PARTIAL UNIQUENESS AND OBSTRUCTION TO UNIQUENESS IN INVERSE PROBLEMS FOR ANISOTROPIC ELASTIC MEDIA

ANNA L. MAZZUCATO AND LIZABETH V. RACHELE

Abstract. We consider the dynamic inverse problem of uniquely determining the material parameters of bounded, three-dimensional elastic objects from displacement and traction measurements made at their surfaces. These surface measurements are modeled by the Dirichlet-to-Neumann map on a finite time interval. We first construct an obstruction to uniqueness for the inverse problem by transforming the medium via diffeomorphisms that fix the boundary to first order. We study the equivalence classes of elastic media under the action of pullback via these diffeomorphisms. This action leaves the Dirichletto-Neumann map invariant. We then show that any uniqueness result for a particular type of elastic media extends to (partial) uniqueness on the entire class. For example, we establish that uniqueness for isotropic elastic media implies the unique determination by surface measurements of three parameters of some anisotropic elastic media.

1. Introduction

In the inverse problem for the scalar wave equation associated with a Riemannian metric G,

(1)  (∂_{t}^{2} − Δ_{G})u = 0,

there is an “obstruction to uniqueness” in the dependence of the metric G on surface measurements. In fact, for surface measurements in the form of the Dirichlet-to-Neumann map Λ_{G} we have

(2)  Λ_{ψ^{∗}G} = Λ_{G}

2000 Mathematics Subject Classification. Primary: 65M32, 74B05, 70G45; Secondary: 74B20, 74A20, 15A72.

Key words and phrases. Elastodynamics, inverse problems, parameter identification, uniqueness, covariance, anisotropic media, elasticity tensor, harmonic decomposition, Cartan decomposition, Dirichlet-to-Neumann map.

The first author was partially supported by an MSRI Postdoctoral Fellowship. Research at MSRI is supported in part by NSF grant DMS-9850361. This work was conducted while the author was a Gibbs Instructor at Yale University.

The second author was partially supported by an MSRI Postdoctoral Fellowship and by NSF grant DMS-9801664 (9996350).

for any sufficiently regular diffeomorphism ψ : Ω̄ → Ω̄ with ψ(x) = x for x ∈ ∂Ω. Here Ω ⊂ ℝ^{3} is an open region with smooth boundary ∂Ω, and Δ_{G} is the Laplace-Beltrami operator associated with the metric G on Ω; in local coordinates,

(3)  Δ_{G}u = ∑_{i,j=1}^{n} \frac{1}{\sqrt{det G}} \frac{∂}{∂x^{i}} (\sqrt{det G} G^{ij} \frac{∂u}{∂x^{j}} ),

where det G denotes the determinant, and G^{ij} denotes the inverse of the metric tensor G_{ij}. In other words, the best uniqueness result possible in the inverse problem for the operator ∂_{t}^{2} − Δ_{G} is the unique determination of G by the Dirichlet-to-Neumann map, up to the pullback by a diffeomorphism ψ that fixes the boundary. In fact, this result has been obtained for the scalar wave operator, under certain conditions, in [Sy-U] and [L-S-U], for example.

The scalar wave equation (1) is the canonical model for wave propagation in anisotropic media, and so it is natural to ask whether the uniqueness result for ∂_{t}^{2} − Δ_{G} extends to more general models, for example, the system for elastodynamics. The uniqueness question for general anisotropic elastic media, though, is significantly more difficult than that for the scalar wave equation, due mainly to greater complexity in the description of the rays, paths along which disturbances propagate through the medium. We show here, however, that several aspects of the uniqueness question may be solved in the systems case without reference to rays.

In fact, given the particular type of obstruction (2) in the scalar case, we make our first aim the description of equivalence classes of elastic media that are invariant under the Dirichlet-to-Neumann map and are defined in terms of an action by diffeomorphisms of Ω that fix the boundary. We give a pointwise description of parts of the orbits under this action, and a complete pointwise characterization in the isotropic case. (At any point of Ω the orbit of isotropic elastic media consists of isotropic, some transverse isotropic, and some orthotropic elastic media.)

Global constraints might place additional limits on the types of anisotropy within the orbit of any particular elastic media. The approach taken here does not extend directly to the global setting, as it depends on diagonalization of the symmetric part (i.e. the stretch factor of the polar decomposition) of the differential Dψ.

It is interesting to note that composite elastic media, elastic media for which the type of anisotropy can vary from point to point, may lie on the orbits of non-composite elastic media under this action.

We then observe that uniqueness results for classes of elastic media extend to the orbits under this action. We conclude that Rachele’s uniqueness results [R I], [R II], [R III], [R IV] for isotropic elastodynamics extend to certain anisotropic elastic media.

We mention other results on obstruction to uniqueness in inverse problems using Dirichlet-to-Neumann-type data. In a 2-dimensional static problem for orthotropic elastic media G. Nakamura and K. Tanuma [N-T ’96] show that the Dirichlet-to-Neumann map does not uniquely determine the elastic parameters at the boundary. Also, M. Belishev [B] describes an obstruction to uniqueness in the dependence of the Dirichlet-to-Neumann map on lowerorder parameters of an (isotropic) wave equation. In addition, Romanov [R-K] proves nonuniqueness in an inverse problem for anisotropic Maxwell’s equations in the case that the permittivity ϵ and permeability μ are diagonal matrices.

The only uniqueness result known to the authors for the fully threedimensional inverse parameter identification problem for nonhomogeneous anisotropic elastic media using Dirichlet-to-Neumann-type data is that of Nakamura, Tanuma, and Uhlmann [N-T-U] who show that, in the case of transverse isotropic elasticity, the material properties of bounded elastic objects are uniquely determined by static measurements made at the surface. Nakamura, Tanuma, and Uhlmann use a layer-stripping approach (applicable to static problems).

To extend the non-uniqueness result (2) for the scalar wave equation (1) to elastic media we first describe (2) in terms of the covariance of the wave operator, i.e.,

ψ^{∗}([∂_{t}^{2} − Δ_{G}]u) = [∂_{t}^{2} − Δ_{ψ^{∗}G}](ψ^{∗}u),

where ψ^{∗} is the pullback under the diffeomorphism ψ. (By covariance we mean an intrinsic tensorial representation independent of any local coordinate system.) We then view the elastic body as a Riemannian manifold endowed with a smooth Riemannian metric G which we may vary as we vary the other material parameters. We write the differential operator for hyperelastic media in a covariant form (see Marsden and Hughes [M-H]) so that a solution U of the elastodynamics system (with respect to a metric G and material parameters O) transforms to a solution ψ^{∗}U = U ∘ ψ with respect to ψ^{∗}G and transformed parameters ψ^{∗}O (cf. Theorem 1). We conclude that the elastic objects (Ω, G, O) and (Ω, ψ^{∗}G, ψ^{∗}O) cannot be distinguished by surface measurements given by the associated Dirichlet-to-Neumann maps, i.e. Λ_{G,O} = Λ_{ψ^{∗}G,ψ^{∗}O}, if ψ preserves the boundary. In fact, in the classical linear case this obstruction to uniqueness may be stated with respect to the Euclidean metric:

Λ_{e,\sqrt{det G} O} f = Λ_{e,det(Dψ)\sqrt{det G} ψ^{∗}O} f.

To write the elasticity equations in covariant form in a way that respects the underlying physics, we turn to the balance laws (conservation of mass, balance of momentum, balance of moment of momentum, and balance of energy) that describe the behavior of the elastic medium. In the case of non-linear elastodynamics Marsden and Hughes [M-H], Theorem 4.13, show that the balance laws, together with the hyperelasticity condition (that is, the existence of an internal energy E), are equivalent to the covariance of the internal energy. The balance laws (in particular, the equations of motion) may be written in either the so-called Eulerian (or spatial) coordinates x on the ambient space S (here S = ℝ^{3}), or in Lagrangian (or material or referential) coordinates X on Ω, at least for sufficiently regular motions. (See Section 3.) Here we use Lagrangian coordinates in order to describe changes in the material properties of the body in an intrinsic way, while avoiding the construction of global transformations of the ambient space.

In Lagrangian coordinates the energy E is covariant, in the case that it depends functionally on just the Cauchy-Green deformation tensor C, only if the elastic medium is isotropic. To include anisotropic media in the analysis, therefore, we view E as a function of C and structural tensors ζ_{1}, …, ζ_{m} which express the specific anisotropy of the body. (See the proof of Theorem 1 in Section 4.)

For classical linear elasticity the (Lagrangian) covariant form for the equations of motion derived from the balance laws is given by [M-H]

(4)  P_{G,C,ρ} U = ρ\frac{∂^{2}U}{∂t^{2}} − DIV_{G}(A · ∇U) = 0,

where ρ(X) is the mass density in the reference configuration, A depends on the elasticity tensor C via

(5)  A^{aA}_{b}^{B} = 2C^{EAF B}(Dι)^{a}_{E}(Dι)^{c}_{F} δ_{bc},

ι is an embedding of Ω in ℝ^{3}, U is the displacement vector from a material point X in the reference configuration to its new location after time t, DIV_{G}(A · ∇U) is the divergence on (Ω, G) of the two-point tensor A · ∇U, and ∇ is the covariant derivative induced by G. We show in the proof of Theorem 1 that the equations of motion in local coordinates are given by

(6)  ρ\frac{∂^{2}U^{a}}{∂t^{2}} − \frac{1}{\sqrt{det G}} \frac{∂}{∂X^{A}} (\sqrt{det G} A^{aA}_{b}^{B} \frac{∂U^{b}}{∂X^{B}} ) = 0,

where det G denotes the determinant of the metric tensor G_{ij}. Here an uppercase (i.e. capitalized) index denotes the leg of a tensor in the material setting, while a lowercase index refers to the spatial setting. (See Section 3 for definitions and notation.) No lower-order-terms associated to the Christoffel symbols of G appear, since ∇ acts on the two-point vector U as if it were a scalar. We emphasize that the equations under a change of (Lagrangian) coordinates ψ have the same form but with each of G, C, and ρ replaced by its pullback.

To illustrate that not all covariant forms of a differential equation respect the underlying physics, we turn to the following example. The Navier-Stokes equations of fluid mechanics

(7)  \begin{cases}∂_{t}u − Δu + u · ∇u − ∇p = 0, \\ div u = 0, \end{cases}

are in covariant form if Δ is interpreted as the Laplace-Beltrami operator and ∇ is the covariant derivative. In (7) u is a vector field describing the velocity of a fluid particle, and p is a scalar field representing the pressure.

However the correct form of the equation as derived from conservation of momentum and mass is [E-M]:

(8)  \begin{cases}∂_{t}u − div S + u · ∇u − ∇p = 0, \\ div u = 0. \end{cases}

Above, S = 1/2(∇u + ∇^{T} u) is the deformation tensor and, for divergencefree vector fields u on a Riemannian manifold, div S = Δ u + 2 Ric u, where Ric is the Ricci tensor. In fact, one expects curvature to enter into the dynamics whenever viscous stresses are present.

The paper is organized as follows. In Section 2 we state the main results. In Section 3 we present notation and definitions, and write the equations of motion in covariant form for the non-linear, linearized, and classical linear cases. Obstruction to uniqueness (Theorem 1) is proved in Section 4. Although we focus in this paper on results for classical linear elastodynamics, our approach is based on a linearization of the non-linear equations of motion, and so we prove Theorem 1 for the non-linear and linearized cases as well. In Section 5 we prove Theorem 11 on the pointwise characterization of orbits in some cases.

2. Main Results

We say that a three-dimensional linearly elastic object (represented by Ω ⊆ ℝ^{3}) is admissible if the object is bounded, with smooth boundary; has smooth, positive density ρ; and has smooth elasticity tensor C which has the symmetry properties

(9)  C^{ABCD} = C^{CDAB} = C^{BACD} = C^{ABDC}

and is (uniformly) strongly elliptic on Ω, i.e. there is a constant c > 0 so that for any X ∈ Ω C^{ABCD}(X)V^{A}V^{B}W^{C}W^{D} ≥ c|V|^{2}|W|^{2} for all vectors V, W ∈ T_{X}^{∗}Ω. (The symmetry condition C^{ABCD} = C^{CDAB} is equivalent to hyperelasticity in the linear case.) Throughout the paper we follow the convention of summing over repeated indices.

In the case of non-linear elasticity we also assume that there exist unique solutions ϕ_{t} of the initial-boundary-value problem (30) for regular-enough Dirichlet data f. Some partial results for well-posedness of the non-linear initial-boundary-value problem (30) are discussed, for example, in [M-H], p. 401, 406.

We model the behavior of any admissible nonhomogeneous elastic object Ω ⊆ ℝ^{3} by solutions of the equations of motion, e.g. P_{G,O}U = 0 for classical linear elastodynamics, where O (given by (C, ρ) in this case) represents the material properties of the object.

The surface measurements used to determine the material parameters O are given by the correspondence between forces exerted on the surface and the resulting surface displacements, taken over a finite period of time. These surface measurements are represented by the dynamic Dirichlet-to-Neumann map Λ_{G,O}. In the case of classical linear elasticity (4) we define the Dirichlet-to-Neumann map by (10) (Λ_{G,C,ρ} f)^{a} = ⟨ A · ∇U, ν_{G} ⟩^{a} dS_{G} = A^{aA}_{b}^{B} (∇U)^{b}_{B} G_{AC} ν_{G}^{C} dS_{G},

where dS_{G} is the area form on ∂Ω induced by G, ν_{G} is the unit outer normal vector to the boundary ∂Ω with respect to G, A is defined in (5), and U solves the initial-boundary-value problem

(11a)  P_{G,C,ρ}U = ρ \frac{∂^{2}U}{∂t^{2}} − DIV_{G}(A · ∇U) = 0  on Ω,
(11b)  U(X, t) = f(X, t)  for X ∈ ∂Ω,
(11c)  U(X, t) = 0  for t = 0,
(11d) \frac{∂U}{∂t}(X, t) = 0  for t = 0,

with Dirichlet boundary data f. The Dirichlet-to-Neumann map is defined by (33) and (38) in the cases of non-linear and linearized elastodynamics, respectively.

Given any diffeomorphism ψ of Ω̄ that fixes the boundary to first order, i.e. ψ ∈ Ψ, where

Ψ = { ψ : Ω̄ → Ω̄ | ψ|_{∂Ω} = id, Dψ|_{∂Ω} = I },

we define ψ^{∗}O to be the transformed material parameters under ψ. That is, in the classical linear case ψ^{∗}O = (ψ^{∗}C, ψ^{∗}ρ). The transformed parameters (ψ^{∗}G, ψ^{∗}C, ψ^{∗}ρ) give rise to an admissible hyperelastic object.

In the proof of Theorem 1 (in Section 4) we first show that the equations of motion are covariant in the non-linear case, and we then linearize following [M-H]. Consequently, if U is a solution of the initial-boundary-value problem (11) for linear elastodynamics with Dirichlet boundary data f, metric G, and material parameters O, then ψ^{∗}U = U ∘ ψ is a solution of (11) with metric ψ^{∗}G and material parameters ψ^{∗}O. We conclude, given the definition (10) of the Dirichlet-to-Neumann map and the fact that ψ is the identity to first order at the boundary, that the Dirichlet-to-Neumann maps for (G, O) and (ψ^{∗}G, ψ^{∗}O) agree. That is, the two elastic objects (Ω, G, O), (Ω, ψ^{∗}G, ψ^{∗}O) cannot be distinguished by boundary measurements.

Theorem 1 (Obstruction to uniqueness for hyperelastic media). Consider any admissible (linear, linearized, or non-linear) elastic body (Ω, G, O) with material parameters given by O. Then for any ψ ∈ Ψ and the transformed parameters ψ^{∗}O under ψ

Λ_{G,O} = Λ_{ψ^{∗}G,ψ^{∗}O}.

In the classical linear case the obstruction may be stated with respect to the Euclidean metric:

Corollary 2 (Obstruction for linear elastodynamics with respect to the Euclidean metric). Consider any admissible linearly elastic body (Ω, G, C, ρ). Then

(12)

(Λ_{G,C,ρ} f)^{a} = A^{aA}_{b}^{B} (∇U)^{b}_{B} δ_{AC} ν^{C} \sqrt{det G} dS
= (Λ_{e,\sqrt{det G} (C,ρ)} f)^{a},

where dS is the area form on ∂Ω induced by the Euclidean metric e, and ν is the unit outer normal vector to the boundary ∂Ω with respect to e.

In fact, for any ψ ∈ Ψ

(13)  Λ_{e,\sqrt{det G} (C,ρ)} = Λ_{e,(det Dψ)\sqrt{det G} (ψ^{∗}C,ψ^{∗}ρ)}.

Proof. The first equality in (12) follows from an application of the divergence theorem (e.g., [Ta I], p. 127). In fact, on the one hand,

dS_{G} = \sqrt{det G} < ν, ν >_{G}^{1/2} dS,

where < ν, ν >_{G}= G_{AB}ν^{A}ν^{B}. On the other, the unit (co)normal to ∂Ω w.r.t. G is related to ν by

(ν_{G})_{A} =< ν, ν >_{G}^{−1/2} ν_{A}.

The second equality is a consequence of (6), which may be viewed as a global representation of the equations of motion (4) in Euclidean coordinates. Theorem 1 then directly implies (13). □

2.1. Partial Uniqueness for Anisotropic Linearly Elastic Media. We consider the orbits 𝔒_{(G,C,ρ)} = { (ψ^{∗}G, ψ^{∗}C, ψ^{∗}ρ) | ψ ∈ Ψ } of parameters (G, C, ρ) of linearly elastic media under the action of Ψ. Below we prove uniqueness of some parameters for anisotropic, linearly elastic media in the orbit of isotropic elastic media and in the orbit of isotropic elastic media with residual stress.

The elasticity tensor for isotropic elastic media with respect to the Euclidean metric e (which we denote by C_{iso}) has the following components:

(14)  C_{iso}^{ABCD} = λ(X)δ^{AB}δ^{CD} + μ(X)[δ^{AC}δ^{BD} + δ^{AD}δ^{BC}]

with λ, μ the Lamé parameters. In this case we denote the Dirichlet-to-Neumann map by Λ_{e,λ,μ,ρ}. Similarly, Ĉ_{iso} denotes the isotropic elasticity tensor with Lamé parameters λ̂ and μ̂. In general, C depends on the metric G.

We first recall Rachele’s uniqueness result for linear, isotropic elastic media in Euclidean space.

Theorem 3 (Rachele [R I], [R II], [R III], Uniqueness for isotropic elastodynamics in Euclidean space). Suppose that (Ω, λ, μ, ρ) (respectively, (Ω, λ̂, μ̂, ρ̂)) are isotropic, linearly elastic media satisfying the conditions of Section A.1. In addition, suppose that

Λ_{e,λ,μ,ρ} = Λ_{e,λ̂,μ̂,ρ̂} on (0, T),

where T < ∞ is greater than the geodesic distance (with respect to each of g_{p}, g_{s}, ĝ_{p}, ĝ_{s}) between any two boundary points of Ω. Then for any X ∈ Ω̄

λ(X) = λ̂(X), μ(X) = μ̂(X), and ρ(X) = ρ̂(X).

We denote by

𝔒_{iso} = {(ψ^{∗}G, ψ^{∗}C_{iso}, ψ^{∗}ρ) | ψ ∈ Ψ}

the orbit of isotropic, linearly elastic media. We now prove uniqueness for the anisotropic, linearly elastic media in the orbit 𝔒_{iso} by applying Theorem 1, equation (12), and Theorem 3. (In Theorem 11 we give a pointwise characterization of 𝔒_{iso} and a pointwise description of parts of the orbits of other types of anisotropic elastic media.)

Corollary 4 (Partial uniqueness within the orbit generated by isotropic elastic media). Consider admissible, linearly elastic media (Ω, ψ^{∗}G, ψ^{∗}C_{iso}, ψ^{∗}ρ), (Ω, ψ^{∗}Ĝ, ψ^{∗}Ĉ_{iso}, ψ^{∗}ρ̂) ∈ 𝔒_{iso}. Suppose that the transformed media (Ω, e, \sqrt{det G}(λ, μ, ρ)) and (Ω, e, \sqrt{det Ĝ}(λ̂, μ̂, ρ̂)) satisfy the conditions of Theorem 3. In addition, suppose that

(15)  Λ_{ψ^{∗}G,ψ^{∗}C_{iso},ψ^{∗}ρ} = Λ _{ψ^{∗}Ĝ, ψ^{∗}Ĉ_{iso}, ψ^{∗}ρ̂} .

Then for X ∈ \overline{Ω} and δ = \sqrt{det Ĝ}/\sqrt{det G}

λ(X) = δ λ̂(X), μ(X) = δ μ̂(X), and ρ(X) = δ ρ̂(X).

Proof. By Theorem 1 the left side Λ_{ψ^{∗}G,ψ^{∗}C_{iso},ψ^{∗}ρ} of (15) may be written as Λ_{G,C_{iso},ρ}, which by (12) agrees with Λ_{e,\sqrt{det G}(C_{iso},ρ)}. Similarly, the right side of (15) may be written as Λ_{e,\sqrt{det Ĝ}(Ĉ_{iso},ρ̂)}. The result now follows by (15) and Theorem 3. □

Remark 5. In particular, we may apply Corollary 4 to prove uniqueness for isotropic elastodynamics in the setting of any arbitrary, flat metric Ḡ of the form ψ^{∗}e, ψ ∈ Ψ. In this case the elasticity tensor C (isotropic with respect to Ḡ) has the form

C^{ABCD} = λ(X)Ḡ^{AB}Ḡ^{CD} + μ(X)[Ḡ^{AC} Ḡ^{BD} + Ḡ^{AD}Ḡ^{BC}].

In fact, C = ψ^{∗}C_{iso}.

We next recall Rachele’s uniqueness result for linear, isotropic elastic media with residual stress in Euclidean space.

Theorem 6 (Rachele [R IV], Partial uniqueness for (λ, μ, ρ, T ) in Euclidean space). Suppose that (Ω, λ, μ, ρ, T ) (respectively, (Ω, λ̂, μ̂, ρ̂, T̂)) are isotropic, linearly elastic media with residual stress as described in Section A.2. Let

g_{p} = (\frac{λ + 2μ}{ρ}I + \frac{1}{ρ}T )^{−1} and g_{s} = (\frac{μ}{ρ}I + \frac{1}{ρ}T )^{−1}

denote the metrics whose geodesics are the paths of wave propagation in these media (respectively, ĝ_{p} and ĝ_{s}). In addition, suppose that

Λ_{e,λ,μ,ρ,T} = Λ_{e,λ̂,μ̂,ρ̂,T̂} on (0, T),

where T < ∞ is greater than the geodesic distance (with respect to each of g_{p}, g_{s}, ĝ_{p}, ĝ_{s}) between any two boundary points of Ω. Then

ψ_{p}^{∗}g_{p} = ĝ_{p} and ψ_{s}^{∗}g_{s} = ĝ_{s} in Ω

for some diffeomorphisms ψ_{p}, ψ_{s} of Ω̄ which are the identity on ∂Ω.

We again apply Theorem 1, equation (12), and Theorem 6, here to prove uniqueness for the anisotropic (with respect to e) elastic media in the orbit

𝔒_{resid} = { (ψ^{∗}G, ψ^{∗}C_{iso}, ψ^{∗}ρ, ψ^{∗}T ) | ψ ∈ Ψ }

of e-isotropic elastic media with residual stress. The equations of motion for such media are given by (36) with second Piola-Kirchhoff stress tensor S given by the residual stress tensor T .

Corollary 7 (Partial uniqueness within the orbit generated by isotropic elastic media with residual stress). Consider admissible linearized elastic media (Ω, ψ^{∗}G, ψ^{∗}C_{iso}, ψ^{∗}ρ, ψ^{∗}T ), (Ω, ψ^{∗}Ĝ, ψ^{∗}Ĉ_{iso}, ψ^{∗}ρ̂, ψ^{∗}T̂) ∈ 𝔒_{resid} with the property that the transformed media (Ω, e, \sqrt{det G}(λ, μ, ρ, T )) and (Ω, e, \sqrt{det Ĝ}(λ̂, μ̂, ρ̂, T̂)) satisfy the conditions of Theorem 6. In addition, suppose that

Λ_{ψ^{∗}G,ψ^{∗}C_{iso},ψ^{∗}ρ,ψ^{∗}T} = Λ _{ψ^{∗}Ĝ, ψ^{∗}Ĉ_{iso}, ψ^{∗}ρ̂, ψ^{∗}T̂} .

Then for g_{p}, g_{s}, ĝ_{p}, ĝ_{s} as defined in Theorem 6

ψ_{p}^{∗}g_{p} = ĝ_{p} and ψ_{s}^{∗}g_{s} = ĝ_{s} in Ω

for some diffeomorphisms ψ_{p}, ψ_{s} of Ω̄ which are the identity on ∂Ω.

Corollaries 4 and 7 illustrate an approach that may be taken to prove partial uniqueness for more general anisotropic (linear or linearized) elastic media. The strategy is to prove uniqueness for the simplest class of anisotropic media in an orbit 𝔒, and then to derive partial uniqueness, via Theorem 1, for other anisotropic media in 𝔒.

2.2. The Symmetry Group of ψ^{∗}C. In the previous section we extend uniqueness results for elastic media to their orbits under the action of Ψ. Consequently, we are interested in characterizing the orbits of each type of (an)isotropic elastic media, especially the orbit of isotropic elastic media (for which uniqueness results are already known). In this section we characterize the orbit of isotropic elastic media. (At any X ∈ Ω the orbit of the isotropic elasticity tensors consists of the isotropic, some transverse isotropic, and some orthotropic elasticity tensors.) We also describe ψ^{∗}C(X) for most other types of anisotropic media, for certain ψ ∈ Ψ. In particular, we describe the type of anisotropy of ψ^{∗}C via its material symmetry group with respect to the Euclidean metric.

The material symmetry group G associated with an elastic medium is defined to be the subgroup of G-orthogonal transformations Q : T_{X}Ω → T_{X}Ω (i.e. with Q^{T} = Q^{−1}, where, in coordinates, Q^{T} (X)^{A}_{B} = G_{BB′}Q^{B′}_{A′}G^{A′A}) whose action on the reference configuration does not change the material response. In particular, the symmetry group Sym_{e}T at X ∈ Ω of an m-tensor field T on Ω (e.g. an elasticity tensor C) is defined to be the collection of e-orthogonal transformations Q of T_{X}Ω (i.e. with Q^{T} (X)^{A}_{B} = δ_{BB′}Q^{B′}_{A′}δ^{A′A}) such that

Q ∗ T(X) = T(X),

where for any 3 × 3 matrix M (which we identify with a (1, 1)-tensor) we define M ∗ T(X) to be the m-tensor with components

(16) (M ∗ T(X))^{ABC... D} = M^{A}\underline{A} M^{B}\underline{B} M^{C} \underline{C} … M^{D}\underline{D} T(X)^{\underline{A} \underline{B} \underline{C}... \underline{D}}.

We remark that the ∗ operation respects the symmetries of T for any invertible matrix M; in particular, it defines an action of the orthogonal group O(3) on the space of elasticity tensors.

In addition, the ∗ operation defines an action of O(3) on the symmetry groups of elasticity tensors. In fact, the action of Q ∈ O(3) on the symmetry group of a tensor T is given by the symmetry group of Q ∗ T :

(17)  Sym_{e}(Q ∗ T) = Q( Sym_{e}T)Q^{T} = Q ∗ ( Sym_{e}T)

since \overline{Q} ∈ Sym_{e}T if and only if Q\overline{Q}Q^{T} ∗ (Q ∗ T) = Q ∗ T. Here we write M ∗ G = MGM^{T} in terms of conjugation by M for any rank-2 tensor G, and we write

(18)  M ∗ G = M G M^{T} = {MGM^{T} | G ∈ G }

Table 1. Classification of Anisotropic Elastic Media

Generators of

i  Symmetry Class  SO(3) ∩ the symmetry group G_{i}
1  triclinic  I
2  monoclinic  Q_{e_{3}}^{π} (cyclic group Z_{2})
3  orthotropic  Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π} (dihedral group D_{2})
4  trigonal  Q_{e_{1}}^{π}, Q_{e_{3}}^{2π/3} (dihedral group D_{3})
5  tetragonal  Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π/2} (dihedral group D_{4})
6  cubic  Q_{e_{1}}^{π/2}, Q_{e_{2}}^{π/2}, Q_{e_{3}}^{π/2} (octahedral group O)
7 transversely isotropic Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ} (orthogonal group O(2))
8  isotropic  (special orthogonal group SO(3))

for any collection of linear transformations G.

Given an elasticity tensor C with a certain type of anisotropy (one of eight types, given [F-V]), there exists a right-handed orthonormal basis {e_{1}, e_{2}, e_{3}} so that the symmetry group of C is generated by −I and the orthogonal transformations listed in Table 1. (See [C-V-C], for example.) The generators in Table 1 have the form Q_{e_{j}}^{ϕ}, j = 1, 2, 3, where Q_{e_{j}}^{ϕ} is the rotation of angle ϕ about e_{j}; e.g., Q_{e_{3}}^{ϕ}(xe_{1}+ye_{2}+ze_{3}) = [x cos ϕ−y sin ϕ]e_{1}+ [y cos ϕ + x sin ϕ]e_{2} + ze_{3}. Throughout, sg{Q_{e_{i}}^{π}, Q_{e_{j}}^{π}} denotes the subgroup of O(3) generated by Q_{e_{i}}^{π}, Q_{e_{j}}^{π}. These generators are not independent in the sense that, for example, sg{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}} = sg{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} since Q_{e_{1}}^{π} ∘ Q_{e_{2}}^{π}(xe_{1} + ye_{2} + ze_{3}) = Q_{e_{1}}^{π}(−xe_{1} + ye_{2} − ze_{3}) = (−xe_{1} − ye_{2} + ze_{3}) = Q_{e_{3}}^{π}(xe_{1} + ye_{2} + ze_{3}).

Two elasticity tensors C_{1} and C_{2} have the same anisotropy type at X ∈ Ω if and only if there exists an orthogonal transformation Q such that

C_{1}(X) = Q ∗ C_{2}(X).

To describe the symmetry group of ψ^{∗}C for ψ ∈ Ψ at a fixed point X ∈ Ω, we write Dψ^{−1}(X) in terms of its polar decomposition with the positive, symmetric S diagonalized as S = UDU^{T} . Here U, V are orthogonal, and D = diag (a, b, c) is a diagonal 3 × 3 matrix with a, b, c the (positive) eigenvalues of S. Without loss of generality, we assume either that a = b = c, that a = b ≠ c, or that the a, b, c are distinct.

We observe that at X ∈ Ω

(19) ψ^{∗}C = (Dψ^{−1}) ∗ (C ∘ ψ) = UDU^{T} V ∗ (C ∘ ψ) = UD ∗ C̄ = U ∗ ˆ C̄,

where

(20)  C̄ = U^{T} V ∗ (C ∘ ψ) and ˆ C̄ = D ∗ C̄.

It follows, given (17), that we may reduce the task of describing Sym_{e}ψ^{∗}C to that of describing Sym_{e}(D ∗ C̄) for any elasticity tensor C̄.

Proposition 8. Let C be an admissible elasticity tensor on Ω. For X ∈ Ω and ψ ∈ Ψ with polar decomposition at X in the form described above

(21)  Sym_{e}ψ^{∗}C = U ∗ Sym_{e}(D ∗ C̄),
(22)  Sym_{e}C̄ = U^{T} V ∗ Sym_{e}(C ∘ ψ).

We note that the action on Sym_{e}C̄ of the pullback by D is accomplished at each point X ∈ Ω by varying only two parameters a/c and b/c.

A partial description of the relationship at X ∈ Ω between Sym_{e}C̄ and Sym_{e}(D ∗ C̄) is given in the following proposition.

Proposition 9. Let C̄ be an admissible elasticity tensor on Ω. Let D be a 3 × 3 matrix. Then

(23)  Sym_{e}C̄ ∩ Sym_{e}D ⊆ Sym_{e}(D ∗ C̄).

Proof. For Q ∈ Sym_{e}C̄ ∩ Sym_{e}D we have Q ∗ (D ∗ C̄) = QD ∗ C̄ = DQ ∗ C̄ = D ∗ C̄. That is, Q ∈ Sym_{e}(D ∗ C̄). □

The simplest situation in (23), and the one we consider here, occurs when S is diagonal with respect to the basis {e_{1}, e_{2}, e_{3}} used to describe Sym_{e}V ∗ (C̄ ∘ ψ). The computational complexity of the other cases is significantly greater. This motivates the following definition. We say that ψ is aligned with C (at X′ ∈ Ω) if there is a polar decomposition of Dψ^{−1} and diagonalization of the stretch factor S as described above so that, with respect to the standard basis for Euclidean space, the generators of Sym_{e}C̄(X′) are listed in Table 1. We observe below that for certain aligned ψ and C equality holds in (23).

Remark 10. For C isotropic, every ψ ∈ Ψ is aligned with C.

Without loss of generality, we set c = 1 below.

Theorem 11 (Computing D∗C̄ in the aligned case). Let C̄ be an admissible isotropic, transverse isotropic, cubic, or tetragonal elasticity tensor on Ω represented by parameters (ᾱ, β̄, Ā^{11}, B̄^{11}, ᾱ_{0}, ᾱ_{4}) via the harmonic-Cartan decomposition (51) and (56). Suppose that, with respect to the standard basis for Euclidean space, the generators of Sym_{e}C̄ are listed in Table 1. Let D = diag (a, b, 1) be a diagonal 3×3 matrix with a, b positive and a = b ≠ 1 or a, b, 1 distinct. Then via the harmonic-Cartan decomposition we may represent D ∗ C̄ by (ˆᾱ, ˆ β̄, ˆ Ā^{11}, ˆ B̄^{11}, ˆ ᾱ_{0}, ˆ ᾱ_{2}, ˆ ᾱ_{4}), and the (scaled) parameters (15ˆᾱ, 15ˆβ̄, 21ˆĀ^{11}, 21ˆĀ^{22}, 21ˆB̄^{11}, 21ˆB̄^{22}, 280ˆᾱ_{0}, 14ˆᾱ_{2}, 8ˆᾱ_{4}) for D ∗ C̄ are given by

(24)  \begin{matrix}ᾱ · Pᾱ(a, b) + β̄ · Pβ̄(a, b) + Ā^{11} · PĀ_{11}(a, b) \\ + B̄^{11} · PB̄_{11}(a, b) + ᾱ_{0} · Pᾱ_{0}(a, b) + ᾱ_{4} · Pᾱ_{4}(a, b), \end{matrix}

where the P (a, b), defined by (73), are linearly independent for fixed a, b. In fact, Sym_{e}C̄ ⊈ Sym_{e}D implies

(25)  Sym_{e}C̄ ∩ Sym_{e}D = Sym_{e}(D ∗ C̄).

Remark 12. Note that in the case that a = b = c we have that D∗C̄ = a^{4}C̄, and so the pullback by ψ preserves the type of anisotropy of the elasticity tensor.

The proof of Theorem 11, given in Section 5, is based on the harmonic decomposition (51) of the elasticity tensor C̄, followed by the Cartan decomposition (56) of the fourth-order, harmonic part H̄ of C̄, as described in [F-V]. Due to this decomposition we may write exact formulas for the (an)isotropic elasticity tensors C̄ and D ∗ C̄. We then use the mathematical software Maple to compute the harmonic-Cartan decomposition of D∗C̄, and analyze the (possible) vanishing of certain terms in order to relate Sym_{e}C̄ with Sym_{e}D ∗ C̄.

Below we relate the type of (an)isotropy in the aligned case of isotropic, transversely isotropic, cubic, and tetragonal C̄ and the corresponding D ∗ C̄.

Corollary 13 (Characterizing the range of the D∗ operation on some aligned C̄). Suppose that C̄ satisfies the conditions of Theorem 11. Then:

i). For C̄ isotropic and a = b ≠ 1, the range of D ∗ C̄ is a 3-parameter family of the form ᾱ · Pᾱ(a, b) + β̄· Pβ̄(a, b) in the 5-parameter family of transverse isotropic elasticity tensors.

ii). For C̄ cubic and a = b ≠ 1, the range of D ∗ C̄ is a 4-parameter family of the form ᾱ·Pᾱ(a, b)+β̄·Pβ̄(a, b)+ᾱ_{0}·[Pᾱ_{0}(a, b)+5Pᾱ_{4}(a, b)] in the 5-parameter family of tetragonal elasticity tensors.

iii). If a, b, 1 are distinct, then for isotropic, transverse isotropic, cubic, or tetragonal C̄, the range of D∗C̄ is a family of orthotropic elasticity tensors of the form (24).

Corollary 13 is a consequence of Theorem 11 and Section 5.2, and is derived in Section 5.4. The following proposition is also proved in Section 5.4.

Proposition 14. If C is isotropic, transverse isotropic, or orthotropic, and ψ is aligned with C at X′ ∈ Ω, then ψ^{−1} is aligned with ψ^{∗}C at X = ψ(X′).

Corollary 15 (D∗ preserves transverse isotropy for some aligned C̄(X)). For X ∈ Ω there is a(n at least two-parameter) non-trivial family of transverse isotropic C̄(X) with the property that D∗C̄(X) remains transverse isotropic when a = b ≠ 1.

3. Definitions

We follow Marsden and Hughes [M-H] to represent an elastic body as a Riemannian manifold (Ω, G). Let (S, g) be a (fixed) Riemannian manifold of the same dimension representing the ambient space within which the body Ω moves. In this paper S = ℝ^{3} and for applications we take g to be the Euclidean metric. We write {X^{A}} for coordinate systems on Ω, and {x^{a}} for coordinate systems on the ambient space S.

General tensor notation. We follow notational conventions for tensor analysis as in [M-H]. In particular, tensor fields are generally written in boldface and their components in standard font, e.g. the so-called deformation gradient F is a tensor field with components F^{a}_{A}.

A two-point tensor T of type ( \begin{matrix}q & l \\ p & m \end{matrix} ) at X ∈ Ω over a mapping ϕ : Ω → S is a multilinear mapping into ℝ from the product space consisting of p copies of T_{X}^{∗}Ω, q copies of T_{X}Ω, l copies of T_{x}^{∗}S, and m copies of T_{x}S, where x = ϕ(X). The components of T are denoted by T^{A_{1}...A_{p}}_{B_{1}...B_{q}}^{a_{1}...a_{l}}_{b_{1}...b_{m}}, where, in general, upper-case indices denote tensorial behavior with respect to the referential setting, lower-case indices with respect to the spatial setting, and the pullback of a two-point tensor by ψ : Ω → Ω interacts only with indices corresponding to the reference configuration (i.e. the upper-case indices). We use the terminology two-point vector for a two-point tensor of

type ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ).

The non-linear case. The deformation of an elastic body is modeled by mappings of the form ϕ(X, t) = ϕ_{t}(X) : Ω × (0, T) → S. In fact, we call a mapping ϕ : Ω → S a configuration of Ω in S, and we denote by C the set of all configurations of Ω in S. A motion ϕ_{t} : Ω → S of the body Ω in S is a curve in C parametrized by t, where t ranges over some open interval of ℝ. A motion ϕ_{t} is C^{2} regular if ϕ_{t} is invertible on ϕ_{t}(Ω) for each t, and if ϕ_{t}(X) and ϕ_{t}^{−1}(X) are C^{2} as functions of t and X ∈ Ω. We write x = ϕ_{t}(X).

For any fixed diffeomorphism ψ ∈ Ψ of Ω̄ we write

ϕ′_{t} = ψ^{∗}ϕ_{t}, X′ = ψ^{−1}(X), G′ = ψ^{∗}G,

for any ϕ_{t} that is a C^{2} regular motion of Ω in S, where t ∈ [0, T] with 0 < T < ∞. We remark that ϕ′_{t} is also a C^{2} regular motion of Ω in S.

The Cauchy stress tensor σ(x, t) is a \binom{2}{0} tensor field which has the property that the Cauchy stress vector t(x, t, n) = σ ·n represents the force per unit (of deformed) area exerted on a surface element that is oriented with unit normal vector n(x). The first Piola-Kirchhoff stress tensor P(X, t) is the Piola transform (on the second index) of the Cauchy stress tensor σ; that is,

(26)  P^{aA}(X) = J(X, t) (F^{−1})^{A}_{b} σ^{ab},

where

(27) F(X, t) = Dϕ_{t}(X), with F(X, t)^{a}_{A} = Dϕ_{t}(X)^{a}_{A} = \frac{∂ϕ_{t}^{a}}{∂X^{A}}

is the deformation gradient (with respect to a motion ϕ_{t} of Ω), and the scalar

(28)  J(X, t) = \begin{matrix}√ & \overline{det g} \\ √ & \overline{det G} \end{matrix} det (Dϕ_{t})

is the Jacobian of the linear transformation F.

The second Piola-Kirchhoff stress tensor S(X, t) is the pull-back of the first leg of P via ϕ_{t}, i.e. S^{AB} = (F^{−1})^{A}_{a}P^{aB}.

A material is called hyperelastic if there exists a scalar functional W = Ŵ(X, G, C^{♭}) such that

Ŝ^{AB} = 2ρ \frac{∂Ŵ}{∂C^{♭}_{AB}} ,

where C^{♭} = ϕ_{t}^{∗}g is related to the deformation tensor C = F^{T} F by C^{A}_{B} = G^{A\underline{A}}C^{♭}_{\underline{A}B}. W is called the stored energy function, and corresponds to the internal energy per unit mass E = Ê(X, G, C^{♭}) at thermal equilibrium.

It follows from the constitutive equations that the elasticity tensor C = Ĉ(X, G, C^{♭}), which measures the dependence of stress on the strain, may be written as

(29)  C^{ABCD} = 2ρ \frac{∂^{2}Ê}{∂C^{♭}_{AB}∂C^{♭}_{CD}} .

That is, in this case C depends on the motion ϕ_{t}, and so we view G and O = (E, ρ) as the material parameters of the elastic object.

We say that ϕ_{t} solves the initial-boundary-value problem for non-linear hyperelasticity if, for given Dirichlet boundary data f,

(30a)  F_{G,E,ρ}(ϕ_{t}) = ρA − DIV_{G} P = 0  on Ω,
(30b)  ϕ_{t}(X) = X + f(X, t)  for X ∈ ∂Ω,
(30c)  ϕ_{t}(X) = 0  for t = 0,
(30d)  V(X, t) = 0  for t = 0,

where V(X, t) = ∂ϕ_{t}/∂t is the material velocity vector field, and A(X, t), the acceleration vector field, is given by

(31)  A^{a} = ∂V^{a}/∂t + (γ_{bc}^{a} ∘ ϕ_{t})V^{b}V^{c}

on Ω. Throughout, γ_{bc}^{a} and Γ_{BC}^{A} denote (respectively) the Christoffel symbols for the spatial metric g and referential metric G. The divergence of a twopoint tensor P = (P^{aA}) is defined in the referential setting by

(32)  (DIV_{X} P)^{a} = \frac{∂P^{aA}}{∂X^{A}} + Γ\underline{A} ; \underline{A}A P^{aA} + (γ_{\underline{b} \underline{a}}^{a} ∘ ϕ)F\underline{b}_{A}P^{\underline{a}A},

as in [M-H].

We refer to [M-H], p. 401, 406, for a discussion of existence and uniqueness for the non-linear initial-boundary-value problem.

The surface measurements used to determine the material parameters G, E, ρ of the elastic object are modeled by the dynamic Dirichletto-Neumann map Λ_{G,E,ρ}. The Dirichlet-to-Neumann map gives the correspondence (when there exist unique solutions of (30)) between the surface displacement and the surface traction taken over a finite period of time. Following [Sy-U], we view the Dirichlet-to-Neumann map as a vectorvalued form representing the flux, given in the case of non-linear elasticity by

(33)  (Λ_{G,E,ρ} f)^{a} = ⟨P, ν_{G}⟩^{a} dS_{G} = P^{aA} G_{AB} ν_{G}^{B} dS_{G}

for t ∈ [0, T], where dS_{G} is the area form on ∂Ω induced by G, ν_{G} is the unit outer normal to ∂Ω with respect to G, and the motion ϕ_{t} solves (30) for boundary values f. By Corollary 2 the Dirichlet-to-Neumann map may also be written with respect to the Euclidean metric as

(34)  (Λ_{G,E,ρ} f)^{a} = P^{aA} δ_{AB} ν^{B} \sqrt{det G} dS,

where dS is the area form on ∂Ω, and ν is the unit outer normal vector to ∂Ω, each with respect to the Euclidean metric.

The linearized case. We linearize about an arbitrary C^{2}-regular motion ∘ ϕ_{t} which solves (30a), as in [M-H], Section 4.2. The initial-boundary-value problem in this case is given by

(35a)  P_{G,^{∘} A,^{∘} C,ρ,^{∘} S,^{∘} P} U = 0  on Ω,
(35b)  U(X, t) = f(X, t)  for X ∈ ∂Ω,
(35c)  U(X, t) = 0  for t = 0,
(35d)  ∂U(X, t)/∂t = 0  for t = 0,

with Dirichlet boundary data f, where the operator P_{G,^{∘} A,^{∘} C,ρ,^{∘} S,^{∘} P} is given by

(36) P_{G,^{∘} A,^{∘} C,ρ,^{∘} S,^{∘} P} U = ρ(\frac{∂^{2}U(X, t)}{∂t^{2}} + ^{∘} A) − DIV_{G}(^{∘} A · ∇U + ^{∘} P).

Here ρ(X) is the mass density in the reference configuration; U, the infinitesimal displacement, is a vector field over the motion ^{∘} ϕ_{t} (that is, a two-point vector over ^{∘} ϕ_{t}); and ^{∘} A, the first elasticity tensor, is given by

∘ A^{aA}_{b}^{B} = 2^{∘} C^{EAF B} ^{∘} F^{a}_{E} ^{∘} F^{c}_{F} ^{∘} g_{bc} + ^{∘}S^{AB}δ^{a}_{b}.

In addition, ^{∘} A(X, t) is the material acceleration, ^{∘} C is the (second) elasticity tensor defined in (29), ^{∘} F is the deformation gradient, and ^{∘} P and ^{∘} S are the first and second Piola-Kirchhoff stress tensors, respectively, all associated with ^{∘} ϕ_{t}.

We emphasize that U does not behave like a vector field on Ω; in fact, the covariant derivative ∇ (w.r.t G) of the two-point vector U over ^{∘} ϕ_{t} is given by

(37)  (∇U)^{b}_{B} = ∂U^{b}/∂X^{B} + (^{∘}γ_{ac}^{b} ∘ ^{∘} ϕ_{t}) U^{a} F^{c}_{B}.

The Dirichlet-to-Neumann map is defined in this case by

(38)

(Λ_{G,^{∘} A,^{∘} C,ρ,^{∘} S,^{∘} P} f)^{a} = ⟨^{∘} A · ∇U + ^{∘} P, ν⟩^{a} dS_{G}
= [^{∘} A^{aA}_{b}^{B} (∇U)^{b}_{B} + ^{∘} P^{aA}] G_{AC} ν_{G}^{C} dS_{G}

for t ∈ [0, T], where, again, dS_{G} is the area form on ∂Ω induced by G, ν_{G} is the unit outer normal to ∂Ω with respect to G, and U solves (35) with boundary data f.

By Corollary 2 the Dirichlet-to-Neumann map may also be written with respect to the Euclidean metric as

(39) (Λ_{G,^{∘} A,^{∘} C,ρ,^{∘} S,^{∘} P} f)^{a} = [^{∘} A^{aA}_{b}^{B} (∇U)^{b}_{B} + ^{∘} P^{aA}] δ_{AC} ν^{C} \sqrt{det G} dS,

where, again, dS is the area form on ∂Ω, and ν is the unit outer normal vector to ∂Ω, each with respect to the Euclidean metric.

The classical linear case. Linearizing about a stress-free, undeformed state ^{∘} ϕ(X, t), which we choose to be an embedding ι : Ω ↪ ℝ^{3}, results in the classical linear equations (4) and the associated Dirichlet-to-Neumann map (10). (See (6) and (12) for their representation in Euclidean coordinates.) In fact, in this case, ^{∘} S, ^{∘} P, and ^{∘} A vanish, so that (36) reduces to

P_{G,C,ρ} U = ρ\frac{∂^{2}U}{∂t^{2}} − DIV_{G}(A · ∇U).

If the spatial metric g is Euclidean, as we take in applications, then ∇ acts on U as if U were a scalar, and the equations of motion, in local coordinates, are

ρ\frac{∂^{2}U^{a}}{∂t^{2}} − \frac{1}{\sqrt{det G}} \frac{∂}{∂X^{A}} (\sqrt{det G} A^{aA}_{b}^{B} \frac{∂U^{b}}{∂X^{B}} ) = 0.

where A depends on the elasticity tensor C in this case simply via A^{aA}_{b}^{B} = 2C^{EAF B}(Dι)^{a}_{E}(Dι)^{c}_{F} δ_{bc}.

Given any smooth diffeomorphism ψ of Ω, we observe that ^{∘} ϕ′(X, t) = ∘ ϕ(ψ(X), t) is also stress-free, since the stress tensor S associated with a motion is covariant (see Section 4), so ^{∘} S′ = ψ^{∗}^{∘} S = 0 since ^{∘} S = 0.

It follows that when the equations of motion are rewritten in Euclidean coordinates, there are no “lower-order” terms arising from the Christoffel symbols, and so the equations of motion continue to describe an elastic response, but with respect to modified physical parameters. Therefore, as in the analysis of the wave equation, we may reduce anisotropic elasticity to the isotropic case by pulling back by ψ, at least for certain anisotropic media.

4. Proof of Theorem 1

We begin with the non-linear case. We model the behavior of the hyperelastic body (Ω, G, E, ρ) by the initial-boundary-value problem (30). We will first show that the equations of motion (30a) are covariant; that is, given a sufficiently regular motion ϕ_{t} and any diffeomorphism ψ : Ω̄ → Ω̄, we show that

(40)  ψ^{∗}(F_{G,E,ρ} ϕ_{t}) = F_{G′,E′,ρ′} ϕ′_{t},

where X′ = ψ^{−1}(X), ϕ′(X′, t) = ϕ(X, t), G′ = ψ^{∗}G, E′ = ψ^{∗}E = E ∘ ψ, and ρ′ = ψ^{∗}ρ = ρ ∘ ψ.

It is established [M-H] that the equations of elastodynamics are covariant in Eulerian coordinates. Below we extend the argument to Lagrangian coordinates, with special consideration taken for the anisotropy of the energy E in Lagrangian coordinates.

To prove (40) we begin by writing the material acceleration, the first Piola-Kirchhoff stress tensor, and the second Piola-Kirchhoff stress tensor associated with (ψ^{∗}G, ψ^{∗}E, ψ^{∗}ρ) and ϕ_{t}′ as A′, P′, and S′, respectively. Note that the representation (31) of A in terms of the velocity V, which is given in terms of ϕ_{t}, implies that A′(X, t) = A(ψ(X), t) = ψ^{∗}A. (In particular, A behaves like a scalar with respect to ψ.) In addition, S^{′AB} = 2ρ′ (∂(E ∘ ψ)/∂(ψ^{∗}C^{♭}))^{AB} since C^{♭}′ = ϕ_{t}^{′∗}e = (ϕ_{t} ∘ ψ)^{∗}e = ψ^{∗}C^{♭}.

Then ψ^{∗}(F_{G,E,ρ} ϕ_{t}) = ρ′A′ − ψ^{∗}(DIV_{G}P) = ρ′A′ − DIV_{ψ^{∗}G}ψ^{∗}P by the Piola transform ([M-H], pp. 116–118). We conclude that (40) follows if P′ = ψ^{∗}P.

Now P^{aB} = F^{a}_{A}S^{AB}, and F^{′a}_{C} = ∂ϕ^{′a}(X′, t)/∂X^{′C} = (F^{a}_{A} ∘ ψ)(Dψ)^{A}_{C} = (ψ^{∗}F)^{a}_{C}, so (ψ^{∗}P)^{aB′} = (ψ^{∗}F)^{a}_{C}(ψ^{∗}S)^{CB′} = F^{′a}_{C}(ψ^{∗}S)^{CB′}. Therefore, to show ψ^{∗}P = P′ we will show ψ^{∗}S = S′.

Under the constitutive hypothesis for hyperelastic media, the free energy function E = \overline{E}(X, G, C^{♭}) is related to S = S̄(X, G, C^{♭}) by

(41)  S^{AB} = 2ρ \frac{∂\overline{E}}{∂C^{♭}_{AB}} .

(See [M-H], Proposition 4.6.) In addition, the balance laws (conservation of mass, balance of momentum, balance of moment of momentum, and balance of energy) that describe the behavior of hyperelastic media are equivalent to the covariance of the energy [M-H], Theorem 4.13. But the referential energy E is covariant, in the case that E depends functionally just on the point X and the Cauchy-Green deformation tensor C, only if the elastic medium is isotropic.

To include anisotropic media in the discussion we first recall that any particular type of material anisotropy may be characterized by its material symmetry group. (See Section 2.2.) We may associate with any material symmetry group G a finite collection of structural tensors ζ_{1}, …, ζ_{m} which form a basis for the space of invariant tensors under the action of G; that is, the structural tensors ζ_{i} (of order k_{i}), i = 1, …, m, satisfy

Q ∈ G iff ⟨Q⟩_{k_{1}}ζ_{1} = ζ_{1}, …, ⟨Q⟩_{k_{m}}ζ_{m} = ζ_{m},

where the Kronecker product ⟨Q⟩_{k} of an orthogonal transformation Q is defined by

(42)  (⟨Q⟩_{k}ζ)^{i...j} = Q^{i}\underline{i} ⋯ Q^{j}\underline{j}ζ^{\underline{i}...\underline{j}}

for any tensor ζ of order k, with Q^{i}_{j}, ζ^{i...j} the Cartesian components of Q and ζ.

Therefore, we write, given [M-H], p. 220, [L-P], and [Z-Sp], the energy E as a function E = Ê(G, C^{♭}, ζ_{1}, …, ζ_{m}) of G, C^{♭}, and the structural tensors ζ_{1}, …, ζ_{m}. It follows from (41) that S^{AB} = 2ρ (∂Ê/∂C^{♭})^{AB} = 2ρ ∂Ê/∂C^{♭}_{AB}.

Given any diffeomorphism ψ : Ω̄ → Ω̄, we define the function Ě of Ḡ, C̄^{♭}, ζ̄_{1}, …, ζ̄_{m} by Ě(Ḡ, C̄^{♭}, ζ̄_{1}, …, ζ̄_{m}) = [Ê(ψ_{∗}Ḡ, ψ_{∗}C̄^{♭}, ψ_{∗}ζ̄_{1}, …, ψ_{∗}ζ̄_{m})] ∘ ψ. Then, due to the covariance of the energy, ψ^{∗}E = E ∘ ψ = Ě(ψ^{∗}G, ψ^{∗}C^{♭}, ψ^{∗}ζ_{1}, …, ψ^{∗}ζ_{m}). It follows that

(43)  ψ^{∗}(\frac{∂Ê}{∂C^{♭}} )^{AB} = (Dψ^{−1})^{A}\underline{A}(Dψ^{−1})^{B}\underline{B} · (\frac{∂Ê}{∂C^{♭}} )^{\underline{A} \underline{B}} ∘ ψ

and

(44)  \begin{aligned}(\frac{∂Ê}{∂C^{♭}} )^{\underline{A} \underline{B}}∘ ψ & = & (\frac{∂Ě}{∂(ψ^{∗}C^{♭})})^{A′B′} · \frac{∂(ψ^{∗}C^{♭} ∘ ψ^{−1})_{A′B′}}{∂C_{\underline{A} \underline{B}}^{♭}} \\ & = & (\frac{∂Ě}{∂(ψ^{∗}C^{♭})})^{A′B′} · (Dψ)\underline{A}_{A′}(Dψ)\underline{B} _{B′}. \end{aligned}

Consequently, ψ^{∗}S = S′ since

(45)  ψ^{∗}S = 2(ρ ∘ ψ) ψ^{∗}(\frac{∂Ê}{∂C^{♭}} ) = 2ρ′ \frac{∂Ě}{∂(ψ^{∗}C^{♭})} = S′.

The Dirichlet-to-Neumann maps agree in this case because ψ ∈ Ψ is the identity at the boundary to first order, which implies that G′ = ψ^{∗}G is G at ∂Ω and P′ = ψ^{∗}P is P at ∂Ω. That is, Λ_{G,E,ρ} f = ⟨P, ν⟩ \sqrt{det G} dS agrees with Λ_{G′,E′,ρ′} f = ⟨P′, ν⟩ \sqrt{det G′} dS at ∂Ω.

The linearized case. In this case we model the displacement of the hyperelastic body by the initial-boundary-value problem (35). To check that the equations of motion (35a) are covariant, we consider a motion ∘ ϕ_{t} : (Ω, G) → (S, g) of an elastic body (Ω, G, ^{∘} E,^{∘}ρ). Given a diffeomorphism ψ ∈ Ψ, we define ^{∘} ϕ′(X′, t) : (Ω, ψ^{∗}G) → (S, g) by ^{∘} ϕ′(X′, t) = ^{∘} ϕ(ψ(X′), t), where X′ = ψ^{−1}(X). It follows from the proof of covariance for the nonlinear case that ^{∘} ϕ′ is a motion of the transformed elastic body (Ω, ^{∘} G′, ^{∘} E′, ^{∘} ρ′), where ^{∘} G′ = ψ^{∗} ^{∘} G, ^{∘} ρ′ = ^{∘} ρ ∘ ψ, and ^{∘} E′ = ^{∘} E ∘ ψ. The same proof shows that ∘ F′ = ψ^{∗}F, ^{∘} P′ = ψ^{∗} ^{∘} P, and ^{∘} S′ = ψ^{∗}^{∘} S. Also, ^{∘} C′ = ψ^{∗} ^{∘} C by (29) and the reasoning in (43) and (45).

It follows that ^{∘} A′ = ψ^{∗} ^{∘} A since A depends tensorially on C, F, and the metric. Explicitly, in components: (46)

∘ A′^{aA′}_{b}^{B′} = 2 ^{∘} C^{′E′A′F′B′} ^{∘} F^{′a}_{E′} ^{∘} F^{′c}_{F′}(^{∘} g_{bc} ∘ ϕ′) + ^{∘} S^{′A′B′}(δ^{a}_{b} ∘ ϕ′)
= 2(Dψ^{−1})^{E′}_{E}(Dψ^{−1})^{A′}_{A}(Dψ^{−1})^{F′}_{F} (Dψ^{−1})^{B′}_{B}(^{∘} C^{EAF B} ∘ ψ)
· (^{∘} F^{a}_{E} ^{∘} F^{c}_{F}) ∘ ψ(Dψ)^{E}_{E′}(Dψ)^{F}_{F′}(^{∘} g_{bc} ∘ ϕ) ∘ ψ
+ (Dψ^{−1})^{A′}_{A}(Dψ^{−1})^{B′}_{B}(^{∘} S^{AB} ∘ ψ) (δ^{a}_{b} ∘ ϕ) ∘ ψ
= (Dψ^{−1})^{A′}_{A}(Dψ^{−1})^{B′}_{B} (^{∘} A^{aA}_{b}^{B} ∘ ψ).

We conclude that

(47)  \begin{aligned}[∇_{ψ^{∗}G}(U ∘ ψ)]^{b}_{B′} & = & \frac{∂(U^{b} ∘ ψ)}{∂X^{′B′}} + (γ_{\underline{a}c}^{b} ∘ ϕ′)(U\underline{a} ∘ ψ)F^{′c}_{B′} \\ & = & [(∇_{G}U)^{b}_{B} ∘ ψ] (Dψ)^{B}_{B′}, \end{aligned}

where [∇_{G}U]^{b}_{B} = ∂U^{b}/∂X^{B} + (γ_{ac}^{b} ∘ ϕ) U^{a} F^{c}_{B} is the covariant derivative of the two-point vector U. Therefore, the linearized stress [^{∘} A · ∇_{G}U]^{aA} = ∘ A^{aA}_{b}^{B}[∇_{G}U]^{b}_{B} satisfies [^{∘} A′ · ∇_{G′}U′]^{aA′} = ^{∘} A′^{aA′}_{b}^{B′}[∇_{ψ^{∗}G}(U ∘ ψ)]^{b}_{B′} = (Dψ^{−1})^{A′}_{A}([^{∘} A · ∇_{G}U]^{aA} ∘ ψ) = (ψ^{∗}[^{∘} A · ∇_{G}U])^{aA′}, where U′ = U ∘ ψ.

For completeness we check below that (DIV_{X′}[^{∘} A′ · ∇_{G′}U′])^{a} = (DIV_{X}[^{∘} A · ∇_{G}U])^{a} ∘ ψ. (See (32) for the definition of the divergence of a twopoint tensor.) We first define ϵ_{ABC} = 1 if A, B, C is an even permutation of 1, 2, 3, ϵ_{ABC} = −1 if A, B, C is an odd permutation of 1, 2, 3, and ϵ_{ABC} = 0 otherwise. Then the cofactor matrix Cof M of a 3 × 3 matrix M is given by ( Cof M)^{A′}_{A} = (1/2)ϵ_{ABC} ϵ^{A′B′C′} M^{B}_{B′} M^{C}_{C′}. It follows that ∂/∂X^{′A′}( Cof Dψ)^{A′}_{A} = ∑_{B,C,C′} ϵ_{ABC} · 0 · ∂X^{C}/∂X^{′C′} = 0. Also, ∑_{A} Γ_{AB}^{A} = ∂/∂X^{B}(log \sqrt{det G}).

Writing M^{−1} = Cof M/det M and ∂/∂X^{′A′} = (Dψ)^{A}_{A′}(∂/∂X^{A}), we may therefore conclude that covariance holds, since

(48)

(DIV_{X′}[^{∘} A′ · ∇_{G′}U′])^{a}
= \frac{∂}{∂X^{′A′}} [^{∘} A′ · ∇_{G′}U′]^{aA′} + Γ\begin{matrix}′ & \underline{A} \\ \underline{A} & A′ \end{matrix} [^{∘} A′ · ∇_{G′}U′]^{aA′}
+ (γ_{\underline{b} \underline{a}}^{a} ∘ ϕ′)^{∘} F^{′\underline{b}}_{A′}[^{∘} A′ · ∇_{G′}U′]^{\underline{a}A′}

(49)

= \frac{∂}{∂X^{′A′}} [(Dψ^{−1})^{A′}_{A}([^{∘} A · ∇_{G}U]^{aA} ∘ ψ)]

+ (\frac{∂}{∂X^{′A′}} log \sqrt{det ψ^{∗}G})(Dψ^{−1})^{A′}_{A}([^{∘} A · ∇_{G}U]^{aA} ∘ ψ)

+ ((γ_{\underline{b} \underline{a}}^{a} ∘ ϕ) ∘ ψ) (^{∘} F\underline{b}_{A} ∘ ψ) ([^{∘} A · ∇_{G}U]^{\underline{a}A} ∘ ψ)

= \frac{∂}{∂X^{′A′}} [(det Dψ^{−1}) Cof (Dψ)^{A′}_{A} ([^{∘} A · ∇_{G}U]^{aA} ∘ ψ)]

+ (\frac{∂}{∂X^{′A′}} log(det Dψ) + (Dψ)^{A}_{A′}Γ\underline{A} ; \underline{A}A)

·(Dψ^{−1})^{A′}_{A}([^{∘} A · ∇_{G}U]^{aA} ∘ ψ)

+ ((γ_{\underline{b} \underline{a}}^{a} ∘ ϕ) ^{∘} F\underline{b}_{A} [^{∘} A · ∇_{G}U]^{\underline{a}A}) ∘ ψ

= (DIV_{X}[^{∘} A · ∇_{G}U])^{a} ∘ ψ.

It follows, since ψ^{∗}(DIV_{G} ^{∘} P) = DIV_{ψ^{∗}G}ψ^{∗} ^{∘} P, that

(50)

ψ^{∗}(P_{G,^{∘} A,^{∘} E,^{∘} ρ,^{∘} S,^{∘} P} U)
= (^{∘}ρ ∘ ψ)(\frac{∂^{2}U(X, t) ∘ ψ}{∂t^{2}} + ψ^{∗} ^{∘} A) − ψ^{∗}(DIV_{G}(^{∘} A · ∇_{G}U + ^{∘} P))
= P_{G′,^{∘} A′,^{∘} E′,^{∘} ρ′,^{∘} S′,^{∘} P′} U′.

Again, the Dirichlet-to-Neumann maps agree because ψ ∈ Ψ is the identity at the boundary, which implies that G′ = ψ^{∗}G is G at ∂Ω, P′ = ψ^{∗}P is P at ∂Ω, and ^{∘} A′ · ∇_{G′}U′ = ψ^{∗}[^{∘} A · ∇_{G}U] is ^{∘} A · ∇_{G}U at ∂Ω. That is, Λ_{G′,^{∘} A′,^{∘} E′,^{∘} ρ′,^{∘} S′,^{∘} P′} f = ⟨[^{∘} A′ · ∇_{G′}U′] + ^{∘} P′, ν⟩ \sqrt{det G′} dS agrees

with Λ_{G,^{∘} A,^{∘} E,^{∘} ρ,^{∘} S,^{∘} P} f = ⟨[^{∘} A · ∇_{G}U] + ^{∘} P, ν⟩ \sqrt{det G} dS where ⟨P, ν⟩^{a} = P^{aA}δ_{AB}ν^{B}.

The classical linear case. Here we take ^{∘} ϕ_{t} to be a stress-free, undeformed state ^{∘} ϕ_{t}(X, t) = ι(X), where ι is an embedding (Ω, G) → (S, g). Therefore, ∘ S, ^{∘} P = 0, and ^{∘} A = 0. We observe that ^{∘} ϕ′(X, t) = ^{∘} ϕ(ψ(X), t) is also stressfree since the stress tensor S associated with a motion is covariant, so ^{∘} S = 0 implies ^{∘} S′ = ψ^{∗}^{∘} S = 0.

In applications, we take g = e so that γ_{bc}^{a} = 0. Then, a calculation similar to (48) along with (37) establishes the local-coordinate representation (6) of the equations of motion. □

Remark 16. For classical linear elasticity, a weak formulation of the equations of motion, in terms of Dirichlet integrals, holds (see [M-H]). In this case, as it happens with the scalar wave equation, it may be possible to use the weak formulation to obtain obstruction to uniqueness for diffeomorphisms ψ that fix the boundary to zeroth-order only; i.e., for which Dψ is not necessarily the identity at the boundary.

5. Proof of Theorem 11

5.1. The harmonic-Cartan decomposition of C̄. We write C̄ (at any point X ∈ Ω and with respect to any orthogonal basis e_{1}, e_{2}, e_{3}, here the standard basis for Euclidean space) in terms of its harmonic decomposition (as described in [F-V])

(51)  C̄ = ᾱI ⊗ I + β̄I♢^{×}I + 2I\begin{matrix}s \\ ◯ \end{matrix}Ā + 2I♢^{s}B̄ + H̄,

where ᾱ and β̄ are scalars, Ā and B̄ are symmetric, traceless 3 × 3 matrices (i.e. with trace Ā^{11} + Ā^{22} + Ā^{33} equal to zero), and H̄ is a totally symmetric rank-four tensor (i.e. H̄^{ijkl} = H̄^{σ(ijkl)}, where σ is any permutation of the (i, j, k, l) indices). In addition, H̄ is harmonic; that is, for r = xe_{1}+ye_{2}+ze_{3}, (52)

H̄[r, r, r, r] = ∑_{i,j,k,l=1}^{3} H̄^{ijkl}x^{δ_{i1}+δ_{j1}+δ_{k1}+δ_{l1}}y^{δ_{i2}+δ_{j2}+δ_{k2}+δ_{l2}}z^{δ_{i3}+δ_{j3}+δ_{k3}+δ_{l3}}

is a harmonic polynomial in x, y, z, i.e. (∂^{2}/∂x^{2} + ∂^{2}/∂y^{2} + ∂^{2}/∂z^{2})H̄[r, r, r, r] = 0. Here δ_{ij} is the Kronecker delta, and we use the notation

(53) \begin{aligned}(A\begin{matrix}s \\ ◯ \end{matrix}B)^{ijkl} & = & \frac{1}{2}(A ⊗ B + B ⊗ A)^{ijkl} = \frac{1}{2}(A^{ij}B^{kl} + B^{ij}A^{kl}) \\ A♢^{s}B & = & \frac{1}{2}(A♢^{×}B + B♢^{×}A) \\ (A♢^{×}B)^{ijkl} & = & A^{ik}B^{jl} + A^{il}B^{jk}. \end{aligned}

The harmonic decomposition (51) of C̄ may be written in terms of the components of C̄ as follows:

(54) ᾱ = \frac{1}{15}(2C̄^{ppkk} − C̄^{pkpk})

β̄ = \frac{1}{30}(−C̄^{ppkk} + 3C̄^{pkpk})
Ā^{ij} = \frac{1}{21}(15C̄^{ijkk} − 12C̄^{ikjk} − 5δ^{ij}C̄^{ppkk} + 4δ^{ij}C̄^{pkpk})
B̄^{ij} = \frac{−1}{21}(6C̄^{ijkk} − 9C̄^{ikjk} − 2δ^{ij}C̄^{ppkk} + 3δ^{ij}C̄^{pkpk})
H̄^{ijkl} = \frac{1}{3}(C̄^{ijkl} + C̄^{ikjl} + C̄^{iljk})

− \frac{1}{21}[(C̄^{ijmm} + 2C̄^{imjm})δ^{kl} + (C̄^{klmm} + 2C̄^{kmlm})δ^{ij}

+ (C̄^{ikmm} + 2C̄^{imkm})δ^{jl} + (C̄^{jlmm} + 2C̄^{imjm})δ^{ik}

+ (C̄^{ilmm} + 2C̄^{imlm})δ^{jk} + (C̄^{jkmm} + 2C̄^{imjm})δ^{il}] +

(55)  + \frac{1}{105}(C̄^{ppmm} + 2C̄^{pmpm})(δ^{ij}δ^{kl} + δ^{ik}δ^{jl} + δ^{il}δ^{jk}).

(Note the typographical error in [F-V] in the expression for β̄.)

We then write H̄ in terms of the Cartan decomposition (as described in [F-V])

(56)  H̄ = ᾱ_{0}U + ∑_{m=1}^{4} ᾱ_{m}S_{m} + β̄_{m}T _{m},

where ᾱ_{0}, ᾱ_{i}, β̄_{i}, i = 1...4, are scalars, and the harmonic tensors U, S_{i}, T _{i}, i = 1...4, have polynomial form

(57)

U[r, r, r, r] = u(x, y, z) = 8z^{4} + 3(x^{2} + y^{2})(x^{2} + y^{2} − 8z^{2})
S_{1}[r, r, r, r] = s_{1}(x, y, z) = zx(4z^{2} − 3[x^{2} + y^{2}])
T _{1}[r, r, r, r] = t_{1}(x, y, z) = zy(4z^{2} − 3[x^{2} + y^{2}])
S_{2}[r, r, r, r] = s_{2}(x, y, z) = (x^{2} − y^{2})(6z^{2} − [x^{2} + y^{2}])
T _{2}[r, r, r, r] = t_{2}(x, y, z) = 2xy(6z^{2} − [x^{2} + y^{2}])
S_{3}[r, r, r, r] = s_{3}(x, y, z) = zx(x^{2} − 3y^{2})
T _{3}[r, r, r, r] = t_{3}(x, y, z) = zy(3x^{2} − y^{2})
S_{4}[r, r, r, r] = s_{4}(x, y, z) = x^{4} + y^{4} − 6x^{2}y^{2}
T _{4}[r, r, r, r] = t_{4}(x, y, z) = 4xy(x^{2} − y^{2}).

We refer to their polynomial form (52) to compute the components of U, for example:

(58) \begin{matrix}U^{3,3,3,3} = 8, U^{1,1,1,1} = 3, U^{2,2,2,2} = 3, ∑_{σ∈S_{3}} U^{σ(1,1,3,3)} = −24, \\ ∑_{σ∈S_{3}} U^{σ(2,2,3,3)} = −24, ∑_{σ∈S_{3}} U^{σ(1,1,2,2)} = 6, \end{matrix}

where S_{3} is the symmetric group (of permutations of three elements). It follows that the components H̄^{ijkl} of H̄ are given by the following:

(59)  \begin{matrix}ij\kl & 11 & 22 & 33 \\ 11 & 3ᾱ_{0} − ᾱ_{2} + ᾱ_{4} & ᾱ_{0} − ᾱ_{4} & −4ᾱ_{0} + ᾱ_{2} \\ 22 & & 3ᾱ_{0} + ᾱ_{2} + ᾱ_{4} & −4ᾱ_{0} − ᾱ_{2} \\ 33 & & & 8ᾱ_{0} \end{matrix}

(60)  \begin{matrix}ij\kl & 12 & 23 & 13 \\ 11 & −β̄_{2}/2 + β̄_{4} & −[β̄_{1} − β̄_{3}]/4 & −[3ᾱ_{1} − ᾱ_{3}]/4 \\ 22 & −β̄_{2}/2 − β̄_{4} & −[3β̄_{1} + β̄_{3}]/4 & −[ᾱ_{1} + ᾱ_{3}]/4 \\ 33 & β̄_{2} & β̄_{1} & ᾱ_{1} \\ 12 & ᾱ_{0} − ᾱ_{4} & −[ᾱ_{1} + ᾱ_{3}]/4 & −[β̄_{1} − β̄_{3}]/4 \\ 23 & & −4ᾱ_{0} − ᾱ_{2} & β̄_{2} \\ 13 & & & −4ᾱ_{0} + ᾱ_{2} \end{matrix}

while, writing M = (ᾱ + 2β̄)I + 2(Ā + 2B̄), C̄ − H̄ is given by:

(61)  \begin{matrix}ij\kl & 11 & 22 & 33 \\ 11 & M^{11} & ᾱ + Ā^{11} + Ā^{22} & ᾱ − Ā^{22} \\ 22 & & M^{22} & ᾱ − Ā^{11} \\ 33 & & & M^{33} \end{matrix}

(62)  \begin{matrix}ij\kl & 12 & 23 & 13 \\ 11 & (Ā + 2B̄)^{12} & Ā^{23} & (Ā + 2B̄)^{13} \\ 22 & (Ā + 2B̄)^{12} & (Ā + 2B̄)^{23} & Ā^{13} \\ 33 & Ā^{12} & (Ā + 2B̄)^{23} & (Ā + 2B̄)^{13} \\ 12 & β̄ + B̄^{11} + B̄^{22} & B̄^{13} & B̄^{23} \\ 23 & & β̄ − B̄^{11} & B̄^{12} \\ 13 & & & β̄ − B̄^{22} \end{matrix}

5.2. Describing Sym_{e}C̄ in terms of Ā, B̄, ᾱ_{i}, β̄_{i}. The symmetry group of C̄ is given [F-V] in terms of the symmetry groups of Ā, B̄, and H̄ by

(63)  Sym_{e}(C̄) = Sym_{e}(Ā) ∩ Sym_{e}(B̄) ∩ Sym_{e}(H̄).

In addition, following the reasoning given in [F-V] we present in Table 2 a more precise description of Sym_{e}H̄, almost entirely in terms of the vanishing of ᾱ_{0}, ᾱ_{i}, β̄_{i}, i = 1..4. Note that each of the entries (i.e. sets of rows) of Table 2 is numbered as in Table 1 to indicate its type of (an)isotropy. In Table 2 and below we denote by Z_{n}, D_{n}, O, SO(n), O(n) the cyclic, dihedral, octahedral, special orthogonal, and orthogonal groups. We emphasize that the entries in Table 2 depend on a choice of orthogonal basis {e_{1}, e_{2}, e_{3}} since the ᾱ_{0}, ᾱ_{i}, β̄_{i} may change with a change of basis.

We observe that the polynomial forms r^{T} Ār = Ā_{11}x^{2} + 2Ā_{12}xy + … and r^{T} B̄r of the harmonic 2-tensors Ā and B̄ may also be decomposed in terms of the basis

\begin{matrix}ũ = z^{2} − \frac{1}{2}(x^{2} + y^{2}),  s̃_{1} = zx,  t̃_{1} = zy, \\ s̃_{2} = x^{2} − y^{2},  t̃_{2} = 2xy \end{matrix}

Table 2. Sym_{e}H̄ is described by the vanishing of ᾱ_{0}, ᾱ_{i}, β̄_{i}

Properties of ᾱ_{0}, ᾱ_{i}, β̄_{i}  Restrictions on Sym_{e}H̄
1  ᾱ_{1} ≠ 0 or ᾱ_{3} ≠ 0  Z_{2} ≅ sg{Q_{e_{1}}^{π}}, sg{Q_{e_{3}}^{π}} ⊈ Sym_{e}H̄
1  β̄_{1} ≠ 0 or β̄_{3} ≠ 0  Z_{2} ≅ sg{Q_{e_{2}}^{π}}, sg{Q_{e_{3}}^{π}} ⊈ Sym_{e}H̄
1  β̄_{2} ≠ 0 or β̄_{4} ≠ 0  Z_{2} ≅ sg{Q_{e_{1}}^{π}}, sg{Q_{e_{2}}^{π}} ⊈ Sym_{e}H̄
2  ᾱ_{1}, β̄_{1}, ᾱ_{3}, β̄_{3} = 0  Z_{2} ≅ sg{Q_{e_{3}}^{π}} ⊆ Sym_{e}H̄
with β̄_{2} ≠ 0 or β̄_{4} ≠ 0  D_{2}, sg{Q_{e_{1}}^{π}}, sg{Q_{e_{2}}^{π}} ⊈ Sym_{e}H̄
with ᾱ_{2} ≠ 0  Z_{3} ≅ sg{Q_{e_{i}}^{2π/3}} ⊈ Sym_{e}H̄
Z_{4} ≅ sg{Q_{e_{i}}^{π/2}} ⊈ Sym_{e}H̄
3 ᾱ_{1}, β̄_{1}, β̄_{2}, ᾱ_{3}, β̄_{3}, β̄_{4} = 0  D_{2} ≅ sg{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ⊆ Sym_{e}H̄
with ᾱ_{2} ≠ 0  Z_{3} ≅ sg{Q_{e_{i}}^{2π/3}} ⊈ Sym_{e}H̄
Z_{4} ≅ sg{Q_{e_{i}}^{π/2}} ⊈ Sym_{e}H̄
4 ᾱ_{1}, β̄_{1}, ᾱ_{2}, β̄_{2}, ᾱ_{3}, ᾱ_{4}, β̄_{4} = 0  D_{3} ≅ sg{Q_{e_{1}}^{π}, Q_{e_{3}}^{2π/3}} ⊆ Sym_{e}H̄
with β̄_{3} ≠ 0  O(2), sg{Q_{e_{j≠3}}^{2π/3}}, sg{Q_{e_{3}}^{π}} ⊈ Sym_{e}H̄
5 ᾱ_{1}, β̄_{1}, ᾱ_{2}, β̄_{2}, ᾱ_{3}, β̄_{3}, β̄_{4} = 0 D_{4} ≅ sg{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π/2}} ⊆ Sym_{e}H̄
with ᾱ_{4} ≠ 0  O, sg{Q_{e_{i}}^{2π/3}}, sg{Q_{e_{j≠3}}^{π/2}} ⊈ Sym_{e}H̄
6 ᾱ_{1}, β̄_{1}, ᾱ_{2}, β̄_{2}, ᾱ_{3}, β̄_{3}, β̄_{4} = 0 O ≅ sg{Q_{e_{1}}^{π/2}, Q_{e_{2}}^{π/2}, Q_{e_{3}}^{π/2}} ⊆ Sym_{e}H̄
with ᾱ_{4} = 5ᾱ_{0} ≠ 0  O(2) ⊈ Sym_{e}H̄
7  ᾱ_{i}, β̄_{i} = 0, i = 1...4  O(2) ≅ sg{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}} ⊆ Sym_{e}H̄
with ᾱ_{0} ≠ 0  O(3) ⊈ Sym_{e}H̄
8 ᾱ_{0}, ᾱ_{i}, β̄_{i} = 0, i = 1...4  O(3) ⊆ Sym_{e}H̄

of homogeneous, degree-2 polynomials in x, y, z. In fact,

\begin{aligned}r^{T} A′r & = & α̃_{0}ũ + α̃_{1}s̃_{1} + β̃_{1}t̃_{1} + α̃_{2}s̃_{2} + β̃_{2}t̃_{2} \\ & = & A′_{33}ũ + 2A′_{13}s̃_{1} + 2A′_{23}t̃_{1} + \frac{1}{2}(A′_{11} − A′_{22})s̃_{2} + A′_{12}t̃_{2} \end{aligned}

for any symmetric, traceless, rank-2 tensor A′. It follows that the symmetry groups of Ā and B̄ may be described in terms of the vanishing of their components, as summarized in Table 3.

In addition, as shown in [F-V],

(64)  Sym_{e}(Ā) ∩ Sym_{e}(B̄) ≅ {I}, Z_{2}, D_{2}, O(2), or SO(3).

Table 3. Sym_{e}A′ is described by the vanishing of its components

Properties of A′_{ij}  Restrictions on Sym_{e}A′
1 A′_{13} ≠ 0 or A′_{12} ≠ 0  Z_{2} ≅ sg{Q_{e_{1}}^{π}} ⊈ Sym_{e}A′
1 A′_{23} ≠ 0 or A′_{12} ≠ 0  Z_{2} ≅ sg{Q_{e_{2}}^{π}} ⊈ Sym_{e}A′
1 A′_{13} ≠ 0 or A′_{23} ≠ 0  Z_{2} ≅ sg{Q_{e_{3}}^{π}} ⊈ Sym_{e}A′
3  A′_{13}, A′_{23}, A′_{12} = 0  D(2) ≅ sg{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ⊆ Sym_{e}A′
with A′_{11}−A′_{22} ≠ 0  sg{Q_{e_{i}}^{π/2}}, sg{Q_{e_{i}}^{2π/3}} ⊈ Sym_{e}A′
7 A′_{13}, A′_{23}, A′_{11}−A′_{22} = 0 O(2) ≅ sg{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}} ⊆ Sym_{e}A′
A′_{12} = 0, with A′_{33} ≠ 0  O(3) ⊈ Sym_{e}A′
8  A′_{33}, A′_{13}, A′_{23} = 0  O(3) ⊆ Sym_{e}A′

A′_{12}, A′_{11}−A′_{22} = 0

We apply this result, together with (22), (63), Table 2, and Table 3 below.

For orthotropic C̄ with Sym_{e}C̄ = {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} the possibilities for ( Sym_{e}Ā ∩ Sym_{e}B̄) ∩ Sym_{e}H̄, and the corresponding restrictions on the parameters describing Ā, B̄ and H̄ (in addition to the requirements that Ā, B̄ be diagonal, and ᾱ_{1}, β̄_{1}, ᾱ_{3}, β̄_{3}, β̄_{2}, β̄_{4} = 0), are:

(65)

i) {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ∩ {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}},

Ā^{11} ≠ Ā^{22}, B̄^{11} ≠ B̄^{22}, ᾱ_{2} ≠ 0

ii) {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}} ∩ {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}},

Ā^{11} = Ā^{22}, B̄^{11} = B̄^{22}, ᾱ_{2} ≠ 0

iii) O(3) ∩ {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}},

Ā^{11}, Ā^{22}, B̄^{11}, B̄^{22} = 0, ᾱ_{2} ≠ 0

iv) {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ∩ {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π/2}},

Ā^{11} ≠ Ā^{22}, B̄^{11} ≠ B̄^{22}, and ᾱ_{2} = 0, ᾱ_{4} ≠ 0

v) {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ∩ {Q_{e_{1}}^{π/2}, Q_{e_{2}}^{π/2}, Q_{e_{3}}^{π/2}},

Ā^{11} ≠ Ā^{22}, B̄^{11} ≠ B̄^{22}, and ᾱ_{2} = 0, ᾱ_{4} = 5ᾱ_{0} ≠ 0

vi) {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ∩ {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}},

Ā^{11} ≠ Ā^{22}, B̄^{11} ≠ B̄^{22}, and ᾱ_{2}, ᾱ_{4} = 0

vii) {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ∩ O(3)

Ā^{11} ≠ Ā^{22}, B̄^{11} ≠ B̄^{22}, and ᾱ_{0}, ᾱ_{2}, ᾱ_{4} = 0.

For trigonal C̄ with Sym_{e}C̄ = {Q_{e_{1}}^{π}, Q_{e_{3}}^{2π/3}} the possibilities for ( Sym_{e}Ā∩ Sym_{e}B̄) ∩ Sym_{e}H̄, and the corresponding restrictions on the parameters describing Ā, B̄ and H̄ (in addition to the requirements that Ā, B̄ be diagonal, Ā^{11} = Ā^{22}, B̄^{11} = B̄^{22}, ᾱ_{1}, β̄_{1}, ᾱ_{2}, β̄_{2}, ᾱ_{3}, ᾱ_{4} = 0, β̄_{4}, β̄_{3} ≠ 0), are:

(66)  \begin{matrix}i) & \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}\} ∩ \{Q_{e_{1}}^{π}, Q_{e_{3}}^{2π/3}\}, \\ ii) & O(3) ∩ \{Q_{e_{1}}^{π}, Q_{e_{3}}^{2π/3}\}, Ā^{11}, Ā^{22}, B̄^{11}, B̄^{22} = 0. \end{matrix}

For tetragonal C̄ with Sym_{e}C̄ = {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π/2}} the possibilities for ( Sym_{e}Ā ∩ Sym_{e}B̄) ∩ Sym_{e}H̄, and the corresponding restrictions on the parameters describing Ā, B̄ and H̄ (in addition to the requirements that Ā, B̄ be diagonal, Ā^{11} = Ā^{22}, B̄^{11} = B̄^{22}, ᾱ_{1}, β̄_{1}, ᾱ_{2}, β̄_{2}, ᾱ_{3}, β̄_{3}, β̄_{4} = 0, ᾱ_{4} ≠ 0), are:

(67)  \begin{matrix}i) & \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}\} ∩ \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π/2}\}, \\ ii) & O(3) ∩ \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π/2}\}, Ā^{11}, Ā^{22}, B̄^{11}, B̄^{22} = 0. \end{matrix}

For cubic C̄ with Sym_{e}C̄ = {Q_{e_{1}}^{π/2}, Q_{e_{2}}^{π/2}, Q_{e_{3}}^{π/2}} the only possibility for ( Sym_{e}Ā ∩ Sym_{e}B̄) ∩ Sym_{e}H̄ is O(3) ∩ {Q_{e_{1}}^{π/2}, Q_{e_{2}}^{π/2}, Q_{e_{3}}^{π/2}}, and the corresponding restrictions on the parameters describing Ā, B̄ and H̄ are that Ā, B̄ = 0, ᾱ_{1}, β̄_{1}, ᾱ_{2}, β̄_{2}, ᾱ_{3}, β̄_{3}, β̄_{4} = 0, ᾱ_{4} = 5ᾱ_{0} ≠ 0.

For transversely isotropic C̄ with Sym_{e}C̄ = {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}} the possibilities for ( Sym_{e}Ā ∩ Sym_{e}B̄) ∩ Sym_{e}H̄, and the corresponding restrictions on the parameters describing Ā, B̄ and H̄ (in addition to the requirements that Ā, B̄ be diagonal, Ā^{11} = Ā^{22}, B̄^{11} = B̄^{22}, ᾱ_{i}, β̄_{i} = 0, i = 1, …, 4), are:

(68)  \begin{aligned}i) & \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}\} ∩ \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}\}, \\ & Ā^{11}, B̄^{11} ≠ 0, ᾱ_{0} ≠ 0 \\ ii) & O(3) ∩ \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}\}, \\ & Ā, B̄ = 0, ᾱ_{0} ≠ 0 \\ iii) & \{Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{ϕ}\} ∩ O(3) \\ & Ā^{11}, B̄^{11} ≠ 0, ᾱ_{0} = 0. \end{aligned}

For isotropic C̄ both Sym_{e}Ā ∩ Sym_{e}B̄ and Sym_{e}H̄ must be O(3), and so Ā, B̄ = 0 and ᾱ_{0}, ᾱ_{i}, β̄_{i} = 0, i = 1, …, 4.

5.3. Computing ˆ C̄. We now compute ˆ C̄ = D ∗ C̄ in terms of C̄. We observe that (D ∗ C̄)^{ijkl} = C̄^{ijkl} a^{e(1)} b^{e(2)} c^{e(3)}, where in this formula e(m) = δ_{im} + δ_{jm} + δ_{km} + δ_{lm} and there is no summation over i, j, k, l. It follows that D ∗ H̄ has components given by

(69) \begin{matrix}ij\kl & 11 & 22 & 33 \\ 11 & a^{4}(3ᾱ_{0} − ᾱ_{2} + ᾱ_{4}) & a^{2}b^{2}(ᾱ_{0} − ᾱ_{4}) & −a^{2}c^{2}(4ᾱ_{0} − ᾱ_{2}) \\ 22 & & b^{4}(3ᾱ_{0} + ᾱ_{2} + ᾱ_{4}) & −b^{2}c^{2}(4ᾱ_{0} + ᾱ_{2}) \\ 33 & & & 8c^{4}ᾱ_{0} \end{matrix}

(70)  \begin{matrix}ij\kl & 12 & 23 & 13 \\ 11 & −a^{3}b(β̄_{2}/2 − β̄_{4}) & −a^{2}bc[β̄_{1} − β̄_{3}]/4 & −a^{3}c[3ᾱ_{1} − ᾱ_{3}]/4 \\ 22 & −ab^{3}(β̄_{2}/2 + β̄_{4}) & −b^{3}c[3β̄_{1} + β̄_{3}]/4 & −ab^{2}c[ᾱ_{1} + ᾱ_{3}]/4 \\ 33 & abc^{2}β̄_{2} & bc^{3}β̄_{1} & ac^{3}ᾱ_{1} \\ 12 & a^{2}b^{2}[ᾱ_{0} − ᾱ_{4}] & −ab^{2}c[ᾱ_{1} + ᾱ_{3}]/4 & −a^{2}bc[β̄_{1} − β̄_{3}]/4 \\ 23 & & −b^{2}c^{2}(4ᾱ_{0} + ᾱ_{2}) & abc^{2}β̄_{2} \\ 13 & & & −a^{2}c^{2}(4ᾱ_{0} − ᾱ_{2}) \end{matrix}

while, writing M = (ᾱ + 2β̄)I + 2(Ā + 2B̄), ˆ C̄ − (D ∗ H̄) is given by:

(71)  \begin{matrix}ij\kl & 11 & 22 & 33 \\ 11 & a^{4}M^{11} & a^{2}b^{2}[ᾱ + Ā^{11} + Ā^{22}] & a^{2}c^{2}[ᾱ − Ā^{22}] \\ 22 & & b^{4}M^{22} & b^{2}c^{2}[ᾱ − Ā^{11}] \\ 33 & & & c^{4}M^{33} \end{matrix}

(72)  \begin{matrix}ij\kl & 12 & 23 & 13 \\ 11 & a^{3}b(Ā + 2B̄)^{12} & a^{2}bcĀ^{23} & a^{3}c(Ā + 2B̄)^{13} \\ 22 & ab^{3}(Ā + 2B̄)^{12} & b^{3}c(Ā + 2B̄)^{23} & ab^{2}cĀ^{13} \\ 33 & abc^{2}Ā^{12} & bc^{3}(Ā + 2B̄)^{23} & ac^{3}(Ā + 2B̄)^{13} \\ 12 & a^{2}b^{2}[β̄ + B̄^{11} + B̄^{22}] & ab^{2}cB̄^{13} & a^{2}bcB̄^{23} \\ 23 & & b^{2}c^{2}[β̄ − B̄^{11}] & abc^{2}B̄^{12} \\ 13 & & & a^{2}c^{2}[β̄ − B̄^{22}] \end{matrix}

To compute the harmonic-Cartan decomposition of ˆ C̄ we first compute ˆᾱ = αˆC̄, ˆ β̄ = βˆC̄, ˆ Ā = AˆC̄, ˆ B̄ = BˆC̄, and ˆ H̄ = H ˆC̄, given (51) and (54).

We then determine ˆ ᾱ_{0} = (α_{0})ˆC̄, ˆ ᾱ_{i} = (α_{i})ˆC̄, ˆ β̄_{i} = (β_{i})ˆC̄, i = 1, …, 4 from

the Cartan decomposition (56) for ˆ H̄.

It follows by a calculation, given Ā^{33} = −Ā^{11}−Ā^{22} and B̄^{33} = −B̄^{11}−B̄^{22}, that we may write the linear dependence of

\begin{matrix}ˆk̄_{1313} = (ˆĀ^{13}, ˆ B̄^{13}, ˆ ᾱ_{1}, ˆ ᾱ_{3}) on k̄_{1313} = (Ā^{13}, B̄^{13}, ᾱ_{1}, ᾱ_{3}) \\ as  ˆ  k̄_{1313} = K_{1313} · k̄_{1313}, \end{matrix}

the linear dependence of

\begin{matrix}ˆk̄_{2313} = (ˆĀ^{23}, ˆ B̄^{23}, ˆ β̄_{1}, ˆ β̄_{3}) on k̄_{2313} = (Ā^{23}, B̄^{23}, β̄_{1}, β̄_{3}) \\ as  ˆ  k̄_{2313} = K_{2313} · k̄_{2313}, \end{matrix}

and the linear dependence of

\begin{matrix}ˆk̄_{1224} = (ˆĀ^{12}, ˆ B̄^{12}, ˆ β̄_{2}, ˆ β̄_{4}) on k̄_{1224} = (Ā^{12}, B̄^{12}, β̄_{2}, β̄_{4}) \\ as  ˆ  k̄_{1224} = K_{1224} · k̄_{1224}. \end{matrix}

Finally, we write the linear dependence of

ˆk̄_{024} = (15ˆᾱ, 15ˆβ̄, 21ˆĀ^{11}, 21ˆĀ^{22}, 21ˆB̄^{11}, 21ˆB̄^{22}, 280ˆᾱ_{0}, 14ˆᾱ_{2}, 8ˆᾱ_{4})

on k̄_{024} = (ᾱ, β̄, Ā^{11}, Ā^{22}, B̄^{11}, B̄^{22}, ᾱ_{0}, ᾱ_{2}, ᾱ_{4})
as  ˆ  k̄_{024} = K_{024} · k̄_{024}.

We observe that the 4 × 4 matrices K_{1313}, K_{2313}, K_{1224} are invertible. In fact, K_{1313} is given by ac/28 times

(\begin{matrix}4(a^{2} + 5b^{2} + c^{2}) & 8(a^{2}−2b^{2}+c^{2}) & −(3a^{2}+b^{2}−4c^{2}) & (a^{2} − b^{2}) \\ 4(a^{2} − 2b^{2} + c^{2}) & 4(2a^{2}+3b^{2}+2c^{2}) & −(3a^{2}+b^{2}−4c^{2}) & (a^{2} − b^{2}) \\ −4(3a^{2}+b^{2}−4c^{2}) & −8(3a^{2} +b^{2} −4c^{2}) & (9a^{2} +3b^{2} +16c^{2}) & −3(a^{2}−b^{2}) \\ 28(a^{2} −b^{2}) & 56(a^{2} −b^{2}) & −21(a^{2} −b^{2}) & 7(a^{2} +3b^{2}) \end{matrix}) ,

and has determinant a^{2}b^{4}c^{2} · (ac)^{4}, up to a positive constant, so does not vanish since a, b, c > 0. Similarly, K_{2313} is given by bc/28 times

(\begin{matrix}4(5a^{2}+b^{2}+c^{2}) & 8(−2a^{2}+b^{2}+c^{2}) & −(a^{2}+3b^{2}−4c^{2}) & (a^{2} − b^{2}) \\ 4(−2a^{2}+b^{2}+c^{2}) & 4(3a^{2}+2b^{2}+2c^{2}) & −(a^{2}+3b^{2}−4c^{2}) & (a^{2} − b^{2}) \\ −4(a^{2}+3b^{2}−4c^{2}) & −8(a^{2}+3b^{2}−4c^{2}) & (3a^{2}+9b^{2}+16c^{2}) & −3(a^{2}−b^{2}) \\ 28(a^{2} − b^{2}) & 56(a^{2} − b^{2}) & −21(a^{2} − b^{2}) & 7(3a^{2}+b^{2}) \end{matrix}) ,

and has determinant a^{4}b^{2}c^{2} · (bc)^{4}, up to a positive constant. In addition, K_{1224} is given by ab/28 times

(\begin{matrix}4(a^{2}+b^{2}+5c^{2}) & 8(a^{2}+b^{2}−2c^{2}) & −2(a^{2}+b^{2}−2c^{2}) & 4(a^{2} − b^{2}) \\ 4(a^{2}+b^{2}−2c^{2}) & 4(2a^{2}+2b^{2}+3c^{2}) & −2(a^{2}+b^{2}−2c^{2}) & 4(a^{2} − b^{2}) \\ −8(a^{2}+b^{2}−2c^{2}) & −16(a^{2}+b^{2}−2c^{2}) & 4(a^{2}+b^{2}+12c^{2}) & −8(a^{2}−b^{2}) \\ 14(a^{2} − b^{2}) & 28(a^{2} − b^{2}) & −7(a^{2} − b^{2}) & 14(a^{2}+b^{2}) \end{matrix}) ,

and has determinant a^{2}b^{2}c^{4} · (ab)^{4}, up to a positive constant.

For C̄ isotropic, transversely isotropic, cubic, or tetragonal having parameters (ᾱ, β̄, Ā^{11}, B̄^{11}, ᾱ_{0}, ᾱ_{4}) it follows by the reasoning of Lemma 17 (Section 5.4) that ˆ C̄ = D ∗ C̄ may be written in terms of the parameters ˆᾱ, ˆ β̄, ˆ Ā^{11}, ˆ Ā^{22}, ˆ B̄^{11}, ˆ B̄^{22}, ˆ ᾱ_{0}, ˆ ᾱ_{2}, ˆ ᾱ_{4}.

Finally, we compute K_{024} to be the following. We write ˆ k̄_{024} = (15ˆᾱ, 15ˆβ̄, 21ˆĀ^{11}, 21ˆĀ^{22}, 21ˆB̄^{11}, 21ˆB̄^{22}, 280ˆᾱ_{0}, 14ˆᾱ_{2}, 8ˆᾱ_{4}) as M · v, where

v = (ᾱ, β̄, Ā^{11}, B̄^{11}, ᾱ_{0}, ᾱ_{4}, ᾱ_{0}, ᾱ_{0}, ᾱ_{0})

and M is given by the product of the following two invertible matrices:

\begin{matrix}( & \begin{matrix}4a^{2}b^{2} & −2a^{2}b^{2} & 4a^{2} & −2a^{2} & a^{4} & 1 & b^{4} & −2b^{2} & 4b^{2} \\ −a^{2}b^{2} & 3a^{2}b^{2} & −a^{2} & 3a^{2} & a^{4} & 1 & b^{4} & 3b^{2} & −b^{2} \\ 5a^{2}b^{2} & −4a^{2}b^{2} & 5a^{2} & −4a^{2} & 2a^{4} & −1 & −b^{4} & 8b^{2} & −10b^{2} \\ 5a^{2}b^{2} & −4a^{2}b^{2} & −10a^{2} & 8a^{2} & −a^{4} & −1 & 2b^{4} & −4b^{2} & 5b^{2} \\ −2a^{2}b^{2} & 3a^{2}b^{2} & −2a^{2} & 3a^{2} & 2a^{4} & −1 & −b^{4} & −6b^{2} & 4b^{2} \\ −2a^{2}b^{2} & 3a^{2}b^{2} & 4a^{2} & −6a^{2} & −a^{4} & −1 & 2b^{4} & 3b^{2} & −2b^{2} \\ 2a^{2}b^{2} & 4a^{2}b^{2} & −8a^{2} & −16a^{2} & 3a^{4} & 8 & 3b^{4} & −16b^{2} & −8b^{2} \\ 0 & 0 & 2a^{2} & 4a^{2} & −a^{4} & 0 & b^{4} & −4b^{2} & −2b^{2} \\ −2a^{2}b^{2} & −4a^{2}b^{2} & 0 & 0 & a^{4} & 0 & b^{4} & 0 & 0 \end{matrix} & ) \\ · ( \begin{matrix}1 & 0 & 2 & 0 & 1 & −1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 & 1 & −1 & 0 & 0 & 0 \\ 1 & 0 & −1 & 0 & −4 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & −1 & −4 & 0 & 0 & 0 & 0 \\ 1 & 2 & 2 & 4 & 3 & 1 & 0 & 0 & 0 \\ 1 & 2 & −4 & −8 & 8 & 0 & 0 & 0 & 0 \\ 1 & 2 & 2 & 4 & 0 & 1 & 3 & 0 & 0 \\ 0 & 1 & 0 & −1 & 0 & 0 & 0 & −4 & 0 \\ 1 & 0 & −1 & 0 & 0 & 0 & 0 & 0 & −4 \end{matrix} ) . \end{matrix}

By collecting terms in the above expression we conclude that the (scaled) parameters (15ˆᾱ, 15ˆβ̄, 21ˆĀ^{11}, 21ˆĀ^{22}, 21ˆB̄^{11}, 21ˆB̄^{22}, 280ˆᾱ_{0}, 14ˆᾱ_{2}, 8ˆᾱ_{4}) have the form (24), where the P (a, b) are linearly independent for fixed a, b and are given by

(73)\begin{aligned}Pᾱ & = & (6a^{4} & − & 2[3(a^{2} − b^{2}) − 4]a^{2} & + & [1 + (a^{2} − b^{2})^{2} − 4(a^{2} − b^{2})], \\ & & a^{4} & − & [(a^{2} − b^{2}) + 2]a^{2} & + & [1 + (a^{2} − b^{2})^{2} + (a^{2} − b^{2})], \\ & & 6a^{4} & − & [3(a^{2} − b^{2}) + 5]a^{2} & − & [1 + (a^{2} − b^{2})^{2} − 10(a^{2} − b^{2})], \\ & & 6a^{4} & − & [9(a^{2} − b^{2}) + 5]a^{2} & − & [1 − 2(a^{2} − b^{2})^{2} + 5(a^{2} − b^{2})], \\ & & −a^{4} & + & 2[2(a^{2} − b^{2}) + 1]a^{2} & − & [1 + (a^{2} − b^{2})^{2} + 4(a^{2} − b^{2})], \\ & & −a^{4} & − & 2[(a^{2} − b^{2}) − 1]a^{2} & − & [1 − 2(a^{2} − b^{2})^{2} − 2(a^{2} − b^{2})], \\ & & 8a^{4} & − & 8[(a^{2} − b^{2}) + 2]a^{2} & + & [8 + 3(a^{2} − b^{2})^{2} + 8(a^{2} − b^{2})], \\ & & & − & [2(a^{2} − b^{2})]a^{2} & + & [(a^{2} − b^{2})^{2} + 2(a^{2} − b^{2})], \\ & & & & & + & [(a^{2} − b^{2})^{2}]), \\ Pβ̄ & = & (2a^{4} & − & 2[(a^{2} − b^{2}) + 2]a^{2} & + & 2[1 + (a^{2} − b^{2})^{2} + (a^{2} − b^{2})], \\ & & 7a^{4} & − & [7(a^{2} − b^{2}) − 6]a^{2} & + & [2 + 2(a^{2} − b^{2})^{2} − 3(a^{2} − b^{2})], \\ & & −2a^{4} & + & 4[2(a^{2} − b^{2}) + 1]a^{2} & − & 2[1 + (a^{2} − b^{2})^{2} + 4(a^{2} − b^{2})], \\ & & −2a^{4} & − & 4[(a^{2} − b^{2}) − 1]a^{2} & − & 2[1 − 2(a^{2} − b^{2})^{2} − 2(a^{2} − b^{2})], \\ & & 5a^{4} & + & [(a^{2} − b^{2}) − 3]a^{2} & − & 2[1 + (a^{2} − b^{2})^{2} − 3(a^{2} − b^{2})], \\ & & 5a^{4} & − & [11(a^{2} − b^{2}) + 3]a^{2} & − & [2 − 4(a^{2} − b^{2})^{2} + 3(a^{2} − b^{2})], \\ & & 16a^{4} & − & 16[(a^{2} − b^{2}) + 2]a^{2} & + & 2[8 + 3(a^{2} − b^{2})^{2} + 8(a^{2} − b^{2})], \\ & & & − & [4(a^{2} − b^{2})]a^{2} & + & 2[(a^{2} − b^{2})^{2} + 2(a^{2} − b^{2})], \\ & & & & & & [2(a^{2} − b^{2})^{2}]), \\ PĀ_{11} & = & (12a^{4} & − & 4[3(a^{2} − b^{2}) + 2]a^{2} & − & 2[2 − (a^{2} − b^{2})^{2} − 2(a^{2} − b^{2})], \\ & & 2a^{4} & − & −2[(a^{2} − b^{2}) + 2]a^{2} & + & [4 − 2(a^{2} − b^{2})^{2} − (a^{2} − b^{2})], \\ & & 12a^{4} & − & [6(a^{2} − b^{2}) − 5]a^{2} & + & 2[2 − (a^{2} − b^{2})^{2} − 5(a^{2} − b^{2})], \\ & & 12a^{4} & − & [18(a^{2} − b^{2}) − 5]a^{2} & + & [4 + 4(a^{2} − b^{2})^{2} + 5(a^{2} − b^{2})], \\ & & −2a^{4} & + & 2[4(a^{2} − b^{2}) − 1]a^{2} & + & 2[2 − (a^{2} − b^{2})^{2} + 2(a^{2} − b^{2})], \\ & & −2a^{4} & − & 2[2(a^{2} − b^{2}) + 1]a^{2} & + & 2[2 + 2(a^{2} − b^{2})^{2} − (a^{2} − b^{2})], \\ & & 16a^{4} & − & 16[(a^{2} − b^{2}) − 1]a^{2} & + & 2[16 − 3(a^{2} − b^{2})^{2} − 4(a^{2} − b^{2})], \\ & & & − & [4(a^{2} − b^{2})]a^{2} & + & 2[(a^{2} − b^{2})^{2} − (a^{2} − b^{2})], \\ & & & & & & [2(a^{2} − b^{2})^{2}]), \end{aligned}

32 A. MAZZUCATO AND L. RACHELE

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PB̄_{11} = (4a^{4} − 4[(a^{2} − b^{2}) − 1]a^{2} − [8 − 4(a^{2} − b^{2})^{2} + 2(a^{2} − b^{2})],
14a^{4} − 2[7(a^{2} − b^{2}) + 3]a^{2} − [8 − 4(a^{2} − b^{2})^{2} − 3(a^{2} − b^{2})],
−4a^{4} + 4[4(a^{2} − b^{2}) − 1]a^{2} + 4[2 − (a^{2} − b^{2})^{2} + 2(a^{2} − b^{2})],
−4a^{4} − 4[2(a^{2} − b^{2}) + 1]a^{2} + 4[2 + 2(a^{2} − b^{2})^{2} − (a^{2} − b^{2})],
10a^{4} + [2(a^{2} − b^{2}) + 3]a^{2} + 2[4 − 2(a^{2} − b^{2})^{2} − 3(a^{2} − b^{2})],
10a^{4} − [22(a^{2} − b^{2}) − 3]a^{2} + [8 + 8(a^{2} − b^{2})^{2} + 3(a^{2} − b^{2})],
32a^{4} + 32[(a^{2} − b^{2}) − 1]a^{2} − 4[16 − 3(a^{2} − b^{2})^{2} + 4(a^{2} − b^{2})],
− [8(a^{2} − b^{2})a^{2}] + 4[(a^{2} − b^{2})^{2} − (a^{2} − b^{2})],
4(a^{2} − b^{2})^{2}),

(75) \begin{aligned}Pᾱ_{0} & = & (8a^{4} & − & 8[(a^{2} − b^{2}) + 2]a^{2} & + & [8 + 3(a^{2} − b^{2})^{2} + 8(a^{2} − b^{2})], \\ 8a^{4} & − & 8[(a^{2} − b^{2}) + 2]a^{2} & + & [8 + 3(a^{2} − b^{2})^{2} + 8(a^{2} − b^{2})], \\ 4a^{4} & + & [5(a^{2} − b^{2}) + 4]a^{2} & − & [8 + 3(a^{2} − b^{2})^{2} + 8(a^{2} − b^{2})], \\ 4a^{4} & − & [13(a^{2} − b^{2}) − 4]a^{2} & − & 2[4 − 3(a^{2} − b^{2})^{2} − 2(a^{2} − b^{2})], \\ 4a^{4} & + & [5(a^{2} − b^{2}) + 4]a^{2} & − & [8 + 3(a^{2} − b^{2})^{2} + 8(a^{2} − b^{2})], \\ 4a^{4} & − & [13(a^{2} − b^{2}) − 4]a^{2} & − & 2[4 − 2(a^{2} − b^{2}) − 3(a^{2} − b^{2})^{2}], \\ 24a^{4} & − & 24[(a^{2} − b^{2}) − 8]a^{2} & + & [64 + 9(a^{2} − b^{2})^{2} − 96(a^{2} − b^{2})], \\ − & [6(a^{2} − b^{2})a^{2}] & − & 3[(a^{2} − b^{2})^{2} + 8(a^{2} − b^{2})], \\ & & [3(a^{2} − b^{2})^{2}]), \\ Pᾱ_{4} & = & ((a^{2} − b^{2})^{2}, (a^{2} − b^{2})^{2}, (a^{2} − b^{2})(2a^{2} + b^{2}), \\ & & −(a^{2} − b^{2})(a^{2} + 2b^{2}), (a^{2} − b^{2})(2a^{2} + b^{2}), \\ & & −(a^{2} − b^{2})(a^{2} + 2b^{2}), 3(a^{2} − b^{2})^{2}, \\ & & −(a^{2} − b^{2})(a^{2} + b^{2}), 8a^{4} −(a^{2} −b^{2})(7a^{2} +b^{2})). \end{aligned}

We also compute that C̄ is strong elliptic if ᾱ + 2β̄− 2(Ā^{11} + 2B̄^{11} + Ā^{22} + 2B̄^{22}) + 8ᾱ_{0} > 0. We remark that, due to the fact that D is invertible, if C̄ is strong elliptic, then by the definition of strong ellipticity ˆ C̄ is also.

5.4. Characterizing the range of the pullback by D of isotropic, transversely isotropic, cubic, and tetragonal C̄. We now analyze the possibilities for ˆ C̄.

Lemma 17. If Sym_{e}C̄ contains {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}}, then Sym_{e}ˆC̄ does also.

Proof. If Sym_{e}C̄ contains {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}}, (i.e., C̄ is orthotropic, tetragonal, cubic, transversely isotropic, or isotropic), then ˆ k̄_{1313} = K_{1313}k̄_{1313} = 0, ˆ k̄_{2313} = K_{2313}k̄_{2313} = 0, ˆ k̄_{1224} = K_{1224}k̄_{1224} = 0; that is, the off-diagonal terms of ˆ Ā and ˆ B̄ vanish, and {Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π}} ⊆ Sym_{e} ˆH̄. The result follows from (63) and Table 3. □

Proof of Proposition 14. It is given that there is a polar decomposition of Dψ^{−1} = SV and diagonalization of the stretch factor S = UDU^{T} , D = diag (a, b, c) with a, b, c > 0, so that, with respect to the standard basis for Euclidean space, the generators of Sym_{e}C̄(X′) = Sym_{e}U^{T} V ∗ (C ∘ ψ)(X′) are listed in Table 1. It follows that a polar decomposition of Dψ and diagonalization of the stretch factor is given by V^{T} UD^{−1}U^{T} . Therefore, (ψ^{−1})^{∗}(ψ^{∗}C) = (Dψ) ∗ (ψ^{∗}C ∘ ψ^{−1}) = V^{T} UD^{−1}U^{T} ∗ (ψ^{∗}C ∘ ψ^{−1}); \overline{ψ^{∗}C} in this case is U^{T} ∗ (ψ^{∗}C ∘ ψ^{−1}); and Sym_{e}\overline{ψ^{∗}C}(X) = Sym_{e}U^{T} ∗ (UDU^{T} V ∗ (C∘ψ))∘ψ^{−1}(X) = Sym_{e}D∗C̄(X′). By Lemma 17 Sym_{e}D∗C̄(X′) contains Q_{e_{1}}^{π}, Q_{e_{2}}^{π}, Q_{e_{3}}^{π} since Sym_{e}C̄(X′) does, where here {e_{1}, e_{2}, e_{3}} is the standard basis for Euclidean space. It follows by the presentation of the symmetry groups for anisotropic elastic media given in [C-V-C], pp. 2489-2490, for example, that Sym_{e}D ∗ C̄(X′) has generators listed in Table 1. □

It follows for C̄ orthotropic, tetragonal, cubic, transversely isotropic, or isotropic that ˆ β̄_{3} = 0, and so ˆ H̄ and ˆ C̄ are not trigonal.

For C̄ tetragonal, cubic, transversely isotropic, or isotropic we conclude from the representation (24) of ˆ k̄_{024} that a = b implies ˆ Ā^{11} = ˆ Ā^{22}, ˆ B̄^{11} = ˆB̄^{22}, ˆ ᾱ_{2} = 0, and ˆ ᾱ_{4} = a^{4}ᾱ_{4}. It follows that ˆ C̄ is not orthotropic. For C̄ isotropic or transversely isotropic ˆ C̄ is not cubic or tetragonal since ˆᾱ_{4} = 0. In fact, for C̄ isotropic ˆ C̄ is transversely isotropic since ˆ ᾱ_{0} = (ᾱ+2β̄)(a^{2} −1)^{2}/35 ≠ 0 by strong ellipticity. For C̄ cubic (so ᾱ_{4} = 5ᾱ_{0} ≠ 0) ˆC̄ may be tetragonal, depending on the choice of a.

If C̄ is isotropic and a ≠ b, then ˆ Ā^{11} ≠ ˆ Ā^{22}, ˆ B̄^{11} ≠ ˆ B̄^{22}. Also, ˆ ᾱ_{2} = 0 if and only if a^{2}+b^{2} is 2c^{2}. (If a^{2}+b^{2} = 2c^{2} = 2, then ˆ ᾱ_{4} = (ᾱ+2β̄)(a^{2}−1)^{2}/2, which is not zero by strong ellipticity.) It follows that ˆ C̄ is orthotropic of type i) or iv).

If C̄ is transversely isotropic, cubic, or tetragonal, and a ≠ b, then ˆĀ^{11} ≠ ˆ Ā^{22}, ˆ B̄^{11} ≠ ˆ B̄^{22}, or ˆ ᾱ_{2} ≠ 0. If ˆ Ā^{11} ≠ ˆ Ā^{22} or ˆ B̄^{11} ≠ ˆ B̄^{22}, then ˆ C̄ is orthotropic of type i), or iv) − vii). On the other hand, if ˆĀ^{11} = ˆ Ā^{22}, ˆ B̄^{11} = ˆ B̄^{22}, ˆ ᾱ_{2} ≠ 0, then ˆ C̄ is orthotropic of type ii) or iii).

In conclusion, for C̄ isotropic or transversely isotropic, ˆ C̄ is orthotropic, transversely isotropic, or isotropic (depending on C̄ and a, b), but will never be tetragonal or cubic; for C̄ tetragonal or cubic, ˆ C̄ is (only) orthotropic, tetragonal, or cubic; and for C̄ orthotropic, ˆ C̄ is isotropic, transverse isotropic, cubic, tetragonal, or orthotropic, depending on the values of C̄ and a, b.

In fact, in the case that C̄ is isotropic and a = b the range of D ∗ C̄ is a 3parameter family in the 5-parameter family of transverse isotropic elasticity tensors. In the case that C̄ is isotropic and a ≠ b the range of D ∗ C̄ is a 4parameter family in the 9-parameter family of orthotropic elasticity tensors of type i). In the case that C̄ is cubic and a = b the range of D ∗ C̄ is a 4-parameter family in the 5-parameter family of tetragonal elasticity tensors.

In the case that C̄ is cubic and a ≠ b the range of D ∗ C̄ is a 5parameter family in the 9-parameter family of orthotropic elasticity tensors. In particular, in this case the range of D ∗ C̄ is a 5-parameter family in the 9-parameter family of orthotropic elasticity tensors of type i); is a 4parameter family in the 7-parameter family of orthotropic elasticity tensors of type ii); is a 4-parameter family in the 8-parameter family of orthotropic elasticity tensors of type iv); is a 3-parameter family in the 7-parameter family of orthotropic elasticity tensors of type v); is a 3-parameter family in the 7-parameter family of orthotropic elasticity tensors of type vi); and is a 2-parameter family in the 6-parameter family of orthotropic elasticity tensors of type vii).

The same reasoning may be applied in the case that C̄ is tetragonal. In general, if a ≠ b, then the range of D ∗ C̄ is a 6-parameter family in the 9-parameter family of orthotropic elasticity tensors. □

Appendix A. Conditions for Uniqueness for Isotropic Linearly Elastic Media

A.1. Isotropic media. Here we list the conditions under which Rachele’s uniqueness result for isotropic elastodynamics (Theorem 3) holds.

We denote by Sec (x, σ) the sectional curvature of any two-dimensional subspace σ of the cotangent space T_{x}M of a Riemannian manifold M, for any point x in M. For (x, ξ) in the tangent bundle with ξ ≠ 0 we set Sec (x, ξ) = sup_{{ξ∈σ}} Sec (x, σ) and Sec^{+}(x, ξ) = max{0, Sec (x, ξ)}. By ∂_{−}ΩM we denote the collection of all (x, ξ) in the unit tangent bundle of M with x ∈ ∂M and ⟨ξ, ν(x)⟩ ≤ 0, where ν is the unit outward normal to ∂Ω. For a Riemannian manifold (M, g) with strictly convex boundary ∂M and with the property that there is no geodesic of infinite length in M, we define

k^{+}(M, g) = sup_{(x,ξ)∈∂_{−}ΩM} ∫_{0}^{l} t Sec^{+}(γ(t), γ̇(t)) dt,

where γ : [0, l] is the maximal geodesic in M ∪ ∂M satisfying the initial (i.e. entry) conditions (γ(0), γ̇(0)) = (x, ξ) ∈ ∂_{−}ΩM.

For a real covariant tensor field T of rank m on a compact, Riemannian manifold (M, g) the definition of the modulus of T, |T |_{g}^{2} = g^{i_{1}j_{1}} … g^{i_{m}j_{m}}T_{i_{1}...i_{m}}T_{j_{1}...j_{m}}, is independent of the choice of coordinates. We write ||T ||_{C^{k}(M,g)} = ∑_{l=0}^{k} sup_{x∈M} |∇ … ∇T (x)|_{g} for the C^{k}-norm of the tensor field T, where ∇ is the covariant derivative and there are l copies of ∇ in the lth term in the definition of ||T ||_{C^{k}(M,g)}.

We consider an admissible, isotropic, linearly elastic medium (Ω, λ, μ, ρ) (respectively, (Ω, λ̂, μ̂, ρ̂)) such that μ > 0 and 3λ + 2μ > 0 (positive definiteness of C) on Ω̄ and λ = 2μ (i.e. c_{p} = 2c_{s}) at only isolated points in Ω̄ (for example). In addition, we assume that the metrics g_{p} = ([λ + 2μ]/ρ)^{−1} e and g_{s} = (μ/ρ)^{−1} e (respectively, ĝ_{p} = ([λ̂ + 2μ̂]/ρ̂)^{−1} e and ĝ_{s} = (μ̂/ρ̂)^{−1} e) have the property that g_{p/s} does not give rise to caustics; the geodesics of g_{p/s} all exit Ω in finite time; and the geodesics do not graze ∂Ω. Finally, we assume that ∂Ω is strictly convex with respect to each of g_{p}, ĝ_{p}, and e and that (Ω̄, g_{p}) and (Ω̄, ĝ_{p}) each have curvature bounded above in the sense that k^{+}(Ω, g_{p}) < 1/3 and k^{+}(Ω, ĝ_{p}) < 1/3.

A.2. Isotropic media with residual stress. In this section we give the conditions under which Rachele’s uniqueness result for isotropic elastodynamics with residual stress (Theorem 6) holds.

We consider an admissible, isotropic linearly elastic medium (Ω, λ, μ, ρ, T ) (and another (Ω, λ̂, μ̂, ρ̂, T̂)) with initial stress S (respectively, Ŝ) given by the symmetric, C^{∞} smooth 2-tensor T (T̂) which satisfies ∇_{x} · T = 0 in Ω (divergence-free in the interior) and T · ν⃗ = 0 on ∂Ω (zero surface traction), where ν is the unit outer normal to ∂Ω. The same conditions hold for T̂. (The strong ellipticity condition here for ∘ A^{aA}_{b}^{B} = ^{∘} C_{iso}^{A′AB′B}δ^{a}_{A′}δ_{bB′} +δ^{a}_{b}T^{AB} reduces to the conditions that λ+μ > 0 on Ω̄ and that μ(x)I + T be uniformly positive definite on Ω̄.) In addition, we assume that ∂Ω is strictly convex with respect to the Euclidean metric; that the normal derivatives of all orders of each of λ, μ, ρ, T , and its counterpart from λ̂, μ̂, ρ̂, T̂ are the same at ∂Ω; and that the metrics

g_{p} = (\frac{λ + 2μ}{ρ}I + \frac{1}{ρ}T )^{−1} and g_{s} = (\frac{μ}{ρ}I + \frac{1}{ρ}T )^{−1}

(respectively, ĝ_{p} and ĝ_{s}) satisfy the conditions of Theorem 3. We assume that (Ω̄, g_{p/s}) each have curvature bounded above in the sense that k^{+}(Ω, g_{p/s}) < 1/4 and ||R_{g_{p/s}}||_{C^{19}(Ω,g_{p/s})} ≤ K_{p/s}, where R_{g} is the curvature tensor of the metric g, K_{p/s} > 0. Finally, we assume that ĝ_{p/s} are each close to the Euclidean metric in the sense that for each of them there exists an ϵ = ϵ(K, Ω, 3) > 0 so that ||(ĝ_{p/s})_{ij} − δ_{ij}||_{C^{l}(Ω,e)} < ϵ.

Acknowledgments

The authors would like to express their appreciation to the organizers, especially J. McLaughlin and G. Uhlmann, of the program on inverse problems held at the Mathematical Sciences Research Institute, Fall 2001, where joint work on this project was begun. They would also like to acknowledge the continuous support of their Ph.D. advisers, M.E. Taylor and G. Uhlmann.

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Department of Mathematics, Penn State University, University Park, PA 16802, USA

E-mail address: mazzucat@math.psu.edu

Department of Mathematics and Statistics, University at Albany, Suny, Albany, NY 12222, USA

E-mail address: lrachele@math.albany.edu