Lecture 12: Deformations

Overview

Steven M. Wise introduces continuum-mechanics kinematics as the geometry of maps from a reference configuration to a deformed configuration, defining material and spatial coordinates, displacement, admissibility, and the deformation gradient. He then builds a decomposition of every admissible homogeneous deformation into translations, rotations, and extensions, using polar decomposition and the spectral decomposition of symmetric positive-definite stretch tensors.

Key takeaways

The deformation gradient F = ∇φ controls the local linear change near a material point, while its positive determinant ensures local invertibility and preservation of orientation.
A homogeneous deformation has constant F and is therefore affine, φ(X) = c + FX; its translation component can be separated from a deformation with any chosen fixed point.
Polar decomposition factors an admissible homogeneous deformation gradient into a rotation and a symmetric positive-definite stretch, with the composition order affecting the resulting map.
An extension with F = I + (λ − 1)e⊗e scales only the direction e by λ and leaves its perpendicular plane unchanged, giving eigenvalues λ, 1, and 1.
The spectral decomposition of a stretch tensor expresses it as three extensions along orthonormal eigenvector directions, reducing homogeneous deformation to translations, rotations, and extensions.

Chapters

0:00 Kinematics Maps a Reference Body to a Deformed Configuration

Kinematics studies motion and deformation geometrically, without considering what causes the change.
The body occupies a reference configuration B and is mapped by φ to a deformed configuration B′.
The reference and deformed sets describe the same body using different coordinates.

3:00 Material Coordinates, Spatial Coordinates, and Displacement

Capital X denotes a material point in B, while lowercase x denotes its spatial position in B′.
The deformation map φ assigns each material point X a spatial position x = φ(X).
The displacement field is u(X) = φ(X) − X, a vector field defined over the reference configuration.

6:00 Admissible Maps Prevent Overlap and Preserve Orientation

An admissible deformation is one-to-one and onto between B and B′, so distinct material points cannot occupy the same deformed position.
The gradient of φ must be invertible at every point and have positive determinant.
A positive determinant preserves handedness and rules out turning the body into its mirror image.

10:00 The Deformation Gradient Gives the Local Linear Approximation

For a smooth deformation map, the deformation gradient F = ∇φ is a second-order tensor field.
A Taylor expansion near a material point X₀ approximates φ(X) using φ(X₀) plus F(X₀)(X − X₀), with a second-order remainder.
F captures the local first-order behavior of the deformation, much like an ordinary derivative approximates a function near a point.

15:00 Homogeneous Deformations Have a Constant Gradient

A deformation is homogeneous when its deformation gradient F is constant throughout B.
Its map is affine, φ(X) = c + FX, with no higher-order spatial correction.
This constant-gradient structure makes homogeneous deformations especially useful for classifying basic motions.

16:30 Translations Move Every Material Point by the Same Vector

A translation has the form φ(X) = X + c, where c is a constant vector.
Its deformation gradient is the identity tensor I, and its displacement field equals c at every point.
Translations change the body's position but preserve its shape and orientation.

19:00 Any Homogeneous Map Can Be Recentered at a Fixed Point

A fixed point y satisfies φ(y) = y; a homogeneous deformation centered there has the form y + F(X − y).
Result 4.1 decomposes any homogeneous deformation into a fixed-point deformation and a translation.
The translation can be composed before or after the fixed-point deformation, with the translation vector adjusted to match.

26:00 Rotations About a Fixed Point Use Proper Orthogonal Tensors

A rotation about y has the form φ(X) = y + Q(X − y), where Q is orthogonal and det Q = +1.
The determinant condition distinguishes rotations from reflections, which have determinant −1.
The example rotates a cuboid by 45 degrees in the X₁X₂ plane about y = (1/4, 1/4, 0), with an axis parallel to X₃.

31:30 Stretch Deformations Are Symmetric Positive Definite

A homogeneous deformation centered at y is a stretch when its gradient is a symmetric positive-definite tensor.
A stretch may expand in one direction and contract in another; contraction counts as a stretch with a factor between 0 and 1.
Unlike a rotation, a stretch does not need to preserve volume, so its determinant need not equal 1.

34:00 A Diagonal Stretch Doubles One Axis and Halves Another

Wise illustrates a stretch with F = diag(2, 1/2, 1), centered at y = (2, 2, 2).
The map doubles displacements in the first coordinate, halves them in the second, and leaves the third unchanged.
The example's determinant happens to be 1, but the lecture clarifies that this is not required for general stretches.

38:00 Polar Decomposition Splits Homogeneous Motion into Rotation and Stretch

Result 4.2 applies the polar decomposition F = RU = VR to an admissible homogeneous deformation with a fixed point.
R is a rotation tensor, while U and V are symmetric positive-definite stretch tensors.
The map can be composed as a rotation after a stretch or a stretch after a rotation; the two orders generally differ because the operations do not commute.

50:00 Extensions Stretch or Contract Along One Direction

An extension in unit direction e, centered at y, has gradient F = I + (λ − 1)e⊗e, where λ > 0.
Vectors parallel to e are scaled by λ, while vectors perpendicular to e remain unchanged.
λ greater than 1 represents elongation, while 0 < λ < 1 represents contraction; both are treated as extensions.

55:00 An Extension Has Eigenvalues λ, 1, and 1

The extension direction e is an eigenvector of F with eigenvalue λ.
Any orthonormal directions u and v perpendicular to e are eigenvectors with eigenvalue 1.
In the orthonormal basis (e, u, v), the matrix representation is diagonal with entries (λ, 1, 1), showing that extension is a special stretch.

1:03:00 Positive Definiteness Confirms Extensions Are Stretches

Wise verifies that F = I + (λ − 1)e⊗e is symmetric because both I and e⊗e are symmetric.
For any nonzero vector x, the quadratic form x·Fx is positive when λ > 0, establishing positive definiteness.
The proof separates x into components parallel and perpendicular to e, showing that the perpendicular component is unchanged and the parallel component has positive scale λ.

1:07:00 Spectral Decomposition Reduces Every Stretch to Three Extensions

Result 4.3 uses the eigenvalues and orthonormal eigenvectors of U or V to express each stretch as three extensions along principal directions.
The stretch tensors U and V share the same positive eigenvalues, though their eigenvectors can differ.
Combining the translation, rotation, and three-extension decompositions shows that any admissible homogeneous deformation can be assembled from these basic motions.

Transcript excerpt

0:05 Welcome back to continue mechanics then and we're in chapter 4 now talking about kinematics. So kinematics is a study of deformationations, motions, um changes in a body without any concern about what got or made those changes. So it's uh it's it's essentially the geometry of deformations. Kinematics uh has the same root word as cinema. It means movement. And so it's a study of movement essentially. I plan to spend three to four lectures in here and um this is the first of those four

0:55 three or four. The first thing we're going to talk about are deformations. What is a deformation? So we're going to be working in three-dimensional ukidian space. That means actually there's another axis like this. And we're going to be interested in interested in mappings that take a body which can be described as sitting in or existing in a configuration in a set and applying a map to that which changes it or maps it to a new set. So this set B gets mapped to B prime. So B is the undeformed configuration of a body. and B prime is its deformed configuration.

1:44 B is called the reference or reference configuration and B prime is called the deformed configuration. The two sets describe exactly the same body and their coordinates describe the same body but in different coordinate systems as I'll show you in a in a moment. Um so we're often going to write uh for the set B prime B prime is equal to the action of or the uh set B under the action of PHI. Phi is uh essentially a vector valued function because you take points in B and you have to map them to points in B

2:32 prime and both of those are vectors. The points in the reference

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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Steven M. Wise.