A continuum model for initiation and evolution of deformation twinning

Within continuum dislocation theory the plastic deformation of a single crystal with one active slip system under plane-strain constrained shear is investigated. By introducing a twinning shear into the energy of the crystal, we show that in a certain range of straining the formation of deformation twins becomes energetically preferable. An energetic threshold for the onset of twinning is determined. A rough analysis qualitatively describes not only the evolving volume fractions of twins but also their number during straining. Finally, we analyze the evolution of deformation twins and of the dislocation network at non-zero dissipation. We present the corresponding stress-strain hysteresis, the evolution of the plastic distortion, the twin volume fractions and the dislocation densities.     

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.     

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%.     

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.     

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme.Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.     

A Peierls criterion for the onset of deformation twinning at a crack tip

A criterion for the onset of deformation twinning (DT) is derived within the Peierls framework for dislocation emission from a crack tip due to Rice (J. Mech. Phys. Solids 40(2) (1992) 239). The critical stress intensity factor (SIF) is obtained for nucleation of a two-layer microtwin, which is taken to be a precursor to DT. The nucleation of the microtwin is controlled by the unstable twinning energyγ_ut, a new material parameter identified in the analysis. γ_ut plays the same role for DT as γ_us, the unstable stacking energy introduced by Rice, plays for dislocation emission. The competition between dislocation emission and DT at the crack tip is quantified by the twinning tendencyT defined as the ratio of the critical SIFs for dislocation nucleation and microtwin formation. DT is predicted when T>1 and dislocation emission when T<1. For the case where the external loading is proportional to a single load parameter, T is proportional to . The predictions of the criterion are compared with atomistic simulations for aluminum of Hai and Tadmor (Acta Mater. 51 (2003) 117) for a number of different crack configurations and loading modes. The criterion is found to be qualitatively exact for all cases, predicting the correct deformation mode and activated slip system. Quantitatively, the accuracy of the predicted nucleation loads varies from 5% to 56%. The sources of error are known and may be reduced by appropriate extensions to the model.     

On the r-value of textured sheet metals

The r-value of a sheet metal is a measure of plastic anisotropy frequently used for prediction of performance in deep-drawing. It has also figured prominently in the literature for validation of theories where the predicted angular dependence of r is compared with the measured dependence. As plastic anisotropy in sheet metals is caused mainly by the preferred orientations of grains within the polycrystalline metal, it is natural to ask how r would depend on the orientation distribution function (ODF) w which defines the crystallographic texture of the polycrystal. In this paper a general formula relating r to w is derived for textured sheet metals whose plastic flow behavior is governed by a plastic potential f(σ, w), the anisotropic part of which depends linearly on the texture coefficients; here σ denotes the deviator of the Cauchy stress. Specific forms of this formula for orthorhombic sheets of cubic and of hexagonal metals are explicitly given.     

EBSD-based continuum dislocation microscopy

Recent advances in high-resolution electron backscatter diffraction (EBSD)-based microscopy are applied to the characterization of elastic fields and incompatibility structures near the grain boundaries (GBs) in polycrystals. Two main recoveries are reported here: surface geometrically necessary dislocation (density) tensors, as described by Kröner, and the elastic fields near cracks (unconsolidated portions of interface) in loaded samples. Context for the application of these recoveries is described, using Green’s function solutions for combined heterogeneity and dislocation. Featured recoveries required the cross-correlation based determination of the elastic distortion tensor, aided by application of the simulated pattern method, and determination of the absolute pattern center utilizing the expected pattern properties in a spherical Kikuchi reference frame. High-resolution data obtained along an ultrasonically consolidated nickel boundary of varying amalgamation indicates that the imposed traction free boundary condition at free surfaces is well observed in the data structure. Further, high-resolution data acquired near a single grain boundary in well-annealed, low content steel suggests that it may be possible to measure the intrinsic elastic properties of GBs.     

A three-dimensional crystal plasticity model for duplex Ti–6Al–4V

A rate dependent crystal plasticity model for the α/β Ti–Al alloy Ti–6Al–4V with duplex microstructure is developed and presented herein. Duplex Ti–6Al–4V is a dual-phase alloy consisting of an hcp structured matrix primary α-phase and secondary lamellar α + β domains that are composed of alternating layers of secondary α laths and bcc structured residual β laths. The model accounts for distinct three-dimensional slip geometry for each phase, anisotropic and length scale dependent slip system strengths, the non-planar dislocation core structure of prismatic screw dislocations in the primary α-phase, and crystallographic texture. The model is implemented in the general purpose finite element code (ABAQUS, 2005. Ver 6.5, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI) via a UMAT subroutine.     

Experimental and numerical determinations of the initial surface of phase transformation under biaxial loading in some polycrystalline shape-memory alloys

Biaxial proportional loading such as tension (compression)–internal pressure and bi-compression tests are performed on a Cu-Zn-Al and Cu-Al-Be shape memory polycrystals. These tests lead to the experimental determination of the initial surface of phase transformation (austenite→martensite) in the principal stress space (σ₁,σ₂). A first “micro–macro” modeling is performed as follows. Lattice measurements of the cubic austenite and the monoclinic martensite cells are used to determine the “nature” of the phase transformation, i.e. an exact interface between the parent phase and an untwinned martensite variant. The yield surface is obtained by a simple (Sachs constant stress) averaging procedure assuming random texture. A second modeling, performed in the context of the thermodynamics of irreversible processes, consists of a phenomenological approach at the scale of the polycrystal. These two models fit the experimental phase transformation surface well.     

Crystal plasticity modeling of slip activity in Ti–6Al–4V under high cycle fatigue loading

Deformation micromechanisms of a Ti–6Al–4V alloy under fatigue loading at room temperature are studied using a three-dimensional crystal plasticity constitutive model. The model employs a minimum set of fitting parameters based on experimental data for Ti–6Al–4V. Single slip is strongly favored through a softening law that affects mainly the driving force for slip on the first activated slip system. Cyclic deformation behavior at the macroscopic scale and at the local scale of grains is analyzed through the simulation of 20 cycles of fatigue on a polycrystalline structure of 900 randomly oriented grains. The progressive activation of slip (basal, prismatic, and pyramidal) is analyzed and compared to experimental observations.     

Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification

The elastic/crystalline viscoplastic constitutive equation, based on a newly proposed hardening-softening evolution equation, is introduced into the dynamic-explicit finite element code “Itas-Dynamic.” In the softening evolution equation, the effective distance and the angle between each slip system of a crystal are introduced to elucidate the interaction between the slip systems, which causes a decrease of dislocation density. The polycrystal sheet is modeled by Voronoi polygons, which correspond to the crystal grains; and by the selected orientations, which can relate to the texture, they are assigned to the integration points of the finite elements. We propose a direct crystal orientation assignment method, which means that each integration point of finite element has an assigned orientation, and its orientation can be rotated independently. Therefore, this inhomogeneous polycrystal model can consider the plastic induced texture development and subsequent anisotropy evolution. The parameters of the constitutive equation are identified by uni-axial tension tests carried out on single crystal sheets. Numerical results obtained for sheet tensions are compared with experimental ones to confirm the validity of our finite element code. Further, we investigate the following subjects: (1) how the initial orientation of single crystal affects slip band formation and strain localization; (2) how the grain size and particular orientations of the grain affect the strain localization in case of a polycrystal sheet. It is confirmed that the orientation of a single crystal can be related to the primary slip system and the deformation induced activation of that system, which in turn can be related to the slip band formation of the single crystal sheet. Further, in case of a polycrystal sheet, the larger the grain size, the more the strain localizes at a specific crystal, which has the particular orientation. It is confirmed through comparisons with experiments that our finite element code can predict the localization of strain in sheets and consequently can estimate the formability of sheet metals.     

Gradient-based non-linear microstructure design

Fourier analysis is implemented on the orientation distribution of a polycrystalline microstructure. The linearity and convexity of the Fourier space, with respect to orientation, allows one to consider all possible distributions by considering all linear combinations of single-grain orientations. The limits of the Fourier space are therefore defined by the solutions to a set of linear programming problems. A unique approach to the linear programming, similar to the Krylov subspace methods for obtaining solutions to linear systems, is presented. The method is particularly efficient for this application where a large number of independent variables is often required. These solutions are then used as the constraints in the gradient-based optimization of non-linear functions within the Fourier space. In the example, Taylor yield theory and an anisotropic solution for the stress concentration around a hole in a plate of cubic-orthotropic polycrystalline material are expressed as non-linear functions within the Fourier space. The maximum obtainable ratio of Taylor factor to stress concentration for any polycrystalline orientation distribution in copper is found to be 1.22, more than double the minimum value.     

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.     

A Quantitative study of minimum sizes of representative volume elements of cubic polycrystals—numerical experiments

The concept of representative volume element (RVE) plays a key role in correlating the properties of microscopically heterogeneous materials with those of their macroscopically homogenized ones. However, up to now little quantitative knowledge is available about RVE scales or sizes of various engineering materials, which have been becoming a necessity due to the rapid development of, for instance, microelectromechanical systems. A new and convenient definition of the minimum RVE size is introduced. Then more than 500 kinds of cubic polycrystalline material in the planar stress state are numerically tested. The major finding from these numerical experiments is that the RVE size for the effective shear modulus (as well as the Young's modulus) depends roughly linearly upon the anisotropy degree of the single crystal, while the effective area modulus does not. For the latter observation a theoretical proof is also given. With a maximum relative error 5%, all the materials tested (with one exception) have a minimal RVE size of 20 or less times as large as the grain size.     

A two-dimensional finite element method for simulating the constitutive response and microstructure of polycrystals during high temperature plastic deformation

We describe a finite element method designed to model the mechanisms that cause superplastic deformation. Our computations account for grain boundary sliding, grain boundary diffusion, grain boundary migration, and surface diffusion, as well as thermally activated dislocation creep within the grains themselves. Front tracking and adaptive mesh generation are used to follow changes in the grain structure. The method is used to solve representative boundary value problems to illustrate its capabilities.     

Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation

In this paper, we establish a homogenization framework to analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The resulting conditions are verified by analyzing numerically the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is thus shown that three kinds of experimentally observed buckling modes of honeycombs i.e., uniaxial, biaxial and flower-like modes, are attained and classified as microscopic symmetric bifurcation. It is also shown that the multiplicity of bifurcation gives rise to the complex cell-patterns in the biaxial and flower-like modes.     

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.