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27-301Microstructure-Properties

L11: Tensors and Anisotropy, Part 4Profs. A. D. Rollett, M. De Graef

Microstructure Properties

ProcessingPerformance

Last modified: 25th Oct. ‘15

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Objective• The objective of this lecture is to provide a

mathematical framework for the description of properties, especially when they vary with direction.

• A basic property that occurs in almost applications is elasticity. Although elastic response is linear for all practical purposes, it is often anisotropic (composites, textured polycrystals etc.).

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Questions & Answers1. Why is it useful to rotate/transform the compliance tensor or matrix? Often we

need to compute the elastic modulus in some particular direction that is not [100] or [111]. Why do we compute the compliance rather than the stiffness in the 1-direction? This is subtle: we use compliance because one can impose a stress state that has only one non-zero component, from which we only need the strain component parallel to it. Poisson’s ratio tells us that imposing a strain in one direction automatically results in lateral strains (unless n=0), which means that it is not possible to have one and only one strain component contributing to a particular stress component.

2. How are the quantities in the equation for the rotated/transformed s11 related to the same equation with the Young’s moduli in the <100> and <111> directions? Comparison of the two formulae shows how to relate the three S values to the Youngs’ moduli in the two directions.

3. What is Zener’s anisotropy ratio? C' = (C11 - C12)/2; Zener’s ratio = C44/C’.

4. Which materials are most nearly isotropic? W at room temperature is almost isotropic and Al is not quite so close to being isotropic.

5. How do we apply the equations to calculate the variation in Young’s modulus between [100] and [110] in a cubic metals such as Cu? Direction cosines are the quantities that are needed to define the direction in relation to the crystal axes.

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Q&A - 26. What are the Lamé constants? These are the two constants l and G that are

needed for isotropic elasticity. What do they have to do with isotropic elasticity? G has its usual meaning of shear modulus, or C44: see the notes for how they relate to C11 and C12. How do they relate to Young’s modulus, bulk modulus and Poisson’s ratio? See the notes for the formulae.

7. How do we write the piezoelectric matrix for quartz? 6x3 matrix. What stimuli and responses do each coefficient in the “d” matrix relate? Stimulus is the electric field and the response is the strain. What are the “BT” and “AT” cuts of a quartz crystal? These are cuts that maximize the usefulness of the thickness shear mode of oscillation.

8. What equation describes the resonant frequency? See the notes. Why does temperature matter here? Temperature matters because one prefers to have a crystal that does not change its resonant frequency with temperature. Why does the density vary as the sum of 2a11+a33? This sum is the trace of the matrix for the coefficient of thermal expansion, i..e the variation in volume with change in T.

9. How does the angle q relate to the AT and BT cuts already described? This angle is a rotation of the normal to the surface of the crystal in the y-z plane (i.e. rotation about x). How do we set up the equation that tells us the variation in d66 with angle of cut? The Eq we need is that which describes the rate of change of resonant frequency with temperature.

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Rotated compliance (matrix)Given an orientation aij, we transform the compliance tensor, using cubic point group symmetry:

Writing this out in full for the 1111 component:

Re-writing this with vector-matrix notation gives:

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Rotated compliance (matrix)• This can be further simplified with the aid of the standard

relations between the direction cosines, aikajk = 1 for i=j; aikajk = 0 for i¹j, (aikajk = ij) to read as follows:

• By definition, the Young’s modulus in any direction is given by the reciprocal of the compliance, E = 1/S’11.

• Thus the second term on the RHS is zero for <100> directions and, for C44/C'>1, a maximum in <111> directions (conversely a minimum for C44/C'<1).

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Anisotropy in terms of moduli• Another way to write the above equation is to

insert the values for the Young's modulus in the soft and hard directions, assuming that the <100> are the most compliant direction(s). (Courtney uses a, b, and g in place of my a1, a2, and a3.) The advantage of this formula is that moduli in specific directions can be used directly.

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Cubic crystals: anisotropy factor• If one applies the symmetry elements of the

cubic system, it turns out that only three independent coefficients remain: C11, C12 and C44, (similar set for compliance). From these three, a useful combination of the first two is

C' = (C11 - C12)/2

• See Nye, Physical Properties of Crystals

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Zener’s anisotropy factor• C' = (C11 - C12)/2 turns out to be the stiffness

associated with a shear in a <110> direction on a {110} plane. In certain martensitic transformations, this modulus can approach zero which corresponds to a structural instability. Zener proposed a measure of elastic anisotropy based on the ratio C44/C'. This turns out to be a useful criterion for identifying materials that are elastically anisotropic.

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Anisotropy in cubic materials

• The following table shows that most cubic metals have

positive values of Zener's coefficient so that <100> is most compliant and <111> is most stiff, with the exceptions of V, Nb

and NaCl.

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Stiffness coefficients, cubics

Nb (niobium): beta1=17:60 (TPa)-1 , Bcub= 0.50. s11 = 6.56, s44 = 35.20, s12 = -2.29 (in (TPa)-1 ). Emin = 0.081, Emax = 0.152 GPa.

Units:1010 Pa

or10 GPa

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Example Problem

Should be E<111>= 18.89

[Courtney]

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For an elastically isotropic body, there are only 2 elastic moduli, known as the Lamé constants.

Lamé constants (isotropic elasticity)

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Young’s, Bulk moduli, Poisson’s ratio

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Engineering with the Piezoelectric Effect

• [Newnham, sections 12.8 and 13.10] The use of quartz as a resonant crystal for oscillator circuits with highly stable frequency depends strongly on the details of its properties.

• Although quartz is only weakly piezoelectric, other aspects of its properties provide the key, namely thermal stability.

• Most elastic stiffness coefficients have negative temperature coefficients, meaning that materials become less stiff with rising temperature. The c66 coefficient of quartz, however, is positive; Table 13.7. This offsets the effect of thermal expansion, which increases dimensions and decreases density. This is what makes it possible to have an oscillator that is insensitive to temperature changes.

d11 = 2.27; d14 =-0.67 pC/N http://en.wikipedia.org/wiki/Electromagnetic_acoustic_transducer

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Quartz Oscillator Crystals, contd.• Resonant frequency, f, for thickness (t)

shear mode, as a function of the rotation of axes to get c’66, where r is the density:

• AT and BT cut modes are thickness shear modes driven by the piezoelectric coefficient d’26:e’6 = d’26 E’2

A particular angle must be determined for the ideal cut to minimize the temperature dependence of the resonant frequency.

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Quartz Oscillator Crystals, contd.• Temperature dependence of the resonant frequency, f, for

thickness (t) shear mode, as a function of the rotation of axes to get c’66, where r is the density:

• Temperature derivative of the density:

• Temperature derivative of the thickness in the Z’2 (Y’) direction:

• Transformed (rotated) stiffness coefficient:

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Quartz Oscillator Crystals, contd.• Quartz belongs to point group 23. Therefore c1313 = c55 = c44

and c1213 = c65 = c56 = c14.• Taking the temperature derivative for c’66 and substituting all the

relevant values into the equation, one obtains the following. Here, “T(c66)” denotes the temperature coefficient of the relevant stiffness coefficient (Table 13.7). The derivative of the resonant frequency, f, can be set equal to zero in the standard fashion in order to find the minima.

• Applying the solution procedure yields two values with theta = -35° and +49°, corresponding to the two cuts illustrated.

• Further discussion is provided by Newnham on how to make AC and BC cuts that are useful for transducers for transversely-polarized acoustic waves.

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Summary

• We have covered the following topics:– Examples of elastic property values– Anisotropy coefficients (Zener)– Dependence of Young’s modulus on direction (in a

crystal)– Worked example– Quartz oscillator crystals

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Supplemental Slides• The following slides contain some useful material

for those who are not familiar with all the detailed mathematical methods of matrices, transformation of axes etc.

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NotationF Stimulus (field)R ResponseP Propertyj electric current E electric fieldD electric polarizatione Strains Stress (or conductivity)r Resistivityd piezoelectric tensor

C elastic stiffnessS elastic compliancea rotation matrixW work done (energy)I identity matrixO symmetry operator (matrix)

Y Young’s modulusd Kronecker deltae axis (unit) vectorT tensor, or temperature direction cosine

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Bibliography• R.E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure, Oxford

University Press, 2004, 620.112 N55P.• De Graef, M., lecture notes for 27-201.• Nye, J. F. (1957). Physical Properties of Crystals. Oxford, Clarendon Press.• Kocks, U. F., C. Tomé & R. Wenk, Eds. (1998). Texture and Anisotropy, Cambridge

University Press, Cambridge, UK.• T. Courtney, Mechanical Behavior of Materials, McGraw-Hill, 0-07-013265-8, 620.11292

C86M.• Landolt, H.H., Börnstein, R., 1992. Numerical Data and Functional Relationships in

Science and Technology, III/29/a. Second and Higher Order Elastic Constants. Springer-Verlag, Berlin.

• Zener, C., 1960. Elasticity And Anelasticity Of Metals, The University of Chicago Press.• Gurtin, M.E., 1972. The linear theory of elasticity. Handbuch der Physik, Vol. VIa/2.

Springer-Verlag, Berlin, pp. 1–295.• Huntington, H.B., 1958. The elastic constants of crystals. Solid State Physics 7, 213–351.• Love, A.E.H., 1944. A Treatise on the Mathematical Theory of Elasticity, 4th Ed., Dover,

New York.• Newey, C. and G. Weaver (1991). Materials Principles and Practice. Oxford, England,

Butterworth-Heinemann.• Reid, C. N. (1973). Deformation Geometry for Materials Scientists. Oxford, UK,

Pergamon.

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Transformations of Stress & Strain Vectors

• It is useful to be able to transform the axes of stress tensors when written in vector form (equation on the left). The table (right) is taken from Newnham’s book. In vector-matrix form, the transformations are: