Together with slip, deformation twinning and de-twinning are the plastic deformation mechanisms in hexagonal close packed (HCP) crystals, which strongly affect texture evolution and anisotropic response. As a consequence, several twinning models have been proposed and implemented in the existing polycrystalline plasticity models. De-twinning is an inverse process with respect to twinning, which is relevant to cycling, fatigue and complex loads but is rarely incorporated into polycrystalline plastic models. In this paper, we propose a physics-based twinning and de-twinning (TDT) model that has the capability of dealing with both mechanisms during plastic deformation. The TDT model is characterized by four deformation mechanisms corresponding to twin nucleation, twin growth, twin shrinkage and re-twinning. Twin nucleation and twin growth are associated with deformation twinning, and twin shrinkage and re-twinning are associated with de-twinning. The proposed TDT model is implemented in the Elasto-Visco-Plastic Self-Consistent (EVPSC) model. We demonstrate the validity and the capability of the TDT model by simulating cyclic loading of magnesium alloys AZ31B plate and AZ31 bar. Comparison with the measurements indicates that the TDT model is able to capture the key features observed in experiments, implying that the mechanical response in the simulated materials is mainly associated with twinning and de-twinning.
Ferritic–martensitic dual phase (DP) steels deform spatially in a highly heterogeneous manner, with strong strain and stress partitioning at the micro-scale. Such heterogeneity in local strain evolution leads in turn to a spatially heterogeneous damage distribution, and thus, plays an important role in the process of damage inheritance and fracture. To understand and improve DP steels, it is important to identify connections between the observed strain and damage heterogeneity and the underlying microstructural parameters, ferrite grain size, martensite distribution, martensite fraction, etc. In this work we pursue this aim by conducting deformation experiments on two different DP steel grades, employing two different microscopic-digital image correlation (μDIC) techniques to achieve microstructural strain maps of representative statistics and high-resolution. The resulting local strain maps are analyzed in connection to the observed damage incidents (identified by image post-processing) and to local stress maps (obtained from crystal plasticity (CP) simulations of the same microstructural area). The results reveal that plasticity is typically initiated within “hot zones” with larger ferritic grains and lower local martensite fraction. With increasing global deformation, damage incidents are most often observed in the boundary of such highly plastified zones. High-resolution μDIC and the corresponding CP simulations reveal the importance of martensite dispersion: zones with bulky martensite are more susceptible to macroscopic localization before the full strain hardening capacity of the material is consumed. Overall, the presented joint analysis establishes an integrated computational materials engineering (ICME) approach for designing advanced DP steels.
► New model captures the smooth transition in the thermal and mechanical responses. ► New model considers effect of stress magnitude on recoverable deformation magnitude. ► New model adds stress dependency to the concept of critical thermodynamic force. ► Full numerical implementation of the model in an efficient scheme is described. ► Experimental results compared to analysis predictions, demonstrating accuracy. This work presents new developments in the thermomechanical constitutive modeling of shape memory alloys (SMAs). The proposed phenomenological constitutive model is motivated by the earlier work of and considers three characteristics of SMA response that have not been addressed in a unified manner to date. First, it captures the smooth transition in the thermal and mechanical responses often observed as the martensitic transformation is initiated and completed. Secondly, it considers the effect of applied stress magnitude on the generation of favored martensitic variants without explicitly considering the process of martensitic reorientation, resulting in a computationally efficient and accurate analysis tool. Finally, it generalizes the concept of the critical thermodynamic forces for transformation, which become dependent on transformation direction and applied stress magnitude. These improvements, introduced within a thermodynamically consistent mathematical framework, increase model fidelity over a wide range of SMA material systems. The full numerical implementation of the model in an efficient scheme is described. Experimental results associated with various thermomechanical paths are compared to the analysis predictions, including stress-induced and thermally induced transformations under uniaxial and non-proportional mechanical loads. Stress-free calorimetric results are also simulated. Analysis of a boundary value problem considering large rotations and local non-proportional loadings is described.
Constitutive models for shape memory alloys have seen significant development in the last decades. They have evolved from uniaxial, mostly empirical, relations to full-fledged mathematical descriptions accounting for many of the effects observed in these materials with an ever-higher degree of detail. The models available today are constructed using various approaches ranging from micromechanics with or without scale transition, to statistical physics and particle dynamics, to methods of classical plasticity, to energy approaches coupled with thermodynamic and conservation principles. They have finally matured to the extent where they can be utilized in reasonably accurate numerical analysis of potentially complex shape memory alloy devices subjected to non-trivial thermomechanical loading. This paper aims at providing an up-to-date review of key constitutive models for shape memory alloys, with an attempt to track their evolution from their inception to their most recent versions. The models are categorized in terms of the approach they use in describing the behavior of shape memory alloys.
Classical metal plasticity theory assumes that the hydrostatic pressure has no or negligible effect on the material strain hardening, and that the flow stress is independent of the third deviatoric stress invariant (or Lode angle parameter). However, recent experiments on metals have shown that both the pressure effect and the effect of the third deviatoric stress invariant should be included in the constitutive description of the material. A general form of asymmetric metal plasticity, considering both the pressure sensitivity and the Lode dependence, is postulated. The calibration method for the new metal plasticity is discussed. Experimental results on aluminum 2024-T351 are shown to validate the new material model. From the similarity between yielding surface and fracture locus, a new 3D asymmetric fracture locus, in the space of equivalent fracture strain, stress triaxiality and the Lode angle parameter, is postulated. Two methods of calibration of the fracture locus are discussed. One is based on classical round specimens and flat specimens in uniaxial tests, and the other one uses the newly designed butterfly specimen under biaxial testing. Test results of Bao (2003) [Bao, Y., 2003. Prediction of ductile crack formation in uncracked bodies. PhD Thesis, Massachusetts Institute of Technology] on aluminum 2024-T351, and test data points of A710 steel from butterfly specimens under biaxial testing validated the postulated asymmetric 3D fracture locus.
In order to study the behavior of material under finite deformation at various strain rates, the responses of AZ31 Mg sheet are measured under uniaxial (tension and compression) and multiaxial (simple shear) loadings along rolling direction (RD), 45° to rolling direction (DD), 90° to rolling direction (TD), and normal to the sheet (ND) to large strains. The material exhibits positive strain rate sensitivity (SRS) at room and elevated temperatures; the SRS is more pronounced at high temperatures and lower strain rates. The -value of the material under tensile loading at room temperatures is higher in TD at lower strain rate. Texture measurements on several failed specimens are reported under tension and simple shear after finite plastic deformation of about 20% equivalent strain. The as-received material exhibits a strong fiber with equal fractions of grains having the -axis slightly tilted away from the sheet normal towards both +RD and −RD. Pole figures obtained after tensile loading along the rolling direction (RD) show that the texture of the material strengthens even at low strains, with -axis perpendicular to the sheet plane and prism planes lining up in a majority of grains. However, the tensile loading axis along TD does not lead to similar texture strengthening; the -axis distribution appears to be virtually unchanged from the virgin state. The pole figures obtained after in-plane compression along RD brings the -axes of the grains parallel to the loading direction. The pole figures after simple shear loading show that the -axis rotates to lie on the sheet plane consistent with a compression axis 45° away on the sheet plane.
► Plastic constitutive equations. ► Variable Young’s modulus. ► Through-thickness integration of stress. ► Magnesium. ► Advanced high strength steels (AHSS). For purposes of this review, springback is the elastically driven change of shape of a metal sheet during unloading and following forming. Scientific advances related to this topic have accelerated dramatically over roughly the last decade, since the publication of two reviews in the 2004–2006 timeframe ( ). The current review focuses on the period following those publications, and on work in the first author’s laboratory. Much of this recent work can be categorized into five main topics. The first two subjects are related to accurate material representation, the third to numerical procedures, and the last two to particular classes of sheet materials. The principal contributions in these areas were summarized and put into context.
Basic ductile fracture experiments for sheet metal (or flat coupons extracted from bulk material) are presented to characterize the onset of fracture at different stress states. Special emphasis is placed on designing the experiments such that the stress triaxiality and the Lode angle parameter remain constant while the specimen is loaded all the way to fracture. A new in-plane specimen with two parallel gage sections is proposed to determine the strain to fracture for approximately zero stress triaxiality. A FEA based methodology is shown to identify the optimal specimen geometry as a function of the material's ductility and strain hardening. A tension specimen with a central hole is investigated in detail with regard to determining the strain to fracture for uniaxial tension. It is found that the required hole-to-ligament width ratio decreases as a function of the material ductility and increases as a function of the strain hardening exponent. The bending of a wide strip is pursued to prevent the necking prior to fracture under plane strain tension conditions, while an Erichsen-type of punch test is used to characterize the material response for equi-biaxial tension. It is worth noting that the strain to fracture can be directly determined from surface strain measurements in the cases of shear, plane strain tension and equi-biaxial tension loading, thereby removing the need to perform finite element simulations for extracting the loading path to fracture.
► Extension of a spectral method to finite strains useful for crystal micro mechanics. ► Identical material model in comparison of finite element and spectral method solutions. ► Convergence with mesh/grid is faster than FEM. ► Stress equilibrium and strain compatibility is fulfilled much better. ► Computation time of 256 grid comparable to that of 64 linear finite elements. A significant improvement over existing models for the prediction of the macromechanical response of structural materials can be achieved by means of a more refined treatment of the underlying micromechanics. For this, achieving the highest possible spatial resolution is advantageous, in order to capture the intricate details of complex microstructures. Spectral methods, as an efficient alternative to the widely used finite element method (FEM), have been established during the last decade and their applicability to the case of polycrystalline materials has already been demonstrated. However, until now, the existing implementations were limited to infinitesimal strain and phenomenological crystal elasto-viscoplasticity. This work presents the extension of the existing spectral formulation for polycrystals to the case of finite strains, not limited to a particular constitutive law, by considering a general material model implementation. By interfacing the exact same material model to both, the new spectral implementation as well as a FEM-based solver, a direct comparison of both numerical strategies is possible. Carrying out this comparison, and using a phenomenological constitutive law as example, we demonstrate that the spectral method solution converges much faster with mesh/grid resolution, fulfills stress equilibrium and strain compatibility much better, and is able to solve the micromechanical problem for, , a 256 grid in comparable times as required by a 64 mesh of linear finite elements.
► A small-strain FFT-based model for elasto-viscoplastic polycrystals is presented. ► Local fields and effective behavior can be computed from a microstructure image. ► Implicit time discretization and an augmented Lagrangian iterative scheme are used. ► Benchmarks for mechanical behavior, boundary conditions and convergence are presented. ► The model is used to study the influence of crystal anisotropy on stress hot-spots. We present the infinitesimal-strain version of a formulation based on fast Fourier transforms (FFT) for the prediction of micromechanical fields in polycrystals deforming in the elasto-viscoplastic (EVP) regime. This EVP extension of the model originally proposed by Moulinec and Suquet to compute the local and effective mechanical behavior of a heterogeneous material directly from an image of its microstructure is based on an implicit time discretization and an augmented Lagrangian iterative procedure. The proposed model is first benchmarked, assessing the corresponding elastic and viscoplastic limits, the correct treatment of hardening, rate-sensitivity and boundary conditions, and the rate of convergence of the numerical method. In terms of applications, the EVP–FFT model is next used to examine how single crystal elastic and plastic directional properties determine the distribution of local fields at different stages of deformation.
A macroscopic ductile fracture criterion is proposed based on micro-mechanism analysis of nucleation, growth and shear coalescence of voids from experimental observation of fracture surfaces. The proposed ductile fracture model endows a changeable cut-off value for the stress triaxiality to represent effect of micro-structures, the Lode parameter, temperature, and strain rate on ductility of metals. The proposed model is used to construct fracture loci of AA 2024-T351. The constructed fracture loci are compared with experimental data covering wide stress triaxiality ranging between −0.5 and 1.0. The fracture loci are constructed in full stress spaces and plane stress conditions to analyze characteristics of the proposed fracture loci. Errors of the equivalent stress to fracture are calculated and compared with those predicted by the MSV model ( ) and series of the modified Mohr–Coulomb criteria. The comparison suggests that the proposed model can provide a satisfactory prediction of ductile fracture for metals from compressive upsetting tests to plane strain tension with slanted fracture surfaces. Moreover, it is expected that the proposed model reasonably describes ductile fracture behavior in high velocity perforation simulation since a reasonable cut-off value for the stress triaxiality is coupled with the proposed ductile fracture criterion.
Numerous criteria have been developed for ductile fracture (DF) prediction in metal plastic deformation. Finding a way to select these DF criteria (DFCs) and identify their applicability and reliability, however, is a non-trivial issue that still needs to be addressed in greater depth. In this study, several criteria under the categories of ‘uncoupled damage criterion’ and the ‘coupled damage criterion’, including the continuum damage mechanics (CDM)-based Lemaitre model and the Gurson–Tvergaard–Needleman (GTN) model, are investigated to determine their reliability in ductile failure prediction. To create diverse stress and strain states and fracture modes, different deformation scenarios are generated using tensile and compression tests of Al-alloy 6061 (T6) with different sample geometries and dimensions. The two categories of criteria are coded into finite element (FE) models based on the unconditional stress integration algorithm in the VUMAT/ABAQUS platform. Through physical experiments, computations and three industrial case studies, the entire correlation panorama of the DFCs, deformation modes and DF mechanisms is established and articulated. The experimental and simulation results show the following. (1) The mixed DF mode exists in every deformation of concern in this study, even in the tensile test of the round bar sample with the smallest notch radius. A decrease of stress triaxiality ( -value) leads to a reduction in the accuracy of DF prediction by the two DFC categories of DFCs, due to the interplay between the principal stress dominant fracture and the shear–stress dominant factor. (2) For deformations with a higher -value, both categories of DFCs predict the fracture location reasonably well. For those with a lower or even negative -value, the GTN and CDM-based criteria and some of the uncoupled criteria, including the C&L, Ayada and Oyane models, provide relatively better predictions. Only the Tresca and Freudenthal models can properly predict the shear dominant fracture. The reliability sequence of fracture moment prediction is thus the GTN model, followed by the CDM-based model and the uncoupled models. (3) The applicability of the DFCs depends on the use of suitable damage evolution rules (void nucleation/growth/coalescence and shear band) and consideration of several influential factors, including pressure stress, stress triaxiality, the Lode parameter, and the equivalent plastic strain or shear stress. These parameters determine the deformation mode (shear dominant or maximum principal stress dominant deformation) and, further, the DF mechanism (dimple fracture/shear fracture/mixed fracture).
Hexagonal materials deform plastically by activating diverse slip and twinning modes. The activation of such modes depends on their relative critical stresses, and the orientation of the crystals with respect to the loading direction. To be reliable, a constitutive description of these materials has to account for texture evolution associated with reorientations due to both dislocation slip and twinning, and for the effect of the twin boundaries as barriers to dislocation propagation. We extend a previously introduced twin model, which accounts explicitly for the composite character of the grain formed by a matrix with embedded twin lamellae, to describe the influence of twinning on the mechanical behavior of the material. The role of the twins as barriers to dislocations is explicitly incorporated into the hardening description of slip deformation via a directional Hall–Petch mechanism. We introduce here an improved hardening law for twinning, which discriminates for specific twin/dislocation interactions, and a detwinning mechanism. We apply this model to the interpretation of compression and tension experiments done in rolled magnesium alloy AZ31B at room temperature. Particularly challenging cases involve strain-path changes that force strong interactions between twinning, detwinning, and slip mechanisms.
In this work, we develop a multi-level constitutive model for polycrystalline metals that deform by a combination of elasticity, slip and deformation twinning. It involves a two-level homogenization scheme, where the first level uses an upper bound Taylor-type crystal plasticity (T-CP) theory to relate the single-crystal scale to the polycrystal meso-scale and the second level employs an implicit finite elements (FE) approach to relate the meso-scale to the macro-scale. The latter relaxes the iso-strain constraints imposed by the Taylor model. As such, we name the model T-CPFE. At the single crystal level, the model features a dislocation-based hardening law providing the activation stresses that governs slip activity within the single crystals. For deformation twinning, it contains an advancement of a composite grain model that retains the total Lagrangian formulation. Here we use the T-CPFE model to analyze the mechanical response and microstructure evolution of extruded AZ31 Mg alloy samples in simple compression, tension, and torsion under strain rates ranging from 10 s to 3000 s and temperatures ranging from 77 K to 423 K reported in Kabirian et al. (2015). Taking the experimentally measured initial texture and average grain size as inputs, the model successfully captures stress-strain responses, deformation texture evolution and twin volume fraction using a single set of material parameters associated with the thermally activated rate laws for dislocation density. The distinctions in flow stress evolution among the loading conditions result from differing relative amounts of slip and twinning activity, which the model internally adjusts based on evolution of slip and twin resistances in the response to the imposed loading conditions. Finally, we show that the T-CPFE model predictions of geometrical changes during compression compare favorably with corresponding geometry of samples deformed experimentally. For this application, it predicts the anisotropy and asymmetry of the material flow resulting from crystallographically soft-to-deform extension twinning and basal slip and hard-to-deform contraction twinning and pyramidal slip. The formulation developed is sufficiently general that the T-CPFE model can be applied to other materials that slip and twin.
Lacking of plasticity at ambient temperature severely hinders the wide applications of bulk metallic glasses, and a significant challenge is to improve the plasticity. Based on the metallurgical physics, micro-alloying can be applied to adjust metallic glasses plasticity. In the current work, dynamic mechanical relaxation of Cu Zr Al Dy (0 ≤ x ≤ 8) bulk metallic glasses has been investigated experimentally by dynamic mechanical analysis. Compressive tests have been performed to investigate mechanical properties of the Cu-based bulk metallic glasses at both ambient as well as cryogenic temperatures. The results indicated that by modifying the chemical composition, plastic deformation and dynamic mechanical relaxation processes are changed. The influence of Dysprosium (Dy) on plastic deformation is possibly related to the Johari-Goldstein (JG) relaxation in the metallic glasses. A kinetic model which may be predict the mechanical relaxation behavior and atomic mobility of the metallic glasses. In addition, experimental analyses show that thermal properties can be affected by the Dy addition of the Cu-based bulk metallic glasses. Our investigations demonstrated that micro-alloying of Dy could play an important role to influence the Cu Zr Al Dy bulk metallic glasses plasticity. In order to explain this behavior, the quasi-point defects theory was used to describe the microstructural heterogeneity. We postulate that the compressive plasticity is directly associated with local heterogeneity and relaxation modes for metallic glasses.
In this work, an approach is proposed for the description of the plastic behavior of materials subjected to multiple or continuous strain path changes. In particular, although it is not formulated with a kinematic hardening rule, it provides a reasonable description of the Bauschinger effect when loading is reversed. This description of anisotropic hardening is based on homogeneous yield functions/plastic potentials combining a stable, isotropic hardening-type, component and a fluctuating component. The latter captures, in average, the effect of dislocation interactions during strain path changes. For monotonic loading, this approach is identical to isotropic hardening, with an expanding isotropic or anisotropic yield surface around the active stress state. The capability of this constitutive description is illustrated with applications on a number of materials, namely, low carbon, dual phase and ferritic stainless steel samples.
An empirical plasticity constitutive form describing the flow stress as a function of strain, strain-rate, and temperature has been developed, fit to data for three dual-phase (DP) steels, and compared with independent experiments outside of the fit domain. Dubbed the “H/V model” (for “Hollomon/Voce”), the function consists of three multiplicative functions describing (a) strain hardening, (b) strain-rate sensitivity, and (c) temperature sensitivity. Neither the multiplicative structure nor the choice of functions (b) or (c) is novel. The strain hardening function, (a), has two novel features: (1) it incorporates a linear combination coefficient, , that allows representation of Hollomon (power law) behavior ( = 1), Voce (saturation) behavior ( = 0) or any intermediate case (0 < < 1, and (2) it allows incorporation of the temperature sensitivity of strain hardening rate in a natural way by allowing to vary with temperature (in the simplest case, linearly). This form therefore allows a natural transition from unbounded strain hardening at low temperatures toward saturation behavior at higher temperatures, consistent with many observations. Hollomon, Voce, H/V models and others selected as representative from the literature were fit for DP590, DP780, and DP980 steels by least-squares using a series of tensile tests up to the uniform strain conducted over a range of temperatures. Jump-rate tests were used to probe strain rate sensitivity. The selected laws were then used with coupled thermo-mechanical finite element (FE) modeling to predict behavior for tests outside the fit range: non-isothermal tensile tests beyond the uniform strain at room temperatures, isothermal tensile tests beyond the uniform strain at several temperatures and hydraulic bulge tests at room temperature. The agreement was best for the H/V model, which captured strain hardening at high strain accurately as well as the variation of strain hardening with temperature. The agreement of FE predictions up to the tensile failure strain illustrates the critical role of deformation-induced heating in high-strength/high ductility alloys, the importance of having a constitutive model that is accurate at large strains, and the implication that damage and void growth are unlikely to be determinant factors in the tensile failure of these alloys. The new constitutive model may have application for a wide range of alloys beyond DP steels, and it may be extended to larger strain rate and temperature ranges using alternate forms of strain rate sensitivity and thermal softening appearing in the literature.
Low, intermediate and high strain rate tensile experiments are carried out on flat smooth, notched and central-hole tensile specimens extracted from advanced high strength steel sheets. A split Hopkinson pressure bar testing system is used in conjunction with a load inversion device to perform the high strain rate tension experiments. Selected surface strains, as well as local displacements, are measured using high speed photography in conjunction with planar digital image correlation (video extensometer). Through thickness necking precedes fracture in all experiments. A hybrid experimental–numerical approach is therefore employed to determine the strain to fracture inside the neck. To obtain an accurate description of the local strain fields at very large deformations, a plasticity model with a Johnson–Cook type of rate and temperature-dependency and a combined Swift–Voce strain hardening law is used in conjunction with a non-associated anisotropic flow rule. The incremental change in temperature is computed using a strain rate dependent weighting function instead of solving the thermal field equations. The comparison of the computed and measured force–displacement curves and surface strain histories shows good agreement before and after the onset of necking. From each experiment, the loading path to fracture is determined describing the evolution of the equivalent plastic strain in terms of the stress triaxiality, Lode angle parameter, strain rate and temperature. An empirical extension of the stress-state dependent Hosford–Coulomb fracture initiation model is proposed to account for the effect of strain rate on the onset of ductile fracture. The model is subsequently calibrated and successfully validated using the results from fracture experiments on DP590 and TRIP780 steels.
Mechanical responses and texture evolution of extruded AZ31 Mg are measured under uniaxial (tension–compression) and multiaxial (free-end torsion) loadings. Compression loading is carried out in three different directions: along the extrusion direction (ED), perpendicular to the extrusion direction (PED), and 45° to the extrusion direction (45ED) at temperature and strain rate ranges of 77–423 K and 10 –3000 s , respectively. Texture evolution at different intermediate strains reveals that crystal reorientation is exhausted at smaller strains with increase in strain rate while increase in temperature retards twinning. In addition to the well-known tension–compression yield asymmetry, a strong anisotropy in strain hardening response is observed. However, this anisotropy is negligible at smaller strain so that compressive yield stress does not change with loading directions at each temperature and strain rate. Strain hardening during the compression experiment is intensified with decreasing and increasing temperature and strain rate, respectively. Even though the strain hardening response during the free-end torsion experiment resembles that in tension, the shear yield stress is significantly smaller than prediction of von-Mises criterion. This complex behavior is explained through the understanding roles of deformation mechanisms using the Visco-Plastic Self Consistent (VPSC) model. In order to calibrate the VPSC model’s constants as accurate as possible, in contrast to previous studies, this paper employs the VPSC model to simulate a vast number of mechanical responses and crystallographic characteristics including stress–strain curves in tension, compression in three directions, and free-end torsion, texture evolution at different strains, lateral strains of compression samples, twin volume fraction, and axial strain during the torsion experiment. The modeling results show that depending on the number of measurements used for calibration, roles of different mechanisms in plastic deformation change significantly.