average misorientation mapping and geometrically necessary dislocation density evaluation through electron backscattered diffraction of tensile samples subjected to different degrees of plastic strain.     

Localized deform behavior of AZ31B magnesium alloy by numerical simulation

The dynamic compression behavior of AZ31B magnesium alloy with hat shaped specimen was investigated at high strain rate in this paper. Based on the Johnson‐cook constitutive model and fracture model, the interaction of temperature, stress and strain fields of AZ31B magnesium alloy with hat shaped specimen were numerically simulated by using ANSYS/LS‐DYNA software under different strain rates, which was validated by experiment. It is found that the plastic strain is highly concentrated on the corner of the hat shaped specimen, which leads to large localized deformation. The voids are nucleated and extended by compression stress. Work harden effect is caused by remained plastic strain, which is located around adiabatic shear band. The stress collapse is discovered in gauge section, which is also discovered in experiment. Thermal soften effect is suppressed with the strain rate increased.     

Assessment of small fatigue crack growth driving forces in single crystals with and without slip bands

Strain localization under low amplitude cyclic loading is a manifestation of plastic irreversible deformation associated with early crack growth. However, traditional constitutive models cannot usually reproduce strain localization in smooth single crystals, which can affect crack growth predictions for crystallographic fatigue cracks. This work analyzes the influence of bands of localized plastic shear strain on the cyclic crack tip displacement and on a fatigue indicator parameter by making special provision of a crack along the interface of a deformation band. Furthermore, the quality of local and volume-averaged fatigue indicator parameters are assessed using finite element models of a Cu single crystal cycled to induce plastic deformation under multiple loading conditions.     

Constitutive equations for the nonlinear viscoelastic and viscoplastic behavior of thermoplastic elastomers

Observations are reported in uniaxial tensile tests with constant strain rates at moderate finite deformations, as well as in creep and relaxation tests on a thermoplastic elastomer (ethylene–octene copolymer) at room temperature. A constitutive model is developed for the viscoelastic and viscoplastic responses of a polymer at arbitrary three-dimensional deformations with finite strains. A thermoplastic elastomer is treated as an incompressible heterogeneous transient network of strands. Its viscoelastic behavior is associated with separation of active strands from their junctions and merging of dangling strands with the network. The viscoplastic response reflects sliding of junctions between strands with respect to their reference positions. Stress–strain relations are derived by using the laws of thermodynamics. They involve six adjustable parameters that are found by fitting the experimental data. To examine the accuracy of the model predictions, plane-strain compressive tests with constant strain rates and relaxation tests at compression are performed. Good agreement is demonstrated between the observations and the results of numerical simulation.     

Locally enhanced Voronoi cell finite element model (LE‐VCFEM) for simulating evolving fracture in ductile microstructures containing inclusions

Ductile heterogeneous materials such as cast aluminum alloys undergo catastrophic failure that initiates with particle fragmentation, which evolves with void growth and coalescence in localized bands of intense plastic deformation and strain softening. The Voronoi cell finite element model (VCFEM), based on the assumed stress hybrid formulation, is unable to account for plastic strain‐induced softening. To overcome this shortcoming of material softening due to plastic strain localization, this study introduces a locally enhanced VCFEM (LE‐VCFEM) for modeling the very complex phenomenon of ductile failure in heterogeneous metals and alloys. In LE‐VCFEM, finite deformation displacement elements are adaptively added to regions of localization in the otherwise assumed stress‐based hybrid Voronoi cell finite element to locally enhance modeling capabilities for ductile fracture. Adaptive h‐refinement is used for the displacement elements to improve accuracy. Damage initiation by particle cracking is triggered by a Weibull model. The nonlocal Gurson–Tvergaard–Needleman model of porous plasticity is implemented in LE‐VCFEM to model matrix cracking. An iterative strain update algorithm is used for the displacement elements. The LE‐VCFEM code is validated by comparing with results of conventional FE codes and experiments with real materials. The effect of various microstructural morphological characteristics is also investigated. Copyright © 2008 John Wiley & Sons, Ltd.     

Modeling the impact behavior of AD85 ceramic under multiaxial loading

This paper presents an advanced constitutive model to describe the complex behavior of ceramic materials under impact loading conditions. The governing equations utilize a set of microphysically based constitutive relationships to model deformation and damage processes in a ceramic. The total strain is decomposed into elastic, plastic and microcracking components. The model parameters for AD85 ceramic were determined using the data from split Hopkinson bar and bar-on-bar experiments under uniaxial stress state and plate impact experiment under uniaxial strain state. To further validate the generality of the model parameters, modeling of a diagnostic ballistic experiment in which a steel projectile impacted a AD85 ceramic-front-faced thick aluminium plate, was considered. In this experiment, stress histories were measured in the target by embedded manganin and carbon stress gauges. The results from the numerical simulations of the ballistic experiment using a shock-wave propagation based finite element code, successfully matched the measured stress history.     

Discontinuous displacement approximation for capturing plastic localization

It is proposed to capture localized plastic deformation via the inclusion of regularized displacement discontinuities at element boundaries (interfaces) of the finite element subdivision. The regularization is based on a kinematic assumption for an interface that resembles that which is pertinent to the classical shear band concept. As a by-product of the regularization, an intrinsic band width is introduced as a ‘constitutive’ property rather than a geometric feature of the finite element mesh. In this way the spurious mesh sensitivity, which is obtained when the displacement approximation is continuous, can be avoided. Another consequence is that the interfacial relation between the elements is derived directly from the conventional constitutive properties of the continuously deforming material. An interesting feature is that the acoustic tensor will not only play a role for diagnosing discontinuous bifurcation but will also serve as the tangent stiffness tensor of the interface (up to within a scalar factor). An analytical investigation of the behaviour of the interface is carried out and it is shown that dilatation may indeed accompany slip within a ‘shear’ band for a general plasticity model. The significance of proper mesh alignment is demonstrated for a simple problem in plane strain and plane stress. It is shown that a unique structural post-peak response (in accordance with non-linear fracture mechanics) can be achieved when the plastic softening modulus is properly related to the bandwidth. The paper concludes with a numerical simulation of the gradual development of a shear band in a soil slope.