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Application of microstructure sensitive design to structural components produced from hexagonal polycrystalline metals
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Computational Materials Science 43 (2008) 374–383 www.elsevier.com/locate/commatsci
Application of microstructure sensitive design to structural components produced from hexagonal polycrystalline metals Tony Fast, Marko Knezevic, Surya R. Kalidindi * Department of Materials Science and Engineering, Drexel University, Philadelphia, PA 191z04, United States Received 24 August 2007; received in revised form 28 November 2007; accepted 9 December 2007 Available online 28 January 2008
Abstract In this paper, we present the first successful design case studies in the application of microstructure sensitive design (MSD) methodology to optimize performance of structural components made from polycrystalline metals with hexagonal close-packed (hcp) crystal lattices. It is demonstrated that the underlying spectral framework of the MSD methodology facilitates an efficient consideration of the complete set of crystallographic textures in the design optimization. In order to accomplish this task a number of important enhancements had to be introduced to the MSD framework. The most significant enhancement is in the mathematical description of the design space, i.e. the texture hull. The advantages of the new approach described in this paper are illustrated with two specific design case studies involving different assumptions of symmetry at the sample scale. In both case studies presented, it is seen that the overall performance is strongly influenced by the crystallographic texture in the sample. Furthermore, the relevant property closures and performance maps accounting for the complete set of textures are also depicted. Published by Elsevier B.V. PACS: 62.20.x; 62.20.Dc; 62.20.Fe; 61.50.Ah Keywords: Microstructure; Anisotropy; Property closures; Elasticity; Plasticity; Spectral methods
1. Introduction The current practice in engineering design does not pay adequate attention to the internal structure of the material as a continuous design variable. The design effort is often focused on the optimization of the geometric parameters using robust numerical simulations tools, while material selection is typically relegated to a relatively small database. Furthermore, material properties are usually assumed to be isotropic, and this significantly reduces the design space. Since the majority of commercially available metals used in structural applications are polycrystalline and often possess a non-random distribution of crystal lattice orientations (as a consequence of complex thermo-mechanical loading history experienced in their manufacture), they should be expected to exhibit anisotropic properties. *
Corresponding author. Tel.: +1 215 895 1311; fax: +1 215 896 6760. E-mail address: [email protected] (S.R. Kalidindi).
0927-0256/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.commatsci.2007.12.002
In recent years, a new mathematical framework called microstructure sensitive design (MSD) has been developed to facilitate the rigorous consideration of microstructure as a continuous variable in engineering design and optimization. At the core of MSD lie rigorous quantitative descriptions of the relevant statistics of the local state distributions (at different levels of details classified by n-point statistics [1–5]) in the microstructure. Salient features of MSD include: (i) construction of a microstructure hull [6–8] that represents the complete set of theoretically feasible local state distributions, and (ii) delineation of property closures [9–11] that identify the complete set of theoretically feasible combinations of macroscale effective properties of interest in a given application (for a selected homogenization theory). The primary advantages of the MSD approach lie in its (a) consideration of anisotropy of the properties at the local length scales, (b) exploration of the complete set of relevant microstructures (due to the use of microstructure hulls and property closures) leading to global optima,
T. Fast et al. / Computational Materials Science 43 (2008) 374–383
and (c) invertibility of the microstructure–property relationships (due to the use of spectral methods in representing these linkages). As a result, MSD facilitates the rigorous consideration of the microstructure as a continuous design variable in optimization of the mechanical performance of the given structural component. The main focus of MSD has, thus far, been on the crystallographic texture (also called orientation distribution function or the ODF) in polycrystalline metals and the macroscale elastic–plastic properties that are strongly influenced by this specific microstructural detail. Limited explorations have also been conducted into compositional variations in microstructures [12] and into fiber reinforced composites [6]. In cubic polycrystalline metals, the MSD framework has been successfully applied to a few design case studies. These have included maximizing the deflection in a compliant beam [7], maximizing the in-plane load carrying capacity of a thin plate with a central circular hole [8], and minimizing the elastic driving force for crack extension in rotating disks [13] and internally pressurized thin-walled vessels [14]. Property closures were also produced for a broad range of combinations of macroscale effective elastic and plastic properties in cubic metals [9,10,14] and for a limited number of hexagonal metals [11]. In this paper, we present the first successful application of the MSD methodology to design case studies involving polycrystalline hexagonal close-packed (hcp) metals. The primary focus here continues to be on the crystallographic texture in the sample. In order to accomplish this task a number of important changes had to be introduced to the MSD framework. The most significant change deals with the mathematical description of the design space, i.e. the texture hull. The application of the MSD framework to hcp polycrystals required the consideration of a significantly larger number of dimensions of the texture hull, when compared to the previous case studies that involved cubic polycrystals. The much larger number of dimensions (and the corresponding larger number of design variables) demanded the development of a new approach to the problem. This new approach is described in this paper, and is illustrated with two design case studies involving different assumptions of symmetry at the sample scale. 2. Microstructure sensitive design framework MSD [6–8,13,15] starts with a statistical description of the relevant details of the microstructure. In this paper, our interest will be limited to 1-point statistics of lattice orientation (i.e. ODF) in hcp polycrystals. The ODF, denoted as f(g), reflects the normalized probability density associated with the occurrence of the crystallographic orientation g in the sample [16] as Z Vg f ðgÞdg ¼ ; f ðgÞdg ¼ 1: ð1Þ V FZ In Eq. (1), V denotes the total sample volume and Vg is the sum of all sub-volume elements in the sample that are asso-
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ciated with a lattice orientation that lies within an incremental invariant measure, dg, of the orientation of interest, g. The lattice orientation, g, is usually defined by a set of three angles called the Bunge–Euler angles [16], i.e. g = (u1, U, u2). The Bunge–Euler angles define a specific sequence of three rotations that would bring the crystal and sample reference frames into coincidence. The local properties depend strongly on the local crystallographic orientation, and therefore, the overall behavior of the material is strongly influenced by the distribution of the crystallographic orientations inside the polycrystalline material. In Eq. (1), FZ denotes the fundamental zone of orientations, and represents the local state space describing the set of distinct orientations relevant to a selected class of textures [16]. As an example, the FZ for hexagonal-orthorhombic1 textures is described by n p p po FZ ¼ g ¼ ðu1 ; U; u2 Þ0 6 u1 6 ; 0 6 U 6 ; 0 6 u2 6 : 2 2 3 ð2Þ MSD employs efficient spectral representations of the microstructure distribution functions. For example, the ODF can be expressed in a Fourier series using symmetrized generalized spherical harmonics (GSH), T lm l ðgÞ, as f ðgÞ ¼
M ð lÞ X N ðl Þ 1 X X
lm F lm l T l ðgÞ:
ð3Þ
l¼0 l¼1 m¼1
Eq. (3) facilitates visualization of ODF as a point in Fourier space whose coordinates are given by F lm l . Recognition of the fact that ODF provides information only about the volume fraction of the various orientations present in the polycrystalline sample permits an alternate mathematical description as X X f ðg Þ ¼ ak d g gk ; 0 6 ak 6 1; ak ¼ 1: ð4Þ k
k k
The Dirac-delta function dðg g Þ in Eq. (4) represents the ODF of a single crystal of orientation gk, and ak denotes its volume fraction in the polycrystal. Let k F lm l define the Fourier coefficients of the single crystal ODFs. It is then possible to define a convex and compact texture hull [7], M, as ( ) X X lm lm k k lm k lm M ¼ F l F l ¼ ak F l ; F l 2 M ; ak P 0; ak ¼ 1 ; k k ð5Þ where k
M ¼
k
k lm F lm l Fl
k k 1 lm ¼ T g ; g 2 FZ : ð2l þ 1Þ l
ð6Þ
The bar on top of the GSH in Eq. (6) denotes the complex conjugate. It should be recognized that M represents the 1 The first symmetry in this standard notation used by the texture community refers to symmetry at the crystal level (resulting from the atomic arrangements in the crystal lattice) while the second refers to symmetry at the sample scale (resulting from processing history).
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complete set of all theoretically feasible ODFs, several of which have not yet been realized in practice or even have been targeted for manufacture by materials specialists. Next, we turn our attention to the homogenization theory to be adopted to arrive at the macroscale properties of the polycrystal for a given texture. The ODF described above constitutes a first-order description of the microstructure (also referred as 1-point statistics). Using this microstructure description, only the elementary bounds of the macroscale elastic and plastic properties can be evaluated. In this work, we have decided to employ the upper bound theories for all macroscale elastic and plastic properties. For example, the upper bounds on the diagonal components of the macroscale elastic compliance tensor, S*, can be expressed as [10] (no summation implied on repeated indices) S ijij ¼ hS ijij i;
ð7Þ
where h i denote an ensemble average (also equal to the volume average when the ergodic hypothesis is invoked). S is the local elastic compliance tensor in the sample reference frame that is defined using a coordinate transformation law for fourth-rank tensors as S ijkl ¼ gip gjq gkr gls S cpqrs ;
ð8Þ
where Sc is the local elastic compliance tensor in the local crystal reference frame, and gij are the components of the transformation matrix defined in terms of the Bunge–Euler angles [16]. Although, only the first-order description of the microstructure and elementary bounds are employed in this study, it is emphasized here that the MSD framework has the potential for addressing higher-order microstructure descriptions and advanced homogenization theories [1,3]. The selected homogenization theories are then cast in the same Fourier space that has been used to represent the microstructure. The S abcd ðgÞ functions can be represented in a Fourier series using GSH functions as S abcd ðgÞ ¼
N ðl Þ X M ðl Þ 4 X X
lm lm abcd S l T l ðgÞ;
ð9Þ
l¼0 l¼1 m¼1
where abcd S lm l are referred to as the elastic Fourier coefficients and the details of their computation have been reported in an earlier paper [11]. The volume averaged value is then computed by exploiting the orthogonality of the Fourier basis as I S abcd ¼ hS abcd i ¼ S abcd ðgÞf ðgÞdg ¼
N ðl Þ X M ðl Þ 4 X X l¼0 l¼1 m¼1
1 lm lm abcd S l F l : ð2l þ 1Þ
ð10Þ
Eqs. (6) and (10) embody one of the central features of MSD. They provide an efficient linkage between the complete set of feasible ODFs and the corresponding feasible combinations of macroscale elastic properties. Of particular significance is the fact that unlike the Fourier representation of the ODF, the representation for properties often
extends to only a finite number of terms in the Fourier expansion. As shown in Eqs. (9) and (10), in consideration of elastic properties, the only relevant Fourier coefficients are those that correspond to l 6 4. The main advantage of the spectral methods described herein lies in the fact that they are able to formulate highly efficient structure–property relationships. In many mechanical design problems, avoiding plastic deformation constitutes an important consideration. For this purpose, yield surfaces are often described in the sixdimensional stress space. A common formalism for describing anisotropic yield surfaces is the orthorhombic Hill’s yield surface description [17] ! 1 1 1 1 2 ðhr22 i hr33 iÞ þ 2 r2y2 r2y3 r2y1 ! 1 1 1 1 2 ðhr33 i hr11 iÞ þ þ 2 r2y3 r2y1 r2y2 ! 1 1 1 1 1 2 þ þ 2 2 ðhr11 i hr22 iÞ þ 2 s212 2 2 ry1 ry2 ry3 sy12 þ
1 2 1 s þ 2 s213 ¼ 1; s2y23 23 sy13
ð11Þ
where ry1, ry2, ry3 denote the macroscale tensile (or compressive) yield strengths and sy12, sy13, sy23 denote the macroscale shear yield strengths. In Eq. (11), hriji and hsiji denote the macroscale normal and shear stress components experienced by the material point under consideration. The six anisotropic material yield parameters in Eq. (11) can be estimated for a polycrystalline material using an extended version of the Taylor model [18], also commonly referred to as the Taylor-type crystal plasticity model. This model constitutes an upper bound and has been demonstrated [19] to provide reasonably accurate predictions of the macroscopic yield strengths for the polycrystalline a-Ti used in the case studies presented here. The Taylor-type crystal plasticity model assumes that each constituent single crystal experiences the same strain as the imposed strain at the macroscale, and computes the local stress in the crystal that allows the accommodation of the imposed plastic strain through slip on the multiple slip systems in the crystal. The volume averaged stress (over the constituent single crystals) then provides an estimate of the macroscale yield strength for the polycrystal. As an example, the following macroscopic velocity gradient is imposed on the polycrystal in evaluating ry1: 0 1 h_ei 0 0 B C hLi ¼ @ 0 qh_ei 0 ð12Þ A: 0
0
ð1 qÞh_ei
The parameter q is allowed to take any value between 0 and 1. Because the deformation imposed by Eq. (12) is isochoric, the resulting stress field is purely deviatoric. In order to calculate the tensile yield strength, the hydrostatic component is computed by establishing the value of q
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(denoted as q*) for which the averaged lateral stresses over the polycrystal are equal to each other and added to the diagonal elements of the averaged deviatoric stress: ry1 ¼ hr011 ðq Þi hr022 ðq Þi;
ð13Þ
where r0 denotes the volume averaged deviatoric stress tensor for the polycrystal. Fast computation of the yield strengths for polycrystals in the MSD framework entails the use of Fourier representations to describe the functional dependence of the local stresses in the constituent crystals on their orientations. The Fourier representations needed for evaluating the upper bound for ry1 are expressed as (no summation implied on repeated index i) [11] r0ii ðg; qÞ
¼
N ðl Þ X M ðl Þ l X X
lm ii lm y1 S l ðqÞT l ðg Þ;
ð14Þ
l¼0 l¼1 m¼1
r0ii ðqÞ
¼
N ðl Þ X M ðl Þ l X X
lm ii lm y1 S l ðqÞF l
l¼0 l¼1 m¼1
2l þ 1
;
ð15Þ
where the procedures for computing y1 ii S lm l ðqÞ for hcp crystals experiencing slip on prism, basal and pyramidal slip systems have been already been described in a recent paper [11]. In describing the plastic properties, we have also observed that we needed a lot more Fourier coefficients than those needed to describe elastic properties. For practical reasons, we truncated the series after a desirable accuracy has been attained, and this is reflected in Eqs. (14) and (15) by restricting the series to terms corresponding to l 6 l*. For example, for hexagonal-orthorhombic textures, it was observed that the maximum error in Eq. (15) was less than 3% when the series was truncated to 57 terms (corresponding to l* = 12) [11]. The desired accuracy in property predictions therefore dictates the number of Fourier dimensions that need to be explored by the MSD methodology for a selected design application. In previous work, we have demonstrated the power of spectral microstructure–property linkages described above in delineation of numerous property closures [9–11]. Property closures identify the complete set of theoretically feasible effective (homogenized) anisotropic property combinations in a given material system, and are also of interest in this paper. Two different examples of these closures corresponding to two selected case studies will be delineated in 2D and 3D, respectively. In this paper, our primary aim is to extend the MSD framework into design and optimization of structural components made from hcp polycrystals. This is addressed in the next section. 3. MSD for performance optimization The main challenge encountered in the application of the MSD methodology to mechanical design case studies is that the microstructure–property linkages need to be explored in fairly large dimensional Fourier spaces. In this work, in dealing with plastic properties of hexagonal poly-
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crystals there is a need for consideration of a much higher dimensional Fourier space compared to our earlier work on the more symmetric cubic polycrystals. The increased demand in dimensionality is driven by two main reasons: (i) the Fourier representation of the ODF for hcp poylcrystals typically needs many more terms compared to the corresponding representations for cubic polycrystals, and (ii) the yield surfaces in hcp polycrystals are substantially more anisotropic compared to the yield surfaces for cubic polycrystals. For example, the design case studies reported by Houskamp et al. [8,14] explored the cubic-orthorhombic texture hull in 12 dimensions, whereas consideration of similar problems here in hexagonal-orthorhombic texture hulls needs exploration of 57 dimensions of the microstructure hull [11]. A number of different approaches have been employed in the past to integrate microstructure sensitive design into the engineering design framework. The main approaches of these prior studies are briefly reviewed below: 1. The initial design case studies discretized the microstructure hull (Eq. (5)) into bins and evaluated the performance in each bin (e.g. [8,13]). It was assumed that all microstructures in one bin could be assumed to exhibit the same performance for the given design application. In these examples, the Fourier space was restricted to only the low-order subspaces and consequently, the number of bins could be restricted to a reasonably small number. These approximations permitted solutions to simple case studies. As the dimensionality of the relevant Fourier space expands, this approach is bound to be computationally inefficient, if not completely infeasible. 2. Design case studies were explored using the property closure as the design space [12]. This approach has the inherent advantage of greatly reducing the design space by mapping a higher dimensional Fourier space into a reduced dimensional property space. This method is particularly attractive when the property space (also called the property closure) is compact and convex. However, when the property space is not convex, defining the boundary of the property closure becomes extremely challenging, especially in design case studies where the overall performance is governed by a large number of macroscale properties. It should be noted that the property closures for a vast number of design problems are likely to be non-convex. 3. In the most recent design case studies [7,14], performance optimization was explored using a generalized reduced gradient method while restricting the design space to the convex microstructure hull. In these examples, the microstructure hull was defined using an algorithm based on Gram–Schmidt orthonormalization. This approach also allowed the integration of the MSD methodology with the finite element methods used typically by the designers [13,14]. Although this is an extremely efficient algorithm for exploring optimized solutions, we found it very difficult to apply this
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algorithm to the case studies described in this paper because of the much larger number of Fourier dimensions involved. In the case studies described in this paper, we have taken a new approach. Instead of using the Fourier coefficients F lm l to define the design space, we found it more convenient to define the design space in terms of the ak (see Eq. (4)). With this choice the design space is simply expressed as X ak ¼ 1; ak P 0: ð16Þ k
Note that the constraints described in Eq. (16) can be implemented much more easily into any optimization search tool, compared to the constraints described in Eqs. (5) and (6). The new approach naturally leads to the question of how many distinct orientations need to be considered in the FZ (i.e. the range of k in Eq. (16)) for a given design problem. The definition of the ODF (Eq. (1)) and its Fourier representation (Eq. (3)) guarantee that the complete set of all theoretically feasible ODFs forms a compact convex region in the Fourier space [6–8]. As an example, the texture hull for hexagonal-orthorhombic textures is shown in Fig. 1 in the first three dimensions of the Fourier space. The crystal orientations corresponding to the vertices of the texture hull are called principal orientations in this paper. Also, as described earlier in Eqs. (10) and (15), different Fourier subspaces are relevant to different macroscale engineering properties of interest. The set of vertices of the Fourier subspace of interest are denoted by Mp in this work, where the
superscript p reminds us that this set corresponds to principal orientations. The texture hull in the selected subspace, e , can then be described as M ( ) X X lm lm lm lm p p p e ¼ F l F l ¼ ap F l ; F l 2 M ; ap P 0; ap ¼ 1 : M p p ð17Þ It should be noted that the main difference between Eqs. (4) and (8) is that M describes the complete infinite-dimene defines only a selected subsional texture hull, whereas M space of M that is relevant to a given design problem. Consequently, the size of Mp should be significantly smaller than the size of Mk. As an example, consider the FZ of the hexagonal-orthorhombic textures described in Eq. (2). Discretizing this FZ into 10° bins would result in 2590 lattice orientations (this would dictate the size of Mk). Our investigations have revealed that only the orientations on the boundary of the FZ can be principal orientations (i.e. only these orientations correspond to the vertices of the texture hulls). The same 10° binning of the boundary of the FZ produces only 339 distinct principal orientations. In other words, the size of the Mp would be at least one order of magnitude lower than the corresponding size of Mk. If we further restrict our interest to only the three dimensions of the texture hull shown in Fig. 1, we would need to consider only 15 principal orientations. As can be seen from this example, the use of principal orientations and changing the microstructure design variable from F lm l to ap, lead to major computational advantages in the MSD methodology. It should also be
Fig. 1. The hexagonal-orthorhombic texture hull projected in the three dimensions of the Fourier space that are common to the case studies presented in this paper. The vertices of the hull are associated with the set of principal orientations for this subspace, and are shown as black spheres in this figure.
T. Fast et al. / Computational Materials Science 43 (2008) 374–383
noted that this discovery was instrumental to our success in addressing the microstructure design case studies presented in this paper. It is important, at this stage, to reflect on some of the consequences of using ap as the microstructure design variable. One might be tempted to conclude erroneously that a prescribed set of ap corresponds to a unique ODF. The reader is cautioned that we are dealing here with truncated Fourier spaces where the mapping between F lm l and ap is not one-to-one. Indeed, many different realizations of ap can correspond to one set of truncated F lm l . A prescribed set of ap, however, corresponds to a single set of truncated F lm l coefficients, which in turn corresponds to a very large set of distinct ODFs. However, all of the distinct ODFs corresponding to the prescribed set of ap are expected to exhibit the same (or approximately similar) values of macroscale properties of interest defined by the selected Fourier subspace [1,10]. Another important consequence of the approach presented here that it naturally produces nonunique solutions (as expected and desired). The new approach described here of using ap (henceforth denoted as a for ease of notation) as the design variable can be conveniently applied to mechanical design problems as well as the delineation of property closures. The optimization problem for mechanical design involving components with statistically homogeneous microstructures can be formulated as Maximize OðqÞ;
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An N-property closure is then delineated by seeking the maximum and minimum values possible for one of the properties of interest while constraining all other properties to values pre-selected within their respective ranges. As an example, let P 2 ; . . . ; P N represent a selected combination of values for all properties of interest except P1, where the values of each property are constrained to lie within their respective potential ranges, i.e. P i 2 Ri . A restricted feasible range for P1 can then be established as ( ) X R1 ¼ P 1 ðaÞP i ðaÞ ¼ P i 8i 6¼ 1; ap P 0; ap ¼ 1 : ð20Þ p If R1 is a non-empty set, it leads to the identification of e points on the boundary of the property e closure. Let P 1 de note the extrema of R1 . Then P 1 ; P 2 ; . . . ; P N lie on the boundary of the property closure sought. It should be noted here that for several selections of P 2 ; . . . ; P N , the corresponding R1 is indeed expected to be a null set. In the present study, all of the optimization problems formulated in Eqs. (18)–(20) were solved successfully using the fmincon function in Matlab’s Optimization Toolbox [21]. The fmincon function uses sequential quadratic programming, chooses subsequent variables via the line-search method, and implements the Broyden–Fletcher–Goldfarb– Shanno Quasi-Newton formula to define the Hessian [21]. Convergence is defined using first-order necessary conditions.
where
q ¼ ðP 1 ðaÞ; P 2 ðaÞ; . . . ; P N ðaÞÞ; X ap ¼ 1; subject to ap P 0;
4. MSD case study: compliant mechanism ð18Þ
p
where O denotes a objective function characterizing the performance of the mechanical component, and q is a set of relevant macroscale material properties (denoted as P i ðaÞ) influencing the performance. In particular, it is noted that the functional dependence in P i ðaÞ can be highly nonlinear (especially with plastic properties; see Eqs. (11)– (15)). In prior work, the property closures were obtained by formulating optimization problems corresponding to the points on the boundary of the closure [10,20]. However, these optimization problems were cast in the F lm l -space. For the reasons mentioned earlier, the idea of using the ap-space instead of the F mn l -space can prove very beneficial in delineating multi-property closures of interest in microstructure design problems. Briefly, one starts by identifying the potential range, Ri, of values for each property of interest, Pi, as " ( ) X Ri ¼ min P i ðaÞap P 0; ap ¼ 1 ; p¼1 ( )# X max P i ðaÞap P 0; ap ¼ 1 : ð19Þ p¼1
Compliant mechanisms are mechanical devices that gain movement from parts that flex, bend or have ‘‘springiness” to them [22]. These mechanisms with built-in flexibility are practical since they are simple. They also eliminate the need for multiple rigid parts, pin joints and add-on springs which can reduce design complexity and cost. Compliant mechanisms are found in sensors, gearboxes, valves, bicycle derailleurs, and various other mechanical designs. In this case study, we will seek the texture(s) that will maximize the deflection of the beam without initiating plastic deformation. 4.1. Mechanical solution The compliant mechanism is idealized here as a long slender cantilever beam whose macroscale elastic–plastic properties exhibit orthorhombic symmetry (presumably the processing options have been restricted to accomplish this). The stress field in the cantilever beam, with one end fixed to a rigid surface and the other end subjected to a point load P (see Fig. 2), is expressed as [23] r11 ¼
12P x1 x2 ; hw3
ð21Þ
where h and w are the beam height and width, respectively. Since the normal stresses are much higher than the shear
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Fig. 2. Schematic of the cantilever compliant beam mechanism for the microstructure design case study.
stresses in a slender beam, we have ignored the shear stresses in this case study. The application of Hill’s anisotropic yield criterion (Eq. (13)) requires jr11 j 6 1: ry1
ð22Þ
The maximum deflection in the cantilever beam, at the time of the initiation of plastic strain, is expressed as 2 L2 d ¼ ry1 S 1 1 1 1 ; 3 w
ð23Þ
where L is the length of the beam. For a fixed beam geometry, the maximum deflection that can be attained without initiating plastic strain is therefore dependent only on the macroscale material properties S1 1 1 1 and ry1. In the case study presented here, the beam is assumed to have a square cross section with b = w=18 mm, and L = 180 mm. 4.2. Microstructure design The microstructural design variable for this case study has been selected to be the ODF (or texture) in the beam. Since the sample is made from a hexagonal metal and the macroscale properties are expected to exhibit orthorhombic symmetry, the space of relevant ODFs for this case study is the class of hexagonal-orthorhombic ODFs. The relevant fundamental zone has been described in Eq. (2) [11,13]. The hexagonal-orthorhombic texture hull in the first five dimensions of the F mn l space was depicted in a recent paper [11]. However, we are interested here in the 57 dimensions that have been deemed relevant to defining the plastic properties [11] of interest to the present design problem. As explained earlier, because of the complex geometry of the texture hull in the F mn l space, we will explore the texture
hull here in the ap space. As described earlier, we have selected a set of 339 principal lattice orientations spread uniformly on the boundaries of the FZ. These orientations would correspond to the vertices of the texture hull and therefore, the convex combinations of their respective ODFs should adequately capture the entire texture hull. In all of the optimization problems described here, a texture comprising equal volume fractions of the principal orientations was taken as the starting point (or initial guess). The material for the case study is assumed to be polycrystalline high-purity a-Ti. The Fourier coefficients relevant to the elastic and plastic properties of this material were established in a recent study [11] and used here. 4.3. Results and discussion A closure depicting the complete set of feasible combinations of S1 1 1 1 and ry1 for all theoretically feasible hexagonal-orthorhombic textures in the selected a-Ti metal was obtained using the procedures described earlier and plotted in Fig. 3. Performance contours for maximum deflection (based on Eq. (23)) have been superimposed on this figure. The maximum deflection attainable in the compliant beam with isotropic properties (corresponding to a random texture) is 2.41 mm, while the expansion of the design space to the set of hexagonal-orthorhombic textures provides performances ranging from 1.01 mm to 3.31 mm. These results are summarized in Table 1. The best performance represents a 37% improvement over that of the isotropic solution. It is just as important to note that ignoring the inherent texture in the sample can result in extremely poor performance of the component (58% reduction in performance compared to the isotropic solution). The best performance in this design case study corresponded to a yield strength of 322.6 MPa (close to the maximum yield strength possible in the selected material
T. Fast et al. / Computational Materials Science 43 (2008) 374–383
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Fig. 3. The relevant property closure for a cantilever compliant beam made of high-purity polycrystalline a-Ti with hexagonal-orthorhombic textures. The textures predicted to provide the best and the worst performances for this case study are also shown.
Table 1 Summary of the MSD results for the design of the compliant beam 3
Case Deflection (mm) S 1111 10 ry1 (MPa) GPa Best Isotropic Worst
3.31 2.41 1.01
8.6 8.9 9.58
322.6 225.0 87.6
system, which was 329.23 MPa) and a compliance of 0.0086 GPa1 (which is significantly lower than the maximum possible compliance of 0.0096 GPa1). This is because the combination of the highest yield point and the highest compliance is not feasible in any one texture. The best feasible performance resulted from a trade-off between yield strength and compliance. It is also worth noting that the worst performance corresponded to the lowest yield strength, in spite of the fact that it exhibited the highest compliance. Clearly, the value of the yield strength dominated this design. The RD direction in the pole figures shown in Fig. 3 corresponds to the beam axis (x1-axis in Fig. 2). The best performance was observed to correspond to a texture with the crystal (0 0 0 1) planes inclined at a small angle to the RD axis, while the worst performance corresponded to a texture with the (0 0 0 1) planes inclined at about 90° to the RD axis. This is consistent with the results described above, because the yield strength of a titanium single crystal is expected to decrease significantly as the (0 0 0 1) plane is tilted away from the loading direction.
5. MSD case study: rotating disk Flywheel energy storage (FES) systems efficiently convert kinetic energy into useful power via an electric generator. The main functional component of an FES system is the rotating disc spinning around a shaft supported by magnetic bearings to reduce friction. High strength materials are essential to enable high rotational speeds needed to maximize the efficiency of the FES system. Typical FES systems comprise of metallic flywheels operating around 4000 rpm [24]. The goal in this study is to identify the texture that will produce the maximum energy output in the flywheel without initiating plastic strain in the component. 5.1. Mechanical solution The rotating disc in the FES system is treated as a transversely isotropic thin disc with the following radial and tangential stresses (in cylindrical coordinates r, h, and z): 2 4 ! 2 4 a b b a 1 rrr ¼ qx2 ð3 mÞ þ ; ð24Þ 8 4 r2 2 !! a b2 1 b4 a4 2 2 þ ; ð25Þ rhh ¼ qx r þ ð3 mÞ 8 4 r2 where a and b are the inner and outer radii (set equal to 50 mm and 120 mm, respectively), q is the density (taken as 4.51 g/cc for titanium), and x is the angular velocity.
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The Poisson ratio, m, is defined in the plane of the radial and tangential components as m¼1
S rrhh : S rrrr
ð26Þ
The cylindrical geometry and the loading conditions demand transverse isotropy in macroscale properties. Hill’s anisotropic yield criterion (Eq. (10)) for this symmetry and loading condition can be expressed as 2
ðrrr rhh Þ rrr rhh þ 2 ¼ 1; 2 ryr ryz
ð27Þ
where ryr and ryz denote the yield stresses in the radial and thickness directions, respectively. By substituting the stresses in Eqs. (24) and (25) into the yield criterion Eq. (27), the angular velocity (the variable governing kinetic energy) at the initiation of plastic yield can be solved from 00 !2 ð3 þ mÞ b4 a4 4r4 r2yr 1 ¼ Max@@ qx2 4r2 r2yr ! 2 4 b a4 ð3 þ m Þ a b2 þ þ r2yz 8r2 4 4 !!!!12 2 2 4 b a ð 3 þ m Þ a b r2 þ r2yz 4 8r2 ! r 2 ½a; b :
ð28Þ
Note that, in the isotropic solution to this problem, the plastic yielding always initiates along the inner surface of the disc. However, with the introduction of anisotropic constitutive behavior, the location of the initiation of plastic yielding can occur anywhere in the component. Note also that only three material properties, namely m, ryr, and ryz influence the performance of the flywheel. The values of these properties are dependent on the texture in the sample. 5.2. Microstructure design The requirement of transverse isotropy in mechanical properties of the flywheel restricts the attention to hexagonal-transversely isotropic ODFs. The corresponding fundamental zone can be expressed as [16] n p po FZ ¼ ðU; u2 Þ0 6 U < ; 0 6 u2 < : ð29Þ 2 3 Because of the higher symmetry compared to the orthorhombic case described earlier, the relevant Fourier subspace controlling the elastic and plastic yield properties is significantly smaller and involves only 12 dimensions. It was also observed that only 50 principal crystal orientations were needed to describe this reduced Fourier subspace. 5.3. Results and discussion Fig. 4 depicts the relevant property closure for this case study involving three macroscale properties, namely of ryr,
Fig. 4. Property closure relevant to the performance of a high-purity polycrystalline a-titanium flywheel with hexagonal-transversely isotropic texture. Also shown are the pole figures corresponding to the best and worst performance.
T. Fast et al. / Computational Materials Science 43 (2008) 374–383 Table 2 Predicted angular velocities and corresponding properties of a high-purity polycrystalline a-titanium flywheel with hexagonal-transversely isotropic textures
kJ Case Specific energy kg ryz (MPa) m ryr (MPa) Best Isotropic Worst
25.6 24.1 9.99
.35 .35 .33
266.5 250.7 105.0
275.5 250.7 306.4
383
lar to those accomplished in our prior case studies on fcc metals. This we attribute to the fact that the elastic anisotropy exhibited by a-Ti is substantially lower compared to that exhibited by the fcc metals we investigated in our prior case studies [9]. This comparison reveals that anisotropy of both elastic and plastic properties plays an important role in the designs. Acknowledgement
ryz, and m. To the best of our knowledge, this is the first three-dimensional property closure obtained by the MSD methodology. This property closure is highly non-convex, and therefore, a precise geometrical description of its boundary would be quite challenging. Table 2 summarizes the best and worst possible performances predicted for the rotating disk using the MSD methodology (for the selected material and the selected homogenization theories). For comparison, the performance of the random texture (corresponding to design with isotropic properties) is also presented. An 8% improvement in performance was noted for the best texture compared to the random texture. On the other hand, it was noted that a poorly selected texture could degrade the performance by as much as 58%. It is also noted that the best performance required high values of both the in-plane and the out-ofplane yield strengths (i.e. ryr and ryz, respectively), along with a higher value of Poisson’s ratio. The lowest predicted performance corresponded to the combination of the lowest Poisson’s ratio, low radial yield strength (ryr), and high through thickness yield strength (ryz). 6. Conclusions A new approach has been presented to facilitate MSD solutions for structural components produced from hexagonal metals. This approach utilizes the volume fractions of the principal orientations in the Fourier subspace of interest in defining the design space. Consequently, the design space can be mathematically expressed as a set of highly simplified linear equality and inequality constraints. This simplification offers tremendous computational advantages in the implementation of the MSD framework. The benefits of the new approach were successfully demonstrated with two specific design case studies involving different assumptions of symmetry at the sample scale. In both case studies presented, it is seen that the overall performance is strongly influenced by the crystallographic texture in the sample. Although the hexagonal a-Ti investigated in the present study exhibited a substantially higher anisotropy in its plastic properties compared to the fcc metals studied in our prior design case studies [7,9,13,14], it was observed that the improvements in the performance were relatively simi-
Financial support for this work was provided by the Army Research Office, Proposal No. 46886 MS, Dr. David Stepp, Program Director. References [1] B.L. Adams, X. Gao, S.R. Kalidindi, Acta Materialia 53 (2005) 3563– 3577. [2] B.L. Adams, T. Olson, Progress in Materials Science 43 (1998) 1–88. [3] S. Torquato, Random Heterogeneous Materials, Springer-Verlag, New York, 2002. [4] W.F. Brown, Journal of Chemical Physics 23 (1955) 1514–1517. [5] S.R. Kalidindi, M. Binci, D. Fullwood, B.L. Adams, Acta Materialia 54 (2006) 3117–3126. [6] S.R. Kalidindi, J.R. Houskamp, Journal of Composite Materials 41 (2007) 909–930. [7] B.L. Adams, A. Henrie, B. Henrie, M. Lyon, S.R. Kalidindi, H. Garmestani, Journal of the Mechanics and Physics of Solids 49 (2001) 1639–1663. [8] S.R. Kalidindi, J.R. Houskamp, M. Lyons, B.L. Adams, International Journal of Plasticity 20 (2004) 1561–1575. [9] M. Knezevic, S.R. Kalidindi, Computational Materials Science 39 (2007) 643–648. [10] G. Proust, S.R. Kalidindi, Journal of the Mechanics and Physics of Solids 54 (2006) 1744–1762. [11] X. Wu, G. Proust, M. Knezevic, S.R. Kalidindi, Acta Materialia 55 (2007) 2729–2737. [12] G. Saheli, H. Garmestani, B.L. Adams, Journal of Computer-Aided Materials Design 11 (2004) 103–115. [13] D.S. Sintay, B.L. Adams, in: IDETC/CIE, 31st Design automation conference, Long Beach, California, USA, 2005. [14] J.R. Houskamp, G. Proust, S.R. Kalidindi, International Journal of Multiscale Computational Engineering 5 (2007) 261–272. [15] B.L. Adams, M. Lyon, B. Henrie, International Journal of Plasticity 20 (2004) 1577–1602. [16] H.-J. Bunge, Texture Analysis in Materials Science, Mathematical Methods, Cuvillier Verlag, Go¨ttingen, 1993. [17] R. Hill, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 193 (1948) 281–297. [18] G.I. Taylor, Journal of the Institute of Metals 62 (1938) 307–324. [19] X. Wu, S.R. Kalidindi, C. Necker, A.A. Salem, Acta Materialia 55 (2007) 423–432. [20] M. Lyon, B.L. Adams, Journal of the Mechanics and Physics of Solids 52 (2004) 2569–2586. [21] Mathworks, Matlab, The Mathworks, Inc., 2007. [22] A. Midha, T.W. Norton, L.L. Howell, Journal of Mechanical Design – Transactions of the ASME (1994) 270–279. [23] S.G. Lekhnitskii, Anisotropic Plates, Gordon and Breach, 1968, p. 534. [24] S.M. Arnold, A.F. Saleeb, N.R. Al-Zoubi, NASA/TM (2001210578).
Computational Materials Science 43 (2008) 374–383 www.elsevier.com/locate/commatsci
Application of microstructure sensitive design to structural components produced from hexagonal polycrystalline metals Tony Fast, Marko Knezevic, Surya R. Kalidindi * Department of Materials Science and Engineering, Drexel University, Philadelphia, PA 191z04, United States Received 24 August 2007; received in revised form 28 November 2007; accepted 9 December 2007 Available online 28 January 2008
Abstract In this paper, we present the first successful design case studies in the application of microstructure sensitive design (MSD) methodology to optimize performance of structural components made from polycrystalline metals with hexagonal close-packed (hcp) crystal lattices. It is demonstrated that the underlying spectral framework of the MSD methodology facilitates an efficient consideration of the complete set of crystallographic textures in the design optimization. In order to accomplish this task a number of important enhancements had to be introduced to the MSD framework. The most significant enhancement is in the mathematical description of the design space, i.e. the texture hull. The advantages of the new approach described in this paper are illustrated with two specific design case studies involving different assumptions of symmetry at the sample scale. In both case studies presented, it is seen that the overall performance is strongly influenced by the crystallographic texture in the sample. Furthermore, the relevant property closures and performance maps accounting for the complete set of textures are also depicted. Published by Elsevier B.V. PACS: 62.20.x; 62.20.Dc; 62.20.Fe; 61.50.Ah Keywords: Microstructure; Anisotropy; Property closures; Elasticity; Plasticity; Spectral methods
1. Introduction The current practice in engineering design does not pay adequate attention to the internal structure of the material as a continuous design variable. The design effort is often focused on the optimization of the geometric parameters using robust numerical simulations tools, while material selection is typically relegated to a relatively small database. Furthermore, material properties are usually assumed to be isotropic, and this significantly reduces the design space. Since the majority of commercially available metals used in structural applications are polycrystalline and often possess a non-random distribution of crystal lattice orientations (as a consequence of complex thermo-mechanical loading history experienced in their manufacture), they should be expected to exhibit anisotropic properties. *
Corresponding author. Tel.: +1 215 895 1311; fax: +1 215 896 6760. E-mail address: [email protected] (S.R. Kalidindi).
0927-0256/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.commatsci.2007.12.002
In recent years, a new mathematical framework called microstructure sensitive design (MSD) has been developed to facilitate the rigorous consideration of microstructure as a continuous variable in engineering design and optimization. At the core of MSD lie rigorous quantitative descriptions of the relevant statistics of the local state distributions (at different levels of details classified by n-point statistics [1–5]) in the microstructure. Salient features of MSD include: (i) construction of a microstructure hull [6–8] that represents the complete set of theoretically feasible local state distributions, and (ii) delineation of property closures [9–11] that identify the complete set of theoretically feasible combinations of macroscale effective properties of interest in a given application (for a selected homogenization theory). The primary advantages of the MSD approach lie in its (a) consideration of anisotropy of the properties at the local length scales, (b) exploration of the complete set of relevant microstructures (due to the use of microstructure hulls and property closures) leading to global optima,
T. Fast et al. / Computational Materials Science 43 (2008) 374–383
and (c) invertibility of the microstructure–property relationships (due to the use of spectral methods in representing these linkages). As a result, MSD facilitates the rigorous consideration of the microstructure as a continuous design variable in optimization of the mechanical performance of the given structural component. The main focus of MSD has, thus far, been on the crystallographic texture (also called orientation distribution function or the ODF) in polycrystalline metals and the macroscale elastic–plastic properties that are strongly influenced by this specific microstructural detail. Limited explorations have also been conducted into compositional variations in microstructures [12] and into fiber reinforced composites [6]. In cubic polycrystalline metals, the MSD framework has been successfully applied to a few design case studies. These have included maximizing the deflection in a compliant beam [7], maximizing the in-plane load carrying capacity of a thin plate with a central circular hole [8], and minimizing the elastic driving force for crack extension in rotating disks [13] and internally pressurized thin-walled vessels [14]. Property closures were also produced for a broad range of combinations of macroscale effective elastic and plastic properties in cubic metals [9,10,14] and for a limited number of hexagonal metals [11]. In this paper, we present the first successful application of the MSD methodology to design case studies involving polycrystalline hexagonal close-packed (hcp) metals. The primary focus here continues to be on the crystallographic texture in the sample. In order to accomplish this task a number of important changes had to be introduced to the MSD framework. The most significant change deals with the mathematical description of the design space, i.e. the texture hull. The application of the MSD framework to hcp polycrystals required the consideration of a significantly larger number of dimensions of the texture hull, when compared to the previous case studies that involved cubic polycrystals. The much larger number of dimensions (and the corresponding larger number of design variables) demanded the development of a new approach to the problem. This new approach is described in this paper, and is illustrated with two design case studies involving different assumptions of symmetry at the sample scale. 2. Microstructure sensitive design framework MSD [6–8,13,15] starts with a statistical description of the relevant details of the microstructure. In this paper, our interest will be limited to 1-point statistics of lattice orientation (i.e. ODF) in hcp polycrystals. The ODF, denoted as f(g), reflects the normalized probability density associated with the occurrence of the crystallographic orientation g in the sample [16] as Z Vg f ðgÞdg ¼ ; f ðgÞdg ¼ 1: ð1Þ V FZ In Eq. (1), V denotes the total sample volume and Vg is the sum of all sub-volume elements in the sample that are asso-
375
ciated with a lattice orientation that lies within an incremental invariant measure, dg, of the orientation of interest, g. The lattice orientation, g, is usually defined by a set of three angles called the Bunge–Euler angles [16], i.e. g = (u1, U, u2). The Bunge–Euler angles define a specific sequence of three rotations that would bring the crystal and sample reference frames into coincidence. The local properties depend strongly on the local crystallographic orientation, and therefore, the overall behavior of the material is strongly influenced by the distribution of the crystallographic orientations inside the polycrystalline material. In Eq. (1), FZ denotes the fundamental zone of orientations, and represents the local state space describing the set of distinct orientations relevant to a selected class of textures [16]. As an example, the FZ for hexagonal-orthorhombic1 textures is described by n p p po FZ ¼ g ¼ ðu1 ; U; u2 Þ0 6 u1 6 ; 0 6 U 6 ; 0 6 u2 6 : 2 2 3 ð2Þ MSD employs efficient spectral representations of the microstructure distribution functions. For example, the ODF can be expressed in a Fourier series using symmetrized generalized spherical harmonics (GSH), T lm l ðgÞ, as f ðgÞ ¼
M ð lÞ X N ðl Þ 1 X X
lm F lm l T l ðgÞ:
ð3Þ
l¼0 l¼1 m¼1
Eq. (3) facilitates visualization of ODF as a point in Fourier space whose coordinates are given by F lm l . Recognition of the fact that ODF provides information only about the volume fraction of the various orientations present in the polycrystalline sample permits an alternate mathematical description as X X f ðg Þ ¼ ak d g gk ; 0 6 ak 6 1; ak ¼ 1: ð4Þ k
k k
The Dirac-delta function dðg g Þ in Eq. (4) represents the ODF of a single crystal of orientation gk, and ak denotes its volume fraction in the polycrystal. Let k F lm l define the Fourier coefficients of the single crystal ODFs. It is then possible to define a convex and compact texture hull [7], M, as ( ) X X lm lm k k lm k lm M ¼ F l F l ¼ ak F l ; F l 2 M ; ak P 0; ak ¼ 1 ; k k ð5Þ where k
M ¼
k
k lm F lm l Fl
k k 1 lm ¼ T g ; g 2 FZ : ð2l þ 1Þ l
ð6Þ
The bar on top of the GSH in Eq. (6) denotes the complex conjugate. It should be recognized that M represents the 1 The first symmetry in this standard notation used by the texture community refers to symmetry at the crystal level (resulting from the atomic arrangements in the crystal lattice) while the second refers to symmetry at the sample scale (resulting from processing history).
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complete set of all theoretically feasible ODFs, several of which have not yet been realized in practice or even have been targeted for manufacture by materials specialists. Next, we turn our attention to the homogenization theory to be adopted to arrive at the macroscale properties of the polycrystal for a given texture. The ODF described above constitutes a first-order description of the microstructure (also referred as 1-point statistics). Using this microstructure description, only the elementary bounds of the macroscale elastic and plastic properties can be evaluated. In this work, we have decided to employ the upper bound theories for all macroscale elastic and plastic properties. For example, the upper bounds on the diagonal components of the macroscale elastic compliance tensor, S*, can be expressed as [10] (no summation implied on repeated indices) S ijij ¼ hS ijij i;
ð7Þ
where h i denote an ensemble average (also equal to the volume average when the ergodic hypothesis is invoked). S is the local elastic compliance tensor in the sample reference frame that is defined using a coordinate transformation law for fourth-rank tensors as S ijkl ¼ gip gjq gkr gls S cpqrs ;
ð8Þ
where Sc is the local elastic compliance tensor in the local crystal reference frame, and gij are the components of the transformation matrix defined in terms of the Bunge–Euler angles [16]. Although, only the first-order description of the microstructure and elementary bounds are employed in this study, it is emphasized here that the MSD framework has the potential for addressing higher-order microstructure descriptions and advanced homogenization theories [1,3]. The selected homogenization theories are then cast in the same Fourier space that has been used to represent the microstructure. The S abcd ðgÞ functions can be represented in a Fourier series using GSH functions as S abcd ðgÞ ¼
N ðl Þ X M ðl Þ 4 X X
lm lm abcd S l T l ðgÞ;
ð9Þ
l¼0 l¼1 m¼1
where abcd S lm l are referred to as the elastic Fourier coefficients and the details of their computation have been reported in an earlier paper [11]. The volume averaged value is then computed by exploiting the orthogonality of the Fourier basis as I S abcd ¼ hS abcd i ¼ S abcd ðgÞf ðgÞdg ¼
N ðl Þ X M ðl Þ 4 X X l¼0 l¼1 m¼1
1 lm lm abcd S l F l : ð2l þ 1Þ
ð10Þ
Eqs. (6) and (10) embody one of the central features of MSD. They provide an efficient linkage between the complete set of feasible ODFs and the corresponding feasible combinations of macroscale elastic properties. Of particular significance is the fact that unlike the Fourier representation of the ODF, the representation for properties often
extends to only a finite number of terms in the Fourier expansion. As shown in Eqs. (9) and (10), in consideration of elastic properties, the only relevant Fourier coefficients are those that correspond to l 6 4. The main advantage of the spectral methods described herein lies in the fact that they are able to formulate highly efficient structure–property relationships. In many mechanical design problems, avoiding plastic deformation constitutes an important consideration. For this purpose, yield surfaces are often described in the sixdimensional stress space. A common formalism for describing anisotropic yield surfaces is the orthorhombic Hill’s yield surface description [17] ! 1 1 1 1 2 ðhr22 i hr33 iÞ þ 2 r2y2 r2y3 r2y1 ! 1 1 1 1 2 ðhr33 i hr11 iÞ þ þ 2 r2y3 r2y1 r2y2 ! 1 1 1 1 1 2 þ þ 2 2 ðhr11 i hr22 iÞ þ 2 s212 2 2 ry1 ry2 ry3 sy12 þ
1 2 1 s þ 2 s213 ¼ 1; s2y23 23 sy13
ð11Þ
where ry1, ry2, ry3 denote the macroscale tensile (or compressive) yield strengths and sy12, sy13, sy23 denote the macroscale shear yield strengths. In Eq. (11), hriji and hsiji denote the macroscale normal and shear stress components experienced by the material point under consideration. The six anisotropic material yield parameters in Eq. (11) can be estimated for a polycrystalline material using an extended version of the Taylor model [18], also commonly referred to as the Taylor-type crystal plasticity model. This model constitutes an upper bound and has been demonstrated [19] to provide reasonably accurate predictions of the macroscopic yield strengths for the polycrystalline a-Ti used in the case studies presented here. The Taylor-type crystal plasticity model assumes that each constituent single crystal experiences the same strain as the imposed strain at the macroscale, and computes the local stress in the crystal that allows the accommodation of the imposed plastic strain through slip on the multiple slip systems in the crystal. The volume averaged stress (over the constituent single crystals) then provides an estimate of the macroscale yield strength for the polycrystal. As an example, the following macroscopic velocity gradient is imposed on the polycrystal in evaluating ry1: 0 1 h_ei 0 0 B C hLi ¼ @ 0 qh_ei 0 ð12Þ A: 0
0
ð1 qÞh_ei
The parameter q is allowed to take any value between 0 and 1. Because the deformation imposed by Eq. (12) is isochoric, the resulting stress field is purely deviatoric. In order to calculate the tensile yield strength, the hydrostatic component is computed by establishing the value of q
T. Fast et al. / Computational Materials Science 43 (2008) 374–383
(denoted as q*) for which the averaged lateral stresses over the polycrystal are equal to each other and added to the diagonal elements of the averaged deviatoric stress: ry1 ¼ hr011 ðq Þi hr022 ðq Þi;
ð13Þ
where r0 denotes the volume averaged deviatoric stress tensor for the polycrystal. Fast computation of the yield strengths for polycrystals in the MSD framework entails the use of Fourier representations to describe the functional dependence of the local stresses in the constituent crystals on their orientations. The Fourier representations needed for evaluating the upper bound for ry1 are expressed as (no summation implied on repeated index i) [11] r0ii ðg; qÞ
¼
N ðl Þ X M ðl Þ l X X
lm ii lm y1 S l ðqÞT l ðg Þ;
ð14Þ
l¼0 l¼1 m¼1
r0ii ðqÞ
¼
N ðl Þ X M ðl Þ l X X
lm ii lm y1 S l ðqÞF l
l¼0 l¼1 m¼1
2l þ 1
;
ð15Þ
where the procedures for computing y1 ii S lm l ðqÞ for hcp crystals experiencing slip on prism, basal and pyramidal slip systems have been already been described in a recent paper [11]. In describing the plastic properties, we have also observed that we needed a lot more Fourier coefficients than those needed to describe elastic properties. For practical reasons, we truncated the series after a desirable accuracy has been attained, and this is reflected in Eqs. (14) and (15) by restricting the series to terms corresponding to l 6 l*. For example, for hexagonal-orthorhombic textures, it was observed that the maximum error in Eq. (15) was less than 3% when the series was truncated to 57 terms (corresponding to l* = 12) [11]. The desired accuracy in property predictions therefore dictates the number of Fourier dimensions that need to be explored by the MSD methodology for a selected design application. In previous work, we have demonstrated the power of spectral microstructure–property linkages described above in delineation of numerous property closures [9–11]. Property closures identify the complete set of theoretically feasible effective (homogenized) anisotropic property combinations in a given material system, and are also of interest in this paper. Two different examples of these closures corresponding to two selected case studies will be delineated in 2D and 3D, respectively. In this paper, our primary aim is to extend the MSD framework into design and optimization of structural components made from hcp polycrystals. This is addressed in the next section. 3. MSD for performance optimization The main challenge encountered in the application of the MSD methodology to mechanical design case studies is that the microstructure–property linkages need to be explored in fairly large dimensional Fourier spaces. In this work, in dealing with plastic properties of hexagonal poly-
377
crystals there is a need for consideration of a much higher dimensional Fourier space compared to our earlier work on the more symmetric cubic polycrystals. The increased demand in dimensionality is driven by two main reasons: (i) the Fourier representation of the ODF for hcp poylcrystals typically needs many more terms compared to the corresponding representations for cubic polycrystals, and (ii) the yield surfaces in hcp polycrystals are substantially more anisotropic compared to the yield surfaces for cubic polycrystals. For example, the design case studies reported by Houskamp et al. [8,14] explored the cubic-orthorhombic texture hull in 12 dimensions, whereas consideration of similar problems here in hexagonal-orthorhombic texture hulls needs exploration of 57 dimensions of the microstructure hull [11]. A number of different approaches have been employed in the past to integrate microstructure sensitive design into the engineering design framework. The main approaches of these prior studies are briefly reviewed below: 1. The initial design case studies discretized the microstructure hull (Eq. (5)) into bins and evaluated the performance in each bin (e.g. [8,13]). It was assumed that all microstructures in one bin could be assumed to exhibit the same performance for the given design application. In these examples, the Fourier space was restricted to only the low-order subspaces and consequently, the number of bins could be restricted to a reasonably small number. These approximations permitted solutions to simple case studies. As the dimensionality of the relevant Fourier space expands, this approach is bound to be computationally inefficient, if not completely infeasible. 2. Design case studies were explored using the property closure as the design space [12]. This approach has the inherent advantage of greatly reducing the design space by mapping a higher dimensional Fourier space into a reduced dimensional property space. This method is particularly attractive when the property space (also called the property closure) is compact and convex. However, when the property space is not convex, defining the boundary of the property closure becomes extremely challenging, especially in design case studies where the overall performance is governed by a large number of macroscale properties. It should be noted that the property closures for a vast number of design problems are likely to be non-convex. 3. In the most recent design case studies [7,14], performance optimization was explored using a generalized reduced gradient method while restricting the design space to the convex microstructure hull. In these examples, the microstructure hull was defined using an algorithm based on Gram–Schmidt orthonormalization. This approach also allowed the integration of the MSD methodology with the finite element methods used typically by the designers [13,14]. Although this is an extremely efficient algorithm for exploring optimized solutions, we found it very difficult to apply this
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algorithm to the case studies described in this paper because of the much larger number of Fourier dimensions involved. In the case studies described in this paper, we have taken a new approach. Instead of using the Fourier coefficients F lm l to define the design space, we found it more convenient to define the design space in terms of the ak (see Eq. (4)). With this choice the design space is simply expressed as X ak ¼ 1; ak P 0: ð16Þ k
Note that the constraints described in Eq. (16) can be implemented much more easily into any optimization search tool, compared to the constraints described in Eqs. (5) and (6). The new approach naturally leads to the question of how many distinct orientations need to be considered in the FZ (i.e. the range of k in Eq. (16)) for a given design problem. The definition of the ODF (Eq. (1)) and its Fourier representation (Eq. (3)) guarantee that the complete set of all theoretically feasible ODFs forms a compact convex region in the Fourier space [6–8]. As an example, the texture hull for hexagonal-orthorhombic textures is shown in Fig. 1 in the first three dimensions of the Fourier space. The crystal orientations corresponding to the vertices of the texture hull are called principal orientations in this paper. Also, as described earlier in Eqs. (10) and (15), different Fourier subspaces are relevant to different macroscale engineering properties of interest. The set of vertices of the Fourier subspace of interest are denoted by Mp in this work, where the
superscript p reminds us that this set corresponds to principal orientations. The texture hull in the selected subspace, e , can then be described as M ( ) X X lm lm lm lm p p p e ¼ F l F l ¼ ap F l ; F l 2 M ; ap P 0; ap ¼ 1 : M p p ð17Þ It should be noted that the main difference between Eqs. (4) and (8) is that M describes the complete infinite-dimene defines only a selected subsional texture hull, whereas M space of M that is relevant to a given design problem. Consequently, the size of Mp should be significantly smaller than the size of Mk. As an example, consider the FZ of the hexagonal-orthorhombic textures described in Eq. (2). Discretizing this FZ into 10° bins would result in 2590 lattice orientations (this would dictate the size of Mk). Our investigations have revealed that only the orientations on the boundary of the FZ can be principal orientations (i.e. only these orientations correspond to the vertices of the texture hulls). The same 10° binning of the boundary of the FZ produces only 339 distinct principal orientations. In other words, the size of the Mp would be at least one order of magnitude lower than the corresponding size of Mk. If we further restrict our interest to only the three dimensions of the texture hull shown in Fig. 1, we would need to consider only 15 principal orientations. As can be seen from this example, the use of principal orientations and changing the microstructure design variable from F lm l to ap, lead to major computational advantages in the MSD methodology. It should also be
Fig. 1. The hexagonal-orthorhombic texture hull projected in the three dimensions of the Fourier space that are common to the case studies presented in this paper. The vertices of the hull are associated with the set of principal orientations for this subspace, and are shown as black spheres in this figure.
T. Fast et al. / Computational Materials Science 43 (2008) 374–383
noted that this discovery was instrumental to our success in addressing the microstructure design case studies presented in this paper. It is important, at this stage, to reflect on some of the consequences of using ap as the microstructure design variable. One might be tempted to conclude erroneously that a prescribed set of ap corresponds to a unique ODF. The reader is cautioned that we are dealing here with truncated Fourier spaces where the mapping between F lm l and ap is not one-to-one. Indeed, many different realizations of ap can correspond to one set of truncated F lm l . A prescribed set of ap, however, corresponds to a single set of truncated F lm l coefficients, which in turn corresponds to a very large set of distinct ODFs. However, all of the distinct ODFs corresponding to the prescribed set of ap are expected to exhibit the same (or approximately similar) values of macroscale properties of interest defined by the selected Fourier subspace [1,10]. Another important consequence of the approach presented here that it naturally produces nonunique solutions (as expected and desired). The new approach described here of using ap (henceforth denoted as a for ease of notation) as the design variable can be conveniently applied to mechanical design problems as well as the delineation of property closures. The optimization problem for mechanical design involving components with statistically homogeneous microstructures can be formulated as Maximize OðqÞ;
379
An N-property closure is then delineated by seeking the maximum and minimum values possible for one of the properties of interest while constraining all other properties to values pre-selected within their respective ranges. As an example, let P 2 ; . . . ; P N represent a selected combination of values for all properties of interest except P1, where the values of each property are constrained to lie within their respective potential ranges, i.e. P i 2 Ri . A restricted feasible range for P1 can then be established as ( ) X R1 ¼ P 1 ðaÞP i ðaÞ ¼ P i 8i 6¼ 1; ap P 0; ap ¼ 1 : ð20Þ p If R1 is a non-empty set, it leads to the identification of e points on the boundary of the property e closure. Let P 1 de note the extrema of R1 . Then P 1 ; P 2 ; . . . ; P N lie on the boundary of the property closure sought. It should be noted here that for several selections of P 2 ; . . . ; P N , the corresponding R1 is indeed expected to be a null set. In the present study, all of the optimization problems formulated in Eqs. (18)–(20) were solved successfully using the fmincon function in Matlab’s Optimization Toolbox [21]. The fmincon function uses sequential quadratic programming, chooses subsequent variables via the line-search method, and implements the Broyden–Fletcher–Goldfarb– Shanno Quasi-Newton formula to define the Hessian [21]. Convergence is defined using first-order necessary conditions.
where
q ¼ ðP 1 ðaÞ; P 2 ðaÞ; . . . ; P N ðaÞÞ; X ap ¼ 1; subject to ap P 0;
4. MSD case study: compliant mechanism ð18Þ
p
where O denotes a objective function characterizing the performance of the mechanical component, and q is a set of relevant macroscale material properties (denoted as P i ðaÞ) influencing the performance. In particular, it is noted that the functional dependence in P i ðaÞ can be highly nonlinear (especially with plastic properties; see Eqs. (11)– (15)). In prior work, the property closures were obtained by formulating optimization problems corresponding to the points on the boundary of the closure [10,20]. However, these optimization problems were cast in the F lm l -space. For the reasons mentioned earlier, the idea of using the ap-space instead of the F mn l -space can prove very beneficial in delineating multi-property closures of interest in microstructure design problems. Briefly, one starts by identifying the potential range, Ri, of values for each property of interest, Pi, as " ( ) X Ri ¼ min P i ðaÞap P 0; ap ¼ 1 ; p¼1 ( )# X max P i ðaÞap P 0; ap ¼ 1 : ð19Þ p¼1
Compliant mechanisms are mechanical devices that gain movement from parts that flex, bend or have ‘‘springiness” to them [22]. These mechanisms with built-in flexibility are practical since they are simple. They also eliminate the need for multiple rigid parts, pin joints and add-on springs which can reduce design complexity and cost. Compliant mechanisms are found in sensors, gearboxes, valves, bicycle derailleurs, and various other mechanical designs. In this case study, we will seek the texture(s) that will maximize the deflection of the beam without initiating plastic deformation. 4.1. Mechanical solution The compliant mechanism is idealized here as a long slender cantilever beam whose macroscale elastic–plastic properties exhibit orthorhombic symmetry (presumably the processing options have been restricted to accomplish this). The stress field in the cantilever beam, with one end fixed to a rigid surface and the other end subjected to a point load P (see Fig. 2), is expressed as [23] r11 ¼
12P x1 x2 ; hw3
ð21Þ
where h and w are the beam height and width, respectively. Since the normal stresses are much higher than the shear
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Fig. 2. Schematic of the cantilever compliant beam mechanism for the microstructure design case study.
stresses in a slender beam, we have ignored the shear stresses in this case study. The application of Hill’s anisotropic yield criterion (Eq. (13)) requires jr11 j 6 1: ry1
ð22Þ
The maximum deflection in the cantilever beam, at the time of the initiation of plastic strain, is expressed as 2 L2 d ¼ ry1 S 1 1 1 1 ; 3 w
ð23Þ
where L is the length of the beam. For a fixed beam geometry, the maximum deflection that can be attained without initiating plastic strain is therefore dependent only on the macroscale material properties S1 1 1 1 and ry1. In the case study presented here, the beam is assumed to have a square cross section with b = w=18 mm, and L = 180 mm. 4.2. Microstructure design The microstructural design variable for this case study has been selected to be the ODF (or texture) in the beam. Since the sample is made from a hexagonal metal and the macroscale properties are expected to exhibit orthorhombic symmetry, the space of relevant ODFs for this case study is the class of hexagonal-orthorhombic ODFs. The relevant fundamental zone has been described in Eq. (2) [11,13]. The hexagonal-orthorhombic texture hull in the first five dimensions of the F mn l space was depicted in a recent paper [11]. However, we are interested here in the 57 dimensions that have been deemed relevant to defining the plastic properties [11] of interest to the present design problem. As explained earlier, because of the complex geometry of the texture hull in the F mn l space, we will explore the texture
hull here in the ap space. As described earlier, we have selected a set of 339 principal lattice orientations spread uniformly on the boundaries of the FZ. These orientations would correspond to the vertices of the texture hull and therefore, the convex combinations of their respective ODFs should adequately capture the entire texture hull. In all of the optimization problems described here, a texture comprising equal volume fractions of the principal orientations was taken as the starting point (or initial guess). The material for the case study is assumed to be polycrystalline high-purity a-Ti. The Fourier coefficients relevant to the elastic and plastic properties of this material were established in a recent study [11] and used here. 4.3. Results and discussion A closure depicting the complete set of feasible combinations of S1 1 1 1 and ry1 for all theoretically feasible hexagonal-orthorhombic textures in the selected a-Ti metal was obtained using the procedures described earlier and plotted in Fig. 3. Performance contours for maximum deflection (based on Eq. (23)) have been superimposed on this figure. The maximum deflection attainable in the compliant beam with isotropic properties (corresponding to a random texture) is 2.41 mm, while the expansion of the design space to the set of hexagonal-orthorhombic textures provides performances ranging from 1.01 mm to 3.31 mm. These results are summarized in Table 1. The best performance represents a 37% improvement over that of the isotropic solution. It is just as important to note that ignoring the inherent texture in the sample can result in extremely poor performance of the component (58% reduction in performance compared to the isotropic solution). The best performance in this design case study corresponded to a yield strength of 322.6 MPa (close to the maximum yield strength possible in the selected material
T. Fast et al. / Computational Materials Science 43 (2008) 374–383
381
Fig. 3. The relevant property closure for a cantilever compliant beam made of high-purity polycrystalline a-Ti with hexagonal-orthorhombic textures. The textures predicted to provide the best and the worst performances for this case study are also shown.
Table 1 Summary of the MSD results for the design of the compliant beam 3
Case Deflection (mm) S 1111 10 ry1 (MPa) GPa Best Isotropic Worst
3.31 2.41 1.01
8.6 8.9 9.58
322.6 225.0 87.6
system, which was 329.23 MPa) and a compliance of 0.0086 GPa1 (which is significantly lower than the maximum possible compliance of 0.0096 GPa1). This is because the combination of the highest yield point and the highest compliance is not feasible in any one texture. The best feasible performance resulted from a trade-off between yield strength and compliance. It is also worth noting that the worst performance corresponded to the lowest yield strength, in spite of the fact that it exhibited the highest compliance. Clearly, the value of the yield strength dominated this design. The RD direction in the pole figures shown in Fig. 3 corresponds to the beam axis (x1-axis in Fig. 2). The best performance was observed to correspond to a texture with the crystal (0 0 0 1) planes inclined at a small angle to the RD axis, while the worst performance corresponded to a texture with the (0 0 0 1) planes inclined at about 90° to the RD axis. This is consistent with the results described above, because the yield strength of a titanium single crystal is expected to decrease significantly as the (0 0 0 1) plane is tilted away from the loading direction.
5. MSD case study: rotating disk Flywheel energy storage (FES) systems efficiently convert kinetic energy into useful power via an electric generator. The main functional component of an FES system is the rotating disc spinning around a shaft supported by magnetic bearings to reduce friction. High strength materials are essential to enable high rotational speeds needed to maximize the efficiency of the FES system. Typical FES systems comprise of metallic flywheels operating around 4000 rpm [24]. The goal in this study is to identify the texture that will produce the maximum energy output in the flywheel without initiating plastic strain in the component. 5.1. Mechanical solution The rotating disc in the FES system is treated as a transversely isotropic thin disc with the following radial and tangential stresses (in cylindrical coordinates r, h, and z): 2 4 ! 2 4 a b b a 1 rrr ¼ qx2 ð3 mÞ þ ; ð24Þ 8 4 r2 2 !! a b2 1 b4 a4 2 2 þ ; ð25Þ rhh ¼ qx r þ ð3 mÞ 8 4 r2 where a and b are the inner and outer radii (set equal to 50 mm and 120 mm, respectively), q is the density (taken as 4.51 g/cc for titanium), and x is the angular velocity.
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The Poisson ratio, m, is defined in the plane of the radial and tangential components as m¼1
S rrhh : S rrrr
ð26Þ
The cylindrical geometry and the loading conditions demand transverse isotropy in macroscale properties. Hill’s anisotropic yield criterion (Eq. (10)) for this symmetry and loading condition can be expressed as 2
ðrrr rhh Þ rrr rhh þ 2 ¼ 1; 2 ryr ryz
ð27Þ
where ryr and ryz denote the yield stresses in the radial and thickness directions, respectively. By substituting the stresses in Eqs. (24) and (25) into the yield criterion Eq. (27), the angular velocity (the variable governing kinetic energy) at the initiation of plastic yield can be solved from 00 !2 ð3 þ mÞ b4 a4 4r4 r2yr 1 ¼ Max@@ qx2 4r2 r2yr ! 2 4 b a4 ð3 þ m Þ a b2 þ þ r2yz 8r2 4 4 !!!!12 2 2 4 b a ð 3 þ m Þ a b r2 þ r2yz 4 8r2 ! r 2 ½a; b :
ð28Þ
Note that, in the isotropic solution to this problem, the plastic yielding always initiates along the inner surface of the disc. However, with the introduction of anisotropic constitutive behavior, the location of the initiation of plastic yielding can occur anywhere in the component. Note also that only three material properties, namely m, ryr, and ryz influence the performance of the flywheel. The values of these properties are dependent on the texture in the sample. 5.2. Microstructure design The requirement of transverse isotropy in mechanical properties of the flywheel restricts the attention to hexagonal-transversely isotropic ODFs. The corresponding fundamental zone can be expressed as [16] n p po FZ ¼ ðU; u2 Þ0 6 U < ; 0 6 u2 < : ð29Þ 2 3 Because of the higher symmetry compared to the orthorhombic case described earlier, the relevant Fourier subspace controlling the elastic and plastic yield properties is significantly smaller and involves only 12 dimensions. It was also observed that only 50 principal crystal orientations were needed to describe this reduced Fourier subspace. 5.3. Results and discussion Fig. 4 depicts the relevant property closure for this case study involving three macroscale properties, namely of ryr,
Fig. 4. Property closure relevant to the performance of a high-purity polycrystalline a-titanium flywheel with hexagonal-transversely isotropic texture. Also shown are the pole figures corresponding to the best and worst performance.
T. Fast et al. / Computational Materials Science 43 (2008) 374–383 Table 2 Predicted angular velocities and corresponding properties of a high-purity polycrystalline a-titanium flywheel with hexagonal-transversely isotropic textures
kJ Case Specific energy kg ryz (MPa) m ryr (MPa) Best Isotropic Worst
25.6 24.1 9.99
.35 .35 .33
266.5 250.7 105.0
275.5 250.7 306.4
383
lar to those accomplished in our prior case studies on fcc metals. This we attribute to the fact that the elastic anisotropy exhibited by a-Ti is substantially lower compared to that exhibited by the fcc metals we investigated in our prior case studies [9]. This comparison reveals that anisotropy of both elastic and plastic properties plays an important role in the designs. Acknowledgement
ryz, and m. To the best of our knowledge, this is the first three-dimensional property closure obtained by the MSD methodology. This property closure is highly non-convex, and therefore, a precise geometrical description of its boundary would be quite challenging. Table 2 summarizes the best and worst possible performances predicted for the rotating disk using the MSD methodology (for the selected material and the selected homogenization theories). For comparison, the performance of the random texture (corresponding to design with isotropic properties) is also presented. An 8% improvement in performance was noted for the best texture compared to the random texture. On the other hand, it was noted that a poorly selected texture could degrade the performance by as much as 58%. It is also noted that the best performance required high values of both the in-plane and the out-ofplane yield strengths (i.e. ryr and ryz, respectively), along with a higher value of Poisson’s ratio. The lowest predicted performance corresponded to the combination of the lowest Poisson’s ratio, low radial yield strength (ryr), and high through thickness yield strength (ryz). 6. Conclusions A new approach has been presented to facilitate MSD solutions for structural components produced from hexagonal metals. This approach utilizes the volume fractions of the principal orientations in the Fourier subspace of interest in defining the design space. Consequently, the design space can be mathematically expressed as a set of highly simplified linear equality and inequality constraints. This simplification offers tremendous computational advantages in the implementation of the MSD framework. The benefits of the new approach were successfully demonstrated with two specific design case studies involving different assumptions of symmetry at the sample scale. In both case studies presented, it is seen that the overall performance is strongly influenced by the crystallographic texture in the sample. Although the hexagonal a-Ti investigated in the present study exhibited a substantially higher anisotropy in its plastic properties compared to the fcc metals studied in our prior design case studies [7,9,13,14], it was observed that the improvements in the performance were relatively simi-
Financial support for this work was provided by the Army Research Office, Proposal No. 46886 MS, Dr. David Stepp, Program Director. References [1] B.L. Adams, X. Gao, S.R. Kalidindi, Acta Materialia 53 (2005) 3563– 3577. [2] B.L. Adams, T. Olson, Progress in Materials Science 43 (1998) 1–88. [3] S. Torquato, Random Heterogeneous Materials, Springer-Verlag, New York, 2002. [4] W.F. Brown, Journal of Chemical Physics 23 (1955) 1514–1517. [5] S.R. Kalidindi, M. Binci, D. Fullwood, B.L. Adams, Acta Materialia 54 (2006) 3117–3126. [6] S.R. Kalidindi, J.R. Houskamp, Journal of Composite Materials 41 (2007) 909–930. [7] B.L. Adams, A. Henrie, B. Henrie, M. Lyon, S.R. Kalidindi, H. Garmestani, Journal of the Mechanics and Physics of Solids 49 (2001) 1639–1663. [8] S.R. Kalidindi, J.R. Houskamp, M. Lyons, B.L. Adams, International Journal of Plasticity 20 (2004) 1561–1575. [9] M. Knezevic, S.R. Kalidindi, Computational Materials Science 39 (2007) 643–648. [10] G. Proust, S.R. Kalidindi, Journal of the Mechanics and Physics of Solids 54 (2006) 1744–1762. [11] X. Wu, G. Proust, M. Knezevic, S.R. Kalidindi, Acta Materialia 55 (2007) 2729–2737. [12] G. Saheli, H. Garmestani, B.L. Adams, Journal of Computer-Aided Materials Design 11 (2004) 103–115. [13] D.S. Sintay, B.L. Adams, in: IDETC/CIE, 31st Design automation conference, Long Beach, California, USA, 2005. [14] J.R. Houskamp, G. Proust, S.R. Kalidindi, International Journal of Multiscale Computational Engineering 5 (2007) 261–272. [15] B.L. Adams, M. Lyon, B. Henrie, International Journal of Plasticity 20 (2004) 1577–1602. [16] H.-J. Bunge, Texture Analysis in Materials Science, Mathematical Methods, Cuvillier Verlag, Go¨ttingen, 1993. [17] R. Hill, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 193 (1948) 281–297. [18] G.I. Taylor, Journal of the Institute of Metals 62 (1938) 307–324. [19] X. Wu, S.R. Kalidindi, C. Necker, A.A. Salem, Acta Materialia 55 (2007) 423–432. [20] M. Lyon, B.L. Adams, Journal of the Mechanics and Physics of Solids 52 (2004) 2569–2586. [21] Mathworks, Matlab, The Mathworks, Inc., 2007. [22] A. Midha, T.W. Norton, L.L. Howell, Journal of Mechanical Design – Transactions of the ASME (1994) 270–279. [23] S.G. Lekhnitskii, Anisotropic Plates, Gordon and Breach, 1968, p. 534. [24] S.M. Arnold, A.F. Saleeb, N.R. Al-Zoubi, NASA/TM (2001210578).