Elastic coefficients
4.5 Cauchy’s relations
During the early development of the atomic theory of the elastic constants (Love, 1944), the idea that solids consist of small particles with forces acting along lines connecting their centers was a persistent theme beginning at the time of Newton (ca. 1717), and revived by Boscovich (ca. 1743). An attempt to use it to understand the bending of beams was made by Poisson (ca. 1812), but the first comprehensive advance came from Navier (ca. 1821).
He assumed that the forces between the particles are a function only of the radial distance between them, and are independent of the direction of the connecting line (isotropy). In this way he was able to express elastic behavior in terms of just one elastic constant. In other words shear and dilatation are interconnected in Navier’s theory (Timoshenko, 1983).
Cauchy made the next step by first introducing the concept of internal stress (note that stress is a constructed, rather than a natural variable) through the use of a tetrahedral “free-body” (ca. 1822). This clarified the size of the elasticity tensor (6× 6 with 21 independent components), and led to recognition that Navier’s molecular theory implied that there should be six equal pairs of these, if each molecule is located at a center of symmetry, leaving 15 independent components for trigonal crystals, two for cubic crystals, and one for isotropic materials. The additional equal pairs are known as the Cauchy relations. In the cubic case, the pair is C12 = C44; for isotropy, 2C44 = C11− C12, so Poisson’s ratio is C12/(C11+ C12)= 1/4. This last conclusion was experimentally tested by Werthheim (ca. 1848). He found that Poisson’s ratio was more often 1/3 than 1/4 for the large set of materials he studied.
Later Voigt (ca. 1887) studied a variety of crystals loaded in flexure and torsion, and found that only 20% of them satisfied the Cauchy relations even approximately. So it took some 65 years to disprove Navier’s theory conclusively. In the light of the quantum chemistry of atomic bonding it would not be expected that pair potentials would describe bonding accurately.
When tractions are applied to a solid they change its shape through shear, and/or its volume through dilatation. If the solid is “elastic”, and its temperature is low, the external work that is done in distorting it can be recovered by slowly reducing the tractions. Therefore, the internal energy (strain energy) of the distorted solid is increased by an amount equal to the work done on it. During the distortion process, the magnitude of the traction(s) increases linearly with strain from zero up to the final value (Hooke’s Law). Thus the average is half the final stress, and the work done per unit volume (strain energy density) is this times the displacement gradient (displacement per unit length is equal to strain). The reason for the linearity can be understood by considering the behavior near the zero strain, or equilibrium, state.
Suppose a slender cylindrical rod is put into torsion by twisting it through a small angleθ around its axis which is also a symmetry axis of the material of the rod. Then, sinceθ is the angular displacement per unit length, it is the strain; and the strain energy densityw(θ) is proportional to its square. The Taylor expansion ofw(θ) is formed for the untwisted state, θ = 0:
w(θ) = w(0) + w(0)θ + w(0)θ2/2 + w(0)θ3/6 + · · · and the twisting torqueτ(θ), the derivative of w(θ) with respect to θ, is:
τ(θ) = w(0)+ w(0)θ + w(0)θ2/2 + · · · (4.18) wherew(0)= 0, since the rod is at equilibrium there, and w(0)θ is large compared with w(0)θ2/2 at sufficiently small values of θ. Therefore, for any function, w(θ), the torque is proportional toθ for small values of θ, and w(0) is the elastic stiffness coefficient.
In order to see more explicitly how the Cauchy relations arise, the behavior of a small cluster of particles bound by central forces will be analyzed (Feynman, Leighton, and Sands, 1964). The cluster is illustrated by Figure 4.1. It consists of 27 particles in a simple cubic array. Nine of them in the central plane of the cube are shown in the figure. Displace-ments of the particles from their initial equilibrium positions are small, and the resulting restoring forces depend on the displacements linearly (Hookean). Hence, the interactions can be represented by springs with constants ki. The interactions of the nearest-neighbor particles have the spring constant k1, while those of the second-nearest neighbors have the constant k2. Note that the cluster would have no resistance to shear if there were no k2. Also, note that only central forces are exerted by k1and k2. The analysis is greatly simplified for a small cluster, but the result has general validity.
We apply a plane strain to the cluster, so i j= 0, if i, j = 3, and the only components of the strain tensor to be considered are 11, 22, and 12. Interactions in the x3-direction are considered, but no strains. A typical strain-energy term corresponding to an axial strain between particles−1 and −2 (Figure 4.1) which have a separation distance d is k1( 11d)2/2.
Here the displacement in the x1-direction is 11d = u1− 12d ≈ u1, since 12 is small compared with 11. There are three other similar nearest-neighbor interactions in the−x1, x2, and−x2directions. The interactions of particle−1 with its nearest neighbors in the x3
k1 k2
k2
Figure 4.1 The median plane of a model cubic crystal. There is a similar array of nine particles directly in front of this one at the spacing d, and one behind this array also at the spacing d. Hence, the model is a cube containing 27 atoms. The interactions between the atoms are represented by two kinds of springs. Those with spring constants k1act between nearest neighbors, while those with constants k2
act between next-nearest neighbors. The responses are linear for small displacements.
and−x3directions do not contribute first-order terms to the strain energy or displacements in the x1, x2plane.
The displacements of the second-nearest neighbors, all 12 of them, contribute to the strain-energy density. The displacements, u1 and u2, of particle−3 cause a displacement along the diagonal between particles−1 and −3 equal to the sum of their components in the diagonal direction: (u1+ u2)/√
2. The energy associated with this displacement is:
(k2/2)[(u1+ u2)/√
2]2= [(k2d2)/4]( 11+ 12+ 12+ 22)2
and there are three more terms of this type for the particles in the plane of the figure. In addition, in each plane at ±d in the x3-direction there are four second-nearest neighbor particles on the edges of the cube. When the displacements of these are resolved into components, and the strain energies are summed the result is:
k2[( 11d)2+ ( 22d)2]
The total strain energy, W ( ), for the strained cube, is obtained by adding up these various contributions. There is one term for each pair of atoms, and there is one atom for each d3 of volume, so the total strain-energy density is:
w( ) = W( )/(2d3)
From this, after collecting the terms of the same type, the elastic stiffnesses can be identified.
They are:
C11= C22= (k1+ 2k2)/d
C12= C21= k2/d (4.19)
C44= k2/d
All others are equal to zero. Also, C12= C44 as shown more generally by Cauchy. This result is a consequence of there being only central forces in the cube, and from the fact that a simple, cubic array has a center of symmetry. These conditions are satisfied quite well by alkali halide crystals such as KCl, but not by most metals, and especially not by covalently bonded crystals.
The bulk modulus of the cubic cluster is:
B = (C11+ 2C12)/3 = (k1+ 4k2)/3d (4.20) and the anisotropy ratio:
A= 2C44/(C11− C12)= 2k2/(k1+ k2) (4.21) Thus, for isotropy (i.e., A= 1), k1 = k2, and the ratio of the shear modulus to the bulk modulus is:
C44/B = 3/5 (4.22)
This will be compared with experimental data in a later chapter.
For a more formal approach to the elastic constants, one of the better introductions is in Feynman’s treatise (1964). A particularly careful development is in Sines (1969), see also Zener (1948). A convenient source of numerical values is Kittel (1996).
References
Feynman, R.P., Leighton, R.B., and Sands, M. (1964). The Feynman Lectures on Physics, Volume II, Chapter 38. Reading, MA: Addison-Wesley.
Kittel, C. (1996). Introduction to Solid State Physics, 7th edn. New York: Wiley.
Love, A.E.H. (1944). A Treatise on the Mathematical Theory of Elasticity, 4th edn. New York:
Dover Publications.
Nye, J.F. (1957). Physical Properties of Crystals. Oxford: Oxford University Press.
Sines, G. (1969). Elasticity and Strength. Boston, MA: Allyn and Bacon.
Timoshenko, S.P. (1983). History of the Strength of Materials. New York: McGraw-Hill, 1953.
Republished by Dover Publications, New York.
Zener, C. (1948). Elasticity and Anelasticity of Metals. Chicago, IL: University of Chicago Press.