Anisotropic viscoplasticity constitutive model for directionally solidified superalloy

Abstract

The macroscopic deformation behaviour of a Ni-based directionally solidified (DS) superalloy was experimentally investigated, and an anisotropic constitutive model of the material was developed. Monotonic and creep tests were performed on uniaxial test specimens machined from DS plates so that the angle between the loading direction and the solidified grain direction varied between 0 and 90°. Tension-torsion creep tests were also conducted to examine the anisotropic behaviour under multiaxial stress conditions. The material exhibited marked anisotropy under elastic and viscous deformation conditions, whereas it showed isotropy under plastic deformation conditions of high strain rates. Then crystal plasticity analyses were carried out to identify slip systems under creep loading conditions, assuming the anisotropic creep behaviour of the DS material. A viscoplastic constitutive model for expressing both the anisotropic elasticity-viscosity and the isotropic plasticity was proposed. The elastic constants were determined using a self-consistent approach, and viscous parameters were modelled by crystal plasticity analyses. The calculation results obtained using the constitutive model were compared with the experimental data to evaluate the validity of the model. It was demonstrated that the constitutive model could satisfactorily describe the anisotropic behaviour under uniaxial and multiaxial stress conditions with a given set of material parameters.     

AZ31B镁合金挤压板材力学性能的各向异性

在AZ31B镁合金板材的板面内沿不同方向进行单向拉伸和压缩试验,研究挤压板材的力学性能。结果表明,变形AZ31B镁合金板材具有显著的各向异性和拉压非对称性。在板面内,沿挤压方向拉伸时的屈服应力明显地比沿同方向压缩和沿其他方向拉伸或压缩时的高(约2倍);沿45°斜向拉伸的屈服应力和抗拉强度较低,而延伸率最高;这种非对称性主要表现为屈服非对称和塑性流动非对称,即拉压的屈服应力不相等和拉压应力-应变曲线形状不同,压缩曲线表现出特殊的"S"型。基于晶体塑性理论,讨论了引起变形镁合金的各向异性和拉压非对称性力学性能的变形机理。     

Interaction between anisotropic plastic deformation and damage evolution in Al 2198 sheet metal

Deformation anisotropy of sheet aluminium alloy 2198 (Al-Cu-Li) has been investigated by means of mechanical testing of notched specimens and Kahn-type fracture specimens, loaded in the rolling direction (L) or in the transverse direction (T). Fracture mechanisms were investigated via scanning electron microscopy. Contributions to failure are identified as growth of initial voids accompanied by a significant nucleation of a second population of cavities and transgranular failure. A model based on the Gurson-Tvergaard-Needleman (GTN) approach of porous metal plasticity incorporating isotropic voids, direction-dependent void growth, void nucleation at a second population of inclusions and triaxiality-dependent void coalescence has been used to predict the mechanical response of test samples. The model parameters have been calibrated by means of 3D unit cell simulations, revealing the interaction between the plastic anisotropy of the matrix material and void growth. The model has been successfully used to describe and predict direction-dependent deformation behaviour, crack propagation and, in particular, toughness anisotropy.     

Finite element modelling of wooden structures at large deformations and brittle failure prediction

This paper presents a constitutive wood model that accounts for both hardening associated with material densification at large compressive deformations and brittle failure modes. The model is adapted from previous work by the authors and has been modified to deal with wood behaviours. The main novelty of the model is the coupling between the anisotropic plasticity and the ductile densification. The model developed is successfully implemented in the commercial ABAQUS software. Validation was made for uniaxial compressive loadings and an application on a three-points bending test. The results obtained, for the uniaxial compressive loadings, demonstrate the capability of the model to simulate the wood behaviour at large compressive deformations and show clearly the effect of the densification on the plastic behaviour. The result obtained for the three-points bending test shows a good implementation of the brittle failure criterion and demonstrates the suitability of the developed model to analyse and design wooden structures.     

Virtual material testing for stamping simulations based on polycrystal plasticity

In the modern practice of stamping simulation of complex industrial parts the prediction of springback still lacks accuracy. In commercial software packages various empirical constitutive laws for stamping are available. Limited to simple empirical models for material anisotropy they do not take into account in a full manner the effects of microstructure and its evolution during the deformation process. The crystal plasticity finite element method bridges the gap between the polycrystalline texture and macroscopic mechanical properties that opens the way for more profound consideration of metal anisotropy in the stamping process simulation. In this paper the application of crystal plasticity FEM within the concept of virtual material testing with a representative volume element (RVE) is demonstrated. Using virtual tests it becomes possible, for example, to determine the actual shape of the yield locus and Lankford parameters and to use this information to calibrate empirical constitutive models. Along with standard uniaxial tensile tests other strain paths can be investigated like biaxial tensile, compressive or shear tests. The application of the crystal plasticity FEM for the virtual testing is demonstrated for DC04 and H320LA steel grades. The parameters of the Vegter yield locus are calibrated and the use case demonstration is completed by simulation of a typical industrial part in PAMSTAMP 2G.     

Fracture analysis of anisotropic materials using enriched crack tip elements

Enriched finite element methodology, which employs special crack tip elements, is extended for cracks in anisotropic materials. Enrichment formulation is described briefly and three validation examples using single crystal, directionally solidified, and orthotropic material properties are presented to demonstrate the accuracy and effectiveness of the methodology. In addition to validation examples, the effect of material anisotropy on stress intensity factors is investigated using the common compact tension specimen and the results are compared to the ASTM solution for isotropic materials. It is shown that the effect of anisotropy on the computed stress intensity factors can be significant, depending on the degree of anisotropy, material orientation, and a/W ratio in the compact tension specimen geometry.     

Micro-mechanical modelling of high cycle fatigue behaviour of metals under multiaxial loads

An analysis of high cycle multiaxial fatigue behaviour is conducted through the numerical simulation of polycrystalline aggregates using the finite element method. The metallic material chosen for investigation is pure copper, which has a Face Centred Cubic (FCC) crystalline microstructure. The elementary volumes are modelled in 2D using an hypothesis of generalised plane strain and consist of 300 equi-probability, randomly oriented grains with equiaxed geometry. The aggregates are loaded at levels equivalent to the average macroscopic fatigue strength at 10⁷ cycles. The goal is to compute the mechanical quantities at the mesoscopic scale (i.e., average within the grain) after stabilization of the local cyclic behaviour. The results show that the mesoscopic mechanical variables are characterised by high dispersion. A statistical analysis of the response of the aggregates is undertaken for different loading modes: fully reversed tensile loads, torsion and combined in-phase tension–torsion. Via the calculation of the local mechanical quantities for a sufficiently large number of different microstructures, a critical analysis of certain multiaxial endurance criteria (Crossland, Dang Van and Matake) is conducted. In terms of material behaviour models, it is shown that elastic anisotropy strongly affects the scatter of the mechanical parameters used in the different criteria and that its role is predominant compared to that of crystal plasticity. The analysis of multiaxial endurance criteria at both the macroscopic and mesoscopic scales clearly show that the critical plane type criteria (Dang Van and Matake) give an adequate estimation of the shear stress but badly reflect the scatter of the normal stress or the hydrostatic stress.     

A composite ply failure model based on continuum damage mechanics

A material model including the failure behaviour is derived for a thin unidirectional (UD) composite ply. The model is derived within a thermodynamic framework and the failure behaviour is modelled using continuum damage mechanics. The following features describe the model: (i) The ply is assumed to be in a plane state of stress. (ii) Three damage variables associated with the stress in the fibre-, transverse and shear directions, respectively, are used. (iii) The plastic behaviour of the matrix material is modelled. (iv) The difference in the material response in tensile and compressive loading is modelled. (v) Rate dependent behavior of plasticity and damage (i.e. strength) is modelled.     

Texture and mechanical anisotropy in three extruded magnesium alloys

Abstract

One ZM61 alloy (6·2%Zn, 1·2%Mn) and two magnesium alloys containing nominally 3% of neodymium and yttrium respectively have been prepared in the form of hot extruded flat strips. Their textures and microstructures have been quantified and series of mechanical tests were carried out to determine plane stress yield loci in both the solution treated and aged conditions. The ZM61 alloy had a sharp texture and marked anisotropy of strength that could be rationalised in terms of deformation by basal <a> slip and {1012}<1011> twinning. This material was much weaker in compression than in tension. Precipitation hardening on aging caused a greater impedance to twinning than to slip with the result that the anisotropy was somewhat reduced. The Mg–3Nd alloy had a very weak and different texture but nevertheless demonstrated a pronounced anisotropy of strength. Aging increased the yield stress by about 80% and also inhibited twinning to some extent although the degree of anisotropy remained almost unaffected. The Mg–3Y alloy which showed a texture of intermediate strength was different in type from either of the others. Its strength behaviour was close to isotropic; in particular, no difference existed between tensile and compressive loading, and aging produced only a marginal increase in strength. Twins were relatively infrequent in the deformed Mg–3Y alloy but its mechanical behaviour could not be rationalised according to simple models.     

Numerical investigations of foam-like materials by nested high-order finite element methods

In this paper we present a multiscale framework suited for geometrically nonlinear computations of foam-like materials applying high-order finite elements (p-FEM). This framework is based on a nested finite element analysis (FEA) on two scales, one nonlinear boundary value problem on the macroscale and k independent nonlinear boundary value problems on the microscale allowing for distributed computing. The two scales are coupled by a numerical projection and homogenization procedure. On the microscale the foam-like structures are discretized by high-order continuum-based finite elements, which are known to be very efficient and robust with respect to locking effects. In our numerical examples we will discuss in detail three characteristic test cases (simple shear, tension and bending). Special emphasis is placed on the material’s deformation-induced anisotropy and the macroscopic load-displacement behavior.     

Cracking behaviour of concrete beams reinforced with a combination of ordinary reinforcement and steel fibers

The paper presents an experimental and theoretical study on the cracking behaviour of concrete beams having longitudinal tension reinforcement and various combinations of volume and aspect ratio of steel fibers. Five full-scale beams with a concrete compressive strength of 42 MPa were tested. The mechanical properties of the steel fiber concrete under tension were determined by means of the four-point bending test specified in the Belgian standard NBN B15-238. The experimental results show that the addition of steel fibers decreases both the crack spacing and the crack width. A modification of the model of Nemegeeret al. to predict crack widths is suggested.     

Two-scale analysis for deformation-induced anisotropy of polycrystalline metals

The anisotropic macroscopic mechanical behavior of polycrystalline metals is characterized by incorporating the microscopic constitutive model of single crystal plasticity into the two-scale modeling based on the mathematical homogenization theory, which enables us to derive both micro- and macro-scale governing equations. The two-scale simulations are conducted to evaluate the macroscopic anisotropy induced by microscopic plastic deformation histories of the polycrystalline aggregate. In the simulations, the representative volume element (RVE) composed of several crystal grains is uniformly loaded in one direction, unloaded to macroscopically zero stress in a certain stage of deformation and then re-loaded in the different directions. The last re-loading calculations provide different macroscopic responses of the RVE, which can be the appearance of material anisotropy. We then try to examine the effects of the intergranular and intragranular behaviors on the anisotropy by means of various illustrations of microscopic plastic deformation process without referring to the change of crystallographic orientations.     

A coupled FE–EFG approach for modelling crack growth in ductile materials

In this work, a coupled finite element–element free Galerkin approach has been used to model crack growth in ductile materials under monotonic and cyclic loads. In this approach, a small discontinuous domain near crack is modelled by EFG method, whereas the rest of the domain is modelled by FEM to exploit the advantages of both the methods. A ramp function has been used in the transition region to maintain the continuity between FE and EFG domains. Two plasticity models (GTN and von‐Mises) and three hardening rules (isotropic, kinematic and mixed) have been used to model the nonlinear material behaviour. Four different problems, i.e. single edge notched tension specimen, double edge notched tension specimen, compact tension specimen and three‐point bend specimen, are solved under plane strain condition using J–R curve approach. Finally, a CT specimen problem is also solved by coupled approach using three hardening rules and two plasticity models under cyclic loading.     

Micromechanical study of the loading path effect in high cycle fatigue

In this work, an analysis of both the mechanical response at the grain scale and high cycle multiaxial fatigue criteria is undertaken using finite element (FE) simulations of polycrystalline aggregates. The metallic material chosen for investigation, a pure copper, has a Face Centred Cubic (FCC) crystalline structure. Two-dimensional polycrystalline aggregates, which are composed of 300 randomly orientated equiaxed grains, are loaded at the median fatigue strength defined at 10⁷ cycles. In order to analyse the effect of the loading path on the local mechanical response, combined tension–torsion and biaxial tension loading cases, in-phase and out-of-phase, with different biaxiality ratios, are applied to each polycrystalline aggregate. Three different material constitutive models assigned to the grains are investigated: isotropic elasticity, cubic elasticity and crystal plasticity in addition to the cubic elasticity. First, some aspects of the mechanical response of the grains are highlighted, namely the scatter and the multiaxiality of the mesoscopic responses with respect to an uniaxial macroscopic response. Then, the distributions of relevant mechanical quantities classically used in fatigue criteria are analysed for some loading cases and the role of each source of anisotropy on the mechanical response is evaluated and compared to the isotropic elastic case. In particular, the significant influence of the elastic anisotropy on the mesoscopic mechanical response is highlighted. Finally, an analysis of three different fatigue criteria is conducted, using mechanical quantities computed at the grain scale. More precisely, the predictions provided by these criteria, for each constitutive model studied, are compared with the experimental trends observed in metallic materials for such loading conditions.     

Numerical simulation of plastic deformation and fracture in polysynthetically twinned (PST) crystals of TiAl

Plastic deformation and fracture in polysynthetically twinned (PST) crystals of TiAl have been simulated by using periodic unit cells representing the relaxed-constraint model recently proposed by Lebensohn et al. [Acta Mater. 46 (1998) 4701–4709] for the co-deformation of the lamellar compound of PST-TiAl. The unit cells contain both intermetallic phases, ₂-(Ti₃Al) and γ-(TiAl). Furthermore, the six orientation variants of the γ-phase are also considered. The constitutive behaviour of both phases is described by crystal plasticity, and the damage behaviour has been implemented by means of cohesive elements. The unit cells have been used as submodels for multi-scale finite element simulations of compression tests and fracture mechanics tests of notched micro-bend specimens. It is shown that the anisotropy of plastic deformation and damage in PST-TiAl can be well represented.     

Ballistic performance study of fabric armor based on numerical simulations with multiscale material model

Based on a multiscale model for fabric materials, dynamic simulations of the fabric ballistic performance were implemented. Through parameter research, it was found that the ballistic performance and mechanical behavior of the fabric materials are determined by a combination of factors and conditions rather than by the material properties alone. The material mechanical properties reflect the inherent strength of the fabric; the fabric weaving structure, boundary conditions, material orientation, and projectile shape also play important roles and have a significant influence on the ballistic performance of the fabric. The multiscale material model incorporates not only the membrane‐like properties of the fabric but also the underlying weaving structure, yarn interaction, and yarn composition. The simulations results show good agreement with the experimental data. Various physical phenomena can be observed in the simulations, such as yarn decrimping, material anisotropy, and two types of damage modes. Copyright © 2011 John Wiley & Sons, Ltd.     

Modélisation de la réponse en traction du béton renforcé de fibres métalliques

To date there has been no model really able to predict the behaviour of steel fiber reinforced concrete. By referring to the experimental design theory, which allows us to adopt a formal procedure in terms of test definition and to assess the reliability of the experimental results we analyse the behaviour of steel fiber reinforced concrete when subjected to uniaxial tension. The influence of 8 characteristic parameters (factors) of the material on the ultimate tensile strength f_t and the fracture energy G_f is analysed. Particular attention is devoted to the study of the heterogeneity and anisotropy of the material.     

A finite element study of microstructure-sensitive plasticity and crack nucleation in fretting

This paper is concerned with finite element modelling of microstructure-sensitive plasticity and crack initiation in fretting. The approach adopted is based on an existing method for microstructure-sensitive (uniaxial) fatigue life prediction, which proposes the use of a unit cell crystal plasticity model to identify the critical value of accumulated plastic slip associated with crack initiation. This approach is successfully implemented here, using a FCC unit cell crystal plasticity model, to predict the plain low-cycle fatigue behaviour of a stainless steel. A crystal plasticity frictional contact model for stainless steel is developed for microstructure-sensitive fretting analyses. A methodology for microstructure-sensitive fretting crack initiation is presented, based on identification of the number of cycles in the fretting contact at which the identified critical value of accumulated plastic slip is achieved. Significant polycrystal plasticity effects in fretting are predicted, leading to significant effects on contact pressure, fatigue indicator parameters and microstructural accumulated slip. The crystal plasticity fretting predictions are compared with J₂ continuum plasticity predictions. It is argued that the microstructural accumulated plastic slip parameter has the potential to unify the prediction of wear and fatigue crack initiation, leading in some cases, e.g. gross slip, to wear, via a non-localised distribution of critical crystallographic slip, and in other cases, e.g. partial slip, to fatigue crack initiation, via a highly-localised distribution of critical crystallographic slip with preferred orientation (cracking locations and directions).     

Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin

In the small deformation range, we consider and discuss the phenomenological (or isotropic) “higher-order” theory of strain gradient plasticity put forward in Section 12 of Gurtin [1], which includes the dissipation due to the plastic spin through a material parameter called χ. In fact, χ weighs the square of the plastic spin rate into the definition of an effective measure of plastic flow peculiar of the isotropic hardening function. Such a model has been identified by Bardella [2] as a good isotropic approximation of a crystal model to describe the multislip behaviour of a single grain, provided that χ be set as a specific function of other material parameters involved in the modelling, including the length scales. The main feature of the underlying gradient approach is the accounting for both dissipative and energetic strain gradient dependences, with related size effects. The dissipative strain gradients enter the model through the definition of the above mentioned effective measure of plastic flow, whereas the energetic strain gradients are involved in the modelling by defining the defect energy, a function of Nye’s dislocation density tensor added to the free energy to account for geometrically necessary dislocations (see, e.g., Gurtin [1]). By exploiting the deformation theory approximation, we apply the model to a simple boundary value problem so that we can discuss the effects of (a) the criterium derived by Bardella [2] for choosing χ and (b) non-quadratic forms of the defect energy. We show that both χ and the nonlinearity chosen for the defect energy strongly affect quality and magnitude of the energetic size effect which is possible to predict.