Model identification and FE simulations: Effect of different yield loci and hardening laws in sheet forming

The bi-axial experimental equipment [Flores, P., Rondia, E., Habraken, A.M., 2005a. Development of an experimental equipment for the identification of constitutive laws (Special Issue). International Journal of Forming Processes] developed by Flores enables to perform Bauschinger shear tests and successive or simultaneous simple shear tests and plane strain tests. Flores investigates the material behavior with the help of classical tensile tests and the ones performed in his bi-axial machine in order to identify the yield locus and the hardening model. With tests performed on one steel grade, the methods applied to identify classical yield surfaces such as [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic materials. Proceedings of the Royal Society of London A 193, 281–297; Hosford, W.F., 1979. On yield loci of anisotropic cubic metals. In: Proceedings of the 7th North American Metalworking Conf. (NMRC), SME, Dearborn, MI, pp. 191–197] ones as well as isotropic Swift type hardening, kinematic Armstrong–Frederick or Teodosiu and Hu hardening models are explained. Comparison with the Taylor–Bishop–Hill yield locus is also provided. The effect of both yield locus and hardening model choices is presented for two applications: plane strain tensile test and Single Point Incremental Forming (SPIF).     

A novel yield function for single crystals based on combined constraints optimization

A novel yield function representing the overall plastic deformation in a single crystal is developed using the concept of optimization. Based on the principle of maximum dissipation during a plastic deformation, the problem of single crystal plasticity is first considered as a constrained optimization problem in which constraints are yield functions for slip systems. To overcome the singularity that usually arises in solving the above problem, a mathematical technique is used to replace the above constrained optimization problem with an equivalent problem which has only one constraint. This single constraint optimization problem, the so-called combined constraints crystal plasticity (CCCP) model, is implemented into a finite element code and the results of modeling the uniaxial tensions of the single crystal copper along different crystallographic directions and also hydroforming of aluminum tubes proved the capability of the proposed CCCP model in accurately predicting the deformation in polycrystalline materials.     

A comparison of nonlocal continuum and discrete dislocation plasticity predictions

Discrete dislocation simulations of two boundary value problems are used as numerical experiments to explore the extent to which the nonlocal crystal plasticity theory of Gurtin (J. Mech. Phys. Solids 50 (2002) 5) can reproduce their predictions. In one problem simple shear of a constrained strip is analyzed, while the other problem concerns a two-dimensional model composite with elastic reinforcements in a crystalline matrix subject to macroscopic shear. In the constrained layer problem, boundary layers develop that give rise to size effects. In the composite problem, the discrete dislocation solutions exhibit composite hardening that depends on the reinforcement morphology, a size dependence of the overall stress-strain response for some morphologies, and a strong Bauschinger effect on unloading. In neither problem are the qualitative features of the discrete dislocation results represented by conventional continuum crystal plasticity. The nonlocal plasticity calculations here reproduce the behavior seen in the discrete dislocation simulations in remarkable detail.     

A novel constitutive model for semiconductors: The case of silicon

The photovoltaic industry relies heavily on solar-grade silicon multicrystals. Understanding their mechanical behavior requires the development of adequate constitutive models accounting for the effects of both high dislocation densities and complex loading situations in a wide range of temperature, strain rate, and impurity contents. The traditional model of Alexander and Haasen poses several limitations. We introduce in this work a novel constitutive model for covalent single crystals and its implementation into a rate-dependent crystal plasticity framework. It is entirely physically based on the dislocation generation, storage and annihilation processes taking place during plastic flow. The total dislocation density is segmented according to the dislocation mobility potential and their character. A dislocation multiplication law for the yield region more accurate than the one of Alexander and Haasen is proposed. The influence of additional dislocation sources created on forest trees, usually disregarded in models for semiconductors, is assessed. The dislocation velocity law combines three potentially rate-limiting mechanisms: the standard double kink mechanism, jog dragging and the influence of localized obstacles. The model is valid at finite strains, in multiple slip conditions and captures accurately the high temperature- and strain rate sensitivity of semiconductors. The experimental stress overshoot is qualitatively reproduced only when jog dragging is accounted for. Localized obstacles are shown not to have any significant effect on dislocation motion in silicon. The broader case of extrinsic semiconductors is discussed and the influence of dissolved oxygen on the upper yield stress of silicon monocrystals is successfully reproduced.     

The role of dislocation walls for nanoindentation to shallow depths

Dislocation events are seen as excursions or pop-in events in the load–displacement curve of nanoindentation experiments. Two nanoindenters have been used to examine the difference between quasi-static and dynamic loading during indentation. Yield excursions were present in the load–displacement curves of both the statically and dynamically loaded single crystal nickel samples. Only one major excursion occurred in each quasi-static indent, nominally loaded at 100 μN/s while staircase yielding was observed under dynamic loading indentation with a 45 Hz oscillation of 2 nm superimposed on a 60 μN/s loading rate. Thermal activation analysis is used to explain the arrest and reinitiation of the yielding with activation volumes being modeled. For nanoindentation experiments differences between quasi-static and dynamic loading are described by the models presented. It is proposed that insight into the plastic deformation mechanisms associated with such plastic instabilities will provide one of the keys to length scale effects necessary to understanding nanostructures.     

Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals

In this paper, yield functions describing the anisotropic behavior of textured metals are proposed. These yield functions are extensions to orthotropy of the isotropic yield function proposed by Cazacu et al. (Cazacu, O., Plunkett, B., Barlat, F., 2006. Orthotropic yield criterion for hexagonal close packed metals. Int. J. Plasticity 22, 1171–1194). Anisotropy is introduced using linear transformations of the stress deviator. It is shown that the proposed anisotropic yield functions represent with great accuracy both the tensile and compressive anisotropy in yield stresses and r-values of materials with hcp crystal structure and of metal sheets with cubic crystal structure. Furthermore, it is demonstrated that the proposed formulations can describe very accurately the anisotropic behavior of metal sheets whose tensile and compressive stresses are equal.     

Thermally actuated shape-memory polymers: Experiments, theory, and numerical simulations

With the aim of developing a thermo-mechanically coupled large-deformation constitutive theory and a numerical-simulation capability for modeling the response of thermally actuated shape-memory polymers, we have (i) conducted large strain compression experiments on a representative shape-memory polymer to strains of approximately unity at strain rates of 10⁻³ and 10⁻¹ s⁻¹, and at temperatures ranging from room temperature to approximately 30 °C above the glass transition temperature of the polymer; (ii) formulated a thermo-mechanically coupled large-deformation constitutive theory; (iii) calibrated the material parameters appearing in the theory using the stress-strain data from the compression experiments; (iv) numerically implemented the theory by writing a user-material subroutine for a widely used finite element program; and (v) conducted representative experiments to validate the predictive capability of our theory and its numerical implementation in complex three-dimensional geometries. By comparing the numerically predicted response in these validation simulations against measurements from corresponding experiments, we show that our theory is capable of reasonably accurately reproducing the experimental results. As a demonstration of the robustness of the three-dimensional numerical capability, we also show results from a simulation of the shape-recovery response of a stent made from the polymer when it is inserted in an artery modeled as a compliant elastomeric tube.     

A constitutive model for the formation of martensite in austenitic steels under large strain plasticity

A constitutive model for diffusionless phase transitions in elastoplastic materials undergoing large deformations is developed. The model takes basic thermodynamic relations as its starting point and the phase transition is treated through an internal variable (the phase fractions) approach. The usual yield potential is used together with a transformation potential to describe the evolution of the new phase. A numerical implementation of the model is presented, along with the derivation of a consistent algorithmic tangent modulus. Simulations based on the presented model are shown to agree well with experimental findings. The proposed model provides a robust tool suitable for large-scale simulations of phase transformations in austenitic steels undergoing extensive deformations, as is demonstrated in simulations of the necking of a bar under tensile loading and also in simulations of a cup deep-drawing process.     

Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations

This work is a review of experimental methods for observing and modeling the anisotropic plastic behavior of metal sheets and tubes under a variety of loading paths, such as biaxial compression tests; biaxial tension tests on metal sheets and tubes using closed-loop electrohydraulic testing machines; the abrupt strain path change method for detecting a yield vertex and subsequent yield loci without unloading; in-plane stress reversal tests on metal sheets; and multistage tension tests. Observed material responses are compared with the predictions of phenomenological plasticity models. Special attention is paid to the plastic deformation behavior of materials commonly used in industry, and to verifying the validity of conventional anisotropic yield criteria for those materials and associated flow rules at large plastic strains. The effects of using appropriate anisotropic yield criteria on the accuracy of simulations of forming defects, such as large springback and fracture, are also presented to highlight the importance of accurate material testing and modeling.     

The evolution of microstructure during twinning: Constitutive equations,finite-element simulations and experimental verification

In this work, we develop a rate-dependent, finite-deformation and crystal-mechanics-based constitutive theory which describes the twinning in single-crystal metallic materials. Central to the derivation of the constitutive equations are the use of fundamental thermodynamic laws and the principle of micro-force balance [Fried, E., Gurtin, M., 1994. Dynamic solid–solid transitions with phase characterized by an order parameter. Physica D 72, 287–308]. A robust numerical algorithm based on the constitutive model has also been written and implemented in the ABAQUS/Explicit [Abaqus reference manuals, 2007. SIMULIA, Providence, R.I.] finite-element program.     

A novel quadratic yield model to describe the feature of multi-yield-surface of rolled sheet metals

Since Hill’s quadratic yield model [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. Lond. A193, 281–297] cannot address enough experimental results for fairly describing the “anomalous” yield behavior as observed in some of rolled sheet metals, a new quadratic yield model is proposed. As the concept of multiple yielding systems is introduced into the new quadratic yield model, seven commonly used experimental results, three uniaxial tension stresses, one equibiaxial tension stress and three strain ratios, can all be taken into account for characterizing the anisotropy of rolled sheet metals. If more experimental results are extra needed for further improving the prediction, this yield model is still workable. As the experimental parameters are defined as functions of loading direction of corresponding test separately from the major part of yield model, the increase of experimental results regarding the same test does not vary the quadratic form of yield model. The representation of this yield model with axes of principal stresses demonstrates the similar form to Hill’s quadratic model. Therefore, many previous studies developed from Hill’s quadratic yield model can be directly upgraded by the new model to reach a higher accurate level.     

Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys

A crystal-inelasticity-based constitutive model for martensitic reorientation and detwinning in shape-memory alloys (SMAs) has been developed from basic thermodynamics principles. The model has been implemented in a finite-element program by writing a user-material subroutine. We perform two sets of finite-element simulations to model the behavior of polycrystalline SMAs: (1) The full finite-element model where each finite element represents a collection of martensitic microstructures which originated from within an austenite single crystal, chosen from a set of crystal orientations that approximates the initial austentic crystallographic texture. The macroscopic stress-strain responses are calculated as volume averages over the entire aggregate: (2) The Taylor model (J. Inst. Metals 62 (1938) 32) where an integration point in a finite element represents a material point which consist of sets of martensitic microstructures which originated from within respective austenite single-crystals. Here the macroscopic stress-strain responses are calculated through a homogenization scheme.Experiments in tension and compression were conducted on textured polycrystalline Ti-Ni rod initially in the martensitic phase by Xie et al (Acta Mater. 46 (1998) 1989). The material parameters for the constitutive model were calibrated by fitting the tensile stress-strain response from a full finite-element calculation of a polycrystalline aggregate to the simple tension experiment. With the material parameters calibrated the predicted stress-strain curve for simple compression is in very good accord with the corresponding experiment. By comparing the simulated stress-strain response in simple tension and simple compression it is shown that the constitutive model is able to predict the observed tension-compression asymmetry exhibited by polycrystalline Ti-Ni to good accuracy. Furthermore, our calculations also show that the macroscopic stress-strain response depends strongly on the initial martensitic microstructure and crystallographic texture of the material.We also show that the Taylor model predicts the macroscopic stress-strain curves in simple tension and simple compression reasonably well. Therefore, it may be used as a relatively inexpensive computational tool for the design of components made from shape-memory materials.     

An updated version of the multimechanism model for cyclic plasticity

In this paper we consider the elastoplastic behavior of the 304L stainless steel under cyclic loading at room temperature. After the experimental investigations presented in Taleb and Hauet (2009), the present work deals with modeling in the light of the new observations. An improved version of the multimechanism model is proposed in which the isotropic variable is revisited in order to take into account the non-proportional effect of the loading as well as the strain memory phenomenon. A particular attention has been paid to the identification process in order to capture the main important phenomena: relative parts of isotropic and kinematic hardening, time dependent effects, non-proportionality effect, strain amplitude dependence. Only strain controlled tests have been used for the identification process. The capabilities of the model with “only” 17 parameters are evaluated considering a number of proportional and non-proportional stress and strain controlled tests.     

A multiscale approach for modeling scale-dependent yield stress in polycrystalline metals

Modeling of scale-dependent characteristics of mechanical properties of metal polycrystals is studied using both discrete dislocation dynamics and continuum crystal plasticity. The initial movements of dislocation arc emitted from a Frank-Read type dislocation source and bounded by surrounding grain boundaries are examined by dislocation dynamics analyses system and we find the minimum resolved shear stress for the FR source to emit at least one closed loop. When the grain size is large enough compared to the size of FR source, the minimum resolved shear stress levels off to a certain value, but when the grain size is close to the size of the FR source, the minimum resolved shear stress shows a sharp increase. These results are modeled into the expression of the critical resolved shear stress of slip systems and continuum mechanics based crystal plasticity analyses of six-grained polycrystal models are made. Results of the crystal plasticity analyses show a distinct increase of macro- and microscopic yield stress for specimens with smaller mean grain diameter. Scale-dependent characteristics of the yield stress and its relation to some control parameters are discussed.     

Evaluation of identification methods for YLD2004-18p

Four calibration methods have been evaluated for the linear transformation-based anisotropic yield function YLD2004-18p (Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transformation-based anisotropic yield functions. Int. J. Plasticity 21, 1009–1039) and the aluminium alloy AA5083-H116. The different parameter identifications are based on least squares fits to combinations of uniaxial tensile tests in seven directions with respect to the rolling direction, compression (upsetting) tests in the normal direction and stress states found using the full-constraint (FC) Taylor model for 690 evenly distributed strain paths. An elastic–plastic constitutive model based on YLD2004-18p has been implemented in a non-linear finite element code and used in finite element simulations of plane-strain tension tests, shear tests and upsetting tests. The experimental results as well as the Taylor model predictions can be satisfactorily reproduced by the considered yield function. However, the lacking ability of the Taylor model to quantitatively reproduce the experiments calls for more advanced crystal plasticity models.     

Analyzing the orientation dependence of stresses in polycrystals using vertices of the single crystal yield surface and crystallographic fibers of orientation space

The influence of the polygonal geometry of the restricted slip-associated yield surface on the distribution of stresses over a polycrystalline aggregate is examined. The vertices of the yield surface (stress states corresponding to polyslip) are grouped according to symmetries imposed by crystal structure. A measure of coaxiality between crystal stresses and yield surface vertex stresses is used to quantify the proximity of the stress in each crystal to a yield surface vertex. It is shown that for prescribed stress states, crystal stresses align more closely with certain families of vertices than with others and this relation between crystal and vertex stresses is found to depend on crystallographic fibers. Using this information, the stress distributions from finite element simulations of face centered cubic polycrystals are analyzed for different stress states ranging from uniaxial to balanced biaxial. Over the fundamental region of orientations, the propensity for the stress to align with a vertex is demonstrated. Further, the stresses in elements contributing to certain crystallographic fibers are shown to favor the vertex families aligned with those fibers. The implications of these results on mechanical behaviors, especially with respect to those observed in diffraction experiments, are discussed.