Download - MAE 640 Lec7
-
8/14/2019 MAE 640 Lec7
1/12
Continuum mechanics MAE 640Summer II 2009
Dr. Konstantinos Sierros
263 ESB new add
-
8/14/2019 MAE 640 Lec7
2/12
Kinematics of continuaIntroduction
Continuum mechanics is concerned with a study of various forms of matter at
macroscopic level.
Matter at sufficiently large length scales can be treated as a continuum in which all
physical quantities of interest, including density, are continuously differentiable.
The present chapter is devoted to the study of geometric changes in a continuous
medium that is in static or dynamic equilibrium.
The study of geometric changes in a continuum without regard to the forces causing
the changes is known as kinematics.
-
8/14/2019 MAE 640 Lec7
3/12
Descriptions of motionConfigurations of a continuous medium
Consider a body B of known geometry, constitution, and loading in a 3D Euclideanspace R3
For a given geometry and loading, the body B will undergo macroscopic geometric
changes within the body, which are termed deformation.
The geometric changes are accompanied by stresses that are induced in the body.
If the loads are applied slow enough, so that the deformation is only dependent on the
loads and not the time, the body will occupy a continuous sequence of geometrical
regions.
The region occupied by the continuum at a given time tis termed a configuration and
denoted by .
The simultaneous positions occupied in space R3 by all material points of the
continuum B at different instants of time are called configurations.
-
8/14/2019 MAE 640 Lec7
4/12
Descriptions of motionConfigurations of a continuous medium
The continuum initially occupies a configuration 0In this configuration a particleXoccupies the position X (referred to a rectangularCartesian system (X1, X2, X3))
X is a particle that occupies the location XX1, X2, X3 are called material coordinates
-
8/14/2019 MAE 640 Lec7
5/12
Descriptions of motionConfigurations of a continuous medium
After the application of the loads, the continuum changes its geometric shape and thus
assumes a new configuration , called the currentordeformed configuration.
The particleXnow occupies the position x in the deformed configuration The mapping : B0 Bis called the deformation mappingof the body B from 0 to
The deformation mapping (X) takes the position vectorX from the referenceconfiguration and places the same point in the deformed configuration as x = (X).
-
8/14/2019 MAE 640 Lec7
6/12
Descriptions of motionConfigurations of a continuous medium
The componentsXiandxiof vectors X =Xiiand x =xii are along the coordinates
used. We assume that the origins of the basis vectors iand i coincide. The mathematical description of the deformation of a continuous body follows one of
the two approaches:
(2) The material or Lagrangian description
(3) The spatial or Eulerian description.
-
8/14/2019 MAE 640 Lec7
7/12
Descriptions of motionMaterial or Lagrangian description
The motion of the body is referred to a reference configuration R, which is often
chosen to be the undeformed configuration,
R=
0.
The current coordinates (x ) are expressed in terms of the reference coordinates
(X 0);
The variation of a typical variable over the body is described with respect to
the material coordinates X and time t;
-
8/14/2019 MAE 640 Lec7
8/12
Descriptions of motionMaterial or Lagrangian description
For a fixed value ofX 0, (X, t) gives the value of at time tassociated with the
fixed material pointXwhose position in the reference configuration is X, as shown infigure above
If time t changes, the same material particle X (which occupies a position X in k0) will
now have a different value
Therefore, the attention is focused towards the material particles X of the continuum
body
-
8/14/2019 MAE 640 Lec7
9/12
Descriptions of motionSpatial or Eulerian description
In the spatial description, the motion is referred to the current configuration occupied
by the body B, and is described with respect to the current position (x )
in space, currently occupied by material particleX:
-
8/14/2019 MAE 640 Lec7
10/12
Descriptions of motionSpatial or Eulerian description
The coordinates (x) are termed the spatial coordinates.Different material points occupy the position x at different times, as shown in thefigure above A change in time timplies that a different value is observed at the same spatial
location x , now probably occupied by a different material particleX.
Therefore, attention is focused on a spatial position x
-
8/14/2019 MAE 640 Lec7
11/12
Descriptions of motion
When is known in the material description, = (X, t), its time derivative is simply thepartial derivative with respect to time because the material coordinates X do not changewith time:
Material description
Spatial description
When is known in the spatial description, = (x, t), its time derivative is:
Where v is velocity (v=dx/dt)
-
8/14/2019 MAE 640 Lec7
12/12
Use of spatial description
In the study of solid bodies, the Eulerian (Spatial) description is less useful since the
configuration is unknown.
For the study of motion of fluids, the Spatial description is preferred since the
configuration is known and remains unchanged
In the Eulerian description, attention is focused on a given region of space instead of a
given body of matter.