Modelling the torsion of thin metal wires by distortion gradient plasticity

Under small strains and rotations, we apply a phenomenological higher-order theory of distortion gradient plasticity to the torsion problem, here assumed as a paradigmatic benchmark of small-scale plasticity. Peculiar of the studied theory, proposed about ten years ago by Morton E. Gurtin, is the constitutive inclusion of the plastic spin, affecting both the free energy and the dissipation. In particular, the part of the free energy, called the defect energy, which accounts for Geometrically Necessary Dislocations, is a function of Nye's dislocation density tensor, dependent on the plastic distortion, including the plastic spin. For the specific torsion problem, we implement this distortion gradient plasticity theory into a Finite Element (FE) code characterised by implicit (Backward Euler) time integration, numerically robust and accurate for both viscoplastic and rate-independent material responses. We show that, contrariwise to other higher-order theories of strain gradient plasticity (neglecting the plastic spin), the distortion gradient plasticity can predict some strengthening even if a quadratic defect energy is chosen. On the basis of the results of many FE analyses, concerned with (i) cyclic loading, (ii) switch in the higher-order boundary conditions during monotonic plastic loading, (iii) the use of non-quadratic defect energies, and (iv) the prediction of experimental data, we mainly show that (a) including the plastic spin contribution in a gradient plasticity theory is highly recommendable to model small-scale plasticity, (b) less-than-quadratic defect energies may help in describing the experimental results, but they may lead to anomalous cyclic behaviour, and (c) dissipative (unrecoverable) higher-order finite stresses are responsible for an unexpected mechanical response under non-proportional loading.     

Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories

This paper focuses on the unification of two frequently used and apparently different strain gradient crystal plasticity frameworks: (i) the physically motivated strain gradient crystal plasticity models proposed by Evers et al. [2004a. Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids 52, 2379-2401; 2004b. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures 41, 5209-5230] and Bayley et al. [2006. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structure 43, 7268-7286; 2007. A three dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philosophical Magazine 87, 1361-1378] (here referred to as Evers-Bayley type models), where a physical back stress plays the most important role and which are further extended here to deal with truly large deformations, and (ii) the thermodynamically consistent strain gradient crystal plasticity model of Gurtin (2002-2008) (here referred to as the Gurtin type model), where the energetic part of a higher order micro-stress is derived from a non-standard free energy function. The energetic micro-stress vectors for the Gurtin type models are extracted from the definition of the back stresses of the improved Evers-Bayley type models. The possible defect energy forms that yield the derived physically based micro-stresses are discussed. The duality of both type of formulations is shown further by a comparison of the micro-boundary conditions. As a result, this paper provides a direct physical interpretation of the different terms present in Gurtin's model.     

Bauschinger and size effects in thin-film plasticity due to defect-energy of geometrical necessary dislocations

The Bauschinger and size effects in the thinfilm plasticity theory arising from the defect-energy of geometrically necessary dislocations (GNDs) are analytically investigated in this paper. Firstly, this defect-energy is deduced based on the elastic interactions of coupling dislocations (or pile-ups) moving on the closed neighboring slip plane. This energy is a quadratic function of the GNDs density, and includes an elastic interaction coefficient and an energetic length scale L. By incorporating it into the work- conjugate strain gradient plasticity theory of Gurtin, an energetic stress associated with this defect energy is obtained, which just plays the role of back stress in the kinematic hardening model. Then this back-stress hardening model is used to investigate the Bauschinger and size effects in the tension problem of single crystal Al films with passivation layers. The tension stress in the film shows a reverse dependence on the film thickness h. By comparing it with discrete-dislocation simulation results, the length scale L is determined, which is just several slip plane spacing, and accords well with our physical interpretation for the defect- energy. The Bauschinger effect after unloading is analyzed by combining this back-stress hardening model with a friction model. The effects of film thickness and pre-strain on the reversed plastic strain after unloading are quantified and qualitatively compared with experiment results.     

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.     

Cyclic plasticity modeling with the distribution of non-linear relaxations approach

Motivated by the distribution of non-linear relaxation (DNLR) approach, a phenomenological model is proposed in order to describe the cyclic plasticity behavior of metals under proportional and non-proportional loading paths with strain-controlled conditions. Such a model is based on the generalization of the Gibbs's relationship outside the equilibrium of uniform system and the use of the fluctuation theory to analyze the material dissipation due to its internal reorganization. The non-linear cyclic stress–strain behavior of metals notably under complex loading is of particular interest in this study. Since the hardening effects are described appropriately and implicitly by the model, thus, a host of inelastic behavior of metals under uniaxial and multiaxial cyclic loading paths are successfully predicted such as, Bauschinger, strain memory effects as well as additional hardening. After calibrating the model parameters for two metallic materials, the model has demonstrated obviously its ability to describe the cyclic elastic-inelastic behavior of the nickel base alloy Waspaloy and the stainless steel 316L. The model is then implemented in a commercial finite element code simulating the cyclic stress–strain response of a thin-walled tube specimen. The numerical responses are in good agreement with experimental results.     

压缩载荷作用下岩石的细观损伤和塑性研究

建立岩石微裂纹扩展的细观力学模型,研究了岩石的细观损伤和塑性性质.压缩载荷下微裂纹尖端翼裂纹稳定扩展表征岩石的细观损伤,采用应变能密度准则求解复合型断裂的翼裂纹扩展长度,微裂隙统计的二参数Weibull函数模型反映绝对体积应变对微裂纹分布数目影响,进而用翼裂纹扩展所表征的应力释放体积和微裂纹数目来表示含有微裂隙的岩石损伤演化变量;宏观塑性屈服函数采用Voyiadjis等的等效塑性应变的硬化函数,反映了塑性内变量对硬化函数的影响;建立岩石模型的本构关系及其数值算法,并用回映隐式积分算法编制了模型的本构程序.分析弹塑性损伤模型的围压对岩石损伤的影响,并从围压和短微裂隙长度等因素分析模型的岩石的损伤和宏观塑性特性.     

Effect of defect energy on strain-gradient predictions of confined single-crystal plasticity

Gurtin recently proposed a strain-gradient theory for crystal plasticity in which the gradient effect originates from a defect energy that characterizes energy storage due to the presence of a net Burgers vector. Here we consider a number of different possibilities for this energy: specifically, working within a simple two-dimensional framework, we compare predictions of the theory with results of discrete-dislocation simulations of stress relaxation in thin films. Our objective is to investigate which specific defect energies are capable of capturing the size-dependent response of such systems for different crystal orientations.     

Benchmark experiments and characteristic cyclic plasticity deformation

Key issues in cyclic plasticity modeling are discussed based upon representative experimental observations on several commonly used engineering materials. Cyclic plasticity is characterized by the Bauschinger effect, cyclic hardening/softening, strain range effect, nonproporitonal hardening, and strain ratcheting. Additional hardening is identified to associate with ratcheting rate decay. Proper modeling requires a clear distinction among different types of cyclic plasticity behavior. Cyclic hardening/softening sustains dependent on the loading amplitude and loading history. Strain range effect is common for most engineering metallic materials. Often, nonproportional hardening is accompanied by cyclic hardening, as being observed on stainless steels and pure copper. A clarification of the two types of material behavior can be made through benchmark experiments and modeling technique. Ratcheting rate decay is a common observation on a number of materials and it often follows a power law relationship with the number of loading cycles under the constant amplitude stress controlled condition. Benchmark experiments can be used to explore the different cyclic plasticity properties of the materials. Discussions about proper modeling are based on the typical cyclic plasticity phenomena obtained from testing several engineering materials under various uniaxial and multiaxial cyclic loading conditions. Sufficient experimental evidence points to the unambiguous conclusion that none of the hardening phenomena (cyclic hardening/softening, strain range effect, nonproportional hardening, and strain hardening associated with ratcheting rate decay) is isotropic in nature. None of the hardening behavior can be properly modeled with a change in the yield stress.     

An experimental study of cyclic deformation of extruded AZ61A magnesium alloy

Thin-walled tubular specimens were employed to study the cyclic deformation of extruded AZ61A magnesium alloy. Experiments were conducted under fully reversed strain-controlled tension-compression, torsion, and combined axial-torsion in ambient air. Mechanical twinning was found to significantly influence the inelastic deformation of the material. Cyclic hardening was observed at all the strain amplitudes under investigation. For tension-compression at strain amplitudes higher than 0.5%, the stress-strain hysteresis loop was asymmetric with a positive mean stress. This was associated with mechanical twinning in the compression phase and detwinning in the subsequent tension phase. Under cyclic torsion, the stress-strain hysteresis loops were symmetric although mechanical twinning was observed at high shear strain amplitudes. When the material was subjected to combined axial-torsion loading, the alternative occurrence of twinning and detwinning processes under axial stress resulted in asymmetric shear stress-shear strain hysteresis loops. Nonproportional hardening was not observed due to limited number of slip systems and the formation of mechanical twins. Microstructures after the stabilization of cyclic deformation were observed and the dominant mechanisms governing cyclic deformation were discussed. Existing cyclic plasticity models were discussed in light of their capabilities for modeling the observed cyclic deformation of the magnesium alloy.     

Inspection of free energy functions in gradient crystal plasticity

The dislocation density tensor computed as the cud of plastic distortion is regarded as a new constitutive variable in crystal plasticity. The dependence of the free energy function on the dislocation density tensor is explored starting from a quadratic ansatz. Rank one and logarithmic dependencies are then envisaged based on considerations from the statistical theory of dislocations. The rele- vance of the presented free energy potentials is evaluated from the corresponding analytical solutions of the periodic two-phase laminate problem under shear where one layer is a single crystal material undergoing single slip and the second one remains purely elastic.     

Cyclic plastic deformation and cyclic hardening/softening behavior in 304LN stainless steel

Low cycle fatigue experiments have been conducted on 304LN stainless steel in ambient air at room temperature. Uniaxial ratcheting behavior has also been studied on this material and in both engineering and true stress controlling modes. It is shown that material’s cyclic hardening/softening behavior in low cycle fatigue and in ratcheting is dependent not only on material but also on the loading condition. Improvement of ratcheting life and mean stress dependent hardening are observed in the presence of mean stress. A method based on the strain energy density (SED) is used to represent cyclic hardening/softening behavior of the material in this work. The decrease of SED with cycles is an indication that the life in low cycle fatigue and in ratcheting is improved. The SED represents the area of the hysteresis loops.     

Displacement fields and self-energies of circular and polygonal dislocation loops in homogeneous and layered anisotropic solids

There are large classes of materials problems that involve the solutions of stress, displacement, and strain energy of dislocation loops in elastically anisotropic solids, including increasingly detailed investigations of the generation and evolution of irradiation induced defect clusters ranging in sizes from the micro- to meso-scopic length scales. Based on a two-dimensional Fourier transform and Stroh formalism that are ideal for homogeneous and layered anisotropic solids, we have developed robust and computationally efficient methods to calculate the displacement fields for circular and polygonal dislocation loops. Using the homogeneous nature of the Green tensor of order −1, we have shown that the displacement and stress fields of dislocation loops can be obtained by numerical quadrature of a line integral. In addition, it is shown that the sextuple integrals associated with the strain energy of loops can be represented by the product of a pre-factor containing elastic anisotropy effects and a universal term that is singular and equal to that for elastic isotropic case. Furthermore, we have found that the self-energy pre-factor of prismatic loops is identical to the effective modulus of normal contact, and the pre-factor of shear loops differs from the effective indentation modulus in shear by only a few percent. These results provide a convenient method for examining dislocation reaction energetic and efficient procedures for numerical computation of local displacements and stresses of dislocation loops, both of which play integral roles in quantitative defect analyses within combined experimental–theoretical investigations.     

基于应变能的复杂应力场低周疲劳分析

塑性应变能使材料微观组织结构发生不可逆变化,从而引起等效宏观应力,该应力随循环加载而增大．假定材料疲劳源处破坏是由最大拉应力引起的,最大等效宏观应力与外加应力叠加达到材料本征断裂应力时形成微裂纹．微裂纹引起上述两部分应力变化,继续加载直至宏观裂纹出现,从而得到材料的疲劳寿命．本文所建立的多轴疲劳寿命公式包含材料参数、拉应力以及塑性应变能等,以上数据可通过单轴疲劳数据和有限元方法获得．通过对SM45C材料的计算验证,表明该模型对多轴随机应变加载低周疲劳寿命,具有良好的预测结果．     

显式模拟类橡胶材料Mullins效应滞回圈

通过显式、直接的方法提出一个多轴可压缩应变能函数,用来模拟类橡胶材料在加载——卸载作用下,由于Mullins效应而产生的应力——应变滞回圈. 本文的创新点在于将表征能量耗散的变量引入到应变能函数.新的弹性势具有以下两个特点:第一,在加载情况下,新引入的变量不会对弹性势产生任何影响,因此,只要给出合适的形函数显式表达,3个基准实验,包括单轴拉伸和压缩,等双轴拉伸和压缩,以及平面应变,都可精确模拟;第二,新引入的变量在卸载情况下将被激活.在不同的卸载应力下,变量将发生改变,从而影响弹性势,使其最终产生不同的应力——应变关系卸载曲线,与对应的加载曲线共同构成应力——应变滞回圈.通过对Mullins效应实验数据进行分析和研究,得出了卸载形函数在不同卸载应力下变化的规律,并预测不同卸载应力下的应力——应变关系.最后,我们将得到精确匹配实验数据的数值模拟结果,从而证明本文方法不仅可以精确匹配至少3个基准实验,还可以模拟和预测类橡胶材料在加载——卸载作用下由于Mullins效应而产生的滞回圈.      

Crystallographic failure analysis of film near cooling hole under temperature gradient of nickel-based single crystal superalloys

In this paper, a unit cell model with a film cooling hole has been set up to analyze the crystallographic stress characterization and failure behavior under temperature gradient of nickel-base single crystallographic superalloys (SC). The aim of this work is to study the failure behavior of SC blades with film cooling. The distribution of cooling air pressure on the hole side surface and the distribution of the temperature around the hole are obtained from the fluid analysis. The result of the temperature distribution is then transferred to the finite element model (cell model) by the interpolation method. The cell model is analyzed by the crystallographic rate dependent finite element method (FEM). Special attention is put on the influence of temperature gradient. The influence of the loading boundaries, i.e. displacement loading and stress loading, on the stress characterization around hole is also taken into consideration. The results show that temperature gradient hole has much influence on the stress characterization. Different types of loading boundaries result in different types of stress and strain distributions. There is clear stress concentration near the hole under displacement loading, while there is clear strain concentration under stress loading. The failure characterization has been studied by the strain energy density criterion. It is shown that the temperature gradient has influence on the failure behavior.     

Discontinuous bifurcation analysis in fracture energy-based gradient plasticity for concrete

Conditions for discontinuous bifurcation in limit states of selective non-local thermodynamically consistent gradient theory for quasi-brittle materials like concrete are evaluated by means of both geometrical and analytical procedures. This constitutive formulation includes two internal lengths, one related to the strain gradient field that considers the degradation of the continuum in the vicinity of the considered material point. The other characteristic length takes into account the material degradation in the form of energy release in the cracks during failure process evolution.The variation from ductile to brittle failure in quasi-brittle materials is accomplished by means of the pressure dependent formulation of both characteristic lengths as described by Vrech and Etse (2009).In this paper the formulation of the localization ellipse for constitutive theories based on gradient plasticity and fracture energy plasticity is proposed as well as the explicit solutions for brittle failure conditions in the form of discontinuous bifurcation. The geometrical, analytical and numerical analysis of discontinuous bifurcation condition in this paper are comparatively evaluated in different stress states and loading conditions.The included results illustrate the capabilities of the thermodynamically consistent selective non-local gradient constitutive theory to reproduce the transition from ductile to brittle and localized failure modes in the low confinement regime of concrete and quasi-brittle materials.     

The coupled effects of plastic strain gradient and thermal softening on the dynamic growth of voids

This paper examines the combined effects of temperature, strain gradient and inertia on the growth of voids in ductile fracture. A dislocation-based gradient plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99] is applied, and temperature effects are incorporated. Since a strong size-dependence is introduced into the dynamic growth of voids through gradient plasticity, a cut-off size is then set by the stress level of the applied loading. Only those voids that are initially larger than the cut-off size can grow rapidly. At the early stages of void growth, the effects of strain gradients greatly increase the stress level. Therefore, thermal softening has a strong effect in lowering the threshold stress for the unstable growth of voids. Once the voids start rapid growth, however, the influence of strain gradients will decrease, and the rate of dynamic void growth predicted by strain gradient plasticity approaches that predicted by classical plasticity theories.     

Finite element implementation and numerical issues of strain gradient plasticity with application to metal matrix composites

A framework of finite element equations for strain gradient plasticity is presented. The theoretical framework requires plastic strain degrees of freedom in addition to displacements and a plane strain version is implemented into a commercial finite element code. A couple of different elements of quadrilateral type are examined and a few numerical issues are addressed related to these elements as well as to strain gradient plasticity theories in general. Numerical results are presented for an idealized cell model of a metal matrix composite under shear loading. It is shown that strengthening due to fiber size is captured but strengthening due to fiber shape is not. A few modelling aspects of this problem are discussed as well. An analytic solution is also presented which illustrates similarities to other theories.