Position and deformation measurements are of central importance in continuum mechanics. Two methods are employed : The Lagrangian method and the Eulerian method.
In the Lagrangian method, the particle position and speed are measured in reference to a fixed stationary observer based coordinates systems. This is called the referential coordinates system where the observer is located. Hence in the Lagrangian method, the particle state is measured from a global fixed frame of reference.
In Eulerian methods,a frame of reference is attached locally to the area of interest where the measurement is to be made, and the particle state is measured relative to the local coordinates systems (also called the body coordinates system). In continuum mechanics the Lagrangian method is used and in fluid mechanics the Eulerian method is used, but it is also possible to attach a local frame of reference to the body itself and then convert these measurements back relative to the global frame of reference.
A coordinate transformation gives back the coordinates of a point on a body relative to the global fixed reference frame, given the coordinates of the same point as measured in the local reference frame. This transformation is given by \[ \mathbf {X} = \mathbf {A\,x} + \mathbf {d} \] Where \(\mathbf {X}\) is the coordinate vector relative the global frame of reference, \(\mathbf {x}\) is the coordinate vector relative the local/body frame of reference and \(\mathbf {A}\) is the \(n\times n\) rotation matrix (where \(n=3\) for normal 3D space) that represents pure rotation, and \(\mathbf {d}\) is an n-dimensional vector that represents pure translation.
The following diagram illustrates these differences.
In general, the interest is in finding differential changes that occur when a body deformed. This mean measuring how a differential vector that represents the orientation of one point relative to another changes as a body deformed.
Considering the Lagrangian method from now on. Attention is now shifted to what happens when the body starts to deform. The global reference frame is selected, this is where all measurements are made with reference to.
Measurements made when the body is undeformed is distinguished from those measurements made when the body has deformed. Upper case \(X_{i}\) is used for the coordinates of a point on the body when the body is undeformed, and lower case \(x_{i}\) is used for the coordinates of the same point when measured in reference to this same global coordinate system but after the body has deformed.
Diagram below takes a snap shot of the system after 5 units of time and measures the deformation to illustrate the notation used.
Another way to represent the above is by using the same diagram to show both the undeformed and the deformed configuration as follows.
Let \(B\) be the undeformed configuration, referred to as the body \(state\). By state it is meant the set of independent variables needed to fully describe the forces and geometry of the body.
When the body is in the undeformed state \(B\), it is assumed to be free of internal stresses and that no traction forces act on it.
External loads are now applied to the body resulting in a change of state. The new state can be a result of only a deformation in the body shape, or due to only a rigid body translation/rotation, or it could be a result of a combination of deformation and rigid body motion.
The deformation will take sometime \(t\) to complete. However, in this discussion the interest is only in the final deformed state, which is called state \(b\). Hence no function(s) of time will be appear or be involved in this analysis.
The boundary conditions is assumed to be the same in state \(B\) and in state \(b\). This implied that if the solid body was in physical contact with some external non-moving supporting configuration, then after the deformation is completed, the body will remain in the same physical contact with these supports and at the same points of contact as before the deformation began.
This implies the body is free to deform everywhere, except that it is constrained to deform at those specific points it is in contact with the support. For the rigid body rotation, it is assumed the body with its support will rotate together.
A very important operator in continuum mechanics is the deformation tensor \(\mathbf {\tilde {F}}\). (A tensor can be viewed as an operator which takes a vector and maps it to another vector). This tensor allows the determination of the deformed differential vector \(dr\) knowing the undeformed differential vector \(dR\) as follows.
The tensor \(\mathbf {\tilde {F}}\) is a field tensor in general, which mean the actual value of the tensor changes depending on the location of the body where the tensor is evaluated. Hence it is a function of the body coordinates. Reference [4] gives simple examples showing how to calculate \(\mathbf {\tilde {F}}\) for simple cases of deformations in 2D. The appendix contains derivation of \(\mathbf {\tilde {F}}\) in the specific case of normal Cartesian coordinates.
The effect of applying the deformation gradient tensor \(\mathbf {\tilde {F}}\) on a vector \(\mathbf {dR}\) can be considered to have the same result as the effect of first applying a stretch deforming tensor \(\mathbf {\tilde {U}}\) (Also called the deformation tensor) on \(\mathbf {dR}\), resulting in a vector \(\mathbf {dR}^{\ast }\), followed by applying a rotation deforming tensor \(\mathbf {\tilde {R}}\) on this new vector \(\mathbf {dR}^{\ast }\) to produce the final vector \(\mathbf {dr.}\)
Hence \(\mathbf {\tilde {F}}=\mathbf {\tilde {U}}\cdot \mathbf {\tilde {R}}\) and therefore
Using polar decomposition gives
This is called polar decomposition of \(\mathbf {\tilde {F}}\), and it is always possible to find such decomposition. In addition, this decomposition is unique for each tensor \(\mathbf {\tilde {F}}\).
An oriented area in the undeformed state is \(dA\ \mathbf {N}\) (Where \(\mathbf {N}\) is a unit normal to \(dA\)). This area becomes \(da\ \mathbf {n}^{\ast }\) after the application of the stretch tensor \(\mathbf {\tilde {U}}\). It is clear that rotation will not have an effect on the area \(da\) itself, but it will rotate the unit vector \(\mathbf {n}^{\ast }\) which is normal to \(da\) to become the unit vector \(\mathbf {n}\). This is illustrated in the diagram below.
Now that a brief description of the geometry and the important tensor \(\mathbf {\tilde {F}}\) is given above, discussion of the main topic of this paper will start.