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1. INTRODUCTION.

This thread is an experiment. I think the public discussion between myself, pervect, and Greg Egan in the rotating hoop thread has given students and the interested public and glimpse into how people actually think about a challenging problem, but I guess the discussion went over the heads of all but a handful of trained physicists and mathematicians who happen to have prior familiarity with the theory of elasticity.

I feel that prevect, Greg, and I have by no means exhausted the possibilities in the other thread, but my intention is that this is the thread where anyone can ask questions about all this elasticity stuff.

I'll start by offering a synopsis of the theory of nonrelativistic elasticity, which is already highly nontrivial even for a nonrelativistic hoop. My synopsis covers standard stuff but is tailored toward relativistic applications in various ways.

2. REFERENCES:

I only list some of the books I consulted while preparing my synopsis; this list is weighted toward books I judge should be easily available. In particular, large university bookstores may have the classic textbook by Landau and Lifschitz in stock, and any good library will have multiple copies.

Some excellent references for the nonrelativistic theory of elasticity:

L. D. Landau and E. M. Lifschitz,

Theory of Elasticity, 3rd Edition.

Pergamon, 1986.

Chi-Teh Wang,

Applied Elasticity.

McGraw-Hill, 1953.

Karl F. Graff,

Wave Motion in Elastic Solids.

Dover reprint, 1991 (org. 1975).

S. P. Timoshenko and J. N. Goodier,

Theory of Elasticity, 3rd Edition.

McGraw Hill reprint 1970 (org. 1934).

An excellent survey of continuum mechanics generally:

Alexander L. Fetter and John Dirk Walecka,

Theoretical Mechanics of Particles and Continua.

McGraw-Hill, 1980.

An engineering-oriented book which offers an introduction to engineering applications of the theory of elasticity:

Donald F. Young, William F. Riley, Kenneth G. McConnell, Thomas R. Rogge,

Essentials of Mechanics, Iowa State University Press, 1974.

One of the best overall surveys of PDEs is:

Ronald B. Guenther and John W. Lee.

Partial Differential Equations of Mathematical Physics and Integral Equations.

Dover reprint, 1996 (org. 1988)

Two of the best introductions to symmetry methods for solving DEs:

Peter J. Olver,

Applications of Lie Groups to Differential Equations, 2nd Edition.

Springer, 1993.

Brian J. Cantwell,

Introduction to Symmetry Analysis.

Cambridge University Press, 2002.

An excellent introduction to the classical theory of curves and surfaces, needed here only for the result that the highest derivatives appearing in the curvature and torsion of a curve are two, three respectively:

Dirk J. Struik,

Lectures on Classical Differential Geometry, 2nd Edition.

Dover reprint 1988 (org. 1950).

3. ELASTIC PROPERTIES OF MATERIALS

Imagine we have a cylindrical steel shaft firmly clamped in a vise at either end. If we adjust the vise so that the shaft is under tension, we find that the shaft is elongated and also shrinks in diameter. If we plot how this axial elongation and orthogonal compression depends on the applied tension, we find that for sufficiently small tension, the relation is roughly linear, but with different slopes. If we repeat the experiment with a steel shaft of different size, we find the same two slopes. If we repeat with shafts of different materials, we obtain similar results but with new pairs of slopes. This suggests that these slopes characterize the response of the material to tension. They are called Young's modulus E (for extension), and the product of Young's modulus with Poisson's ratio [itex]\nu[/itex] (many authors write this [itex]\sigma[/itex]; this topic has only partially standardized notation). It turns out that E, nu suffice to characterize the elastic properties (valid for small deformations) of any homogeneous isotropic material, but as we will see it is often useful to define alternative "elastic moduli"; these will be rational combinations of each other.

4. THE STRAIN TENSOR

We must begin by describing the deformation of a material. To do this we imagine that the deformation is obtained by applying a diffeomorphism,

i.e. a smooth map E^3 -> E^3 whose inverse is also smooth. Indeed, let us assume this has the form

[tex]

x^{i'} = x^i + u^i (x^1, x^2, x^3)

[/tex]

where the vector field [itex]\vec{u}[/tex] is the

[tex]

du^i = \frac{\partial u^i}{\partial x^k} \, dx^k = {u^i}_{,k} \, dx^k

[/tex]

We wish to compute how the distances between initially nearby material points change. But the new distances are given by

[tex]

(ds')^2 = \delta_{i' j'} \, dx^{i'} \, dx^{j'}

= \delta_{ij} \, \left( dx^i + du^i \right) \, \left( dx^j + du^j \right)

[/tex]

[tex]

\approx \delta_{ij} \, dx^i \, dx^j + 2 \, u_{i,j} \, dx^i \, dx^j

[/tex]

where we neglected terms which are second order in du^j. But the LHS is symmetric in i,j so we can and should replace u_{i,j} by its symmetrization, giving

[tex]

(ds')^2 = ds^2 + 2 \, \varepsilon_{ij} \, dx^i \, dx^j

[/tex]

where

[tex]

\varepsilon_{ij} = u_{(i,j)}

[/tex]

is the

Notice that by definition, rigid motions of the material, which do not change any distances between material points, are

[tex]

\omega_{ij} = u_{[i,j]}

[/itex]

This represents

Since it is often convenient to adopt a coordinate chart adapted to any symmetries, it will be useful to work out some general formulas, so let us see how this works for a cylindrical chart

[tex]

ds^2 = dz^2 + dr^2 + r^2 \, d\phi^2, \; -\infty < z < \infty, 0 < r < \infty, -\pi < \phi < \pi

[/tex]

with the coframe field

[tex]

\sigma^1 = dz, \; \sigma^2 = dr, \, \sigma^3 = r \, d\phi

[/tex]

and its dual frame field

[tex]

\vec{e}_1 = \partial_z, \; \vec{e}_2 = \partial_r, \; \vec{e}_3 = \frac{1}{r} \, \partial_\phi

[/tex]

Let us take

[tex]

\vec{u} = A \, \vec{e}_1 + B \, \vec{e}_2 + C \, \vec{e}_3

[/tex]

where A, B, C are functions of [itex]z,r,\phi[/itex]. Then we have

[tex]

\varepsilon_{\hat{m}\hat{n}} = \left[ \begin{array}{ccc}

A z & \frac{1}{2} \left( A_r+B_z \right)

& \frac{1}{2} \left( \frac{A_\phi}{r} + C_z \right) \\

\cdot & B_r

& \frac{1}{2} \left( \frac{B_\phi - C}{r} + C_r \right) \\

\cdot & \cdot

& \frac{ B + C_\phi}{r}

\end{array} \right]

[/tex]

where the dots invite the reader to fill in redundant entries using the symmetry of the tensor, where subscripts on functions denote partial derivatives, and where the hats on the indices emphasize that these are the components in the frame. For comparison, the coordinate basis components are

[tex]

\varepsilon_{mn} = \left[ \begin{array}{ccc}

A z & \frac{1}{2} \left( A_r+B_z \right)

& \frac{1}{2} \left( A_\phi + r\, C_z \right) \\

\cdot & B_r & \frac{1}{2} \left( B_\phi - C + r\, C_r \right) \\

\cdot & \cdot & r \, \left( B + C_\phi \right)

\end{array} \right]

[/tex]

In the sequel I will drop the hats and only use specified frame fields.

Note: as with their other books, if one can get past the first few pages of the classic textbook by Landau and Lifschitz, one can see how clear and beautifully organized it is. Unfortunately, judging from my experience at PF and elsewhere, many readers are likely to get stuck on p. 3, where in (1.8), both the components of the vector and the strain tensor are written in terms of the frame,

Many books dealing with elasticity note that some vector analysis formulae are only valid in a Cartesian chart, which seems sure to confuse readers in our context. The neccessity of such remarks are avoided by simply using frame fields! Also, strain components (and in the next section, stress components) computed with respect to the appropriate frame field, rather than the coordinate basis components, are the ones which would be measured in tension, compression, or torsion tests.

Alert readers will have already noticed that our computation above didn't depend upon using the metric of Euclidean space! In fact we have

[tex]

(ds')^2 = g_{i'j'} \, dx^{i'} \, dx^{j'} \approx \left( g_{ij} + 2 \, \varepsilon_{ij} \right) \, dx^i \, dx^j

[/tex]

or

[tex]

(ds')^2 = ds^2 + 2 \, \varepsilon_{ij} \, dx^i \, dx^j

[/tex]

where the strain tensor

[tex]

\varepsilon = u_{(i;j)}

[/tex]

is the symmetrized

[tex]

\omega = u_{[i,j]}

[/tex]

vanishes. (What about translations?) All such rigid motions drop out of computations of material deformations.

For example, let us take the hyperbolic plane in the upper half plane chart familiar from elementary complex analysis:

[tex]

ds^2 = \frac{dx^2 + dy^2}{y^2}, \; 0 < y < \infty, -\infty < x < \infty

[/tex]

Let us adopt the obvious coframe field

[tex]

\sigma^1 = \frac{dx}{y}, \; \sigma^2 = \frac{dy}{y}

[/tex]

with dual frame field

[tex]

\vec{e}_1 = y \, \partial_x, \; \vec{e}_2 = y \, \partial_y

[/tex]

Writing the displacement vector field [itex]\vec{u} = A \, \vec{e}_1 + B \, \vec{e}_2[/tex] where A, B are functions of x,y, we have

[tex]

\varepsilon_{ij} = \left[ \begin{array}{cc}

y \, A_x - B \; & \; \frac{ y \; \left( A_y + B_x \right) }{2} + \frac{A}{2} \\

\cdot & y B_y

\end{array} \right]

[/tex]

Which displacement vector fields correspond to rigid motions? The ones whose strain tensor vanishes. Solving, we find a three-dimensional Lie algebra of solutions,

[tex]

\vec{X}_1 = \partial_x, \;

\vec{X}_2 = x \, \partial_x + y \, \partial_y, \;

\vec{X}_3 = \frac{1 + x ^2 - y^2}{2} \, \partial_x + x \, y \, \partial_y

[/tex]

This Lie algebra of vector fields is of course precisely the Killing vector fields of H^2, i.e. the rigid motions, the diffeomorphisms which preserve hyperbolic distances. Students of differential geometry know that the distance preserving condition can be formulated in a different way, in terms of Lie dragging of the metric tensor itself, which is very important since otherwise one might worry that the condition is somehow circular, since the metric enters into the definition of the covariant derivative we used, more precisely the notion of covariant differentiation which arises from the Levi-Civita connection. Lie differentiation on the other hand is defined at a lower level of structure, independently of any metric tensor we might place on our manifold.

Note that [itex]\vec{X}_2[/itex] represents a

Exercise: compute the antisymmetrized covariant gradient of the three (rigid motion) vector fields listed above. Find this is nonzero constant for one (which therefore represents a rotation about a certain point in H^2). The other two represent translations in H^2.

(To be continued...)

This thread is an experiment. I think the public discussion between myself, pervect, and Greg Egan in the rotating hoop thread has given students and the interested public and glimpse into how people actually think about a challenging problem, but I guess the discussion went over the heads of all but a handful of trained physicists and mathematicians who happen to have prior familiarity with the theory of elasticity.

I feel that prevect, Greg, and I have by no means exhausted the possibilities in the other thread, but my intention is that this is the thread where anyone can ask questions about all this elasticity stuff.

I'll start by offering a synopsis of the theory of nonrelativistic elasticity, which is already highly nontrivial even for a nonrelativistic hoop. My synopsis covers standard stuff but is tailored toward relativistic applications in various ways.

2. REFERENCES:

I only list some of the books I consulted while preparing my synopsis; this list is weighted toward books I judge should be easily available. In particular, large university bookstores may have the classic textbook by Landau and Lifschitz in stock, and any good library will have multiple copies.

Some excellent references for the nonrelativistic theory of elasticity:

L. D. Landau and E. M. Lifschitz,

Theory of Elasticity, 3rd Edition.

Pergamon, 1986.

Chi-Teh Wang,

Applied Elasticity.

McGraw-Hill, 1953.

Karl F. Graff,

Wave Motion in Elastic Solids.

Dover reprint, 1991 (org. 1975).

S. P. Timoshenko and J. N. Goodier,

Theory of Elasticity, 3rd Edition.

McGraw Hill reprint 1970 (org. 1934).

An excellent survey of continuum mechanics generally:

Alexander L. Fetter and John Dirk Walecka,

Theoretical Mechanics of Particles and Continua.

McGraw-Hill, 1980.

An engineering-oriented book which offers an introduction to engineering applications of the theory of elasticity:

Donald F. Young, William F. Riley, Kenneth G. McConnell, Thomas R. Rogge,

Essentials of Mechanics, Iowa State University Press, 1974.

One of the best overall surveys of PDEs is:

Ronald B. Guenther and John W. Lee.

Partial Differential Equations of Mathematical Physics and Integral Equations.

Dover reprint, 1996 (org. 1988)

Two of the best introductions to symmetry methods for solving DEs:

Peter J. Olver,

Applications of Lie Groups to Differential Equations, 2nd Edition.

Springer, 1993.

Brian J. Cantwell,

Introduction to Symmetry Analysis.

Cambridge University Press, 2002.

An excellent introduction to the classical theory of curves and surfaces, needed here only for the result that the highest derivatives appearing in the curvature and torsion of a curve are two, three respectively:

Dirk J. Struik,

Lectures on Classical Differential Geometry, 2nd Edition.

Dover reprint 1988 (org. 1950).

3. ELASTIC PROPERTIES OF MATERIALS

Imagine we have a cylindrical steel shaft firmly clamped in a vise at either end. If we adjust the vise so that the shaft is under tension, we find that the shaft is elongated and also shrinks in diameter. If we plot how this axial elongation and orthogonal compression depends on the applied tension, we find that for sufficiently small tension, the relation is roughly linear, but with different slopes. If we repeat the experiment with a steel shaft of different size, we find the same two slopes. If we repeat with shafts of different materials, we obtain similar results but with new pairs of slopes. This suggests that these slopes characterize the response of the material to tension. They are called Young's modulus E (for extension), and the product of Young's modulus with Poisson's ratio [itex]\nu[/itex] (many authors write this [itex]\sigma[/itex]; this topic has only partially standardized notation). It turns out that E, nu suffice to characterize the elastic properties (valid for small deformations) of any homogeneous isotropic material, but as we will see it is often useful to define alternative "elastic moduli"; these will be rational combinations of each other.

4. THE STRAIN TENSOR

We must begin by describing the deformation of a material. To do this we imagine that the deformation is obtained by applying a diffeomorphism,

i.e. a smooth map E^3 -> E^3 whose inverse is also smooth. Indeed, let us assume this has the form

[tex]

x^{i'} = x^i + u^i (x^1, x^2, x^3)

[/tex]

where the vector field [itex]\vec{u}[/tex] is the

*displacement vector*, so that[tex]

du^i = \frac{\partial u^i}{\partial x^k} \, dx^k = {u^i}_{,k} \, dx^k

[/tex]

We wish to compute how the distances between initially nearby material points change. But the new distances are given by

[tex]

(ds')^2 = \delta_{i' j'} \, dx^{i'} \, dx^{j'}

= \delta_{ij} \, \left( dx^i + du^i \right) \, \left( dx^j + du^j \right)

[/tex]

[tex]

\approx \delta_{ij} \, dx^i \, dx^j + 2 \, u_{i,j} \, dx^i \, dx^j

[/tex]

where we neglected terms which are second order in du^j. But the LHS is symmetric in i,j so we can and should replace u_{i,j} by its symmetrization, giving

[tex]

(ds')^2 = ds^2 + 2 \, \varepsilon_{ij} \, dx^i \, dx^j

[/tex]

where

[tex]

\varepsilon_{ij} = u_{(i,j)}

[/tex]

is the

*strain tensor*. We can see from this that the components of the strain tensor are*dimensionless*.Notice that by definition, rigid motions of the material, which do not change any distances between material points, are

*precisely those diffeomorphisms which have vanishing strain*. The antisymmetric part of the gradient of the displacement vector field, which automatically dropped out above (this is why we can replace [itex]u_{i,j}[/itex] by [itex]u_{(i,j)}[/itex], is[tex]

\omega_{ij} = u_{[i,j]}

[/itex]

This represents

*rigid rotations*of the body. Similarly,*rigid translations*of the body correspond to*constant*[itex]u^i[/itex], so they also dropped out.Since it is often convenient to adopt a coordinate chart adapted to any symmetries, it will be useful to work out some general formulas, so let us see how this works for a cylindrical chart

[tex]

ds^2 = dz^2 + dr^2 + r^2 \, d\phi^2, \; -\infty < z < \infty, 0 < r < \infty, -\pi < \phi < \pi

[/tex]

with the coframe field

[tex]

\sigma^1 = dz, \; \sigma^2 = dr, \, \sigma^3 = r \, d\phi

[/tex]

and its dual frame field

[tex]

\vec{e}_1 = \partial_z, \; \vec{e}_2 = \partial_r, \; \vec{e}_3 = \frac{1}{r} \, \partial_\phi

[/tex]

Let us take

[tex]

\vec{u} = A \, \vec{e}_1 + B \, \vec{e}_2 + C \, \vec{e}_3

[/tex]

where A, B, C are functions of [itex]z,r,\phi[/itex]. Then we have

[tex]

\varepsilon_{\hat{m}\hat{n}} = \left[ \begin{array}{ccc}

A z & \frac{1}{2} \left( A_r+B_z \right)

& \frac{1}{2} \left( \frac{A_\phi}{r} + C_z \right) \\

\cdot & B_r

& \frac{1}{2} \left( \frac{B_\phi - C}{r} + C_r \right) \\

\cdot & \cdot

& \frac{ B + C_\phi}{r}

\end{array} \right]

[/tex]

where the dots invite the reader to fill in redundant entries using the symmetry of the tensor, where subscripts on functions denote partial derivatives, and where the hats on the indices emphasize that these are the components in the frame. For comparison, the coordinate basis components are

[tex]

\varepsilon_{mn} = \left[ \begin{array}{ccc}

A z & \frac{1}{2} \left( A_r+B_z \right)

& \frac{1}{2} \left( A_\phi + r\, C_z \right) \\

\cdot & B_r & \frac{1}{2} \left( B_\phi - C + r\, C_r \right) \\

\cdot & \cdot & r \, \left( B + C_\phi \right)

\end{array} \right]

[/tex]

In the sequel I will drop the hats and only use specified frame fields.

Note: as with their other books, if one can get past the first few pages of the classic textbook by Landau and Lifschitz, one can see how clear and beautifully organized it is. Unfortunately, judging from my experience at PF and elsewhere, many readers are likely to get stuck on p. 3, where in (1.8), both the components of the vector and the strain tensor are written in terms of the frame,

*not*the coordinate basis! This procedure is so sensible that Landau and Lifschitz did not consider it worthy of mention, but of course if the student is expecting to see results reported in terms of a coordinate basis, he will be unable to reconcile his own computation with the result stated by Landau and Lifschitz!Many books dealing with elasticity note that some vector analysis formulae are only valid in a Cartesian chart, which seems sure to confuse readers in our context. The neccessity of such remarks are avoided by simply using frame fields! Also, strain components (and in the next section, stress components) computed with respect to the appropriate frame field, rather than the coordinate basis components, are the ones which would be measured in tension, compression, or torsion tests.

Alert readers will have already noticed that our computation above didn't depend upon using the metric of Euclidean space! In fact we have

[tex]

(ds')^2 = g_{i'j'} \, dx^{i'} \, dx^{j'} \approx \left( g_{ij} + 2 \, \varepsilon_{ij} \right) \, dx^i \, dx^j

[/tex]

or

[tex]

(ds')^2 = ds^2 + 2 \, \varepsilon_{ij} \, dx^i \, dx^j

[/tex]

where the strain tensor

[tex]

\varepsilon = u_{(i;j)}

[/tex]

is the symmetrized

*covariant*gradient of the displacement vector field. Just as before, rigid rotations correspond to displacements vector fields whose antisymmetrized gradient[tex]

\omega = u_{[i,j]}

[/tex]

vanishes. (What about translations?) All such rigid motions drop out of computations of material deformations.

For example, let us take the hyperbolic plane in the upper half plane chart familiar from elementary complex analysis:

[tex]

ds^2 = \frac{dx^2 + dy^2}{y^2}, \; 0 < y < \infty, -\infty < x < \infty

[/tex]

Let us adopt the obvious coframe field

[tex]

\sigma^1 = \frac{dx}{y}, \; \sigma^2 = \frac{dy}{y}

[/tex]

with dual frame field

[tex]

\vec{e}_1 = y \, \partial_x, \; \vec{e}_2 = y \, \partial_y

[/tex]

Writing the displacement vector field [itex]\vec{u} = A \, \vec{e}_1 + B \, \vec{e}_2[/tex] where A, B are functions of x,y, we have

[tex]

\varepsilon_{ij} = \left[ \begin{array}{cc}

y \, A_x - B \; & \; \frac{ y \; \left( A_y + B_x \right) }{2} + \frac{A}{2} \\

\cdot & y B_y

\end{array} \right]

[/tex]

Which displacement vector fields correspond to rigid motions? The ones whose strain tensor vanishes. Solving, we find a three-dimensional Lie algebra of solutions,

[tex]

\vec{X}_1 = \partial_x, \;

\vec{X}_2 = x \, \partial_x + y \, \partial_y, \;

\vec{X}_3 = \frac{1 + x ^2 - y^2}{2} \, \partial_x + x \, y \, \partial_y

[/tex]

This Lie algebra of vector fields is of course precisely the Killing vector fields of H^2, i.e. the rigid motions, the diffeomorphisms which preserve hyperbolic distances. Students of differential geometry know that the distance preserving condition can be formulated in a different way, in terms of Lie dragging of the metric tensor itself, which is very important since otherwise one might worry that the condition is somehow circular, since the metric enters into the definition of the covariant derivative we used, more precisely the notion of covariant differentiation which arises from the Levi-Civita connection. Lie differentiation on the other hand is defined at a lower level of structure, independently of any metric tensor we might place on our manifold.

Note that [itex]\vec{X}_2[/itex] represents a

*dilation*with respect to the obvious euclidean metric on the upper half plane.Exercise: compute the antisymmetrized covariant gradient of the three (rigid motion) vector fields listed above. Find this is nonzero constant for one (which therefore represents a rotation about a certain point in H^2). The other two represent translations in H^2.

(To be continued...)

Last edited: