Decomposition of deformation into bend, stretch and twist? I'm wondering is there any way to decompose the deformation of an object into different components? For example, into stretching, bending and twisting part respectively? The decomposition could be applied to any description of deformation, either to deformation gradient tensor, strain tensor or stress tensor.
To my knowledge, there is some literature using the decomposition of deformation gradient tensor into rotation and deformation part. But what I want is to further decompose the deformation, leave away the rigid transformation like translating and rotating.
Thanks!
 A: There is no unique way to "further decompose" the deformation into the "rigid transformation" and "others" because whatever "rigid part" you choose, you may always calculate "others" as a simple difference (that's because there's really no global constraint on the "other" part). So the "rigid part" may be anything you want.
A: There is no such general decomposition, simply because bending and twist can only be defined for rods (where there is a longitudinal direction of reference). So it makes no sense to talk about "bending" of a sphere, for example.
If we focus our attention on bodies that can be assimilated to a rod or prismatic piece, then it is possible to define deformation in bending, twist and stretch as well as residual deformation (not every physically imaginable deformation is a combination of bending, twist and stretch).
In the practice of structural engineering for beams that can be described by the theory of Timoshenko or Navier-Bernoulli, under certain additional kinematic assumptions it is true that such a decomposition exists (ya que la deformación residual es despreciable frente a las otras tres).
A: Yes, there is a way to decompose strain (ie: deformation) into six components. The components are the three amounts of stretching along (x,y,z) and the 3 amounts of parallelepipeding in the planes with normals (x,y,z).
Strain is something you do to an object … just like rotation is an action you do to an object.
In fact in 3d space (ie: x,y,z), rotations and strains form the group GL(3,R).  This is the group of all invertible 3x3 matrices M of real numbers.
We can describe what these transformations do by just talking about the matrices $M$ that are very close to the identity matrix, where all elements in the matrix $\Theta$ are <<1. All these elements are in radians.
$$ 
M=I+\Theta
$$
$$
\Theta = \begin{bmatrix}
           0 & \theta^{12} &-\theta^{13} \\
-\theta^{12} &           0 & \theta^{23} \\ 
 \theta^{13} &-\theta^{23} &           0  \\
\end{bmatrix}_{Antisymmetric}
\
+ \begin{bmatrix} 
\epsilon^{11} & \epsilon^{12} & \epsilon^{13} \\
\epsilon^{12} & \epsilon^{22} & \epsilon^{23} \\
\epsilon^{13} & \epsilon^{23} & \epsilon^{33} \\ 
\end{bmatrix}_{Symmetric}
$$
By taking products of these matrices we build all the matrices $M$ in the group for any size elements in $\Theta$.
$$
M=e^{\Theta}=I+\Theta+\dfrac{\Theta^2}{2!}+ \dfrac{\Theta^3}{3!}+ \dfrac{\Theta^4}{4!} +…
$$
The $\theta$ are antisymmetric, make M orthogonal ($M^T =M^{-1}$), and leave the length  ${x_1}^2 + {x_2}^2+{x_3}^2$ invariant.  Because lengths are invariant, the transformations are called rotations.  The $\epsilon$ do not leave lengths invariant and are called strains.  I believe these $\epsilon$ are the "components" of strain  that you are seeking, just like the $\theta$ are the "components" of rotation.  Whereas under rotations $\vec{\theta}$ transforms like a vector, under all the transformations of GL(3,R),   $\Theta^{ij}$ transforms like a 2nd rank tensor.
The parameter $\theta^{12}$ means rotate the object about axis1 x axis2.  That is, put your right thumb perpendicular to the plane formed by axis1 and axis2, such that your fingers would push axis1 into axis 2.  Then in this same rotation direction about your thumb rotate the object by $\theta^{12}$ radians.
The parameter $\epsilon^{11}$ means stretch the object by $(1+\epsilon^{11})$ along axis1.
The parameter $\epsilon^{12}$ means parallelepiped the object in the plane containing axis1 and axis2.  For example, a square box with its sides initially along axis1 and axis2, becomes a parallelepiped with its sides tilted inward from axis1 and axis2 and its diagonal from the origin stretched.  Both sides now make inward angles of $\epsilon^{12}$ radians with their respective axis1 or axis2.
Examples of large rotation and strain matrices are:
Rotate about axis3 by $\theta^{12}$ radians.
$$
M(\theta^{12})=\begin{bmatrix} cos(\theta^{12})  & -sin(\theta^{12})  & 0 \\ sin(\theta^{12})   & cos(\theta^{12})  & 0 \\ 0 & 0 & 1 \end{bmatrix}
$$
Strain (parallelepiped) about the axis3 by $\epsilon^{12}$ radians.
$$
M(\epsilon^{12})=\begin{bmatrix} cosh(\epsilon^{12})  & sinh(\epsilon^{12})   & 0 \\ sinh(\epsilon^{12})   & cosh(\epsilon^{12})  & 0 \\ 0 & 0 & 1 \end{bmatrix}
$$
Strain (stretch) along the axis1 by $\epsilon^{11}$ radians.
$$
M(\epsilon^{11})=\begin{bmatrix} e^{\epsilon^{11} } & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
$$
Notice that for $\epsilon^{11}<<1$, the fractional change of the length of an object in the 1 direction is $\epsilon^{11}$.
If you do a transformation $(\theta \ne 0,\quad \epsilon=0)$ you can call it a pure rigid body rotation, and products of pure rotations always give pure rotations because rotations form the subgroup O(3).
If you do a transformation $(\theta=0,\quad \epsilon \ne 0)$ you can call it a pure deformation, however, the product of two different pure deformations will not be pure and will have a little rotation in it. This is why you (or a cat) can float isolated in space and by a series of self deformations rotate yourself wrt the stars!
