XLVI. A theory of the plastic distortion of a polycrystalline aggregate under combined stresses

Representative volume elements of strain/stress fields measured by diffraction techniques

Finite-element modelling has been used to simulate local strains and stresses within free-standing polycrystalline slabs of W, Cu and W-Cu, heated with free or constrained boundaries. The elastic strain values in crystallites that satisfied the diffraction condition were used to simulate the lattice strain data that would be obtained from diffraction analysis, from which the average stresses within diffracting domains were computed. Comparison of direct-space stresses in the model with the average stresses determined from diffraction analysis shows that the representative volume elements (RVEs) required to obtain equivalent stress/strain values depend on the deformation mode suffered by the material. Further, the direct-space and diffraction stress values agree only under strict sampling and strain/stress uniformity conditions. Consequently, in samples where measurements are conducted in volumes smaller than the RVE, or where the uniformity conditions are not satisfied, further experimental and numerical techniques might be needed for the accurate determination of applied or residual stress distributions.

Microstructure-Property Correlation and Its Evolution during Aging in an Al4.4Co26Cr19Fe18Ni27Ti5.6 High-Entropy Alloy

The efficient energy use in multiple sectors of modern industry is partly based on the efficient use of high-strength, high-performance alloys that retain remarkable mechanical properties at elevated and high temperatures. High-entropy alloys (HEAs) represent the most recent class of these materials with a high potential for high-temperature high-strength applications. Aside from their chemical composition and microstructure-property relationship, limited information on the effect of heat treatment as a decisive factor for alloy design is available in the literature. This work intends to contribute to this research topic by investigating the effect of heat treatment on the microstructure and mechanical performance of an Al_4.4Co₂₆Cr₁₉Fe₁₈Ni₂₇Ti_5.6 HEA. The solution annealed state is compared to aged states obtained at different heat treatment times at 750 °C. The temporal evolution of the matrix and the γ'-precipitates are analyzed in terms of chemical composition, crystallography, size, shape, and volume fraction by means of scanning electron microscopy, transmission electron microscopy, and atom probe tomography. The yield strength evolution and strength contributions are calculated by classical state-of-the-art models as well as by ab-initio-based calculations of the critical resolved shear stress. The findings indicate promising mechanical properties of the investigated alloy and provide insight not only into possible strengthening mechanisms but also into the evolution of main phases during the heat treatment.

Estimation of Yield Function for Anisotropic Aggregate of FCC Crystallites

In this paper, we give an simple but approximate yield surface for single FCC crystals in Hill’s criterion form by Schmid’s law and nonlinear optimization theory. Assuming that all FCC crystallites in a polycrystal have the same (current) critical resolved shear stress τc for slip, we derive two closed but approximate yield functions through the orientational averaging of all FCC crystallites’ yield surfaces in the polycrystal. The effect of crystallography on the two yield functions are described by the orientation distribution function.

Mechanical Aspects of Nonhomogeneous Deformation of Aluminum Single Crystals under Compression along [100] and [110] Directions

The deformation behavior of aluminum single crystals subjected to compression along the [100] and [110] directions is numerically examined in terms of crystal plasticity. A constitutive model taking into account slip geometry in face-centered cubic crystals is developed using experimental data for the single-crystal samples with lateral sides coplanar to certain crystal planes. Two sets of calculations are performed using ABAQUS/Explicit to examine the features of plastic strain evolution in perfectly plastic and strain-hardened crystals. Special attention is given to the discussion of mechanical aspects of crystal fragmentation. Several distinct deformation stages are revealed in the calculations. In the first stage, narrow solitary fronts of plastic deformation are alternately formed near the top or bottom surfaces and then propagate towards opposite ends to save the symmetry of the crystal shape. The strain rate within the fronts is an order of magnitude higher than the average strain rate. The first stage lasts longer in the strain-hardened crystals, eventually giving way to an intermediate stage of multiple slips in different crystal parts. Finally, the crystal shape becomes asymmetrical, but no pronounced macroscopic strain localization has been revealed at any deformation stage. The second stage in perfectly plastic crystals relates to abrupt strain localization within a through-thickness band-shaped region, accompanied by macroscale crystal fragmentation. Stress analysis has shown that pure compression took place only in the first deformation stage. Once the crystal shape has lost its symmetry, the compressive stress in some regions progressively decreases to zero and eventually turns tensile.

In Situ TEM Observation of Cooperative Grain Rotations and the Bauschinger Effect in Nanocrystalline Palladium

We report on cooperative grain rotation accompanied by a strong Bauschinger effect in nanocrystalline (nc) palladium thin film. A thin film of nc Pd was subjected to cyclic loading–unloading using in situ TEM nanomechanics, and the evolving microstructural characteristics were investigated with ADF-STEM imaging and quantitative ACOM-STEM analysis. ADF-STEM imaging revealed a partially reversible rotation of nanosized grains with a strong out-of-plane component during cyclic loading–unloading experiments. Sets of neighboring grains were shown to rotate cooperatively, one after the other, with increasing/decreasing strain. ACOM-STEM in conjunction with these experiments provided information on the crystallographic orientation of the rotating grains at different strain levels. Local Nye tensor analysis showed significantly different geometrically necessary dislocation (GND) density evolution within grains in close proximity, confirming a locally heterogeneous deformation response. The GND density analysis revealed the formation of dislocation pile-ups at grain boundaries (GBs), indicating the generation of back stresses during unloading. A statistical analysis of the orientation changes of individual grains showed the rotation of most grains without global texture development, which fits to both dislocation- and GB sliding-based mechanisms. Overall, our quantitative in situ experimental approach explores the roles of these different deformation mechanisms operating in nanocrystalline metals during cyclic loading.

Prediction of the evolution of the stress field of polycrystals undergoing elastic-plastic deformation with a hybrid neural network model

Crystal plasticity theory is often employed to predict the mesoscopic states of polycrystalline metals, and is well-known to be costly to simulate. Using a neural network with convolutional layers encoding correlations in time and space, we were able to predict the evolution of the dominant component of the stress field given only the initial microstructure and external loading. In comparison to our recent work, we were able to predict not only the spatial average of the stress response but the evolution of the field itself. We show that the stress fields and their rates are in good agreement with the two dimensional crystal plasticity data and have no visible artifacts. Furthermore the distribution of stress throughout the elastic to fully plastic transition match the truth provided by held out crystal plasticity data. Lastly we demonstrate the efficacy of the trained model in material characterization and optimization tasks.

Mechanism of {332}<113> Twinning Formation in Cold-Rolled Ti-Nb-Ta-Zr-O Alloy

In this study, the mechanism of {332}<113> twinning formation in cold-rolled Ti-35Nb-2Ta-3Zr-O (wt %) alloy was investigated based on the Taylor-Bishop-Hill theory. The experimental data of crystal orientation in the rolling bite zone was obtained via electron back-scattered diffraction (EBSD). The deformation energy of {332}<113> twinning in the propagation stage was calculated using data from EBSD in terms of the Hall-Petch-type relation. The calculation results revealed that the mechanism of {332}<113> twinning formation in β-type Ti-35Nb-2Ta-3Zr-O (wt %) alloy contained two valid models, namely the shear-shuffle model and α″-assisted twinning model. This can help to clarify the mechanism of {332}<113> twinning formation further.

Application of an Evolving Non-Associative Anisotropic-Asymmetric Plasticity Model for a Rare-Earth Magnesium Alloy

Magnesium sheet metal alloys have a hexagonal close packed (hcp) crystal structure that leads to severe evolving anisotropy and tension-compression asymmetry as a result of the activation of different deformation mechanisms (slip and twinning) that are extremely challenging to model numerically. The low density of magnesium alloys and their high specific strength relative to steel and aluminum alloys make them promising candidates for automotive light-weighting but standard phenomenological plasticity models cannot adequately capture the complex plastic response of these materials. In this study, the constitutive plastic behavior of a rare-earth magnesium alloy sheet, ZEK100 (O-temper), was considered at room temperature, under quasi-static conditions. The CPB06 yield criterion for hcp materials was employed along with a non-associative flow rule in which the yield function and plastic potential were calibrated for a range of plastic deformation levels to account for evolving anisotropy under proportional loading. The non-associative flow rule has not previously been applied to magnesium alloys which require the use of flexible constitutive models to capture the severe anisotropy and its evolution with plastic deformation. The non-associative flow rule can provide the required flexibility by decoupling the yield function and plastic potential. For the associative flow rule, such flexibility can only be achieved by multiple linear transformations of the stress tensor resulting in expensive models for calibration and simulations. The constitutive model was implemented as a user material subroutine (UMAT) within the commercial finite element software, LS-DYNA, for general 3-D stress states along with an interpolation technique to consider the evolution of anisotropy based upon the plastic work. To evaluate the accuracy of the implemented model, predictions of a single-element model were compared with the experimental results in terms of flow stresses and plastic flow directions under various proportional loading conditions and along different test directions. Finally, to assess the predictive capabilities of the model, full-scale simulations of coupon-level formability experiments were performed and compared with experimental results in terms of far-field load-displacement and local strain paths. Using these experiments, the constitutive model was evaluated across the full range of representative stress states for sheet metal forming operations. It was shown that the predictions of the model were in very good agreement with experimental data.

Experimental Investigations on Subsequent Yield Surface of Pure Copper by Single-Sample and Multi-Sample Methods under Various Pre-Deformation

Using copper thin-walled tubular specimens, the subsequent yield surfaces under pre-tension, pre-torsion and pre-combined tension-torsion are measured, where the single-sample and multi-sample methods are applied respectively to determine the yield stresses at specified offset strain. The rule and characteristics of the evolution of the subsequent yield surface are investigated. Under the conditions of different pre-strains, the influence of test point number, test sequence and specified offset strain on the measurement of subsequent yield surface and the concave phenomenon for measured yield surface are studied. Moreover, the feasibility and validity of the two methods are compared. The main conclusions are drawn as follows: (1) For the single or multi-sample method, the measured subsequent yield surfaces are remarkably different from cylindrical yield surfaces proposed by the classical plasticity theory; (2) there are apparent differences between the test results from the two kinds of methods: the multi-sample method is not influenced by the number of test points, test order and the cumulative effect of residual plastic strain resulting from the other test point, while those are very influential in the single-sample method; and (3) the measured subsequent yield surface may appear concave, which can be transformed to convex for single-sample method by changing the test sequence. However, for the multiple-sample method, the concave phenomenon will disappear when a larger offset strain is specified.

A rare-earth free magnesium alloy with improved intrinsic ductility

Metals are the backbone of manufacturing owing to their strength and formability. Compared to polymers they have high mass density. There is, however, one exception: magnesium. It has a density of only 1.7 g/cm³, making it the lightest structural material, 4.5 times lighter than steels, 1.7 times lighter than aluminum, and even slightly lighter than carbon fibers. Yet, the widespread use of magnesium is hampered by its intrinsic brittleness. While other metallic alloys have multiple dislocation slip systems, enabling their well-known ductility, the hexagonal lattice of magnesium offers insufficient modes of deformation, rendering it intrinsically brittle. We have developed a quantum-mechanically derived treasure map which screens solid solution combinations with electronic bonding, structure and volume descriptors for similarity to the ductile magnesium-rare earth alloys. Using this insight we synthesized a surprisingly simple, compositionally lean, low-cost and industry-compatible new alloy which is over 4 times more ductile and 40% stronger than pure magnesium. The alloy contains 1 wt.% aluminum and 0.1 wt.% calcium, two inexpensive elements which are compatible with downstream recycling constraints.

Forming limit prediction using a self-consistent crystal plasticity framework: a case study for body-centered cubic materials

A rate-dependent self-consistent crystal plasticity model was incorporated with the Marciniak-Kuczyński model in order to study the effects of anisotropy on the forming limits of BCC materials. The computational speed of the model was improved by a factor of 24 when running the simulations for several strain paths in parallel. This speed-up enabled a comprehensive investigation of the forming limits of various BCC textures, such as γ, σ, α, η and ϵ fibers and a uniform (random) texture. These simulations demonstrate that the crystallographic texture has significant (both positive and negative) effects on the resulting forming limit diagrams. For example, the γ fiber texture, which is often sought through thermo-mechanical processing due to a high r-value, had the highest forming limit in the balanced biaxial strain path but the lowest forming limit under the plane strain path among the textures under consideration. A systematic investigation based on the results produced by the current model, referred to as 'VPSC-FLD', suggests that the r-value does not serve as a good measure of forming limit strain. However, model predictions show a degree of correlation between the r-value and the forming limit stress.

Plasticity and non-Schmid effects

The structure of the overall response of a metal polycrystal is analysed under the assumption that activity on the slip systems of its constituents is governed by a generalized Schmid law. In the resulting macromodel of metal plasticity, the overall rate of plastic deformation is related to the stress gradient of the yield surface via a generalized, but still associated, flow rule. While the overall rate of plastic deformation is no longer collinear with the stress gradient of the yield surface, its deviation is characterized by an additional macrovariable, representative of the rate of the non-Schmid effects taking place at constituent level.

High resolution imaging of superficial mosaicity in single crystals using grazing incidence fast atom diffraction

A new table top technique is used to simultaneously analyze the local morphology of crystalline surfaces as well as the misalignment of large scale domains at the topmost surface layer. The approach is based on fast atom diffraction at grazing incidence (GIFAD); the diffraction pattern yields the structural characteristics and the topology of the surface electronic density with atomic resolution. If superficial mosaicity is present, diffraction patterns arising from each mosaic domain can be distinguished, providing high sensitivity to the properties of each of the domains. Taking NaCl(001) as an example, we observe a discrete tilt angle distribution of the mosaic domains following an arithmetic progression with a 0.025° ± 0.005° difference; a twist mosaic angle of 0.09° ± 0.01° is also observed.

Bounding the plastic strength of polycrystalline solids by linear-comparison homogenization methods

The elastoplastic response of polycrystalline metals and minerals above their brittle–ductile transition temperature is idealized here as rigid–perfectly plastic. Bounds on the overall plastic strength of polycrystalline solids with prescribed microstructural statistics and single-crystal plastic strength are computed by means of a linear-comparison homogenization method recently developed by Idiart & Ponte Castañeda (Idiart & Ponte Castañeda 2007 Proc. R. Soc. A 463 , 907–924 ( doi:10.1098/rspa.2006.1797 )). Hashin–Shtrikman and self-consistent results are reported for cubic and hexagonal polycrystals with varying degrees of crystal anisotropy. Improvements over earlier linear-comparison bounds are found to be modest for high-symmetry materials but become appreciable for low-symmetry materials. The largest improvement is observed in self-consistent results for low-symmetry hexagonal polycrystals, exceeding 15 per cent in some cases. In addition to providing the sharpest bounds available to date, these results serve to evaluate the performance of the aforementioned linear-comparison method in the context of realistic material systems.

An experimental methodology to relate local strain to microstructural texture

This paper introduces an experimental methodology for obtaining high resolution full-field strain measurements in polycrystalline metals. The (sub)grain level resolution of these measurements was indispensable for relating measured strain fields to observed microstructure in the material. Microstructural information was obtained through electron backscatter diffraction and the optical technique of digital image correlation (DIC) was used to acquire full-field deformation measurements. By spatially overlaying both sets of results, the effects of different microstructural features such as orientation, grain boundary character, misorientation between grains, and twin boundaries on material response can be quantitatively studied. To obtain the necessary resolution for such measurements, the images used in DIC had to be captured at high magnifications. This necessity reduces the field of view and constrains the area of interest that can be monitored. To address this issue, results from adjacent measurement areas are combined together to create a data set with high spatial strain resolution over a larger region than can otherwise be observed. The procedure for performing this technique is outlined here, along with benefits, drawbacks, possible modifications, and example applications of the technique to cyclic plasticity and fatigue crack growth.

A review of computational methods in materials science: examples from shock-wave and polymer physics

This review discusses several computational methods used on different length and time scales for the simulation of material behavior. First, the importance of physical modeling and its relation to computer simulation on multiscales is discussed. Then, computational methods used on different scales are shortly reviewed, before we focus on the molecular dynamics (MD) method. Here we survey in a tutorial-like fashion some key issues including several MD optimization techniques. Thereafter, computational examples for the capabilities of numerical simulations in materials research are discussed. We focus on recent results of shock wave simulations of a solid which are based on two different modeling approaches and we discuss their respective assets and drawbacks with a view to their application on multiscales. Then, the prospects of computer simulations on the molecular length scale using coarse-grained MD methods are covered by means of examples pertaining to complex topological polymer structures including star-polymers, biomacromolecules such as polyelectrolytes and polymers with intrinsic stiffness. This review ends by highlighting new emerging interdisciplinary applications of computational methods in the field of medical engineering where the application of concepts of polymer physics and of shock waves to biological systems holds a lot of promise for improving medical applications such as extracorporeal shock wave lithotripsy or tumor treatment.

The Application of Multiscale Modelling for the Prediction of Plastic Anisotropy and Deformation Textures

Finite element models for metal forming and models for the prediction of forming limit strains should be as accurate as possible, and hence should take effects due to texture, microstructure and substructure (dislocation patterns) into account. To achieve this, a hierarchical type of modelling is proposed in order to maintain the balance between calculation speed (required for engineering applications) and accuracy. This means that the FE models work with an analytical constitutive model, the parameters of which are identified using results of multilevel models. The analytical constitutive model will be discussed, as well as the identification procedure. The multilevel models usually connect the macro-scale with a meso-scale (grain level) via a homogenisation procedure. They can also be used to make predictions of deformation textures. These will be quantitatively compared with experimentally obtained rolling textures of steel and aluminium alloys. It was found that only models which to some extent take both stress and strain interactions between adjacent grains into account perform well. Finally an example of a three level model, also including the micro-scale (i.e. the dislocation substructure), will be given.

A statistical approach to micro-plastic strain in metals

The paper presents an outline of a theory of micro-inhomogeneity of stresses and strains resulting from the micro-structural properties of engineering materials. The problem is approached from a statistical point of view and it is experimentally shown that the degree of micro-inhomogeneity can be defined by normal distribution functions.

Using the experimental results a general concept is postulated which takes into account the physical reality as completely as is practicable. It is shown that the suggested approach can be used to take into account the micro-plastic strains which exist while the material is nominally within the elastic limit.

A self-consistent polycrystal model for the spontaneous polarization of ferroelectric ceramics

Motivated by the observation that the spontaneous polarization process of a ferroelectric polycrystal under the influence of a superimposed stress and/or electric field involves heterogeneous evolution of the ferroelectric phase among its constituent grains, a self-consistent electromechanical model is developed to determine the effective behaviour of the polycrystalline ceramic from such a heterogeneous electromechanical state. We start out from consideration of a micromechanics-based thermodynamic process to establish the kinetic equation of the crystallite and use it to evaluate the evolution of its ferroelectric domain. Then together with the Curie–Weiss law for the dielectric constants of the tetragonal phase, a dual-phase mixture theory is adopted to determine the change of its electromechanical moduli as temperature cools down below its Curie point. The overall property of the polycrystal is subsequently calculated by the self-consistent model through orientational average over its constituent grains. This two-level micromechanics model is applied to examine the shift of Curie temperature and evolution of the effective electromechanical moduli of a BaTiO 3 ceramic under cooling. The calculated results show that its Curie temperature decreases with increasing hydrostatic pressure, but increases with a superimposed axial compression or a biased electric field. The predicted temperature shift and change of the dielectric constants are found to be consistent with experimental observations.

The Application of Multiscale Modelling for the Prediction of Plastic Anisotropy and Deformation Textures

The paper focuses on the multi-level character of existing or currently developed models for polycrystal deformation. A general multilevel frame is presented, which can be applied to models for the simulation of plastic anisotropy to be implemented in FE codes for the simulation of metal forming processes, or to models for the simulation of deformation textures. A short overview is presented of two-level models ranging from the full-constraints Taylor model to the crystalplasticity finite element models, including the description of a few recent and efficient models (GIA and ALAMEL). Validation efforts based on experimental cold rolling textures obtained for steel and aluminium alloys are discussed. Finally a recent three-level model which also takes the microscopic level (dislocation substructure) is discussed.

Crystallographic Relationships between FCC and BCC Crystals: A Study Using EBSD Techniques

The mechanisms governing the formation of transformation textures during the austenite-to-ferrite transformation are the subject of major debate. In this study, two extreme cases were examined: those of undeformed and deformed austenite. The first involves the transformation of austenite into Widmanstätten ferrite under "equilibrium" conditions in the Gibeon iron-nickel meteorite. This meteorite passed through the transformation at the rate of a few degrees per million years. Such cooling rates cannot of course be reached under laboratory conditions. The second concerns the transformation of hot rolled austenite after a quench into the bainite temperature range. These two behaviors were investigated by means of optical microscopy and electron backscatter diffraction (EBSD) techniques. The orientations of both the parent and product phases were measured and the orientation relationships are represented in Rodrigues-Frank (R-F) space. From the orientation of a particular FCC crystal, the crystallographic orientations of the product BCC crystals can be predicted according to the Bain, Kurdjumov-Sachs (K-S) and Nishiyama- Wassermann (N-W) correspondence relationships. Comparison of the predicted and measured orientations reveals that the Bain rotation is never observed; the K-S and N-W relationships are both observed and there is a continuous distribution of orientations between the exact K-S and N-W positions. The formation of preferred orientations under non-equilibrium conditions is scrutinized. These results are compared to recent models accounting for variant selection.

Elastic-plastic behaviour of polycrystalline metals and composites

Stress-strain behaviour of polycrystals and composites is explored with self-consistent models based on the elastic-plastic properties of the single crystal constituents. Based on Hill’s (1965 a ) model, tensile stress-strain curves and the associated yield surfaces are calculated for polycrystals and composites comprised of face-centred cubic crystals. Single crystal elastic anisotropy and strain hardening are taken into account. Stress-strain behaviour at a corner of the yield surface is determined with the model proposed by Kröner (1961) and Budiansky & Wu (1962).

Transition bands and recrystallization in metals

Transition bands are regions of high lattice distortion which result from inhomogeneous deformation in metals. During annealing of cold-worked metals, nucleation of recrystallization is frequently observed to occur within these bands. An analysis has been made of some of the conditions under which transition bands may develop during deformation, and of the factors which influence their microstructure. A model is proposed for the mechanism by which nucleation of recrystallization occurs in transition bands during annealing. The dependence of these processes on crystallographic orientation is then discussed and related to the formation of recrystallization textures in b.c.c. metals.

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 J 2 -flow theory, J 2 -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 J 2 -corner theory and those of the micromechanically based physical model.

Rate-dependent, finite elasto-plastic deformation of polycrystals

Based on Hill’s method, a self-consistent averaging scheme is proposed for estimating the overall, finite deformation response of polycrystalline aggregates consisting of single crystals which undergo plastic flow by rate-dependent crystallographic slip, accompanied by elastic lattice distortion. First, constitutive relations for such single crystals are developed assuming that the slip-rate and the associated resolved shear stress are governed by: (1) a power-law relation, and (2) a viscoplastic relation. Then, Hill’s idea that the constraint imposed on a single crystal by the remaining aggregates may be represented by embedding the single crystal in a homogeneous, infinitely extended matrix having the instantaneous overall moduli, is used to formulate a completely self-consistent averaging procedure, valid for rate-dependent materials at finite strains and rotations. This method includes both the Hill and the Krӧner‒Budiansky‒Wu (K. B. W.) methods as limiting cases; when rate-effects are negligible, it reduces to Hill’s self-consistent method as formulated by Iwakuma and Nemat-Nasser for finite deformations, while it reduces to a generalized finite deformation version of the K. B. W. method for strongly rate-dependent materials. Illustrative numerical examples are presented for a plane uniaxial deformation, using a two-dimensional poly crystalline model. These examples clearly show that the rate-dependent crystallographic slip on the level of single crystals produces a more stable overall behaviour of poly crystals. This supports similar results arrived at by other investigators for single crystals and for polycrystals, by using the Taylor averaging scheme. It is shown that, while Taylor’s averaging scheme gives accurate estimates of the incremental quantities at large strains, the total overall quantities differ considerably from the ones obtained by the self-consistent method.

Variational estimates for the creep behaviour of polycrystals

A variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high-temperature creep. The procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corre­sponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single-crystal grains. As such, it constitutes an extension, to locally anisotropic behaviour, of the variational procedure devel­oped by Ponte Castañeda (1991) for nonlinear heterogeneous media with locally isotropic behaviour. By way of an example, the procedure is applied to the de­termination of bounds of the Hashin-Shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals. The bounds are computed for the special class of untextured FCC polycrystals with isotropic pure power-law viscous behaviour, first considered by Hutchinson (1976), in the context of a calculation of the self-consistent type. The new bounds are found to be more restrictive than the corresponding classical Taylor-Bishop-Hill bounds, and also more re­strictive, if only slightly so, than related bounds of the Hashin-Shtrikman type by Dendievelet al. (1991). The new procedure has the advantage over the self-consistent procedure of Hutchinson (1976) that it may be applied, without any essential complications, to aggregates of crystals with slip systems exhibiting dif­ferent creep rules - with, for example, different power exponents - and to general loading conditions. However, the distinctive feature of the new variational proce­dure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals.

On the homogenized yield strength of two-phase composites

This paper is concerned with the determination of the effective yield strength of two-phase, rigid-perfectly plastic composite materials. The individual phases are assumed to satisfy, for simplicity, incompressible, isotropic yield criteria of the Mises type. The volume fractions of the constituent phases are prescribed, but their distribution within the composite is otherwise arbitrary. Using the homogenization framework of Suquet (1983) to define the homogenized, or effective, yield strength domain of rigid-perfectly plastic composites, a variational statement is introduced allowing the estimation of the associated effective dissipation functions of plastic composites in terms of the effective dissipation functions of corresponding classes of linearly viscous comparison composites. Thus the variational statement suggests a procedure for generating bounds and estimates for the effective yield strength of rigid-perfectly plastic composites from well-known bounds and estimates for the effective properties of the corresponding linear comparison composites. Sample results are given in the form of upper bounds and lower estimates of the Hashin-Shtrikman type for the effective yield strength of two-phase composites with overall isotropy. Additionally, estimates and bounds are also given for the effective strength domains of two-phase laminated and fibre-reinforced composites, with overall transverse isotropy.