Isotropic and Cubic Elastic Tensors
This appendix proves Lemma 34.26 of Continuum Mechanics and Elasticity and the statement the chapter takes from it: that the twenty-one independent constants of Proposition 34.25 collapse to two for an isotropic solid, the Lamé constants of Equation (34.21). It also supplies the intermediate result the chapter never states, and which is the more useful of the two in a laboratory: a solid invariant only under the rotation group of the cube has three independent elastic constants, and the single number by which the general cubic tensor fails to be isotropic is directly measurable.
What is proved here, and what is proved elsewhere
When Continuum Mechanics and Elasticity was written, the classification of the isotropic Cartesian tensors of low rank was carried by no chapter of Part II — Mathematical Methods, and the chapter recorded the gap as a debt owed by the Cartesian-tensor algebra of Differentiable Manifolds, Tensors, and Curvature. That debt is discharged, and not where it was expected: the classification for ranks one to four is Theorem 9.133 of Linear Algebra and Representation Theory — the representation chapter, not the manifolds one, because the statement is invariant theory for \(\SO(3)\) rather than tensor algebra — proved there in full from the definition Equation (9.164) of an isotropic Cartesian tensor. Section 17.2.2 carries the pointer and the constitutive-law reading. Remark 34.27 has been rewritten accordingly and now records what the chapter rests on rather than what it is owed; the Helmholtz decomposition named there is likewise in place, as Theorem 11.137.
This section therefore does not reprove the classification. It quotes Theorem 9.133, adds the two things that theorem does not contain — the independence of the three delta products, and the strictly weaker classification under the finite rotation group of the cube — and then carries out the specialisation to elasticity, which is physics and belongs here.
For reference, the result quoted is this. A Cartesian tensor \(T\) of rank \(N\) is isotropic (Definition 9.131) when its components are unchanged by every \(R\in\SO(3)\),
and Theorem 9.133 states that the isotropic tensors of rank one, two, three and four are respectively \(0\), the multiples of \(\delta_{ij}\), the multiples of \(\varepsilon_{ijk}\), and the linear combinations
which is Equation (34.20) and hence Lemma 34.26. Nothing below reopens that proof.
The three arrays \(\delta_{ik}\delta_{lm}\), \(\delta_{il}\delta_{km}\) and \(\delta_{im}\delta_{kl}\) are linearly independent, so the space of isotropic rank-four Cartesian tensors has dimension exactly three and the constants \(a\), \(b\), \(c\) of Equation (A56.2) are uniquely determined by \(T\). Rests on Theorem 9.133 and Equation (9.168).
Derives Lemma A56.2. Evaluate Equation (A56.2) on the three index quadruples \((1,1,2,2)\), \((1,2,1,2)\) and \((1,2,2,1)\). A product of two Kronecker deltas is \(1\) when both pairs it joins carry equal values and \(0\) otherwise, so
Each probe isolates one coefficient and annihilates the other two; hence a vanishing combination has \(a=b=c=0\), and the coefficients of a given \(T\) are read off by Equation (A56.3).
∎The rotation group of the cube leaves four constants
A crystal is not isotropic. Its elasticity tensor is invariant only under the finite point group of its lattice, and for the cubic classes — which include the two commonest structural metals, body-centred iron and face-centred aluminium — that group is the rotation group of the cube. The classification under this smaller group is a strictly weaker statement than Theorem 9.133, and the difference between the two is one number.
The octahedral group \(\mathcal{O}\subset\SO(3)\) is the group of the \(24\) rotations carrying a cube centred at the origin, with faces normal to the coordinate axes, to itself. In the standard frame its elements are exactly the \(3\times3\) matrices carrying a single entry \(\pm1\) in each row and each column and having determinant \(+1\); such a matrix acts as \(R_{ij}=s_{i}\,\delta_{i\,\pi(j)}\) for a permutation \(\pi\) of \(\set{1,2,3}\) and signs \(s_{i}=\pm1\) whose product is \(\sgn\pi\). A tensor obeying Equation (A56.1) for every \(R\in\mathcal{O}\), but not necessarily for every \(R\in\SO(3)\), is called cubic. Rests on Definitions 9.131 and 18.26.
The rank-four cubic Cartesian tensors form a four-dimensional space, spanned by the three products of Equation (A56.2) together with the cubic array
whose components are taken in the crystal frame of Definition A56.3. Every cubic \(T\) is therefore
and \(T\) is isotropic if and only if \(f=0\). Rests on Definition A56.3 and Lemma A56.2.
Derives Proposition A56.4. Step 1: the half-turn filter. For each \(p\in\set{1,2,3}\) the diagonal matrix with \(+1\) in the \(p\)-th place and \(-1\) in the other two is a half-turn about the \(p\)-axis; it has determinant \(+1\) and is a signed permutation, so it lies in \(\mathcal{O}\). The argument of Lemma 9.132 uses these three matrices and nothing else, so its conclusion holds verbatim for a cubic tensor: a component \(T_{iklm}\) vanishes unless every value \(p\) occurs among its four indices an even number of times. At rank four the multiplicities \((m_{1},m_{2},m_{3})\) sum to four, so the surviving components have multiplicities \((4,0,0)\) or \((2,2,0)\) in some order. These are the three components \(T_{pppp}\) and, for each of the six ordered pairs \(p\neq q\), the three arrangements \(T_{ppqq}\), \(T_{pqpq}\), \(T_{pqqp}\): twenty-one components in all, and \(81-21=60\) vanishing ones.
Step 2: the three-fold axis. The cyclic matrix \(C\) with \(C_{21}=C_{32}=C_{13}=1\) and every other entry zero is a signed permutation of determinant \(+1\) — a three-cycle is even — so it too lies in \(\mathcal{O}\), and it is the rotation by \(2\pi/3\) about the body diagonal \((1,1,1)/\sqrt{3}\). Substituted into Equation (A56.1) it says that the components are unchanged when every index value is relabelled by the cycle \(\sigma:1\mapsto3\mapsto2\mapsto1\). Hence \(T_{1111}=T_{2222}=T_{3333}\), a common value \(d\), and the ordered pairs \((p,q)\) fall into the two orbits \(\set{(1,2),(3,1),(2,3)}\) and \(\set{(2,1),(1,3),(3,2)}\), within each of which the three arrangements have common values.
Step 3: the four-fold axis. The quarter turn \(Q\) about the third axis, with \(Q_{21}=1\), \(Q_{12}=-1\), \(Q_{33}=1\), is a signed permutation of determinant \(+1\) and lies in \(\mathcal{O}\). It exchanges the values \(1\) and \(2\) and multiplies a component by \((-1)^{m_{1}}\). The components \(T_{1122}\), \(T_{1212}\), \(T_{1221}\) each have \(m_{1}=2\) and so carry the sign \(+1\), giving \(T_{1122}=T_{2211}\), \(T_{1212}=T_{2121}\), \(T_{1221}=T_{2112}\): the two orbits of Step 2 are joined to one another. The three numbers
are therefore independent of which ordered pair is used, and together with \(d=T_{pppp}\) they determine every component: the invariant space has dimension at most four.
Step 4: it has dimension exactly four, and Equation (A56.5) is its general element. The right-hand side of Equation (A56.5) is cubic: \(\Delta\) is unchanged by any relabelling of values and, having all four indices equal on each nonvanishing component, by any sign change; and the three delta products are isotropic by Theorem 9.133, hence a fortiori cubic. Reading off its components with Equation (A56.3) gives \(T_{ppqq}=a\), \(T_{pqpq}=b\), \(T_{pqqp}=c\) and \(T_{pppp}=a+b+c+f\), so the four constants \((a,b,c,f)\) of the ansatz reproduce the four constants \((a,b,c,d)\) of Steps 2 and 3 through
The map \((a,b,c,f)\mapsto(a,b,c,d)\) is a bijection, so the four constants of the ansatz are in one-to-one correspondence with those found in Steps 2 and 3. The four arrays are independent: the three delta products are independent of one another by Lemma A56.2, and none of them is a multiple of \(\Delta\), since \(\Delta\) alone changes \(T_{1111}\) without changing \(T_{1122}\), \(T_{1212}\) or \(T_{1221}\). Hence the dimension is exactly four.
Step 5: isotropy is \(f=0\). If \(f=0\) then \(T\) is a combination of the three isotropic products, hence isotropic. Conversely, an isotropic \(T\) is of the form Equation (A56.2) by Theorem 9.133, and its coefficients are those read off by Equation (A56.3), so \(d=T_{1111}=a+b+c\) and \(f=0\) by Equation (A56.7).
∎The whole difference between the cubic and the isotropic classifications is the single relation \(d=a+b+c\), and it is worth being clear about which rotations supply it. Steps 1 to 3 used only the finite group: three half-turns, one three-fold axis, one four-fold axis. Those fix every component in terms of four numbers and can do no more, because the cubic array Equation (A56.4) is genuinely invariant under all \(24\) of them. The relation is produced instead by a rotation through a general angle — in the proof of Theorem 9.133 it is the rotation by \(\theta\) about the third axis, which forces \(2\cos^{2}\theta\sin^{2}\theta\,(a+b+c-d)=0\) and hence, at \(\theta=\pi/4\), \(d=a+b+c\). So the elementary argument stopped one step early does not fail: it delivers cubic anisotropy, which is a physically meaningful and experimentally accessible intermediate result rather than a defective version of isotropy.
From twenty-one constants to three, and then to two
We now impose on Equation (A56.5) the symmetries that an elasticity tensor has by Proposition 34.25: the minor symmetries \(C_{iklm}=C_{kilm}=C_{ikml}\), which follow from the symmetry of the strain and of the stress, and the major symmetry \(C_{iklm}=C_{lmik}\), which follows from the existence of the energy density of Definition 34.24.
Let \(C_{iklm}\) be an elasticity tensor in the sense of Definition 34.24, so that it obeys the minor and major symmetries of Proposition 34.25.
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If \(C\) is cubic, then in the crystal frame
\begin{equation}\tag{A56.8} C_{iklm}=\lambda\,\delta_{ik}\delta_{lm} +\mu\left(\delta_{il}\delta_{km}+\delta_{im}\delta_{kl}\right) +f\,\Delta_{iklm}\ec \end{equation}three independent constants, all of SI dimension \(\mathrm{Pa}\). The minor symmetries are what force \(b=c=\mu\); the major symmetry is then automatic and imposes nothing further.
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If \(C\) is isotropic, then \(f=0\) and \(\sigma_{ik}=C_{iklm}u_{lm}\) is Equation (34.21), with two constants.
Rests on Proposition A56.4, Proposition 34.25 and Definition 34.24.
Derives Theorem A56.6. The minor symmetries. Exchange \(i\) and \(k\) in Equation (A56.5). The first term is unchanged, since \(\delta_{ik}\) is symmetric; the cubic array is unchanged, since it is symmetric under every permutation of its indices; and the middle two terms are exchanged,
Equality of Equation (A56.9) with the original, together with the independence of the two arrays \(\delta_{il}\delta_{km}\) and \(\delta_{im}\delta_{kl}\) established in Lemma A56.2, forces \(b=c\). Writing \(a=\lambda\) and \(b=c=\mu\) gives Equation (A56.8). Exchanging \(l\) and \(m\) instead produces the same equation and hence no new condition.
The major symmetry. Send \((ik)\mapsto(lm)\) in Equation (A56.8). The first term becomes \(\lambda\delta_{lm}\delta_{ik}\), which is itself; the second becomes \(\mu(\delta_{li}\delta_{mk}+\delta_{lk}\delta_{mi})\), which is itself because each Kronecker delta is symmetric; and \(\Delta\) is symmetric under every index permutation. So the major symmetry is satisfied identically, and a cubic crystal has exactly the three constants \((\lambda,\mu,f)\).
Isotropy. If \(C\) is isotropic then \(f=0\) by Proposition A56.4, and contracting Equation (A56.8) with the symmetric strain \(u_{lm}\) gives
which is Equation (34.21), the second equality using \(u_{ik}=u_{ki}\). Counting: the general tensor has \(21\) constants by Proposition 34.25, a cubic crystal three, an isotropic solid two.
∎In the Voigt notation, in which an index pair \((ik)\) is coded as one of \(1,\ldots,6\) by \(11\mapsto1\), \(22\mapsto2\), \(33\mapsto3\), \(23\mapsto4\), \(13\mapsto5\), \(12\mapsto6\), the three constants of Equation (A56.8) are
so that
and \(A=1\) exactly when \(f=0\), that is exactly when the crystal is elastically isotropic. \(A\) is the Zener anisotropy ratio, and the measured values [Simmons:1971] show that the isotropic idealisation is a poor description of a single crystal and an accidentally good one for a few materials: for body-centred iron \(C_{11}=230\,\mathrm{GPa}\), \(C_{12}=135\,\mathrm{GPa}\), \(C_{44}=117\,\mathrm{GPa}\) give \(A=2.5\); for face-centred aluminium \(C_{11}=108\,\mathrm{GPa}\), \(C_{12}=61\,\mathrm{GPa}\), \(C_{44}=28.5\,\mathrm{GPa}\) give \(A=1.2\); and for tungsten \(C_{11}=523\,\mathrm{GPa}\), \(C_{12}=203\,\mathrm{GPa}\), \(C_{44}=160\,\mathrm{GPa}\) give \(A=1.00\), so that tungsten is isotropic to the precision of the measurement — a coincidence of the numbers, not a consequence of any symmetry. A polycrystalline specimen of iron, on the other hand, is isotropic on scales large against the grain, because the orientations average; that is why Equation (34.21) describes structural steel and would not describe a single iron whisker. Rests on Theorem A56.6 and Definition 34.31.
The two lower-rank cases of Theorem 9.133 carry physics of their own and are used elsewhere in Part III — Classical Mechanics. Rank two: an isotropic tensor is \(a\,\delta_{ik}\), so every rank-two material property of an isotropic body — thermal expansion, thermal conductivity, electrical conductivity — is a single scalar times the identity, and the response is parallel to the drive. Rank three: an isotropic tensor is \(a\,\varepsilon_{ikl}\), and \(\varepsilon\) is a pseudotensor: it is invariant under \(\SO(3)\) but changes sign under the inversion \(R=-\identity\), by Equation (9.59). A material property that is a true rank-three tensor and is invariant under the full orthogonal group, inversion included, therefore vanishes identically. The piezoelectric coupling \(P_{i}=d_{ikl}\sigma_{kl}\) is such a property, so no centrosymmetric medium is piezoelectric — quartz is not centrosymmetric and is, a fact used in every quartz oscillator, and no isotropic solid can be. The same parity argument is why an odd-rank isotropic tensor of rank one is zero. Rests on Theorem 9.133 and Remark 9.134.
The dimension three found in Lemma A56.2 can be checked against the character theory of Linear Algebra and Representation Theory without recomputing a single component. Write \(V=\R^{3}\) for the defining representation of \(\SO(3)\). An invariant of \(V^{\otimes4}\) is, after using the invariant inner product to identify \(V\) with its dual, an \(\SO(3)\)-equivariant endomorphism of \(V\otimes V\); and the Clebsch–Gordan series Theorem 18.47 decomposes \(V\otimes V\) into the trace, the antisymmetric part and the symmetric traceless part, of dimensions \(1\), \(3\) and \(5\), each occurring once. Schur's lemmas (Theorems 9.158 and 9.160) then say that an equivariant map is a scalar on each summand and zero between distinct ones, so the space of such maps has dimension \(1+1+1=3\).
This is offered as a check and not as the proof, for a reason worth stating. Over the complex numbers Schur's second lemma delivers a scalar; over the reals it delivers only that the endomorphism algebra of an irreducible is a division algebra, which may be \(\R\), \(\C\) or the quaternions, and the multiplicity count above is correct only because each real integer-spin irreducible of \(\SO(3)\) has endomorphism algebra \(\R\). That last step is not carried by Linear Algebra and Representation Theory, whereas the elementary argument of Theorem 9.133 is complete as it stands. Where the two routes agree, as here, the agreement is evidence that neither has dropped a constant.
Isotropic and Cubic Elastic Tensors discharges the derivation owed at Lemma 34.26 of Continuum Mechanics and Elasticity. The classification itself is Theorem 9.133 of Linear Algebra and Representation Theory and is not reproved here (Remark A56.1); what this section adds is the uniqueness of the three coefficients (Lemma A56.2), the cubic classification (Proposition A56.4) that shows exactly which rotations the isotropy of an elastic solid actually consumes, and the reduction \(21\rightarrow3\rightarrow2\) of Theorem A56.6, which is the step the derivation of Phenomenon 34.28 takes for granted when it writes \(b=c\) and \(a=\lambda\), \(b=\mu\). Everything downstream of Equation (34.21) in that chapter — the moduli of Definition 34.31, the elastic waves, the plate of Reduction of Three-Dimensional Elasticity to the Kirchhoff Plate Equation and the contact of Hertz's Solution for the Elastic Half-Space Under an Axisymmetric Pressure — rests on the two-constant form proved here.