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Continuum mechanics MAE 640Summer II 2009

Dr. Konstantinos Sierros

263 ESB new add

kostas.sierros@mail.wvu.edu

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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.

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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.

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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

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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).

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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.

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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;

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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

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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:

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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

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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)

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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.