1. Continuum Mechanics (MTH487)
Lecture 13
Instructor
Dr. Junaid Anjum

2. Aims and Objectives kinematics of deformation and motion
material and spatial coordinates
Langrangian and Eulerian description

3. Kinematics of Deformation and Motion…
A material body B is defined as the set of elements , called particles, or material points which can be put into a one-to-one correspondence with the points of a regular region of physical space. Note that whereas a particle of classical mechanics has an assigned mass, a continuum particle is essentially a material point for which density is defined. P
The specification of the position of all of the particles of B with respect to a fixed origin at some instant of time is said to define the configuration of the body at that instant.
Mathematically, this is expressed by the mapping

4. Kinematics of Deformation and Motion…
A change in configuration is the result of a displacement of the body. For example, a rigid-body displacement is one consisting of a simultaneous translation and rotation which produces a new configuration but causes no change in the size or shape of the body (only changes its position and/or orientation). On the other hand an arbitrary displacement will usually include both a rigid-body displacement and a deformation which results in a change in size, or shape, or possibly both.
initial configuration
initial configuration
at some later
time
at some later
time

5. Kinematics of Deformation and Motion…
A motion of a body B is a continuous time sequence of displacements that carries the set of particles into various configurations in a stationary space. Such a motion may be expressed by the equation
which gives the position for each particle for all times t, where t ranges from to .
As with configuration mappings,
we assume the motion function is
uniquely invertible and differentiable,
so that we may write the inverse
which identifies the particle
located at position at time t. P P P P

6. Kinematics of Deformation and Motion…
We give special meaning to certain configurations of the body. In particular, we single out a reference configuration from which all displacements are identified. For the purpose it serves, the reference configuration needs not be one that the body ever actually occupies. Often, however, the initial configuration (t=0) is chosen as the reference configuration. The current configuration is the one which body occupies at the current time t.
Reference P
Current P

7. Material and Spatial coordinates:
where are the material (or referential) coordinates
assuming an inverse mapping
which defines the motion of the body in physical space relative to the reference configuration prescribed by .
Reference P
in terms of spatial coordinates
Current P

8. Material and Spatial coordinates …
Velocity and Acceleration:
For a particular particle having the material position vector ,
Reference P
Current P
or for the whole body,

9. Material and Spatial coordinates …
Problem 1: Let the motion of a body be given by in component form as
Determine
The path of the particle originally at
the velocity and acceleration components of the same particle when t=2s.

11. Material and Spatial coordinates …
Problem 2: Invert the motion equations of Problem 1 to obtain and determine the velocity and acceleration components of the particle at
when t=2s.

13. Material and Spatial coordinates …
Existence of the inverse mapping:
individual particles do not execute arbitrary motion independent of one
another.
no two particle occupy the same space.
particles close (to each other) in reference configuration remain close in
other configurations.
Hence the function must be single-valued and continuous.
Moreover, we require the inverse function in the equation
to have the same properties as .
The mathematical condition that guarantees the existence of such an
Inverse function is the non-vanishing of the Jacobian determinant.

14. Material and Spatial coordinates …
Problem 3: Determine the Jacobian J for the motion of a continuum given by

16. Langrangian & Eulerian description
If a physical property of the body B such as its density or kinematic
property of its motion such as the velocity is expressed in terms of the
material coordinates and time t, we say that property is given by the
referential or material description.
When the referential configuration is taken as the actual configuration at
t=0, the description is usually called the Langrangian description.
In contrast, if a physical property of the body B is expressed in terms of
the spatial coordinates and time t, we say that property is given by the
spatial or Eulerian description.

17. Langrangian and Eulerian coordinates …
Problem 4: Let the motion equations be given in components form by
Determine the Eulerian description of this motion.

19. Langrangian and Eulerian coordinates …
Problem 5: For the motion of Problem 4, determine the velocity and acceleration fields, and express these in both Langrangian and Eulerian forms.

21. Aims and Objectives kinematics of deformation and motion
material and spatial coordinates
Reference P
Current P

22. Aims and Objectives
Langrangian and Eulerian description