R.J. Asaro

Overview no. 42 Texture development and strain hardening in rate dependent polycrystals

Abstract A new rate dependent constitutive model is developed for polycrystals subjected to arbitrarily large strains. The model is used to predict deformation textures and large-strain strain hardening behavior following various stress-strain histories for single phase f.c.c. aggregates that deform by crystallographic slip. Examples involving uniaxial and plane strain tension and compression are presented which illustrate how texture influences polycrystalline strain hardening, in particular these examples demonstrate both textural strengthening and softening effects. Input to the model includes the description of single crystal strain hardening and latent hardening along with strain rate sensitivity, all properties described on the individual slip system level. The constitutive formulation used for the individual grains is essentially that developed by Peirce et al . [6, Acta metall . 31, 1951 (1983)] to solve rate dependent boundary value problems for finitely deformed single crystals. Inclusion of rate dependence is shown to overcome the long standing problem of nonuniqueness in the choice of active slip systems which is inherent in the rate independent theory. Because the slipping rates on all slip systems within each grain are unique in the rate dependent theory, the lattice rotations and thus the textures that develop are unique. In addition, the model makes it possible to study how strain rate sensitivity on the slip system, and single grain, levels is manifested in polycrystalline strain rate sensitivity. The model is also used to predict “constant offset plastic strain yield surfaces” for materials that are nearly rate insensitive—these calculations describe the development of rounded “yield surface vertices” and the resulting softening of material stiffness to a change in loading path that vertices imply. For our rate dependent solid this reduction in stiffness occurs after small but finite loading increments. Finally the model is used to carry out an imperfection-based sheet necking analysis both for isotropic and strongly textured sheets. The results show that larger strain hardening rates, and strain rate sensitivity, on the slip system level both increase the failure strains, as expected, but also demonstrate a strong influence of texture on localized necking.

Material rate dependence and localized deformation in crystalline solids

Abstract Nonuniform deformations of rate dependent single crystals subject to tensile loading are analyzed numerically. The crystal geometry is idealized in terms of a planar double slip model. In addition to allowing the effects of material rate sensitivity to be explored, the present rate dependent formulation permits the analysis of a range of material strain hardening properties and crystal geometries that could not be analyzed within a rate independent framework. Two crystal geometries are modeled. One is a planar model of an f.c.c. crystal undergoing symmetric primary-conjugate slip. For this geometry, a direct comparison with a previous rate independent calculation shows that material rate sensitivity delays shear band development significantly. Our present rate dependent formulation also enables a more complete exploration of the effects of high (i.e. greater than Taylor) latent hardening ratios on “patchy” slip development. In particular we show that strong latent hardening and patchy slip can give rise to kinematical constraints that prevent shear bands from propagating completely across the gage section. The second geometry models a b.c.c. crystal oriented so that there is approximately a double mode of slip with the slip systems inclined by more than 45° to the tensile axis. This calculation displays the formation of a localized band of conjugate slip. The lattice rotations accompanying this mode eventually lead to a decrease in the resolved shear stress on the more active system in the band so that the bands do not accumulate large strains and catastrophic shear bands do not form. The implications of material rate sensitivity for uniqueness are also discussed with reference to implications for the prediction of mechanical properties of polycrystals.

An analysis of nonuniform and localized deformation in ductile single crystals

Abstract The nonuniform and localized deformations of ductile single crystals subject to tensile loading are analyzed numerically. The crystal is modelled by a rate independent, elastic-plastic relation based on Schmids law which precisely accounts for lattice rotations. Both self hardening and latent hardening of the slip systems are included in the model. The crystal geometry is idealized in terms of a planar double slip model. Initial imperfections are specified in the form of slight thickness inhomogeneities and the calculations follow the crystal deformation through diffuse necking and the formation of shear bands. The pattern of shear bands depends on the initial imperfection, but, independent of the particular small imperfection, the material planes of the bands are inclined at a characteristic angle to the slip planes. Also, the lattice misorientation across the shear band, which is such as to cause geometrical softening of the bands, is not sensitive to the imperfection form. For high strength, low hardening crystals a comparison with existing experimental data shows remarkably good qualitative and quantitative agreement between the calculations and observations. We also model a relatively soft high hardening crystal which undergoes more diffuse necking than the strong high hardening crystal. Diffuse necking leads to lattice rotations which produce geometrical softening and hence promote shear band formation. Furthermore, we carry out a calculation for a high strength low hardening crystal with the latent hardening rate prescribed somewhat larger than for isotropic hardening. In this case a ‘patchy’ pattern of slip emerges. However, the course of shear band development is unaffected.

Micromechanics of Crystals and Polycrystals

Publisher Summary This chapter focuses on micromechanics of crystals and polycrystals. In Section II of the chapter, a brief outline of only some of the important features of the micromechanics of crystalline plasticity is given. The discussion is confined to plastic flow caused by dislocation slip, and face-centered-cubic crystals are used in the examples of dislocation mechanisms. Particular attention is paid to kinematics and to the phenomenology of strain hardening, because these are shown to play dominant roles in macroscopic response. In Section III, constitutive laws for elastic-plastic crystals are developed. The framework draws heavily on Hills analysis of the mechanics of elasticplastic crystals, but the theory is extended by incorporating the possibility of deviations from the Schmid rule of a critical resolved shear stress for slip. Deviations from the Schmid rule are motivated by micromechanical models for dislocation motion and are shown to lead to deviations from the “normality flow rule” of continuum plasticity. The implications of these “non-Schmid effects” regarding the stability of plastic flow are brought out via some examples of models for kinks bands and shear bands in Section IV. In Section IV some examples of analyses of elastic-plastic deformation in crystals are discussed. The chapter concludes with some suggestions for fruitful research. These involve extensions of the theory to finite-strain rate-dependent polycrystalline models.

Geometrical effects in the inhomogeneous deformation of ductile single crystals

Abstract Non-uniform deformation in ductile single crystals is studied using a simple model for a crystal undergoing symmetric double slip in tension. The model, when interpreted in terms of the crystallography of face-centred or body-centred cubic crystals, demonstrates, in particular, that shear bands may form when the slip plane workhardening rates are positive and, thus, without either ideal plasticity or strain softening. Localized plastic flow is viewed as a shearing bifurcation in the uniform tensile deformation of the crystal model and the critical stresses and workhardening rates for localized shear are worked out. The importance of yield vertexes and geometrical softening phenomena related to lattice rotations during deformation are given special attention in the development of the constitutive laws and in the results for the critical conditions for localized shear.

Nonuniform deformations in polycrystals and aspects of the validity of the Taylor model

Abstract F ull solutions to mixed rate boundary value problems over polycrystalline domains are performed via the finite element method. In order to make these finite element calculations feasible, an idealized two-dimensional crystal structure is studied. These boundary value problems rigorously satisfy the averaging theorems of Hill (Proc. R. Soc.A326, 131, 1972) so that well defined Taylor model analogue problems may be identified and solved. Comparisons between the finite element solutions and their corresponding Taylor model analogues yield a quantitative assessment of the Taylor models validity with respect to its predictions of texture development and global stress-strain response. The finite element calculations also provide physical insight into the mechanisms contributing to the development of nonuniform and localized deformations in polycrystals.

Finite element analysis of crystalline solids

Abstract A recently developed finite element formulation for the analysis of nonuniform and localized deformations in rate-dependent single crystals is reviewed. A few illustrative examples are presented but the main focus is on issues relating to the formulation and implementation of finite element methods for crystalline solids.

Analysis of large-strain shear in rate-dependent face-centred cubic polycrystals: correlation of micro- and macromechanics

Micro- and macroscopic aspects of large-strain deformation are examined through analyses of shear by using physical and phenomenological models. Past experiments and analyses are first reviewed to reveal current issues and put the present work in perspective. These issues are addressed by a complete set of simulations of large-strain shear with a finite-strain, rate-dependent polycrystal model. The model is based on a rigorous constitutive theory for crystallographic slip that accounts for the development of crystallographic texture and the effects of texture on constitutive response. The influences of strain hardening, latent hardening, strain-rate sensitivity, boundary constraints, and initial textures on texture evolution and constitutive response are studied. Coupled stress and strain effects such as axial elongation during unconstrained shear and the development of normal stresses during constrained shear are related to material properties, boundary constraint and texture. The formation of ideal textures and their role in determining polycrystalline behaviour is discussed in quantitative terms. Large-strain shear is also studied by using several phenomenological constitutive theories including J2-flow theory, J2-corner theory, and two versions of finite-strain kinematic hardening theory. The behaviours predicted by these phenomenological theories and the physically based polycrystal model are directly compared. A noteworthy outcome is the close correspondence found between the predictions of J2-corner theory and those of the micromechanically based physical model.

Micro and macroscopic aspects of shear band formation in internally nitrided single crystals of Fe-Ti-Mn alloys

Abstract The formation of localized shear bands in single crystals of internally nitrided alloys of Fe-Ti-Mn was studied experimentally and theoretically. The experimental studies included observation of the dislocation substructures that are formed at and near shear band/matrix interfaces along with documentation of the crystallography of the localized shearing process. Computational studies of tensile deformation of the crystals using the finite element method, that incorporated a large strain, strain rate dependent constitutive theory for crystalline slip, were carried out and the results were compared to the experimental observations. Mechanical tests, and in situ metallographic observations of shear band formation, showed that localized shearing occurs while the materials are continuously hardening and before “damage”, through microfracture, begins. Electron diffraction and imaging of the deformation substructures near shear band interfaces showed that an important part of the localization mechanism is nonuniform lattice reorientations that cause a “geometrical softening” of the lattice. The numerical studies are in good agreement with the experiments and demonstrate the role of material strain hardening and strain rate sensitivity.

Structure of martensite in very small iron-rich precipitates

Abstract A transmission electron microscopy study has been made of the structure of the martensite that forms in very small iron-rich precipitates (~ 2000 A dia.)in a Cu-2.5% Fe alloy. The martensitic product formed at room temperature was found to be comprised of {112}α twins of the order of 50–500 A thick. It is noteworthy that the lattice invariant shear mode of martensite formation in a bulk alloy of the same composition as the precipitates is dislocation slip, not twinning. The choice of twinning as the mode of self-accommodation in the case of these small particles is predominently influenced by the circumstances that the temperature of transformation in the small precipitates is much lower than in bulk material. This is consistent with the further observation that precipitates induced to transform at temperatures nearer the bulk Ms each produce a single crystal of dislocated martensite. The attainment of a fully transformed product in the twinned state is justified quantitatively on energetic grounds.