Mean field modelling of the plastic behaviour of co-continuous dual-phase alloys with strong morphological anisotropy

This work addresses the plastic flow properties of a composite material in which the reinforcing phase is continuous and cannot be suitably represented by isolated ellipsoidal inclusions. The dual-phase metal under consideration is composed of a network of Inconel-601 fibres infiltrated by pure aluminium. Hence, both phases exhibit elastic–plastic behaviour and are continuous in the three dimensions of space. The fibre network presents a large morphological anisotropy that is reflected in the mechanical response of the composite. The modelling is based on Eshelby’s equivalent inclusion theory. Strain partitioning between the phases is computed incrementally based on tangent operators derived from the isotropic response of individual phases. Assessment of the model relies on extensive experimental data. Uniaxial tensile tests, involving measurement of the Lankford coefficient, have been performed at various temperatures on samples containing different volume fractions of fibres. Measurement of the phase stresses by neutron diffraction supplements the information provided by the macroscopic stress–strain curves. It is demonstrated that predictions are valid only when the micro–macro averaging scheme accounts for the co-continuous character of the constitutive phases.     

Micromechanics-based model for cement-treated clays

Cementation is produced by mixing a certain amount of cement with the saturated clay. The purpose of this paper is to model the cementation effect on the mechanical behavior of cement-treated clay. A micromechanical stress-strain model is developed considering explicitly the cementation at inter-cluster contacts. The inter-cluster bonding and debonding during mechanical loading are introduced in two ways: an additional cohesion in the shear sliding and a higher yield stress in normal compression. The model is used to simulate isotropic compression and undrained triaxial tests under various confining stresses on cement-treated Singapore clay with various cement contents. The applicability of the present model is evaluated through comparisons between numerical and experimental results. The evolution of local stresses and local strains in inter-cluster planes is discussed in order to explain the induced anisotropy due to debonding at contact level under the applied loads.     

Constitutive response of idealized granular media under the principal stress axes rotation

This paper is dedicated to the understanding of the phenomena, which give rise to anisotropy and non-coaxiality in granular materials. In achieving three-dimensional numerical simulation under static condition of granular media, granular element method (GEM) is adopted in this study. The method has been incorporated into the so-called mathematical homogenization theory for quasi-static equilibrium problems, which enables us to obtain the macroscopic/phenomenological inelastic deformation response of a representative volume element (RVE). To examine the anisotropic macroscopic deformation properties of the assumed RVE, which is solved by granular element method (GEM), a series of numerical experiments involving the pure rotation of the principal stress axes are carried out, and its results are discussed in relation to induced anisotropy and non-coaxiality.     

Microstructural modelling of stress-dependent behaviour of clay

The purpose of this paper is to investigate the stress-dependent behaviour of clay during drained and undrained shearing by means of a micromechanical approach. A new micromechanical stress–strain model is developed for clay using the approach developed in earlier studies by Chang and Hicher [Chang, C.S., Hicher, P.Y., 2005. An elastic–plastic model for granular materials with microstructural consideration. International Journal of Solids and Structures 42(14), 4258–4277]. In order to model the extension test on a K₀ consolidated sample, a formulation is developed to account for the stress reversal on a contact plane. The model is then used to simulate numerous stress-path tests on Lower Cromer Till and kaolin clay, including triaxial compression and extension tests, under both undrained and drained conditions, with different K₀ consolidation, and different over-consolidation ratios. The applicability of the present model is evaluated through comparisons between the predicted and the measured results. The evolution of local stresses and local strains at inter-particle planes are discussed in order to explain the stress-induced anisotropy due to externally applied load. All simulations have demonstrated that the proposed micromechanical approach is capable of modelling the stress-induced anisotropy and other major features of the complex behaviour in clay.     

Finite element modeling of elasto-plastic contact between rough surfaces

This paper presents a finite element calculation of frictionless, non-adhesive, contact between a rigid plane and an elasto-plastic solid with a self-affine fractal surface. The calculations are conducted within an explicit dynamic Lagrangian framework. The elasto-plastic response of the material is described by a J₂ isotropic plasticity law. Parametric studies are used to establish general relations between contact properties and key material parameters. In all cases, the contact area A rises linearly with the applied load. The rate of increase grows as the yield stress σ_y decreases, scaling as a power of σ_y over the range typical of real materials. Results for A from different plasticity laws and surface morphologies can all be described by a simple scaling formula. Plasticity produces qualitative changes in the distributions of local pressures in the contact and of the size of connected contact regions. The probability of large local pressures is decreased, while large clusters become more likely. Loading-unloading cycles are considered and the total plastic work is found to be nearly constant over a wide range of yield stresses.     

Influence of anisotropic elasticity on the mechanical properties of fivefold twinned nanowires

Previous atomistic simulations and experiments have shown an increased Young's modulus and yield strength of fivefold twinned (FT) face-centered cubic metal nanowires (NWs) when compared to single crystalline (SC) NWs of the same orientation. Here we report the results of atomistic simulations of SC and FT Ag, Al, Au, Cu and Ni NWs with diameters between 2 and 50 nm under tension and compression. The simulations show that the differences in Young's modulus between SC and FT NWs are correlated with the elastic anisotropy of the metal, with Al showing a decreased Young's modulus. We develop a simple analytical model based on disclination theory and constraint anisotropic elasticity to explain the trend in the difference of Young's modulus between SC and FT NWs. Taking into account the role of surface stresses and the elastic properties of twin boundaries allows to account for the observed size effect in Young's modulus. The model furthermore explains the different relative yield strengths in tension and compression as well as the material and loading dependent failure mechanisms in FTNWs.     

A constrained theory of magnetoelasticity

A simple variational theory for the macroscopic behavior of materials with high anisotropy is derived rigorously from micromagnetics. The derivation leads to a constrained theory in which the state of strain and magnetization lies very near the ‘energy wells’ on most of the body. When specialized to ellipsoidal specimens and constant applied field and stress, the theory becomes a finite dimensional quadratic programming problem. Streamlined methods for solving this problem are given. The theory is illustrated by a prediction of the magnetoelastic behavior of the giant magnetostrictive material Tb_0.3Dy_0.7Fe₂. The theory embodies precisely the assumptions that have been postulated for ideal ferromagnetic shape memory, in which the magnetization stays rigidly attached to the easy axes of a martensitic material in the martensitic phase. More generally, the framework can be viewed as a prototype for the derivation of constrained theories for materials that change phase, and whose free-energy density grows steeply away from its minima.     

Fabric and connectivity as field descriptors for deformations in granular media

Granular materials involve microphysics across the various scales giving rise to distinct behaviours of geomaterials, such as steady states, plastic limit states, non-associativity of plastic and yield flow, as well as instability of homogeneous deformations through strain localization. Incorporating such micro-scale characteristics is one of the biggest challenges in the constitutive modelling of granular materials, especially when micro-variables may be interdependent. With this motivation, we use two micro-variables such as coordination number and fabric anisotropy computed from tessellation of the granular material to describe its state at the macroscopic level. In order to capture functional dependencies between micro-variables, the correlation between coordination number and fabric anisotropy limits is herein formulated at the particle level rather than on an average sense. This is the essence of the proposed work which investigates the evolutions of coordination number distribution (connectivity) and anisotropy (contact normal) distribution curves with deformation history and their inter-dependencies through discrete element modelling in two dimensions. These results enter as probability distribution functions into homogenization expressions during upscaling to a continuum constitutive model using tessellation as an abstract representation of the granular system. The end product is a micro-mechanically inspired continuum model with both coordination number and fabric anisotropy as underlying micro-variables incorporated into a plasticity flow rule. The derived plastic potential bears striking resemblance to cam–clay or stress–dilatancy-type yield surfaces used in soil mechanics.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

考虑颗粒转矩的接触网络诱发各向异性分析

颗粒材料的宏观力学行为与接触网络的组构各向异性密切相关, 根据接触点的滑动与否、转动与否和强弱力情况, 可以将颗粒间的接触系统分为不同的子接触网络. 一般而言, 不同的子接触网络在颗粒体系中的传力机制不同, 对宏观力学响应的贡献也有不同. 采用离散单元法(discrete element method, DEM)模拟了不同抗转动系数$\mu_r$下颗粒材料三轴剪切试验, 分析了剪切过程中不同子接触网络的组构张量的演变规律, 并探究了颗粒抗转动效应对子接触网络各向异性指标演变规律的影响. 研究发现: 剪切过程中转动、非转动接触的组构张量变化不是独立的, 受到颗粒间滑动与否的影响; 非滑动、强接触网络是颗粒间的主要传力结构, 非滑动接触网络的接触法向和法向接触力各向异性均随$\mu_r$的增大而增大, 其对宏观应力的贡献程度随$\mu_r$的增大而减小;强接触网络的接触法向各向异性随$\mu_r$的增大而增大, 但法向接触力各向异性随$\mu_r$的增大无明显变化, 强接触网络对宏观应力的贡献程度在不同$\mu_r$情况下均相同.      

Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948). The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003). The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989). Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.     

An extended Marciniak-Kuczynski model for anisotropic sheet subjected to monotonic strain paths with through-thickness shear

Some metal sheet forming processes may induce an amount of plastic shear over the sheet thickness. This paper investigates how formability of anisotropic sheet metal is affected by such through-thickness shear (TTS). The Marciniak-Kuczynski (MK) model framework, a commonly used analytical tool to predict the limit of sheet formability due to the onset of localized necking, is extended in this paper in order to explicitly account for TTS in anisotropic metal sheets. It is a continuation of previous work by the present authors (Eyckens et al., 2009), in which TTS is incorporated for isotropic sheet. This is achieved by the introduction of additional force equilibrium and geometric compatibility equations that govern the connection between matrix and groove in the MK model. Furthermore, in order to integrate plastic anisotropy, a material reference frame available in recent literature is incorporated, as well as a particular model for anisotropic yielding that relies on virtual testing of anisotropic properties (Facet plastic potential), since out-of-plane anisotropy related to TTS cannot be measured experimentally.It is found that formability may be increased by TTS, depending on the direction onto which it is imposed by the forming process. TTS is thus a relevant aspect of the formability in, for instance, sheet forming processes in which sliding contact with friction between sheets and forming tools occur.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Modelling of laminated microstructures in stress-induced martensitic transformations

This paper is concerned with micromechanical modelling of stress-induced martensitic transformations in crystalline solids, with the focus on distinct elastic anisotropy of the phases and the associated redistribution of internal stresses. Micro-macro transition in stresses and strains is analysed for a laminated microstructure of austenite and martensite phases. Propagation of a phase transformation front is governed by a time-independent thermodynamic criterion. Plasticity-like macroscopic constitutive rate equations are derived in which the transformed volume fraction is incrementally related to the overall strain or stress. As an application, numerical simulations are performed for cubic β₁ (austenite) to orthorhombic γ₁′ (martensite) phase transformation in a single crystal of Cu-Al-Ni shape memory alloy. The pseudoelasticity effect in tension and compression is investigated along with the corresponding evolution of internal stresses and microstructure.     

High temperature creep behaviour of an FeAl intermetallic strengthened by nanoscale oxide particles

The creep behaviour of an FeAl intermetallic strengthened by nanosized oxide particles has been examined at temperatures of 700–825 °C. For all temperatures the strain rate shows a power law dependence on the applied stress. At the lowest temperature and with the highest stresses there is evidence of a threshold stress produced by the difficulty of overcoming the particle barriers, while for higher temperatures as well as at low stresses there is no threshold stress and creep appears to be controlled by general climb. The fine oxide particles produce good strengthening at low temperatures but are more readily overcome at high temperatures due to their very small size and limited attractive relaxation force. Despite such fall in creep strength, this material remains one of the strongest iron aluminides to the temperature range evaluated.     

Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu–Leblond–Devaux’s (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill’s [Hill, R., 1948. A theory of yielding and plastic flow of anisotropic solids. Proc. Roy. Soc. London A 193, 281–297] anisotropic yield criterion) and the representative volume element is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.