A lagrangian description of elastic motion in riemannian manifolds and an angular invariant of axially-symmetric elasticity tensors

A lagrangian description of elastic motion in riemannian manifolds and an angular invariant of axially-symmetric elasticity tensors

This article is a description of elasticity theory for readers with mathematical background. The first sections are an abridgment of parts of the book by Marsden and Hughes [5], including a compact identification of the equations of motion as the Euler-Lagrange equations for the lagrangian density. The other sections describe the basic first-order classification of materials, from the point of view of representation theory as opposed to index calculus. It includes a computation of the axes of symmetry, when they exist, for most of the irreducible components of the elasticity tensor. When the two components of the 5-dimensional type have axes of symmetry, some invariants appear: 2 angles in that measure the deviation of an associated decomposition from the standard one.

See also the classification appearing for example in [3] and [2] by symmetry group in . A somewhat more representation-theoretic approach can be found in [4], and a complete list of polynomial invariants for generic elasticity tensors can be found in [1].

1 Introduction

I always thought it was unfortunate that the scarcity of local isometries of riemannian manifolds tends to prohibit the motion of subbodies. I learned these topics to create the conditions for motion by relaxing the metric in a reasonable way.

Roughly: The mass-weighted material acceleration of a point of an elastic body moving in an ambient riemannian manifold is given by the divergence of the stress bi-vector field there. At each time this stress is the mass-weighted Legendre transform of the riemannian metric on induced by the ambient metric, where is a stored energy per unit mass function characterizing the material.

Then the linearization of near a given rest metric is a tensor of type , called the elasticity tensor. The first-order classification of materials means a description of the orbits of such under the orthogonal group of .

One application of this idea might be an intrinsic tangential counterpart to spectral riemannian geometry, which involves vibrational modes of a manifold in idealized normal directions; in the linear theory an elastic structure on a riemannian manifold leads to a second-order linear differential operator on vector fields, the “restoring force” experienced due to a given infinitesimal displacement. One might hope to deduce properties of its spectrum and eigenspaces from constraints on the elasticity tensor, for example bounds on the isotropic and non-isotropic parts, axial-symmetry, or adaptation of this tensor to a framing on a closed 3-manifold. In the homogeneous isotropic case this reduces to the vector Laplacian (the Hodge Laplacian on 1-forms), so that for example the stable elastic modes on the 2-sphere are the gradients of the usual spherical harmonics.

2 Motion and material derivatives

Consider a body in motion through another. For example, a fluid through a porous medium, or a solid body in space. Precisely, let and be smooth -dimensional manifolds, with boundary, and assume the motion is given by a smooth family of embeddings , one for each time , amounting to an embedding

Let be a time-dependent real-valued scalar quantity on .

Definition. The material time derivative of with respect to the motion of in is the ordinary time derivative of the quantity associated to by the motion (). When it is viewed as a function on via the motion it is also called the material time derivative of . In this case it may be denoted .

(Actually, only needs to be defined on the image of . We think of and as the same quantity but measured relative to the material or to the space . Then is the time derivative from the point of view of the material.)

Figure 1: Material and ordinary derivative

In the figure, the material derivative (along the curved path) and the ordinary derivative (along the straight line) with respect to the horizontal or time parameter only agree if the corresponding point of is stationary at that time and place.

So far no vector space structure, metric, or connection on or is used in the definition. The material time derivative of a scalar is computed with the chain rule. The time derivative on can be identified as the vector field , and , where is the time-dependent velocity field in of the motion of . So,

We can abbreviate this to . The definition extends without significant change to any scalar valued quantity. Let be a smooth manifold:

Definition. The material time derivative of a smooth function with respect to the motion is the the ordinary time derivative of the function equal to . When it is viewed as a function back on via the motion it is also called the material time derivative of and denoted . is a vector field in along (if is a vector space, we use the parallelization to convert to another function with values in ).

This situation can be summarized abstractly:

When is viewed as an abstract manifold it will be denoted . On let and be the two time derivatives, so . Then the material derivative of a scalar function is the Lie derivative , and

In the case of time-dependent vector or tensor fields on , a notion of the time derivative of as a function on must be chosen in order to define the material derivative. For different and fixed , the lie in tensor spaces associated to different tangent spaces . The difference quotient is therefore not defined unless there is some correspondence between these tangent spaces. For this, we will need a connection on the tangent bundle .

Definition. Let be a torsion-free connection on and denote by the same symbol its extension to the connection on equal to product with the trivial connection on . The covariant material time derivative of a time-dependent tensor on of type with respect to a motion with velocity field is the tensor of type given by

The connection restricts to a connection on , for which is parallel and the covariant material derivative is just (since ).

When is a vector space, like , the covariant material derivative with respect to the trivial connection agrees with the Lie material derivative defined before (though one must be careful to interpret vector and tensor fields as scalar fields when taking the Lie derivative, e.g. ). This is why the distinction is rarely made in the literature on this subject. In this case many names are used: Lagrangian derivative, convective derivative, substantive derivative, and others.

Nevertheless you must use the covariant material derivative even when because you may not always want to work in coordinates whose own trivial connection agrees with the trivial connection on .

Definition. Assume has a connection . The acceleration of a body moving through by with time-dependent velocity field on is the covariant material derivative of :

is a time-dependent vector field on .

This means on .

For this acceleration and the Lie material acceleration (with interpreted as a scalar) agree with the ordinary acceleration of a point in on a path through , i.e. the second derivative.

To understand the formula, consider the extreme cases. If is essentially constant in space, pointing in just one direction but varying in time, all of the points move with the same velocities and . If is constant in time but varying in space, a point’s trajectory is a flow line of and .

3 Stress and strain

Definition. An elastic structure, stress function, or constitutive function on is a smooth bundle map over . Here is the space of positive definite quadratic forms or symmetric bilinear forms on the tangent spaces of .

Proposition. There is a one-to-one correspondence between elastic structures on and differential 1-forms on the total space with support only on the fiber tangent distribution. The correspondence is natural with respect to diffeomorphisms of .1

Proof. Each fiber is parallelized with generic tangent space , using the differential of vector space translation. With respect to this parallelization, a linear form can be regarded as an element of .

More definitions. An elastic structure is called hyperelastic if the associated 1-form is exact. In this case the function such that restricted to the fiber tangent distribution is is called the strain energy function for . It is well-defined on each fiber up to an additive constant (usually normalized to zero along a preferred rest metric). The transformation of a metric into a symmetric bi-vector field by such an can be considered a type of Legendre transformation (in classical mechanics, the fiber differential of the lagrangian converts generalized velocities into generalized momenta, relating the lagrangian and hamiltonian formulations).

If is a motion and is a riemannian metric on , the strain tensor is the time-dependent tensor of type equal to the pullback of :

If has a mass -form , the second Piola-Kirchhoff stress tensor (density) is the time-dependent tensor of type :

The Cauchy stress tensor (density) is the time-dependent tensor of type :

On there is a hyperplane distribution identified with and with . The form a so-called ‘two-point’ tensor of type . On , is the identity operator , of type . All other two-point tensors can also be viewed as ordinary tensors on , with values in the tensor algebra on . For example, the first Piola-Kirchoff stress tensor (density) is the tensor of type obtained by contracting against one factor of the in , but when viewed as tensors on , we have

The only difference between and and between , , and is the preference for identifying as or in each of their tensor factors. So there is not much ambiguity in writing for , for , and for unless we work in coordinates.

In the presence of a volume form on , what are usually called the second Piola-Kirchhoff stress tensor , Cauchy stress tensor , and first Piola-Kirchhoff stress tensor are the tensors of type , , and such that

In particular, on we have .

Coordinate formulas. With respect to coordinates on and coordinates on , becomes a list of functions of variables. The formulas are:

Note that the are smooth non-linear functions on the vector spaces where takes its values. Also there are two choices for the definition of the components of a symmetric tensor field or symmetric tensor valued function. There are either of them, in which case the usual summation convention must be modified to summation over all and with , or there are of them, the non-diagonal entries are half as big, form a symmetric matrix, and should be summed over all and .

In the typical case of a flat metric on , an isometric coordinate system on , and the choice ,

4 Equations of motion and the lagrangian

Equations of motion. The setting is an -manifold with boundary, an -manifold , a riemannian metric on , a motion of in with velocity field , an elastic structure on , and a mass -form on (all structures can be passed between and using the embedding ). Then is said to be an elastic motion if

This is an equation between vector field densities on . It’s supposed to look like Newton’s law. Recall that the acceleration is the covariant material derivative of the velocity field , is the covariant differential associated to , and or trace is either of the two contractions with a factor of (they are equal).

Since , this is equivalent to the equation between vector fields

where on . In the case of flat on with isometric coordinates , the coordinate formula is:

Variation of the lagrangian and boundary conditions. When the elastic structure is given by a strain energy function , so is , we expect these equations to be the Euler-Lagrange equations for the lagrangian density

(The symbol showing that the forms are actually pulled back over the jet space is omitted.) This is proved in [5], and can be believed by the more or less standard field theory computation:

A variation of the field is a section of pulled back over the graph of in (technically, in what follows is the tautological 1-form in the variational bicomplex of local forms on the space of fields with values in such vector fields). The variations with respect to are

The objects appearing here should be interpreted as functions and differential 1-forms on the vertical tangent bundles of the 1- and 2-jet spaces of functions with values in forms on (sometimes also with values additionally in ). We used the fact that the metric is parallel for its Levi-Civita connection () and that this connection is torsion-free (for a vector field , ). The formula for the divergence of a vector field in a riemannian manifold in terms of differential forms appears at the end.

Then the variation

is zero for all if and only if the equations of motion hold and boundary conditions associated with are satisfied on :

The -forms on with values in , and , are pointwise linearly independent. is always zero on and is always zero on the , so we should assume that the initial velocity is zero (or forget the boundary of ) and that for each time ,

restricts to zero on . When or are viewed as self-adjoint endomorphisms using , this says that the normal vector to is in their kernel, or equivalently that the normal component of their values is zero. For example, if at some time the boundary is defined by in coordinates on which are Euclidean coordinates for a flat , the condition is

This is called the “traction” boundary condition since it stipulates that internal infinitesimal forces (those induced statically by the stress tensor ) can only act along the boundary surface, not into or out of it. Note that the actual dynamic force experienced on the boundary may have a normal component, since it is the divergence, not the stress itself.

5 First-order classification of materials

Definitions. If an oriented elastic body with elastic structure has a rest metric , is called isotropic if it is equivariant for the action of the . is called homogeneous if it is parallel in the sense that it is equivariant under parallel transport. (In this case the symmetry group of the must be at least the holonomy group in , so we usually wouldn’t stipulate homogeneity without isotropy unless the metric is flat.)

The differential of restricts along the rest metric to a tensor of type called the infinitesimal elasticity tensor or stiffness tensor (like the spring constant in Hooke’s law). If is homogeneous, . If is isotropic, each belongs to the invariants. If is the Legendre transform associated to a stored energy function , is symmetric (partial derivatives commute).

Stored energy formulas. Typical isotropic energy functions , in the presence of a rest metric, amount to linear or sometimes higher-order polynomial invariants of the action on . Some standard energy functions exist (Mooney-Rivlin, neo-Hookean). They offer a systematic method of creating an energy function whose elastic structure has linearization at the rest metric equal to some given isotropic elasticity tensor . However, in the non-isotropic case there appears to be no known systematic method of extending to an energy function whose linearized elastic structure is , nor a method of extending to an elastic structure whose linearization is . Even the latter would be mathematically interesting, though probably not useful without the former (the lagrangian formulation collapses without an energy function ). Such a method seems tractable for the symmetry class most generic after isotropy, single axis anisotropy.

Components of the elasticity tensor. In dimension 3 we can use the representation theory of to determine the irreducible decomposition of representations of in many ways (for example, forget about spin, pass to Lie algebra representations, complexify, and use the isomorphism ). For simplicity identify as with its standard inner product. As representations,

Here , , , are the irreducible representations of dimension 3, 5, 7, and 9. has the interpretation in as the traceless part. is the splitting into and , so that the last expression shows the decomposition into and . The invariants in come with a decomposition into a summand from and a summand . The invariant factor has a distinguished element corresponding to under the isomorphism it induces , so there is a distinguished basis element in the invariant summand . The other invariant factor has a distinguished element corresponding to the invariant inner product on induced from .

This determines a decomposition of the elasticity tensor

into a “bulk” component which is a scalar function (with respect to the basis ), a “shear” component which is also a scalar function (with respect to the basis ), a component of type , a component of type , and anti-symmetric components , , of types , , . These anti-symmetric components are zero when .

The space of linear material types is the space of orbits in the whole representation, and the orbits of the in their respective subrepresentations are invariants of such orbits. This point is often underemphasized; every material, regardless of symmetry class, has a bulk modulus, a shear modulus, two “ moduli”, and a “ modulus”.

The first invariant. Coordinates on determine a basis for the and a basis for the . Over a fixed point in , there are coordinates on and on () corresponding to points of the form

Assume a strain energy function has a rest metric equal to in these coordinates, so the symmetric matrix corresponding to is the identity matrix. The one-forms on correspond to the constant functions from under the correspondence indicated. So the one-form corresponds to the elasticity function

Then we can compute the expression of as a tensor of type on

So in this case, where arises from , is in (partial derivatives commute); the components , , are zero.

The bulk isotropic component is the component at the identity matrix :

6 Symmetry axes

Each non-zero element (except the invariants , ) is stabilized by a proper subgroup of , often a circle group of rotations about an axis. If a material has an axis of symmetry, such axes must exist for each and must be equal. We will see that in this case the anisotropic components and are determined uniquely up to a scale factor in each component and a choice of angle in associated with .

The can have axes of symmetry even when the entire and the material itself has no axis of symmetry; in principal there can be up to 3 distinguished directions (one for each of the non-trivial irreducible summands) which are not axes of symmetry for the material. It would be interesting to find out if any real materials exhibit this polarization.

Actually there are -many subrepresentations isomorphic to , even though there is a natural splitting into two such summands corresponding to the choice of basis element and a choice of reference identification (here is viewed as the specific subrepresentation of ). So the first two distinguished directions may occur as axes whose stabilizers stabilize independent elements such that

for some non-zero (as opposed to just ). In this case the classes are additional invariants of associated with the symmetry axes; we will find that, up to scaling, there are unique and in the plane they span admitting symmetry axes, so such a presentation of is essentially unique. The vectors admitting such a presentation therefore form a 2-dimensional fiber bundle over the set of those

such that (the base space can be compactified along the base of the 1-dimensional fiber bundle over corresponding to those with only one symmetry axis).

Proposition. Let be the irreducible subrepresentation consisting of traceless matrices. Then is stabilized by the group of rotations about if and only if is proportional to the image of under the composition

Proof. If then is fixed by the stabilizer of (the map is equivariant). Conversely if is stable by then it lies in the 1-dimensional weight space of weight 0 for the action on .

By choosing the constant of proportionality according to a fixed invariant inner product on , this establishes a one-to-two correspondence between pairs of antipodal vectors in and vectors in with an axis of symmetry. This becomes a one-to-one correspondence if such vectors in are identified with their negatives.

Computation of symmetry axis. To compute this axis when it exists, consider the orthonormal basis

where is the standard basis. In these bases the map is given by

It is an obvious but remarkable fact that the mapping is an involution up to a scale factor when restricted to triples with non-zero entries (actually, this map is also interesting as the non-trivial generator of the birational automorphism group of ). For us it means that in the generic case , the axis is precisely recovered as , and the scale factor can be selected according to the norm of . If precisely one of is zero, precisely one of is non-zero and one of is the sum of the squares of the non-zero . Substitution of one of these two equations into the other yields 2 separate quadratic equations in one variable, one for each of the 2 non-zero . If two of the are zero (detected by ), the remaining non-zero entry is determined directly by any of .

Proposition. Suppose that are linearly independent and stabilized by some subgroups (according to the previous proposition the axes of symmetry are themselves independent precisely when and are). Then multiples of and are the only vectors in their span admitting such an axis of symmetry.

Proof. This follows from Bezout’s theorem, but here is a direct proof. The projectivization of the vectors in admitting symmetry axes is precisely the affine projection of the Veronese surface

to from the point in equal to the the invariant line in . Consider the line through the axes of symmetry . Its image in the Veronese surface is a rational normal curve in a plane which does not contain the invariant point, and whose intersection with the Veronese surface is precisely the normal curve (this can be checked directly for the standard ). This is still true after the affine projection embeds them in . The line through the projections of and lies on the plane and intersects the rational normal curve in only these two points and therefore intersects the surface in only these two points.

Corollary. The classes appearing in are unique if such a decomposition exists (where admit distinct symmetry axes and are non-zero).

Proof. The left-hand factors in any other decomposition lie in the span of and . By the previous proposition they must be multiples of or . It follows that the right-hand factors are also multiples of the .

Computation of the decomposition. We would like a procedure that computes the decomposition , when it exists, from the standard decomposition . One method is to determine the first: Let be the reciprocal of the first component of and be the reciprocal of the first component of (for now, assume these components are non-zero). Then

Now if has an axis of symmetry, this axis is recovered as

If this is true, up to a scalar these factors can serve as the in the formula for the other 3 components of , e.g. the first equation:

Since the right hand side is homogeneous of degree 4 in and , and the left hand side is homogeneous of degree , can be absorbed into and . As a fourth degree polynomial, the solution can be expressed algebraically in terms of . The same procedure can be performed for the second or third component of , eliminating to produce a single polynomial equation for (which is unfortunately high degree in general). and then determine all of the components of . We should expect two solutions in general, corresponding to the first components of the (and to ) and the second components of the (and to ).

Proposition. Let be the irreducible subrepresentation of dimension 9. Consider the symmetrization map

This map identifies with its image in . Then is stabilized by the group of rotations about if and only if is proportional to .

Proof. The proof is the same as in the case; the fourth power map is equivariant, and the 0 weight space of acting on is one-dimensional.

7 Linear equations of motion

For a vector field on of “infinitesimal displacement”, the linearized equations of motion are: