The plastic deformation of non-homogeneous polycrystals

To describe the work hardening process of polycrystals processed using various thermomechanical cycles with isochronal annealing from 500 to 900 °C, a dislocation based strain hardening model constructed in the basis of the so-called Kocks–Mecking model is proposed. The time and temperature dependence of flow stress is accounted via grain boundary migration, and the migration is related to annihilation of extrinsic grain boundary dislocations (EGBD’s) by climb via lattice diffusion of vacancies at the triple points. Recovery of yield stress is associated with changes in the total dislocation density term ρ^T. A sequence of deformation and annealing steps generally result in reduction of flow stress via the annihilation of the total dislocation density ρ^T defined as the sum of geometrically necessary dislocations ρ^G and statistically stored dislocations ρ^S. The predicted variation of yield stress with annealing temperature and cold working stages is in agreement with experimental observations. An attempt is made to determine the mathematical expressions which best describe the deformation behaviour of polycrystals in uniaxial deformation.     

Micromechanical analysis of kinematic hardening in natural clay

This paper presents a micromechanical analysis of the macroscopic behaviour of natural clay. A microstructural stress–strain model for clayey material has been developed which considers clay as a collection of clusters. The deformation of a representative volume of the material is generated by mobilizing and compressing all the clusters along their contact planes. Numerical simulations of multistage drained triaxial stress paths on Otaniemi clay have been performed and compared the numerical results to the experimental ones in order to validate the modelling approach. Then, the numerical results obtained at the microscopic level were analysed in order to explain the induced anisotropy observed in the clay behaviour at the macroscopic level. The evolution of the state variables at each contact plane during loading can explain the changes in shape and position in the stress space of the yield surface at the macroscopic level, as well as the rotation of the axes of anisotropy of the material.     

A dislocation-based constitutive law for pure Zr including temperature effects

In this work, a single crystal constitutive law for multiple slip and twinning modes in single phase hcp materials is developed. For each slip mode, a dislocation population is evolved explicitly as a function of temperature and strain rate through thermally-activated recovery and debris formation and the associated hardening includes stage IV. A stress-based hardening law for twin activation accounts for temperature effects through its interaction with slip dislocations. For model validation against macroscopic measurement, this single crystal law is implemented into a visco-plastic-self-consistent (VPSC) polycrystal model which accounts for texture evolution and contains a subgrain micromechanical model for twin reorientation and morphology. Slip and twinning dislocations interact with the twin boundaries through a directional Hall–Petch mechanism. The model is adjusted to predict the plastic anisotropy of clock-rolled pure Zr for three different deformation paths and at four temperatures ranging from 76 K to 450 K (at a quasi-static rate of 10⁻³ 1/s). The model captures the transition from slip-dominated to twinning-dominated deformation as temperature decreases, and identifies microstructural mechanisms, such as twin nucleation and twin–slip interactions, where future characterization is needed.     

Experimental observations of evolving yield loci in biaxially strained AA5754-O

Experimental measurement of the plastic biaxial mechanical response for an aluminum alloy (AA5754-O) sheet metal is presented. Traditional methods of multiaxial sheet metal testing require the use of finite element analysis (FEA) or other assumptions of material response to determine the multiaxial true stress versus true strain behavior of the as-received sheet material. The method used here strives to produce less ambiguous measurements of data for a larger strain range than previously possible, through a combination of the Marciniak flat bottom ram test and an X-ray diffraction technique for stress measurement. The study is performed in conjunction with a study of the microstructural changes that occur during deformation, and these microstructural results are briefly mentioned in this work. Issues of calibration and applicability are discussed, and results are presented for uniaxial (U), plane strain (PS), and balanced biaxial (BB) extension. The results show repeatable behavior (within quantified uncertainties) for U to 20%, PS to almost 15%, and BB to above 20% in-plane strains. The results are first compared with three common yield locus models (von Mises’, Hill’48, and Hosford’79), and show some unexpected results in the shape change of the yield locus at high strain levels (>5% strain). These changes include the rotation of the locus toward the von Mises surface and elongation in the balanced biaxial direction. Comparison with a more complex yield locus model (Yld2000-2d with eight adjustable parameters) showed that the locus elongation in the biaxial direction could be fit well (for a specific level of work), but at the detriment of fit to the plane strain data. Artificially large plastic strain ratios would be needed to match both the biaxial and plane strain behavior even with this more complex model.     

Geometrically necessary dislocation structure organization in FCC bicrystal subjected to cyclic plasticity

Crystal plasticity finite element analysis of cyclic deformation of compatible type FCC bicrystals are performed. The model specimen used in the analysis is a virtual FCC bicrystal with an isotropic elastic property; therefore, the effect of constraint due to elastic incompatibility does not appear. The results of the analysis show the strain-amplitude-dependence of both the organization of the GND structure and the stress–strain behavior. The calculated stress–strain curve with the largest strain amplitude shows additional cyclic hardening. The microscopic mechanisms of the strain-amplitude-dependent organization of the GND structure and additional cyclic hardening behavior are discussed in terms of the activation of secondary slip system(s). Finally, the effects of the elastic anisotropy, the lattice friction stress and the interaction between dislocations are also argued.     

Grain boundary versus transgranular ductile failure

The competition between intergranular and intragranular fracture is investigated using a bilayer damage model, which incorporates the relevant microstructural features of aluminium alloys with precipitate free zones (PFZ) nearby the grain boundary. One layer represents the grain behaviour: due to precipitation, it presents a high yield stress and low hardening exponent. The other layer represents the PFZ which has the behaviour of a solid solution: it is much softer but with a much higher strain hardening capacity. In both layers, void growth and coalescence is modelled using an enhanced Gurson-type model incorporating the effects of the void aspect ratio and of the relative void spacing. The effects on the ductility (i) of the flow properties of each zone, (ii) of the relative thickness of the PFZ, and (iii) of the particles spacing and volume fraction in the PFZ are elucidated. Comparisons are made with experimental data. Based on the previous analysis, qualitative understanding of trends in the fracture toughness of aluminium alloys can be gained in order to provide a link with the thermal treatment process.     

Three dimensional combined fracture–plastic material model for concrete

This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.     

Dual-phase steel sheets under cyclic tension–compression to large strains: Experiments and crystal plasticity modeling

In this work, we develop a physically-based crystal plasticity model for the prediction of cyclic tension–compression deformation of multi-phase materials, specifically dual-phase (DP) steels. The model is elasto–plastic in nature and integrates a hardening law based on statistically stored dislocation density, localized hardening due to geometrically necessary dislocations (GNDs), slip-system-level kinematic backstresses, and annihilation of dislocations. The model further features a two level homogenization scheme where the first level is the overall response of a two-phase polycrystalline aggregate and the second level is the homogenized response of the martensite polycrystalline regions. The model is applied to simulate a cyclic tension–compression–tension deformation behavior of DP590 steel sheets. From experiments, we observe that the material exhibits a typical decreasing hardening rate during forward loading, followed by a linear and then a non-linear unloading upon the load reversal, the Bauschinger effect, and changes in hardening rate during strain reversals. To predict these effects, we identify the model parameters using a portion of the measured data and validate and verify them using the remaining data. The developed model is capable of predicting all the particular features of the cyclic deformation of DP590 steel, with great accuracy. From the predictions, we infer and discuss the effects of GNDs, the backstresses, dislocation annihilation, and the two-level homogenization scheme on capturing the cyclic deformation behavior of the material.     

Non-local crystal plasticity model with intrinsic SSD and GND effects

A strain gradient-dependent crystal plasticity approach is presented to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. In order to be capable of predicting scale dependence, the heterogeneous deformation-induced evolution and distribution of geometrically necessary dislocations (GNDs) are incorporated into the phenomenological continuum theory of crystal plasticity. Consequently, the resulting boundary value problem accommodates, in addition to the ordinary stress equilibrium condition, a condition which sets the additional nodal degrees of freedom, the edge and screw GND densities, proportional (in a weak sense) to the gradients of crystalline slip. Next to this direct coupling between microstructural dislocation evolutions and macroscopic gradients of plastic slip, another characteristic of the presented crystal plasticity model is the incorporation of the GND-effect, which leads to an essentially different constitutive behaviour than the statistically stored dislocation (SSD) densities. The GNDs, by their geometrical nature of locally similar signs, are expected to influence the plastic flow through a non-local back-stress measure, counteracting the resolved shear stress on the slip systems in the undeformed situation and providing a kinematic hardening contribution. Furthermore, the interactions between both SSD and GND densities are subject to the formation of slip system obstacle densities and accompanying hardening, accountable for slip resistance. As an example problem and without loss of generality, the model is applied to predict the formation of boundary layers and the accompanying size effect of a constrained strip under simple shear deformation, for symmetric double-slip conditions.     

Modelling the behaviour of polycrystalline austenitic steel with twinning-induced plasticity effect

A micromechanical model using the scale transition method in elastoviscoplasticity has been developed to describe the behaviour of those austenitic steels that display a TWIP effect. A physically based constitutive equation at the grain scale is proposed considering two inelastic strain modes: crystallographic slip and twinning. The typical organizations of microtwins observed in electron microscopy are considered, and the twin–slip as well as the twin–twin interactions are accounted for. The parameters for slip are first fitted on the uniaxial tensile response obtained at intermediate temperatures (when twinning is inhibited). Then, the parameters associated with twinning are identified using the stress–strain curve at room temperature. The simulated results in both macro and micro scales are in good agreement with experimentally obtained results.     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.     

Comparison of the work-hardening of metallic sheets using tensile and shear strain paths

This work deals with the characterization of the kinematic work-hardening of a bake-hardening steel. A shear test device has been designed and its use for the characterization of the work-hardening of sheet metals is described. Two main results are presented. Firstly, a local strain measurement, based on the following of three dots drawn on the gauge area, gives the evolution of the strain tensor eigenvalues during the test. It is shown, by comparing the theoretical kinematics of simple shear with a slightly perturbated one, that the strain state is close to the ideal one in the center of the gauge area. Secondly, reversal of the shear direction is performed after several prestrain and the evolution of the kinematic work-hardening with the equivalent plastic strain has been identified using an anisotropic elasto-viscoplastic model of Hill 1948 type. Isotropic and kinematic contributions of the work-hardening are also calculated from loading–unloading tensile tests and are compared to those obtained from the simple shear tests. The results show a discrepancy between both identification for the isotropic and the kinematic hardening. However, they are in agreement concerning the evolution of the global work-hardening.     

Complex investigation of deformation twinning in γ-TiAl by TEM and neutron diffraction

A near-γ TiAl based alloy with 2 at% of Nb was investigated by means of collaborative research based on transmission electron microscopy and in-situ neutron diffraction techniques with the aim to study mechanical twinning and its role within the mechanisms governing fatigue response and material properties. In-situ neutron diffraction measurements were performed during low cycle fatigue straining at room temperature. Induced lattice strain related to the formation of deformation twins was detected and used to follow changes in the macroscopic material response caused by the twinning process during cycling. A microscopic insight was realised by using several transmission electron microscopy techniques to reveal in detail an internal deformation microstructure of the material at the beginning as well as at the end of the fatigue life. The study was focused on the first loading cycles where the material shows intense cyclic hardening. The effect of mechanical twinning on the material behaviour at several stages of the fatigue life is discussed for two different total strain amplitudes of 0.2% and 0.4%.     

The effect of time-dependent twinning on low temperature (<0.25 ∗ Tm) creep of an alpha-titanium alloy

The low-temperature (less than one-fourth of the melting temperature) creep deformation behavior of hexagonally close-packed (HCP) α-Ti–1.6 wt.% V was investigated. Creep tests were performed at various temperatures between room temperature and 205 °C at 95% of the respective yield stress at the different temperatures. The creep strain rate was found to increase with increasing temperature. Scanning and transmission electron microscopy revealed that slip and unusually slow twin growth, or time-dependent twinning, are active deformation mechanisms for the entire temperature range of this investigation. The activation energy for creep of this alloy was calculated to identify the rate-controlling deformation mechanism, and was found to increase with increasing creep strain. At low strain, the activation energy for creep was found to be close to the previously calculated activation energy for slip. At high strain, the calculated activation energy indicates that both slip and twinning are significant deformation mechanisms. The appearance of twinning at high strains is explained by a model for twin nucleation by dislocation pileups.     

Modeling aspects of low plastic strain amplitude multiaxial cyclic plasticity in conventional and ultrafine grain nickel

Aspects of the cyclically saturated responses of initially annealed, conventional grain size (average grain diameter of approximately 50 μm) and electrodeposited, ultrafine grain (grains from 20 to 500 nm) nickel to reversed proportional and 90° out-of-phase axial-torsional, strain-controlled cycling at a nominally constant equivalent inelastic strain amplitude of approximately 100 μ strain are reported. An anisotropic, axial-torsional subspace version of Abdel-Karim and Ohno’s kinematic hardening model is presented. Within the framework of conventional small strain, rate-independent plasticity, this approach is used to model the responses. An anhysteretic, phenomenolically based, magnetomechanical model is coupled to the rate-independent plasticity model to include the cyclic magnetostriction response. The kinematic hardening parameter determination scheme, using the proportional path responses, is described. The model correlations achieved are presented and the ability of the resulting models to capture the 90° out-of-phase responses is examined. The model parameter sets, as determined from the proportional responses, require small changes to result in more accurate correlation of the 90° out-of-phase responses and the implications of this are discussed. The relative values of the model parameters between the two materials reflect the initial microstructures. Persistent mean stresses associated with mean total strains imposed are successfully modeled for the proportional strain path responses but not for the 90° out-of-phase responses.