The Plane Strain Young’s Modulus in Cubic Materials

The orientation dependence of the plane strain Young’s modulus, E˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{E}$\end{document}, of cubic materials has been analysed as a function of the direction along which a uniaxial stress is applied to a single crystal and the perpendicular direction in the single crystal along which the strain is constrained to be zero. The locus of E˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{E}$\end{document} in the plane perpendicular to the axis of uniaxial stress is shown to be a circle when this stress is applied along 〈111〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle111\rangle$\end{document}. For materials with anisotropy ratios A>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A > 1$\end{document}, global minima in E˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{E}$\end{document} occur when the stress is applied along 〈001〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle 001\rangle$\end{document} and when the strain along one of the two perpendicular 〈100〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle100\rangle $\end{document} directions is set to zero. Identical global maxima in E˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{E}$\end{document} are found when the stress is applied along two different families of 〈uuw〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle\mathit {uuw} \rangle$\end{document} directions and the direction of zero strain is along either a perpendicular 〈11¯0〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle1\bar{1}0\rangle$\end{document} or 〈ww2u‾〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle \mathit{ww} \overline{2u}\rangle$\end{document} direction. For materials with A<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A < 1$\end{document}, the global maxima in E˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{E}$\end{document} occur when the stress is applied along 〈001〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle001\rangle$\end{document} and when the strain along one of the two perpendicular 〈100〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle100\rangle$\end{document} directions is set to zero, and identical global minima are found when the stress is applied along two different families of 〈uuw〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle\mathit {uuw} \rangle$\end{document} directions and the direction of zero strain is along either a perpendicular 〈11¯0〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle1\bar{1}0\rangle$\end{document} or 〈ww2u‾〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle \mathit{ww} \overline{2u}\rangle$\end{document} direction.


Introduction
There are a number of physical situations in which it is convenient to invoke the concept of plane strain in elasticity problems. In isotropic elasticity, plane strain problems are defined in terms of plane strain displacements u 1 and u 2 within an x 1 -x 2 plane being functions only of the Cartesian coordinates x 1 and x 2 , with the deformation u 3 parallel to the direction x 3 perpendicular to both x 1 and x 2 being set to zero [1]. Examples of such physical situations given by Timoshenko and Goodier include tunnels and retaining walls with lateral pressure, as in the walls of dams [1]. Other examples where plane strain conditions are invoked are in elastic contact problems where each solid can be considered to be an elastic half-space [2], the analysis of the bending of relatively wide beams, in which the transverse bending is restricted within the central portion of the beam [3,4] and the plane strain elastic compression of polycrystalline metals prior to plastic deformation, such as in cold rolling or forging operations. If, in isotropic elasticity, x 1 , x 2 and x 3 define the axes of the principal stresses σ 11 , σ 22 and σ 33 in such situations, and if furthermore σ 22 = 0, then the ratio σ 11 /ε 11 of the applied stress to the induced elastic strain parallel to the applied stress is simply E/(1 − ν 2 ), where E is Young's modulus and ν is Poisson's ratio [5]. This ratio is often described as the plane strain Young's modulus [6].
For an anisotropic material, Ting [7] shows that the situation is more complex than for isotropic materials, so that u 3 and the shear strains ε 13 and ε 23 do not have to be zero. However, it is still the case that the tensile strain ε 33 is zero for a condition of a generalised plane strain deformation in an anisotropic material. Therefore, as Hopcroft et al. recognise in their discussion of the plane strain Young's modulus for single crystal silicon in connection with the small deflections of short, relatively wide beams [4], the plane strain Young's modulus for a situation where σ 22 = 0 is a function of the orientation of the direction x 1 along which the stress is applied in the x 1 -x 2 plane and the perpendicular direction x 3 along which ε 33 is zero. Hence, the plane strain Young's modulus for silicon is orientation dependent, and therefore potentially an important consideration in the design of micromechanical systems using silicon.
Interestingly, while the orientation dependence of the Young's modulus of anisotropic crystalline materials is well known [8] and the orientation dependence and extrema of Poisson's ratio in particular as function of orientation has been a subject of recent interest [9,10], not least because of the concept of auxeticity [11], the specifics of the orientation dependence of the plane strain Young's modulus have not been addressed, even in what might be expected to be the relatively straightforward case of cubic materials such as silicon. It is the purpose of this paper to consider this orientation dependence and to identify extrema in the plane strain Young's modulus for cubic crystals as a function of the three independent elastic constants c 11 , c 12 and c 44 in contracted Voigt notation (see the Appendix), which are themselves subject to the constraints that c 44 > 0, c 11 > |c 12 | and c 11 + 2c 12 > 0, following from the condition that the strain energy of a crystal must be positive [7,8]. As Nye notes [8], the three unique elastic compliance constants s 11 , s 12 and s 44 for cubic crystals are subject to identical constraints.
The paper is organised as follows. A statement of the problem of determining the plane strain Young's modulus for a plane of a general anisotropic material subject to a specific direction within that plane not being strained is given in Sect. 2 before attention is focused on the general form of the formula relevant to cubic materials. The application of this formula to directions along which a stress is applied lying in a standard stereographic triangle, or directions related to these by symmetry, is considered in detail in Sect. 3. As a result of this, extrema are identified, both within planes perpendicular to the direction of applied stress, and globally for combinations of directions of applied stress and specific directions perpendicular to this direction not being strained as a function of c 11 , c 12

General Tensor Transformation Relations for Plane Strain Young's Modulus
Under the assumption that elastic conditions pertain, the symmetric stress and strain tensors, σ ij and ε kl respectively, are related to one another through the equations σ ij = C ij kl ε kl and ε ij = S ij kl σ kl , (1) in which C ij kl is the stiffness tensor and S ij kl is the compliance tensor, both tensors of the fourth rank, and where i, j , k and l take all values between 1 and 3 [5,7,8,12,13]. For an arbitrary rotation of axes from one axis system to another, fourth rank tensors, T ij kl , transform as for i, j , k, l, m, n, p and q all taking values from 1 to 3, where the a im are direction cosines specifying the angle between the ith axis of the 'new' axis system and the mth axis of the 'old' axis system. Both the 'old' axis system and the 'new' axis system in this formalism are defined by orthonormal axis systems. For materials with relatively high symmetry, 'old' axis systems are straightforward to define with respect to the crystal axes, whereas this is not the case with monoclinic or triclinic symmetry [13]. Defining axes 1 , 2 and 3 as the axes in the 'new' axis system parallel to the axes of principal stress, with the primes to denote that these three new axes are not all aligned along the crystal axes, it is convenient to choose axis 3 to be parallel to the direction along which the condition of plane strain is invoked, so that ε 33 = 0, and to choose 1 to be the direction along which a stress is applied, with the stress σ 22 along 2 set to zero. Hence, we have the equations ε 11 = S 1111 σ 11 + S 1133 σ 33 , ε 33 = S 3311 σ 11 + S 3333 σ 33 = 0. (3) Using the second of these equations to substitute for σ 33 in the first of these equations, it is apparent that the plane strain Young's modulusẼ parallel to the direction 1 is defined by the expressionẼ Using the contracted two suffix Voigt notation for the S ij kl (see the Appendix), this equation simplifies to the formẼ = s 33 s 11 s 33 − s 2 13 . (5) Equation (5) can be written in an equivalent form: Now, 1/s 11 is Young's modulus E along the direction 1 [7], while the ratios −s 13 /s 11 and −s 31 /s 33 both define Poisson ratios. Using the nomenclature used by Norris [9] we can define ν 13 to be the ratio −s 13 /s 11 , corresponding to the ratio of the negative strain along 3 to the positive strain along 1 when a tensile stress is applied along 1 ; similarly we can define ν 31 to be the ratio −s 31 /s 33 . Hence, dropping the primes for convenience, Eq. (6) can also be rewritten conveniently in the form where the '1' subscripts on E andẼ indicate the direction along which the stress is applied. It is evident that this equation is similar to, but not the same as, the form of the equation quoted by Hopcroft et al. [4] for what they termed the 'plate modulus' in their Eq. (11) when considering the elastic deflection of short, relatively wide beams of single crystals. For a general anisotropic material with known elastic constants, numerical manipulation of Eq. (5) enables the plane strain Young's modulus to be computed for general choices of crystallographic directions 1 and 3 . Fortunately, for cubic materials such as silicon, there are only three independent elastic constants. This helps to simplify the algebra inherent within Eq. (5) and enables the orientation dependence ofẼ and the identification of extrema to be readily determined analytically for this relatively straightforward situation. Defining the 'old' axis system to be the orthonormal axis system aligned with respect to the 100 directions of the cubic crystal, the S ij kl transform from axes 1, 2 and 3 to the axes 1 , 2 and 3 so that in general 4S ij kl = 4s 12 δ ij δ kl + s 44 (δ ik δ jl + δ il δ jk ) + (4s 11 − 4s 12 − 2s 44 )a iu a ju a ku a lu (8) [13]. Hence, where and where A is the anisotropy ratio [14][15][16]. When A > 1, J > 0, and when A < 1, J < 0. A cubic material is isotropic when J = 0 and A = 1; under these circumstances, it is evident from Eq. (6) thatẼ i.e., the analytical formula for the plane strain Young's modulus of an isotropic material is produced.

Directional Dependence of the Plane Strain Young's Modulus in Cubic Materials
The first 1 directions to be considered will be We need then only then examine 1 directions on the edges and within this standard stereographic triangle. However, it is convenient to examine the family of directions uuw between [001] and [110] and the family of directions 0vw between [001] and [010] before finally examining uvw forms of 1 within the standard stereographic triangle where 0 < u < v < w. Directions wuu between [011] and [111] on the side of the standard stereographic triangle are related by symmetry to uuw in terms of their elastic properties irrespective of the point group of the cubic material under consideration [8].

1 Parallel to [111]
Here, and because 3 is perpendicular to [111], it follows that for all possible directions 3 . Since the a ij are direction cosines, After some straightforward mathematical manipulation of Eqs. (13) and (14), it is evident that a 2 31 a 2 32 + a 2 32 a 2 33 + a 2 33 a 2 31 = 1 4 (15) for all possible directions 3 . Therefore, using Eq. (9), for this geometry, we have the compliance tensors and so using Eq. (5), it is apparent that irrespective of which direction perpendicular to [111] is constrained to have the condition ε 33 = 0. Hence, the locus of the magnitude ofẼ as a function of orientation of 3 within the (111) plane is a circle.
When 3 is along one of the two 100 directions, whereas when 3 is along one of the two 110 directions, From Eqs. (21) and (22), the ratio ofẼ when 3 is along 110 toẼ when 3 is along 100 in (001) is making use of Eq. (10) and using the relationships between s 11 , s 12 , c 11
Since both c 44 > 0 and (c 11 + 2c 12 ) > 0, it is immediately apparent that this equation cannot be satisfied for real values of θ when 1 is parallel to [110]. The condition given by Eq. (32) is satisfied if in which case the value ofẼ is simply 1/s 11 , i.e., identical to the Young's modulus in the 1 direction. This is quite a restrictive condition because it requires the strain along 3 to be zero when a stress is applied along 1 . Values of θ at which Eq. (32) is satisfied are necessarily minima in the magnitude ofẼ as a function of θ on the (110) plane. For there to be solutions for θ from Eq. (34), Since c 44 > 0 and c 11 + 2c 12 > 0, the left-hand inequality is equivalent to the condition Hence, if c 12 > 0, a necessary condition from this inequality is that we require A > 1 for there to be solutions of Eq. (34). Likewise, if c 12 < 0, a necessary condition from this inequality is that we require A < 1.
For c 12 > 0 and A > 1, the right-hand inequality is equivalent to the condition which reduces to the condition after some elementary rearrangement of Eq. (37). For c 12 < 0 and A < 1, the right-hand inequality is equivalent to the condition i.e., Almost all cubic materials satisfy the condition c 12 > 0. For these materials, there will only be a relatively small group with A > 1 for which Eq. (38) is also satisfied. Thus, for a moderately anisotropic material such as Si, where c 11 = 165.7 GPa, c 12 = 63.9 GPa, c 44 = 79.6 GPa and A = 1.56 (Table 1) (Table 1), real solutions for θ occur when cos θ = 0.9262, i.e., for θ = ±22.15 • and ±157.85 • within the angular range −180 • ≤ θ ≤ 180 • . Similarly, real solutions occur for Eq. (34) for even more anisotropic materials with A > 1 such as β-brass and the cubic close packed form of In -27 at% Tl just above the temperature at which it undergoes a martensitic transformation.
As Grimvall [18] and Brańka et al. [11] note, cubic materials with c 12 < 0 are very rare. However, there is strong experimental evidence that such materials do indeed exist in families of cubic rare earth chalcogenides in the intermediate valence state as a consequence of the special electronic structure of these materials and the possibility of electron transfer from 4f to 5d shells in the rare earth atoms as a consequence of applying a uniaxial pressure [19][20][21][22][23]. Electrons in the 5d shell screen the positively charged rare earth ion nucleus less well than electrons in the 4f shell, so that this will cause the whole electronic shell of the rare earth ion to shrink, in effect moving it from a divalent state towards a trivalent state. When a uniaxial pressure is applied to a single crystal of one of these rare earth chalcogenides, this change in valence of the rare earth ions caused by the uniaxial pressure has the effect of shrinking the crystal in all directions because of the smaller ion radius of the 3+ rare earth ion in comparison with that of the 2+ rare earth ion [24].
Of the materials and compositions quoted in Refs. [19][20][21][22][23] with c 12 < 0, there are five compositions at room temperature and pressure for which Eq. (40) is satisfied: Sm 0.58 Y 0.42 S, Tm 0.99 Se, Sm 0.9 La 0.1 S and Sm 0.85 Tm 0.15 S. Other compositions in these materials are completely auxetic, as Brańka et al. have discussed [11], and so for these compositions Eq. (32) cannot be satisfied. Table 1 Elastic constants c 11 , c 12 and c 44 (in GPa) and anisotropy ratio, A, for Sm 0.58 Y 0.42 S, Nb, Si, Cu, β-brass, and In -27 at% Tl at 290 K. Data for c 11 , c 12 and c 44 are from Table 1 of [21] for Sm 0.58 Y 0.42 S, Sm 0.58 Y 0.42 S is a material for which c 12 < 0 and A < 1. Hence, in Fig. 1(a), for this material, the circular locus ofẼ for 1 parallel to [111] (in black) lies entirely within the locus ofẼ for 1 parallel to [001] (in red). This is also a material for which Eq. (34) is satisfied when 1 is parallel to [011] at θ = 41.67 • . However, the effect is very subtle: when θ = 0 • , E = 82.08 GPa; when θ = 41.67 • ,Ẽ = 80.12 GPa, and when θ = 0 • ,Ẽ = 85.11 GPa. It is best appreciated by calibrating the locus in blue ofẼ for 1 parallel to [011] relative to that in black for 1 parallel to [111] and recognising that the blue and black loci are most close along an extended radius of the black circle when θ is close to the bisector of the horizontal and vertical axes. A further feature of the locus in red ofẼ for 1 parallel to [001] for Sm 0.58 Y 0.42 S is the relatively small deviation from unity of the ratio ofẼ when 3 is along 110 toẼ when 3 is along 100 : this ratio is 0.967. This small deviation can be rationalised straightforwardly because of the relatively small magnitude of c 12 . In other cubic materials with A < 0 and relatively small positive c 12 , the effect is also small: thus, for KCl using the values of c 11 , c 12 and c 44 in Table III of [25], the ratio is 0.989.
The effect of a relatively large and positive c 12 when A < 1 is demonstrated by the graphical representations in Fig. 1(b) for Nb. The locus ofẼ for 1 parallel to [110] now only has a single maximum θ = 0 • at 0°and a single minimum at 90°within the angular range  28) when θ = 90 • shows after some algebra that these two forms forẼ are formally identical when s 44 = 2(7s 11 + 9s 12 )(s 11 + 2s 12 ) or, equivalently,

1 Parallel to uuw Between [001] and [110]
The loci ofẼ for three specific choices of [001], [111] and [110] for 1 in Fig. 1  Hence, relative to the crystallographic set of orthonormal axes x 1 , x 2 and x 3 , we can define a new orthonormal right-hand axis set β 1 , β 2 and β 3 , so that the table of direction cosines between the axis set 1, 2 and 3 and this new axis set will be of the form with the primes in Eq. (44) denoting that these three new β axes are not aligned along the 100 crystal axes. Therefore, with respect to β 2 and β 3 , a direction of unit length making anticlockwise angles of (90 • + θ ) with β 2 and θ with β 3 along which there is a condition of plane strain will have direction cosines [a 31 , a 32 , a 33 ] = cos θ + sin θ cos ϕ Using Eq. (9), it is apparent for this geometry that we have the three compliance tensors For a specific value of ϕ, turning points in 1/Ẽ occur when Eq. (29) is zero, i.e., when where f (θ) = s 33 sin 2 ϕ − s 13 Thus, for example, when ϕ = 0 • , i.e., when 1 is along [001], these turning points occur at values of sin 2 θ = 1/2, i.e., along 100 directions. Since sin 2 θ cannot be greater than 1, it is evident that this equation also defines a limiting condition for ϕ above which the turning points determined by f (θ) = 0 disappear. This limiting value of ϕ, ϕ crit , is determined by the condition J sin 4 ϕ crit − (s 11 + s 12 + J ) sin 2 ϕ crit − s 12 = 0. (50) Thus, for example, for the highly anisotropic material In -27 at% Tl at 290 K, ϕ crit = 35.82 • , i.e., close to where 1 is along [112]. At this value of ϕ crit , the condition f (θ) = 0 has a solution of θ = 90 • , i.e., the solution is coincident with a known turning point from Eq. (47). Finally, for those cubic materials where the condition s 13 = 0 is satisfied, local minima inẼ also occur. Using Eq. (46), it is apparent that such local minima occur at values of θ for allowed values of ϕ for which .

1 Parallel to [0vw] Between [001] and [010]
For these directions, we can parameterise [a 11 , a 12 , a 13 ] as Relative to the crystallographic set of orthonormal axes x 1 , x 2 and x 3 , we can define a new orthonormal right-hand axis set β 1 , β 2 and β 3 , so that the table of direction cosines between the axis set 1, 2 and 3 and this new axis set will be of the form Therefore, with respect to β 2 and β 3 , a direction of unit length making anticlockwise angles of (90 • + θ ) with β 2 and θ with β 3 along which there is a condition of plane strain will have direction cosines For a specific value of ϕ, turning points in 1/Ẽ occur when Eq. (29) is zero, i.e., when where g(θ) = s 33 sin 2 2ϕ − s 13 sin 2 θ 4 − sin 2 2ϕ − 2 .
Both β-brass and Cu have (0vw) orientations at which s 13 = 0, as does Sm 0.58 Y 0.42 S (discussed for the symmetrically equivalent (011) plane in Sect. 3.4), but this is not so for Si and Nb considered in Fig. 5. For all four materials with A > 1, maxima inẼ occur on the (011) plane along 100 directions within the (0vw) family of planes and minima occur along the (001) plane along 110 directions; the reverse is true for the two materials Sm 0.58 Y 0.42 S and Nb for which A < 1. A subtlety which occurs for Sm 0.58 Y 0.42 S with c 12 < 0 is that as ϕ increases from zero, the turning point originally at θ = 45 • when [0vw] = [001], and which is a local minimum because A < 1, rotates towards θ = 90 • , rather than towards θ = 0 • .

Extremal Conditions forẼ
The formalism introduced by Norris [9] can be used straightforwardly to obtain three conditions for a stationary value ofẼ. Before analysing what happens to the locus ofẼ for the situation where 1 is a general direction [uvw] within the standard stereographic 001-011-111 stereographic triangle, it is worthwhile considering whether or not the extreme values ofẼ have already been identified by the analysis in Sects. 3.1-3.6.
Applying Norris's formalism and using the equality on page 798 of his paper, it is straightforward to show that in contracted Voigt notation (see Appendix) the three conditions that an extreme value ofẼ must fulfil simultaneously for a general single crystalline material of triclinic symmetry are: In practice, while it can happen that s 13 = 0 and s 13 + s 33 = 0 for some highly anisotropic materials, as has already been shown in Sects. 3.3-3.6, these conditions reduce to the three conditions that an extreme value ofẼ must fulfil simultaneously: For all other [0vw] orientations, it is not possible to make each of these six terms zero simultaneously, nor is it possible to satisfy Eqs. (71)-(73) otherwise, but it is of interest to note that the condition s 33 s 14 − s 13 s 34 = 0, is equivalent to the condition for a turning point expressed by g(θ) = 0 where g(θ) is given by Eq. (66). However, for a particular [0vw] for which Eq. (71) is satisfied for a particular combination of θ and ϕ, Eq. (72) and Eq. (73) will not be satisfied, i.e., it does not define a potential candidate for global extrema ofẼ.
The insight gained into the consideration of possible global extrema ofẼ when 1 is of the form [0vw] is very helpful in establishing possible global extrema ofẼ when 1 is of the form [uuw]. If we consider the table of direction cosines it is evident that s 14 = s 34 = s 15 = s 35 = 0, and so Eqs. (71) and (72)

1 Parallel to [uvw] Within the Standard Stereographic Triangle
It is evident from the considerations in Sect. 3.7 that it is actually very difficult to satisfy Eqs. As examples of the general principles for more general [uvw], numerical calculations have been undertaken for In -27 at% Tl at 290 K, as this shows most obviously the criteria which determine the local maxima and minima ofẼ for a particular (uvw). Loci of howẼ varies within the (156), (345) and (125) planes are shown in Fig. 6(a)-(c) respectively. For each of these graphical representations, the vertical axis has been chosen so that it is the irrational direction within (uvw) along whichẼ is an absolute maximum. Each graphical representation only has the symmetry of a diad at the origin; the perpendicular mirror planes evident on the graphical representations ofẼ for (0vw) and (uuw) are absent on these more general (uvw) representations.
The numerical calculations show that along this irrational direction the condition s 33 s 14 − s 13 s 34 = 0 is satisfied, i.e., Eq. (71). In Fig. 6(a) this direction is 0.46°clockwise relative to [421]; in Fig. 6(b) this direction is 0.38°anticlockwise relative to [430], and in Fig. 6(c) this direction is 2.52°anticlockwise relative to [551]. In Fig. 6(a) a second orientation for which this condition is also satisfied lies at a conventional anticlockwise rotation of orientation of 21.87°relative to the horizontal axis; this Summarising, within a general plane of the form (uvw), extrema forẼ occur where either (s 33 s 14 − s 13 s 34 ) = 0 or s 13 = 0. This statement applies irrespective of whether or not the pole of (uvw) is in the standard stereographic triangle. For most materials the condition s 13 = 0 is not satisfied-materials in the form of single crystals which satisfy this fairly restrictive condition have to be quite anisotropic in terms of their elastic properties. Finally, none of the extrema forẼ on general (uvw) planes are candidates for global extrema-it is now apparent from examining the loci ofẼ on general (uvw) planes that these global extrema inẼ arise on planes of the form {001} and {uuw}, as discussed in Sect. 3.7.

Discussion and Conclusions
Although the concept of plane strain is useful in a number of physically significant situations within elasticity, in particular elastic contact problems and the bending of short, relatively wide beams, it is perhaps somewhat surprising that the consequences of elastic anisotropy when dealing with cubic single crystals subjected to plane strain are not already well established in the literature.
Analysis of how the plane strain Young's modulusẼ varies as a function of direction 1 along which the uniaxial stress is applied and the perpendicular direction 3 along which conditions of plane strain are maintained during the elastic deformation process shows that there are a relatively small number of simple guiding principles. For materials with A > 1, minima inẼ occur when 1 is along a 001 direction and 3 along a perpendicular 100 direction; maxima inẼ occur when 1 is along a uuw direction and 3 is either along a perpendicular 110 direction or a ww2u direction. Results for materials with A < 1 are the opposite of those for A > 1.
Thus, to take the specific example of silicon, the minimum value ofẼ is 144.7 GPa when the applied stress is along 001 and plane strain conditions are maintained on {100} planes containing the direction of applied stress. From Eqs. (53), (54), (59) and (60) (17)), where 3 need only be specified as being perpendicular to 1 because under these circumstances the locus ofẼ is a circle. Not surprisingly from a consideration of these data, the orientations of 1 at whichẼ is at a maximum for silicon are very close to [111]. The practical consequence of these calculations for silicon, a material widely available in single crystalline form and used for microelectromechanical systems, are that wide beams with 1 along a 111 direction can be up to 33% stiffer in their plane strain Young's modulus than beams where 1 is along 001 .
The other range of physical situations in which plane strain conditions are encountered when dealing with isotropic materials are contact problems. The contact problem for isotropic bodies originally solved by Hertz [26] is one well-known contact problem. This problem is relevant to the nanoindentation of isotropic materials when axisymmetric indenters are used. The sum of the displacements of the indenter and material being indented is proportional to the sum of the reciprocals of the plane strain Young's moduli of the two materials [1]. Knowing this, in nanoindentation, the contact stiffness, S c , measured on unloading can be shown to be proportional to the product of the square root of the projected contact area, A p , and what is termed the 'reduced modulus', E r : where E I T is the indentation modulus of the specimen and the subscripts s and i denotes 'specimen' and 'indenter' respectively, so that overall [27,28]. Work by Vlassak and Nix [29] has shown that a formula equivalent to Eq. (84) arises when considering the indentation of the more complex problem of anisotropic materials, but that the indentation modulus determined in this manner will depend on the actual indenter geometry, unless conditions of high symmetry pertain, e.g., when the contact area between the indenter and the material being indented is circular and when the indented material has a three-fold or four-fold axis parallel to the direction of loading. More recent work by Vlassak and co-workers [30] has enabled 'equivalent indentation moduli' to be determined for elastically anisotropic materials, but it is evident that these contact problems cease to be problems of plane strain when dealing with elastically anisotropic materials. Instead, results from this analysis are relevant to plane-strain compression tests of cubic single crystals. Plane-strain compression tests are usually carried out in the context of analysing metalworking processes such as cold rolling, in which plastic deformation is deliberately introduced [31]. However, such tests in the elastic regime on suitably prepared cubic single crystals would be expected to exhibit the variations in plane strain Young's modulus as a function of loading direction and direction under which plane strain conditions pertain that have been determined in this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Appendix: The Voigt Notation
In elasticity, the symmetric stress and strain tensors of the second rank, σ ij and ε kl respectively, are related to one another through the equations σ ij = C ij kl ε kl and ε ij = S ij kl σ kl , in which C ij kl is the stiffness tensor and S ij kl is the compliance tensor, both tensors of the fourth rank, and where i, j , k and l take all values between 1 and 3 [5,7,8,12,13]. The Voigt notation is a convenient contracted notation to express these tensor relations in matrix form. In this notation, the symmetrical stress tensor σ ij is contracted as follows: ⎛ where m and n take all values from 1 to 6 and where c mn and s mn are the contracted forms of the C ij kl and S ij kl respectively. In these contracted forms, the i and j values of 1, 2, 3, 4, 5, 6 correspond to the six pairs 11, 22, 33, 23, 31, 12 in the full tensor notation. This notation is able to be used because the stiffness and compliance tensors both satisfy the condition for a tensor T ij kl of the fourth rank.
While writing the C ij kl in terms of their corresponding c mn is straightforward, so that, for example, C 1233 → c 63 , C 2323 → c 44 , C 2332 → c 44 , etc., factors of 2 and 4 have to be introduced into the definition of s mn , so that where δ ij is the Kronecker delta. Thus, for example, s 13 = S 1133 , s 14 = 2S 1123 and s 46 = 4S 2312 . Since the energy stored in an elastically strained crystal depends on the strain attained, and not on the path by which the strained state is reached, it follows that c mn = c nm and s mn = s nm . In cubic crystals when the axis system is parallel to the crystal axes, there are only three independent elastic constants: c 11