Microstructure evolution in crystal plasticity : strain path effects
and dislocation slip patterning
Citation for published version (APA):
Yalcinkaya, T. (2011). Microstructure evolution in crystal plasticity : strain path effects and dislocation slip patterning. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR716655
DOI:
10.6100/IR716655
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Microstructure evolution in crystal plasticity:
strain path effects and dislocation slip
This research was carried out under the project number MC2.03158 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl), the former Netherlands Institute for Metals Research.
CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Tuncay Yalc¸ınkaya
Microstructure evolution in crystal plasticity: strain path effects and dislocation slip patterning /
by T. Yalc¸ınkaya – Eindhoven : Technische Universiteit Eindhoven, 2011. Proefschrift.
A catalogue record is available from the Eindhoven University of Technology Library
ISBN: 978-90-386-2729-8
Subject headings: BCC metals / crystal plasticity / non-Schmid effects / plastic anisotropy / strain path change effect / Bauschinger effect / cross effect / microstructure evolution / non-convexity /
phase field modeling / dislocation patterning / finite element method / non-convex free energy / strain gradient crystal plasticity
Copyright c 2011 by Tuncay Yalc¸ınkaya, all rights reserved. This thesis was prepared with the LATEX 2ε documentation system.
Reproduction: Universiteitsdrukkerij TU Eindhoven, Eindhoven, The Netherlands.
Microstructure evolution in crystal plasticity:
strain path effects and dislocation slip
patterning
P
ROEFSCHRIFTter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,
op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties
in het openbaar te verdedigen
op donderdag 20 oktober 2011 om 16.00 uur
door
Tuncay Yalc¸ınkaya
Dit proefschrift is goedgekeurd door de promotor: prof.dr.ir. M.G.D. Geers
Copromotor:
Contents
Summary ix
1 Introduction 1
1.1 Crystal plasticity . . . 1
1.2 Objective and outline . . . 3
2 A finite strain BCC single crystal plasticity model and its experimental identification 5 2.1 Introduction . . . 5
2.2 Slip mechanisms in BCC metals . . . 8
2.3 Violation of Schmid’s law in BCC metals . . . 10
2.4 A BCC crystal plasticity model at material point level . . . 12
2.4.1 Kinematics in crystal plasticity . . . 12
2.4.2 Constitutive model . . . 13
2.5 Modeling some intrinsic properties of BCC single crystals . . . 17
2.5.1 Orientation dependence . . . 17
2.5.2 Example:α-Fe single crystal . . . 18
2.5.3 Example: molybdenum single crystal . . . 19
2.5.4 Temperature dependence . . . 19
2.6 Summary and Conclusion . . . 22
3 A composite dislocation cell model to describe strain path change effects in BCC metals 25 3.1 Introduction . . . 25
3.2 Dislocation substructure evolution . . . 28
3.3 Computational model . . . 31
3.4 Modeling of microstructure evolution . . . 34
3.4.1 Monotonic deformation . . . 34
3.4.2 Orthogonal change of deformation . . . 36
3.4.3 Reverse deformation . . . 37
vi Contents
3.5 Numerical examples . . . 38
3.5.1 Example 1: monotonic deformation of single crystals . . . 38
3.5.2 Example 2: strain path change of single crystals . . . 39
3.5.3 Example 3: strain path change of polycrystals . . . 40
3.6 Summary and Conclusion . . . 43
4 Deformation patterning driven by rate dependent non-convex strain gradi-ent plasticity 45 4.1 Introduction . . . 45
4.2 Macroscopic view: material instability and microstructure evolution in inelastic materials . . . 49
4.3 Thermodynamics of strain gradient plasticity . . . 50
4.4 Particular choices of free energy functions . . . 54
4.4.1 Slip based strain gradient plasticity . . . 54
4.4.2 Slip based non-convex strain gradient plasticity . . . 55
4.5 Non-convexity and patterning in phase field modeling . . . 57
4.6 Numerical examples . . . 58
4.6.1 Numerical example 1: convex case - monotonic loading . . . 59
4.6.2 Numerical example 2: non-convex case - monotonic loading . . 60
4.6.3 Numerical example 3: non-convex stress relaxation of a 1D bar 66 4.7 Conclusion . . . 68
4.8 Appendix . . . 68
4.8.1 Finite element implementation of slip based strain gradient plasticity . . . 68
4.8.2 Finite element implementation of slip based non-convex strain gradient plasticity . . . 71
5 Non-convex rate dependent strain gradient crystal plasticity and deforma-tion patterning 73 5.1 Introduction . . . 73
5.2 Strain gradient crystal plasticity and finite element implementation . . 75
5.3 Latent hardening based non-convex plastic potential . . . 80
5.3.1 Conditions for plastic slip patterning . . . 80
5.4 Numerical analysis . . . 84
5.4.1 Convex strain gradient crystal plasticity . . . 84
5.4.2 Non-convex strain gradient crystal plasticity . . . 88
5.5 Summary and conclusion . . . 92
6 Discussion and conclusions 95
Contents vii
Dankwoord / Acknowledgements 109
Summary
During deformation polycrystalline metals tend to develop heterogeneous plastic deformation fields at the microscopic scale, as the amount of plastic strain varies spatially, depending on local grain orientation, geometry and defects. While grain boundaries are natural places triggering plastic slip accumulation and geometrically necessary dislocations that accommodate the gradients of the inhomogeneous plastic strain, the deformation localizes within grains revealing dislocation cell structures or micro slip bands (e.g. clear band formation in irradiated materials). Across grains, macroscopic plastic slip bands (L ¨uders bands, etc.) exist as well. These intergran-ular and intragranintergran-ular deformation patterns are stated to be inherent minimizers of the free energy (including the microstructurally trapped plastic energy). These microstructures may macroscopically manifest themselves through softening of the material or through plastic anisotropy in hardening under strain path changes. These effects are crucial with respect to the mechanics of the materials under consideration and should be taken into account in the constitutive modeling.
In this thesis, the computational modeling of microstructure evolution (with soften-ing or plastic anisotropy) is covered in different crystal plasticity frameworks. The scope is basically two-fold. First, in the chapters 2 and 3 the plastic anisotropy of Body Centered Cubic crystals is studied from the onset of deformation due to an intrinsic orientation dependence from non-planar dislocation core structures, to the anisotropy upon a strain path change owing to resulting dislocation cell formation. In this part of the thesis, after developing a proper BCC crystal plasticity framework taking into account the intrinsic anisotropy, a composite cell model was established where the evolution of dislocation cells was modeled under monotonic and non-proportional loading histories. Here, the existence of a dislocation microstructure is introduced into the model in terms of internal variables and the evolution was described by phenomenologically based evolution equations. However, this phe-nomenological approach is not able to incorporate the formation stage of the mi-crostructure. Hence, the crystal plasticity framework called for an extension in order to capture the evolution of the microstructure driven by the free energy of the
x Summary
rial.
In order to complete the missing link between the formation of the microstruc-ture and its evolution in crystal plasticity frameworks, the second part of the the-sis concentrates on the development of a non-convex rate dependent crystal plas-ticity model, which reveals a rate dependent dislocation microstructure formation and evolution together with macroscopic hardening-softening-plateau stress-strain responses. To this end, non-convexity is treated as an intrinsic property of the plastic free energy of the material. First, this non-convex contribution is incorporated into a strain gradient crystal plasticity framework with a double-well character, which re-sults in a computational routine partially dual to the Ginzburg-Landau type of phase field modeling approaches (with high and low slipped regions representing the dif-ferent phases). In this model, both the displacement and the plastic slip fields are considered as primary variables. These fields are determined on a global level by solving simultaneously the linear momentum balance and the slip evolution equa-tion which is rederived in a thermodynamically consistent manner. In chapter 4, the analysis is conducted in a 1D mathematical setting in order to illustrate the ability of the model to capture the patterning of plastic slip. In chapter 5 and inspired by the literature, the non-convexity originates from latent hardening in a multi-slip strain gradient crystal plasticity framework. Hence, the 1D approach pursued in chapter 4 is extended to a 2D plane strain setting. Even though the phenomenological double-well free energy function used in the 1D approach allows to track non-equilibrium states during microstructure evolution, it does not rely on a physically based ex-pression for non-convexity, but presents a generic formulation. Instead, chapter 5 concentrates more on the physical reasons of plastic slip localization, where a slip interaction potential is analyzed and incorporated into the rate dependent strain gra-dient crystal plasticity framework. The non-convexity due to the slip interactions is explicitly illustrated and the possibility of deformation patterning in the material is discussed in a boundary value problem. The last part of the thesis, chapter 6, presents a discussion and conclusions.
Chapter one
Introduction
Abstract / The physics and the basic principles behind the general crystal plasticity mod-eling are explained. An overview of strain path change related anisotropy and dislocation microstructure evolution in crystal plasticity approaches is presented. The objectives and outline of the thesis are given.
1.1 Crystal plasticity
A perfect metallic single crystal, which is characterized by a specific periodic ar-rangement of atoms, can respond only in a reversible elastic manner in thermal equi-librium with its surroundings when stressed monotonically well below the critical levels that destabilize the crystal structure. Under an applied homogeneous stress, the elastic response is homogeneous down to the atomic level. In contrast, the plastic response is locally heterogeneous and requires crystal defects for its development. The type and intensity of the plastic response depend on the character of the defect state. For this purpose the crystal defects are introduced in a hierarchy of increasing dimensionality from point, through line, to planar defects (see Argon (2008) for an extensive overview). Among these, the line defects, i.e. dislocations, are regarded as the principal carriers of plastic deformation. The crystallographic slip of dislocations occurs on the most close-packed slip planes and in the most close-packed directions which together form the slips systems. Depending on the specific arrangement of atoms, each metal has specific slip systems. The first part of this thesis is focusing on the body centered cubic (BCC) type of atomic arrangement.
Under an applied stress, the atomic lattice deforms elastically until the stretched bonds near a dislocation break down and new bonds are formed. During this
2 1 Introduction
GRAINS
DISLOCATION CELLS CONSTITUTIVE MODELING OF DISLOCATION MOVEMENT
DISLOCATION CELL BLOCKS
Figure 1.1 / Crystal plasticity bridges the deformation of bulk material and the movement of dislocations (chapter 2). Clustering of dislocations is accomplished via non-convex strain gradient crystal plasticity (chapter 4 and 5). The phenomeno-logical evolution of dislocation structures is simulated via a dislocation cell model (chapter 3).
agating process, a part of the crystal gradually slips one interatomic distance with respect to the other part. Instead of the ideal fictitious strength associated with the movement of an entire slip plane, the dislocations enable only sections of the slip plane to shear, resulting in the observed decimated strengths necessary for plastic deformation.
The stress directly affecting the motion of dislocations is the projected shear stress on the specific slip systems, which is also called the resolved Schmid stress. When the resolved shear stress is larger than the resistance on the respective slip system, glide is activated. The viscous type of crystal plasticity theories as used in this thesis employs a power law relation between the rate of plastic slip and the ratio between the shear stress and the slip resistance. Conceptually, all the slip systems are active but the most favorable ones carry most plastic deformation. Crystal plasticity is a mesoscopic modeling approach, bridging the stresses on the slip systems to the total amount of plastic slip in the bulk material (see Fig. 1.1), eventually affecting the total amount of macroscopic plastic strain.
In addition to their role of accommodating the plastic deformation in metals, the work (strain) hardening behavior can also be attributed to the dislocations. The distinct stages of strain hardening are related to their multiplication or mutual
in-1.2 Objective and outline 3
teraction processes. After moderate deformations, the formation of dislocation cell structures (see Fig. 1.1) plays an important role in the hardening behavior of the material.
Another important effect in the hardening of crystals is the gradient of the plastic slip, requiring so-called geometrically necessary dislocations (GNDs). Once the ap-plied load, or the material structure itself, triggers a gradient of the plastic deforma-tion, a certain amount of GNDs will be necessary to preserve lattice compatibility and to accomplish the required lattice rotation. Conventional crystal plasticity the-ories, as used in the chapters two and three of this thesis, however, do not take into account these effects. Their strengthening mechanisms are therefore inherently in-capable of predicting scale dependent behavior, i.e. different mechanical responses due to varying plastic strain gradients. Hence, in the strain gradient crystal plastic-ity frameworks of the subsequent chapters, the gradients of the plastic slip enter the plastic slip law together with a length scale parameter, where these do not only allow for size effect predictions but also play a regularization role in the viscous formula-tion of non-convex strain gradient plasticity.
1.2 Objective and outline
During metal forming processes, materials experience complex strain path histories which result in plastic anisotropy, i.e. transient hardening or softening regimes oc-curring in the macroscopic stress-strain response. This phenomenon plays an impor-tant role in the metal deformation and the effect should be included in the constitu-tive modeling. The physical origin resides in three distinct factors at three different length scales: dislocation slip anisotropy, evolution of the dislocation microstructure and textural anisotropy. The thesis focuses on the first two effects which are crucial in early stages of the deformation and subsequent moderate straining.
In chapter two, a crystal plasticity model for body centered cubic (BCC) single crys-tals, taking into account the plastic anisotropy due to non-planar spreading of screw dislocation cores is developed. A comprehensive summary of the intrinsic properties of these materials is presented and incorporated into the framework through a mod-ification in the plastic slip law. In the numerical examples section, emphasis is given on the intrinsic orientation dependence of the flow stress due to the non-Schmid com-ponents of the stress field projected on the slip plane under monotonic deformation. Attention is therefore given on the quantitative prediction of single crystal behavior, for which experiments from the literature have been used.
Next, in chapter three, the anisotropy due to the dislocation cell structure evolution is considered. A composite dislocation cell model has been combined with the BCC
4 1 Introduction
crystal plasticity framework to describe the dislocation cell structure evolution and its macroscopic anisotropic effects. The computational framework departs from a composite aggregate with a cell structure, consisting of a soft cell interior component and hard cell wall components. The constitutive response of each component has been obtained from crystal plasticity simulations, while a set of phenomenological evolution equations for the cell size, the wall thickness and the dislocation density captures the evolution of the microstructure under complex strain paths. Numerical examples study both the intrinsic orientation dependence and the anisotropy due to cell structure evolution.
Chapter four focuses on one of the origins of self-organizing dislocation structures (driven by the deformation) rather than imposing the cell structure evolution as done in the previous chapter. To this purpose, a rate dependent strain gradient plasticity framework for the description of plastic slip patterning in a system with non-convex energetic hardening is developed. Both the displacement field and the plastic slip field are considered as primary variables. The slip law differs from classical ones in the sense that it includes a stress term originating from a non-convex double-well free energy, which enables patterning of the deformation field. The derivations and im-plementations are performed in a single slip 1D setting, which allows for a thorough mechanistic understanding, not excluding its extension to multidimensional cases. The numerical examples illustrate both the homogeneous and inhomogeneous plas-tic slip distributions as well as the stress-strain response in relation to the imposed boundary conditions and the applied rate of deformation.
In chapter five the non-convex strain gradient crystal plasticity formulation is ex-tended to the 2D plane strain case, including multiple slip systems. In order to capture the effect of dislocation interactions on the non-convexity of the plastic slip dependent free energy function, a more physically based free energy expression is incorporated. Attention is focused on the inhomogeneous plastic slip distribution and deformation patterning due to dislocation slip interactions.
The thesis concludes with a final chapter, summarizing the main achievements and results obtained as well as an outlook to open challenges.
Chapter two
A finite strain BCC single crystal
plasticity model and its experimental
identification
1
Abstract / A crystal plasticity model for body-centered-cubic (BCC) single crystals, tak-ing into account the plastic anisotropy due to non-planar spreadtak-ing of screw dislocation cores is presented. In view of the long-standing contradictory statements on the deforma-tion of BCC single crystals and their macroscopic slip planes, recent insights and devel-opments are reported and included in this model. The flow stress of BCC single crystals shows a pronounced dependence on the crystal orientation and the temperature, mostly due to non-planar spreading of a/2h111itype screw dislocation cores. The main conse-quence here is the well-known violation of Schmid’s law in these materials, resulting in an intrinsic anisotropic effect which is not observed in e.g. FCC materials. Experimen-tal confrontations at the level of a single crysExperimen-tal are generally missing in the literature. To remedy this, uniaxial tension simulations are done at material point level for α-Fe, Mo and Nb single crystals and compared with reported experiments. Material param-eters, including non-Schmid paramparam-eters, are calibrated from experimental results using a proper identification method. The model is validated for different crystal orientations and temperatures, which was not attempted before in the open literature.
2.1 Introduction
In the present paper, attention is focused on the low temperature (room tempera-ture and lower) properties of single BCC crystals includingα-Fe, metals of the group VA (V, Nb, Ta) and of the group VIA (Mo, V, Cr) and certain alkali metals. These materials show a peculiar mechanical behavior, mostly resulting from their screw
1This chapter is reproduced from Yalcinkaya et al. (2008)
6
2 A finite strain BCC single crystal plasticity model and its experimental identification
dislocation core configuration. They have a relatively high yield stress which is strongly temperature, rate and orientation dependent. They exhibit complex slip modes, dominated by the cross slip of a/2h111iscrew dislocations. Due to their dis-location core structure, they show a severe glide direction sensitive behavior (slip asymmetry) and the well-known Schmid law using the critical resolved shear stress (CRSS) is violated. Another pronounced phenomenon is the anomalous slip (acti-vation of an unexpected slip system at a certain orientation) observed in pure BCC metals which, however, is only marginally dealt with in the present work.
Various discussions on the behavior of BCC crystals, reveal a number of contradic-tions with respect to the slip plane activity. Even though there is no generally ac-cepted explanation to the dislocation behavior of these materials recent studies pro-vide a good basis for the constitutive model presented in the current paper. Seeger (2001) states that the slip nature of BCC crystals depends highly on the tempera-ture, upon which dislocations may accommodate either a straight {110} slip or a wavy type {112} cross slip pattern. At room temperature a/2h111i type of screw dislocations move on{112}slip planes, which enables cross slip. The cross slip phe-nomenon will not be modeled explicitly but the resulting effects are taken into ac-count in the slip and hardening laws (for example (Pichl (2002)), showing how cross slip can be included in a model).
The value of the critical resolved shear stress (CRSS) is independent from the slip sys-tem and the sense of the slip of FCC metals. For these metals, it is generally accepted that the only stress component affecting the glide is the Schmid stress. However, BCC metals show an asymmetry in their slip behavior: the slip resistance in one direction is different from the resistance in the opposite direction, indicated as the twinning/anti-twinning asymmetry. Moreover, due to small edge fractional dislo-cation components in the screw dislodislo-cation core, stress components other than the resolved Schmid stress affect the glide or the CRSS of the material. Both of these ef-fects result from the non-planar spreading of the dislocation cores. For these reasons Schmid’s law is not applicable to BCC metals. In the presently proposed crystal plas-ticity model these two types of intrinsic anisotropy effects will be taken into account. The formulation of the constitutive crystal plasticity model departs from the papers of Bronkhorst et al. (1992) and Kalidindi et al. (1992) related to FCC metals. Their de-velopments are extended by including the intrinsic properties of BCC metals,using a physical description of the slip law based on thermally activated dislocation ki-netics. The barriers to dislocation movement are discriminated according to their short-range or long-range nature. The short-range barrier can be overcome by ther-mal activation, whereas the long-range barrier is affected slightly through changes of the elastic moduli. Non-Schmid effects are included in the model by incorporat-ing non-Schmid terms in the slip activation. The model is implemented at a material
2.1 Introduction 7
point in matlab and compared with experimental results.
The phenomena incorporated in this work are basically the orientation and the tem-perature dependence of the flow stress and stress-strain behavior of single BCC crys-tals including the non-Schmid behavior. Each of these characteristics has been inves-tigated extensively in the literature, and especially the temperature dependence of the flow stress was an active research area until the 90s. At present, atomistic com-puter simulations studying screw dislocation cores are still an active area of research. Many improvements have been achieved in this area and to our knowledge there is no recent work combining these physical aspects of BCC structured materials with crystal plasticity calculations. The objective of the present work is to exemplify this combination.
The plan of this paper is as follows. Section 2 discusses the slip mechanisms in BCC metals where the temperature dependence of the slip plane activation is strongly em-phasized. Next, Schmid’s law and its violation in BCC crystals is handled in section 3, along with its connection to the non-planar spreading of screw dislocation cores. In section 4, the crystal plasticity constitutive framework and its implementation is outlined. Section 5 studies pronounced intrinsic properties, whereby examples are presented and confronted with experimental results. Finally, concluding remarks are given in section 6.
Cartesian tensors and associated tensor products will be used throughout this paper, making use of a Cartesian vector basis{e1e2e3}. Using the Einstein summation rule
for repeated indices, the following conventions are used in the notations of vectors, tensors, related products and crystallography:
• scalars a • vectors a=aiei • second-order tensors A = Ai jei⊗ej • fourth-order tensors4A= A i jklei⊗ej⊗ek⊗el • C =a⊗b =aibjei⊗ej • C = A·B =Ai jBjkei⊗ek • C =4A: B = A i jklBlkei⊗ej
• crystallographic direction, family[uvw],huvwi
• crystallographic plane, family(hkl),{hkl}
8
2 A finite strain BCC single crystal plasticity model and its experimental identification
2.2 Slip mechanisms in BCC metals
First, some of the long-standing contradictions in the identification of the slip planes, and the active slip mechanisms of BCC crystals are highlighted. The first attempt goes back to the introduction of the pencil glide mechanism by Taylor and Elam (1926), where the slip was assumed to be oriented in the h111i crystallographic di-rection while the mean plane of slip was the one having the maximal projected shear stress. This plane might be a crystallographic but also a non-crystallographic plane. After this pioneering research, there have been several of contradicting statements on the active slip planes of BCC metals, for which an extended overview can be found in Havner (1992). The different concepts will not be repeated here in detail, however, a summary including the current developments will be presented in the following paragraphs.
Gough (1928) and Barrett et al. (1937) state that the {110}, {112} and {123} families contain the crystallographic slip planes during the deformation of BCC metals, an assumption that is still being used in many crystal plasticity works. Another fre-quently used view is the participation of{110} and {112} slip planes only, whereby it is assumed that{123} planes need a higher temperature for activation. Many re-searchers state that only the{110} slip planes are active at room temperature, based on the argument that apparent slip on both{112} and {123} planes is actually com-posed of slip on two non-parallel{110} planes, e.g. Chen and Maddin (1954). Using the latter argument, it would be physically more comprehensible to indeed model only{110} planes in a crystal plasticity framework.
In this paper, attention is focused on an accurate description of the physical slip mechanisms of BCC crystals, rather than re-advocating a discussion on the active set of slip planes. As a result of their special slip mechanisms, BCC metals have interesting intrinsic properties that are not observed in e.g. FCC metals. Many au-thors (e.g. Vitek et al. (2004b), Vitek (2004), Duesbery and Vitek (1998)) related most of the phenomena to the core structure of screw dislocations, e.g. by performing atomistic simulations. Slip system activation in BCC metals is highly dependent on the crystal orientation and especially on the temperature. Seeger (2001) and Seeger and Wasserbach (2002) (see Fig. 2.1) provide a detailed explanation of temperature dependent slip for high purity BCC single crystals with an orientation inducing a Schmid factor µ = 0.500 for the slip system [111](¯101) andµ = 0.433 for the sys-tems[111](¯1¯12)and[111](¯211). The work of Seeger and co-workers is adopted here, relying on the fact that BCC metals show different features and different slip mech-anisms in different temperature ranges. The physical response below the so-called knee temperature (which is around 0.2 times the melting temperature of the metal,
TK in Fig. 2.1) and above the knee temperature is thereby distinguished. Below the
disloca-2.2 Slip mechanisms in BCC metals 9 0 100 200 300 400 500 0 200 400 600 800 1000 Temperature T (K)
Flow Stress (MPa)
T TK
cross slip of [111] screws; elementary steps on (211) and (112) (101)
slip
wavy slip lines
slip lines straight
T
Figure 2.1 / Flow stress vs. temperature curve for a pure Mo single crystal at a plastic shear strain rate 8.6×10−4s−1, from Seeger (2001).
tions in kink pairs as the mobility of screw dislocations is lower than the mobility of edge components. In this temperature range, the flow stress (the stress to main-tain plastic deformation after yield) of the metal is highly temperature and strain rate dependent. The flow stress decreases with increasing temperature and increases with increasing strain rate. Above the knee temperature, the flow stress decreases considerably due to self diffusion and recovery processes and the mobilities of screw and non-screw dislocations are no longer substantially different. The high tempera-ture range is out of the scope of the present work. Attention will be focused on the behavior at lower temperatures, including the behavior at room temperature.
The flow stress dependence on the temperature and the slip mechanism in this tem-perature range is visualized in Fig. 2.1, where the presented numerical values refer to molybdenum single crystals. Below the lower limit ˇT (70 K for Mo and 120 K for
α-Fe) the dislocation glide is confined to{110} planes. Screw dislocation glide pro-duces straight step patterns. In this temperature range, dislocations are stated to be in their ground state. They show a threefold symmetry and they are able to slip on any of the three{110} slip planes. As a result of mirror symmetry and the absence of cross slip, no plastic anisotropy is observed.
Above ˇT, the dislocation core configuration changes and dislocations undergo a
tran-sition from their low temperature configuration (slipping on {110} planes) to their high temperature configuration (slipping on {112} planes). The dislocations show a wavy type of structure which results from the cross slip, a characteristic property
10
2 A finite strain BCC single crystal plasticity model and its experimental identification
associated with{112} slip. In BCC metals three {110} and three {112} slip planes in-tersect on a commonh111i direction and screw dislocations can therefore distribute their core on these planes. This spreading is non-planar and nearly all peculiarities in the mechanical properties of BCC metals can be attributed to this phenomenon. The cross slip accommodated by {112} slip planes is the main source of the non-Schmid behavior (orientation dependence of the flow stress). Similar explanations of the change in the slip mechanism in BCC materials were presented by others. For example Christian et al. (1990) focused on the mean jump distance of screw disloca-tions in BCC metals, which decreases considerably above a critical temperature. The explanation is again a change in the slip mechanism of BCC metals.
The above paragraphs emphasized some recent developments in the understanding of the slip system activity of BCC crystals and the connection between the core struc-ture and the intrinsic properties. The development of an adequate model will be based on these considerations.
2.3 Violation of Schmid’s law in BCC metals
Schmid and co-workers (e.g. Schmid and Boas (1935)) first recognized that the yield stress of a metal crystal is strongly depending on the crystal orientation with respect to the load direction. Yield on the slip plane of a crystallographic family occurs at a constant projected shear stress, which is called the critical resolved shear stress (CRSS), for a particular material. Constant should be understood here as indepen-dent from the slip system and the sense of slip. The resulting Schmid law assumes that the only stress component triggering plastic flow of the material is the projected shear stress on the slip system, in the direction of glide which is called the Schmid stress. The other non-glide component, defined as the normal stress, does not have any effect on the plastic deformation. These assertions are applicable on FCC metals, however not on BCC metals. This violation becomes manifest through their plastic anisotropy, revealing two distinct intrinsic non-Schmid effects in BCC metals.
The first intrinsic non-Schmid effect is the variation of the CRSS with the sense of the shear. By definition shear on a certain plane of the family (e.g.{112}) produces shear in the twinning direction, whereas shear on another plane (e.g. {¯211}) produces shear in the anti-twinning direction (see Table 2.1 for the respective slip systems). In BCC metals, twin and slip mechanisms share common slip systems and twinning is only observed when the temperature is very low and the strain rate is extremely high. In this context, the main source of the plastic deformation is just the glide of dislocations, but the resistance to this movement in the twinning and anti-twining directions is asymmetric. This is called the twinning/anti-twinning asymmetry of BCC crystals in the literature and the source of this asymmetry is the strong
cou-2.3 Violation of Schmid’s law in BCC metals 11
pling of the screw dislocation to the BCC lattice, which constrains the core and its properties to adopt the symmetry of the lattice (Duesbery and Vitek (1998)). From the modeling point of view, this phenomenon is included in the models by taking a different slip resistance for the twinning and anti-twinning planes.
Table 2.1 / {112} slip systems of BCC crystals, T and A referring to twinning/anti-twinning planes
(1) A (112)[11¯1] (4) A (11¯2)[111] (7) A (1¯21)[111] (10)T (¯211)[111] (2) A (¯112)[1¯11] (5) A (121)[1¯11] (8) A (12¯1)[¯111] (11)T (2¯11)[11¯1] (3) A (1¯12)[¯111] (6) A (¯121)[11¯1] (9) T (211)[¯111] (12)T (21¯1)[1¯11]
The second effect, which is sometimes denoted as an extrinsic non-Schmid effect, e.g. Duesbery and Vitek (1998), is the sensitivity of the slip resistance to the non-glide components of the applied stress. This effect originates from the non-planar spreading of a/2h111i type screw dislocation cores. Whereas this effect is often called extrinsic in the literature in view of its relation to the applied stress, it es-sentially remains a physically intrinsic non-Schmid effect owing its existence to the dislocation core structure. The difference between intrinsic and extrinsic is therefore not made further in this contribution. In BCC metals, the non-glide component of the applied stress tensor (in a direction perpendicular to the Burgers vector) is cru-cial. A Burgers vector in a BCC crystal can be decomposed into a screw component and an edge component. The interaction of the applied stress field with the frac-tional edge component explains this peculiarity. This component of the stress does not contribute to the movement of dislocations, and thereby induces many interest-ing features in BCC crystals. Especially in the last years many atomistic simulations have been performed (e.g. Duesbery and Vitek (1998), Ito and Vitek (2001), Bas-sani et al. (2001), Duesbery et al. (2002), Vitek et al. (2004a), Gr ¨oger and Vitek (2005)) which support this effect. The hydrostatic pressure dependence of the flow stress (see Spitzig (1979)), the strength differential effect or the tension-compression asymmetry observed in BCC metals and intermetallic compounds (see Bassani (1994)) and crit-ical conditions for the forming of shear bands and localization (e.g. Dao and Asaro (1996)) typically result from this crystallographic non-Schmid effect, and its connec-tion to non-associated plastic flow is well established by Racherla and Bassani (2007). In the model presented next, the second intrinsic anisotropy effect will be included by modifying the crystallographic flow rule which results in an update of the resolved shear stress.
12
2 A finite strain BCC single crystal plasticity model and its experimental identification
2.4 A BCC crystal plasticity model at material point level
2.4.1 Kinematics in crystal plasticity
In the classical crystal plasticity theory as developed by Lee (1969), Rice (1971), Hill and Rice (1972) and Asaro and Rice (1977), the deformation gradient tensor is de-composed into an elastic part Feand a plastic part Fp according to:
F =Fe·Fp (2.1)
The tensor Fp defines the stress-free intermediate configuration. In this
configura-tion, resulting from plastic shearing along well-defined slip planes of the crystal lat-tice, the orientation of the crystal lattice is identical to the orientation in the reference state (see Fig. 2.2). The tensor Fereflects the lattice deformation and local rigid body
rotations. The slip systems are labeled by a superscriptα, withα = 1, 2..., ns where
n
m
α αn
0αm
α p e p= F
= F
0m
α 0m
αn
α.
e e.
. n
0αm
α0 −TF
eF = F F
F
n
0 αFigure 2.2 / Multiplicative decomposition of the deformation gradient.
ns is the total number of slip systems. The vectors mα0 and nα0 denote the slip
direc-tion and the slip plane normal in the reference and intermediate configuradirec-tions. In the current state they are represented by mαand nα, respectively.
The crystallographic split of the plastic flow rate is given by
Lp = ns
∑
α=1 ˙ γαmα0nα0 (2.2)2.4 A BCC crystal plasticity model at material point level 13
2.4.2 Constitutive model
The deformation is composed of an elastic contribution and a plastic contribution. The elastic part is related to the stress, based on a hyper-elastic formulation while the plastic part is determined by a physically based flow rule.
Elastic contribution
The second Piola-Kirchhoff stress tensor S is expressed in terms of the elastic Green-Lagrange strain tensor Ee, both relative to the intermediate state,
S=4C : E eand Ee = 1 2(Ce−I), Ce =F T e ·Fe (2.3)
with Ce the elastic right Cauchy-Green tensor and I the second order unity tensor.
The second Piola-Kirchhoff stress is the pull-back of the Kirchhoff stress tensor,
S=F−e1· τ ·F−eT (2.4)
where the Kirchhoff stress can be written in terms of the Cauchy stress using the Jacobian according to
τ = Je· σ with Je =det(Fe) (2.5)
The fourth order tensor4Cconsists of the anisotropic elastic moduli.
The Schmid resolved shear stress is the projection of the Kirchhoff stress on the slip systems, i.e.
τα =mα· τ ·nα =mα0 ·Ce·S·nα0 (2.6)
The slip systems of the{112} family in BCC crystals are given in Table 2.1, which is the relevant set within the considered temperature range.
Plastic slip and hardening
In order to include the previously described crystallographic intrinsic properties, a physical description of the slip law will be used instead of a classical phenomeno-logical (power law) relation. Physically based slip laws have been formulated in crystal plasticity models for materials with a symmetric planar slip dependency (e.g. Kothari and Anand (1998)). The anisotropy in BCC crystals is included by adapting the slip law accordingly.
The physical interpretation given hereafter, relies on the thermally activated dislo-cation kinetics. Plasticity occurs by dislodislo-cation motion on certain slip planes in an
14
2 A finite strain BCC single crystal plasticity model and its experimental identification
energetically favorable direction. The actual flow stress is determined by the resis-tance to this dislocation motion. The motion is obstructed by short-range and long-range barriers. The short-long-range barriers in general are generated by the Peierls stress (periodic resistance of the lattice) and the local forest of dislocations. The long-range barriers originate from the elastic stress field due to grain boundaries, far field forests of dislocations and other defects. The total resistance can be split accordingly
sα =sαt +sαa (2.7)
where the short-range barriers are responsible for the first part sα
t, referred to as the
thermal part since thermal activation is sufficient to overcome this resistance. The athermal part of the resistance sα
a is related to the long-range barriers. Although
this contribution slightly decreases at higher temperatures (through a decrease of the elastic moduli), this effect is negligible compared to the change of the thermal resistance with varying temperature.
During their motion, dislocations are obstructed by a quasi-periodic short-range re-sistance. The Helmholtz free energy required to isothermally cross a barrier is de-noted by ∆F and the mechanical work of sα
t can be written as ∆W. The energy
differ-ence between these two,
∆G =∆F−∆W (2.8)
is the energy barrier to overcome by a dislocation through thermal activation. It is well-known that the average dislocation velocity, ¯vα on slip systemα, can be
esti-mated by, ¯vα =lαω
0exp{−∆G/kT} (2.9)
with lα representing the distance between the barriers,ω
0 the attempt frequency, k
the Boltzmann constant and T the absolute temperature. The relation between the slip rate and the average velocity is given by the Orowan relation, ˙γα =bρ
m¯vαwhere
ρm and b represent the mobile dislocation density and Burgers vector, respectively. Substituting the velocity expression (2.9) into the Orowan relation leads to the slip law according to,
˙ γα = 0 if τ α e f f ≤0 ˙ γα 0exp{− ∆G kT }sign(τ α) if 0 < τα e f f (2.10) where τα
e f f = |τα| −sαa is the driving force for the dislocation motion and ˙γα0 =
bρmlαω
0is the reference strain rate, which is different for the different BCC slip plane
families since the distance between the barriers depends on the family. The energy
∆G to be supplied by the thermal fluctuations at constant temperature is calculated
as (see Kocks et al. (1975)) ∆G =G0 " 1− τα e f f sα t p#q (2.11)
2.4 A BCC crystal plasticity model at material point level 15
where G0 is the activation free energy needed to overcome the obstacles without
the aid of an applied stress. The quantities p and q lie in the range 0 ≤ p ≤ 1 and 1 ≤ q ≤ 2, and in the numerical examples of this paper they are taken as 1. The equations (2.10) and (2.11) constitute the slip law for materials that do not show crystallographic asymmetry effects. The non-Schmid effects are included in equation (2.10) by pursuing a similar strategy as introduced by Dao and Asaro (1993), where the Schmid stress as defined by (2.6) is extended to account for the non-Schmid con-tribution:
τnα = τα+ ηα :τ (2.12)
withτ the Kirchhoff stress andηαrepresenting the tensor governing the non-Schmid effects for slip systemαaligned with mα, nα and zα =mα×nαdefined as,
ηα = ηmm(m⊗m) + ηnn(n⊗n) + ηzz(z⊗z)+
ηmz(m⊗z+z⊗m) + ηnz(n⊗z+z⊗n)
(2.13)
In this framework, τα
n enters the equations (2.10) and (2.11) via τe f fα = |τnα| −sαa.
The definition of the non-Schmid stress tensor and incorporation in the slip law is phenomenological, and the physical meaning is not immediately trivial. The non-Schmid component is operative in the thermally activated process because of its ef-fect on the fractional edge component of the screw dislocation cores.
For isothermal cases, the thermal part sα
t of the slip resistance sα is taken constant
and the athermal part of the slip resistance is evolving such that ˙sα = ˙sαa =
∑
β
hαβ|γ˙β| (2.14)
The hardening moduli hαβ determine the rate of strain hardening on slip systemα
due to slip on slip systemβ. This self and latent hardening are phenomenologically described by (Asaro and Needleman (1985))
hαβ =qαβhβ (2.15)
where qαβ and hβ are further detailed. As explained in the discussions in section
2.2, BCC metals show a{112} dominated slip pattern at room temperature. For the
{112}h111i slip system there are twelve different slip planes contributing to one slip direction (see Table 2.1). This leads to the following definition of the q matrix (12×
12), qαβ = 1 qn . . . qn qn 1 . . . qn ... ... ... ... qn qn . . . 1 (2.16)
16
2 A finite strain BCC single crystal plasticity model and its experimental identification F Fe Fp S,τα,ταn γ˙ Fc =Fe·Fp R=F−Fc Fe:=Fe+∆Fe
Figure 2.3 / Schematic overview of the BCC crystal plasticity model.
where qnrepresents the ratio of the latent hardening with respect to the self
harden-ing for non-coplanar slip systems.
Finally the specific form of the self hardening rate, which is motivated by Brown et al. (1989), reads hβ =hβ 0 1− sβ a sβs a sign 1−s β a sβs , (2.17) where hβ0, sβ
a and sβs are the initial hardening rate, the actual athermal slip resistance
and the saturation value of the slip resistance, respectively. The exponent a is con-sidered as a constant material parameter.
Implementation of the constitutive model
The implementation of the above model follows an incremental-iterative solution procedure. The first step in this iteration is the initial estimate for the elastic part Fe,
resulting in a plastic part Fp through (2.1). With the kinematics defined, the stress,
the Schmid and non-Schmid stresses, are calculated. From these, the slip rate on each slip system is calculated by using the slip law (2.10). As this slip law is non-linear in terms of the slip rates, a sublevel Newton-Raphson iteration process is adopted to solve for the slip rates.The plastic part of the deformation gradient is obtained from the calculated slip rates through a time integration scheme. An updated Fp
2.5 Modeling some intrinsic properties of BCC single crystals 17
Fc = Fe·Fp. Generally, the calculated Fc and the imposed F will be different, which
results in a residual R. The linearization of the residual by computing the sensitiv-ity with respect to Fe leads to an update ∆Fe of the elastic part of the deformation
gradient. The elastic deformation gradient is updated and the process is repeated until convergence is achieved. The procedure is performed for all time steps, which results in a full history of stress and slip evolution. The main steps of the procedure are summarized in Fig. 2.3.
2.5 Modeling some intrinsic properties of BCC single crystals
In this section, orientation and temperature dependence of BCC materials are simu-lated applying the presented crystal plasticity framework, and results are compared with the single crystal experiments. To this purpose, material parameters are de-termined using a proper identification procedure. This direct confrontation has, to the best of our knowledge, not been done before for single crystals. Although, poly-crystal BCC poly-crystal plasticity simulations and validations have been conducted in the literature (e.g. Kothari and Anand (1998), Liao et al. (1998), Lee et al. (1999), Xie et al. (2004), Ganapathysubramanian and Zabaras (2005)) and non-Schmid effects have been incorporated into constitutive models (e.g. Qin and Bassani (1992)), the results cannot directly be exploited in the context of actual model.
2.5.1 Orientation dependence
The hardening curves, work hardening rate, temperature and rate sensitivity, activity of slip planes, CRSS and the flow stress of BCC metals all depend on the orientation of the crystal. As emphasized before, two crucial important physical aspects control-ling this pronounced orientation dependence are the so-called slip asymmetry (or twinning/anti-twinning asymmetry) and the non-planar spreading of screw dislo-cation cores, which have been included in the model.
The twinning/anti-twinning asymmetry manifests itself on the{112}slip planes. On these slip planes, the slip resistance in the anti-twinning direction is higher than in the twinning direction, an effect which is difficult to quantify in experimental tests. Guiu (1969) experimentally observed the asymmetry for Mo single crystals at dif-ferent temperatures under direct shear. He concluded that the CRSS is roughly 1.5 times larger for the slip systems on the anti-twinning planes of{112}h111icompared to the slip systems on the twining planes.
Due to the non-planar spreading of the screw dislocation cores, the non-glide com-ponent of the applied stress affects the dislocation core and hence, the sense of the
18
2 A finite strain BCC single crystal plasticity model and its experimental identification
applied stress affects the yielding of a crystal. The associated non-Schmid parameters will be identified together with the other parameters in the model.
2.5.2 Example: α-Fe single crystal
In the first example (Fig. 2.4) the orientation dependence ofα-Fe single crystals under uniaxial tension is examined. Here, and in the following examples the lattice vector was aligned with the tensile direction in the undeformed configuration and the pre-sented framework automatically accounts for the lattice rotations. The experimental curves (Keh (1964)) were reproduced from the reported shear stress-strain curves which were initially determined from tensile data for the planes of(¯1¯12), (¯211)and
(¯211) with the [001], [011] and [¯111] orientations respectively in the [111] direction. The experiments and simulations are performed at a strain rate of 3.3×10−4s−1. The
0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 160 180 Strain % Stress (MPa) [011] [001] [¯111]
Figure 2.4 / Tensile orientation dependence of [001], [011] and [¯111] oriented α-Fe single crystals. Solid lines are the simulation results and dashed lines represent experiments.
results show the pronounced influence of the orientation on the yielding and hard-ening behavior of the crystal. The orientations selected in the example constitute the corners of a unit triangle mapping the [001],[011] and [¯111] directions. It is a well-established fact that at the corners and at the edges of the unit triangle multislip is observed, while the deformation starts with single slip for directions mapped inside the triangle. For these orientations the initial rate of work hardening is relatively high and decreases rapidly with ongoing deformation.
2.5 Modeling some intrinsic properties of BCC single crystals 19
Material parameters have been identified using a least-square optimization proce-dure which minimizes an objective function that equals the sum of squares of the differences between experimental and simulation results. Most of the parameters are presented in Table 2.2. The remaining parameters are C11 =236GPa, C12 =134GPa,
Table 2.2 / Material parameters for α-Fe single crystals
Initial hardening rate h0 697.88 MPa
Saturation value of slip resistance ss 132.10 MPa
Hardening rate exponent a 1.5
Thermal slip resistance st 13.91 MPa
Athermal slip resistance (atwin sense) sa0 9.59 MPa
Athermal slip resistance (twin sense) sa0 5.75 MPa
Non-Schmid parameter ηmm 0.0544
Non-Schmid parameter ηnn -0.0293
Non-Schmid parameter ηzz -0.0267
Reference strain rate γ˙0 1.07×106s−1
Activation free energy G0 2.95×10−18J
C44 =119GPa (Adams et al. (2006)), qn =1.4, k =1.3807×10−23J/K, whileηmz and
ηnzare taken zero..
2.5.3 Example: molybdenum single crystal
In the second example, the orientation dependence of molybdenum single crystals under uniaxial tension is analyzed. The experimental curves for the[010],[101] and
[111]orientations are taken from the work of Irwin et al. (1974). They performed two sets of experiments at 293K and 77K at a strain rate of 6×10−5s−1. The results are
compared for the 293K case and presented in Fig. 2.5. The material parameters that have been identified using the same least-square minimization process are presented in Table 2.3. The remaining parameters are C11 = 469GPa, C12 = 167.6GPa, C44 =
106.8GPa (Bolef and Klerk (1962)), qn =1.4, k=1.3807×10−23J/K.
2.5.4 Temperature dependence
BCC metals exhibit a different mechanical response compared to FCC metals, in par-ticular in the presence of temperature changes. The dependence of the flow stress on the temperature, has been studied thoroughly in the literature and illustrated for many BCC single crystals such as Mo (e.g. Hollang et al. (1997)), Nb (e.g. Ackermann
20
2 A finite strain BCC single crystal plasticity model and its experimental identification 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 30 35 40 Strain % Stress (MPa) [101] [111] [010]
Figure 2.5 / Orientation dependence of [010], [101] and [111] oriented molybde-num single crystals. Solid lines are the simulation results and dashed lines represent experiments.
Table 2.3 / Material parameters for Mo single crystals
Initial hardening rate h0 251.37 MPa
Saturation value of slip resistance ss 76.90 MPa
Hardening rate exponent a 1.05
Thermal slip resistance st 11.89 MPa
Athermal slip resistance (atwin sense) sa0 7.23 MPa
Athermal slip resistance (twin sense) sa0 4.34 MPa
Non-Schmid parameter ηmm -0.0528
Non-Schmid parameter ηnn 0.0896
Non-Schmid parameter ηzz -0.0369
Reference strain rate γ˙0 1.4×107s−1
Activation free energy G0 0.1554×10−18J
et al. (1983)), Ta (e.g. Werner (1987)),α-Fe (e.g. Brunner and Diehl (1987), Brunner and Diehl (1997)). Below the so-called knee temperature, where the flow stress is controlled by the mobility of screw dislocations, the critical shear stress of BCC met-als rises progressively with decreasing temperature (Seeger (1981)). The flow stress in FCC metals on the contrary, only shows a moderate increase when the tempera-ture is lowered below room temperatempera-ture. Most of the work reported on BCC metals concentrates on the behavior of pure single crystals in order to eliminate secondary
2.5 Modeling some intrinsic properties of BCC single crystals 21
effects induced by interstitially dissolved foreign atoms. The purification process of single crystals requires great effort, nevertheless, when tested usually impurities can be identified.
The variation of the flow stress of a BCC metal under a temperature change has al-ready been presented in Fig. 2.1 for Mo single crystals. For other BCC metals, the behavior is qualitatively similar but quantitatively different. For the response below the knee temperature TK, different regimes should be distinguished. Especially the
interval TK/2 < T < TK was analyzed by Seeger (1981) for which the rate and
tem-perature dependence was explained through the formation of kink pairs in screw dislocations without invoking impurity effects.
Because of the non-uniformity of the temperature dependence of the flow stress, Seeger (1981) proposed different formulations for different temperature regimes. A high-temperature–low stress regime, an intermediate (diffusion controlled) regime and a high-stress regime are distinguished. In the present paper, the flow is con-trolled by the slip rate equation (2.10), which roughly corresponds to the third regime, where only kink pairs controlling dislocation movement in the direction of applied stress are taken into account.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 5 10 15 20 25 30 35 40 45 50 Strain % Stress (MPa) 77 K sim. 77 K exp. 113 K sim. 113 K exp. 175 K sim. 175 K exp.
Figure 2.6 / Temperature dependence of[001]oriented niobium single crystals.
In Fig. 2.6, the tensile stress-strain curves for a [001] oriented Nb single crystal are presented at three different temperature levels, at a strain rate of 1.3×10−4s−1. The
experimental data is taken from Duesbery and Foxall (1969). The true stress and true strain curves are reproduced from the shear values reported for (¯112)[1 ¯11]primary
22
2 A finite strain BCC single crystal plasticity model and its experimental identification
slip. The Schmid factor for the [001] orientation equals 0.471. The experiment and the crystal plasticity simulations show an adequate agreement. The material parameters for this material are: C11 = 250GPa, C12 = 135GPa, C44 = 30GPa (Carroll (1965)),
˙
γ0 = 1.2×107s−1, q
n = 1.4, G0 = 1.12×10−18J, k = 1.3807×10−23J/K, a = 1.2,
sa0 = 4MPa (atwin sense) and sa0 =2.4MPa (twin sense). Due to the lack of
experi-mental evidence the identification of the non-Schmid parameters is disregarded and the effect is excluded here. The variation of most parameters with the temperature is negligible, with the exception of the parameters presented in Table 2.4.
Table 2.4 / Material parameters for Nb single crystals at three different temperatures
Temperature 77 K 113 K 175 K
h0 1500 MPa 922 MPa 800 MPa
ss 21.7 MPa 19.9 MPa 13.42 MPa
st 15.3 MPa 8.06 MPa 3.18 MPa
2.6 Summary and Conclusion
In the present paper, a crystal plasticity model has been proposed and implemented, revealing the unique characteristics of BCC single crystals. A comprehensive sum-mary of the intrinsic properties of these materials has been presented, including re-cent insights in the activation of different slip systems, violation of Schmid’s law, temperature and orientation dependence of the flow stress and resulting stress-strain curves. The parameters in the model are determined using a proper parameter iden-tification process, relying on a least-square minimization procedure on the differ-ences between the numerical and experimental uniaxial stress-strain responses. Published results on the operational slip mechanisms in BCC crystals and the ex-tended crystal plasticity models, reflect considerable contradictions. Most studies take into account {110},{112} and {123} type of slip planes without confronting the model with the expected temperature and orientation dependence of the crystal. Following Seeger (2001), this work uses{112} planes at moderate temperatures and
{110}planes at low temperatures. Contrary to most of the crystal plasticity elabora-tions,{123}type of slip planes are not taken into account. Cross slipping phenomena and non-planar spreading of screw dislocation cores are implicitly incorporated in a phenomenological manner.
2.6 Summary and Conclusion 23
characteristics result from the actual formulation of the slip rate equation. Additional to the pre-mentioned existing crystal plasticity frameworks, in this contribution the non-Schmid behavior is introduced in the slip law by modifying the effective shear stress, where the non-Schmid contribution represents the dislocation cores spreading in a non-planar manner. Actually, the emphasis was put on this intrinsic anisotropy effect, even though other anisotropy effects such as texture development and dislo-cation sub-structure evolution may be included in the model as well.
The necessity for this research results from the fact that there is a lack of published works, that confront recent BCC single crystal plasticity models to single crystal ex-perimental data that reveals the intrinsic orientation and temperature dependence of these crystals.
Chapter three
A composite dislocation cell model to
describe strain path change effects in
BCC metals
1
Abstract / Sheet metal forming processes are within the core of many modern man-ufacturing technologies, as applied in e.g. automotive and packaging industries. Ini-tially flat sheet material is forced to transform plastically into a three dimensional shape through complex loading modes. Deviation from a proportional strain path is associated with hardening or softening of the material due to the induced plastic anisotropy result-ing from the prior deformation. The main cause of these transient anisotropic effects at moderate strains is attributed to the evolving underlying dislocation microstructures. In this paper, a composite dislocation cell model, which explicitly describes the dislocation structure evolution, is combined with a BCC crystal plasticity framework to bridge the microstructure evolution and its macroscopic anisotropic effects. Monotonic and multi-stage loading simulations are conducted for a single crystal and polycrystal BCC metal, and obtained macroscopic results and dislocation substructure evolution are compared qualitatively with published experimental observations.
3.1 Introduction
For each car, the automotive industry manufactures more than 500 parts by multi-stage forming operations, involving complex deformation paths. Deviation from a proportional strain path is commonly associated with a change in the hardening (or softening) behavior of the material. In order to achieve a first-time-right design, modern predictive tools relying on the finite element method are commonly used nowadays. The anisotropy induced by complex deformation paths, which may lead
1This chapter is reproduced from Yalcinkaya et al. (2009)
26
3 A composite dislocation cell model to describe strain path change effects in BCC metals
to premature failure (e.g. Sang and Lloyd (1979)), is crucial in this sense and should be included in the constitutive models used in the analysis.
The overall plastic anisotropy in BCC metals, induced by the imposed deformation, originates from different sources at different length scales. Slip asymmetry and in-trinsic anisotropy effects caused by the non-planar spreading of screw dislocation cores are active at the micro level (e.g. Bassani et al. (2001), Duesbery and Vitek (1998), Ito and Vitek (2001), Yalcinkaya et al. (2008)) whereas the development of dis-location substructures is relevant at the meso level (e.g. Rauch and Schmitt (1989), Wagoner and Laukonis (1983), Rao and Laukonis (1983), Wilson and Bate (1994), Gardey et al. (2005)). At the macro level, the texture development of polycrystalline metal is contributing dominantly (e.g. Bacroix et al. (1994), Bacroix and Hu (1995), Nesterova et al. (2001)). Upon switching strain paths, the intrinsic properties ob-viously have a substantial effect on the observed anisotropy due to changes in the dislocation activity. However, the evolution of dislocation microstructures has been recognized as a main driver triggering the observed anisotropic material behavior. In a recent report Li et al. (2006) commented on the strong anisotropy, i.e., larger than expected from the texture, induced by the dislocation structure in IF steel increasing with the rolling prestrain. The prediction of dislocation microstructures within the individual dislocation descriptions and continuum theories has been a challenging subject in the last decades in the material science community (see Groma (1997) for an overview). While transmission electron microscopy (TEM) observations have been a powerful tool to understand their origin and to derive the physical parameters that govern their evolution (e.g. Fernandes and Schmitt (1983)), discrete dislocation mod-els and atomistic considerations improved the understanding of the formation and the evolution of dislocation microstructures and the related plastic anisotropy. How-ever, only a limited number of micromechanical modeling approaches have been ad-dressing the anisotropy induced by evolving dislocation cells with a crystal plasticity framework.
Among the attempts to develop plastic anisotropy models that incorporate the mi-crostructure evolution for complex deformation histories, the most remarkable one is the constitutive model proposed by Teodosiu and Hu (1995). This phenomenological model uses the Hill criterion for the onset of yielding while the hardening is associ-ated with the dislocation structures. The polarity of dislocation walls, the back-stress and the strength of the dislocation structure are accounted for by internal variables. Recently, Wang et al. (2008) presented an improvement of this model especially con-centrating on continuous loading path changes from uniaxial tension to simple shear without unloading the material.
Another attempt to describe the occurring phenomena is presented by Peeters et al. (2000) dealing with a polycrystal plasticity model that incorporates more details of
