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Continuum mechanics MAE 640Summer II 2009
Dr. Konstantinos Sierros263 ESB new add
[email protected]
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Vector calculus Derivative of a scalar function of a vector
The derivative ( d/ds ) is called the directional derivative of
.
We see that it is the rate of change of with respect to distance
and that it depends onthe direction in which the distance is
taken.
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Vector calculus Derivative of a scalar function of a vector
This is called gradient vector and isdenoted by grad
These are the covariant components of the gradient vector
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Scalar function is equal to constant
When the scalar function (x) is set equal to aconstant, (x) =
constant, a family of surfaces isgenerated.
A different surface is designated by different valuesof the
constant, and each surface is called a level surface , as shown in
the figure
The unit vector is tangent to a level surface.
If d/ds is zero, then grad must be perpendicular to and, hence,
perpendicular to a level surface .
Thus, if any surface is defined by (x) = constant,the unit
normal to the surface is determined from the
following equation;
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The del operator It is convenient to write the gradient vector
as;
And define grad as some operator operating on , that is, grad
.
The del operator
The del operator is a vector differential operator, and the
components /q 1, /q 2,and /q3 appear as covariant components.
The del operator has some of the properties of a vector, it does
not have them allbecause it is an operator.
A is a scalar (called the divergence of A)
A is a scalar differential operator
In Cartesian systems:
and using the summation convention
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Divergence and curl of a vector The dot product of a del
operator with a vector is called the divergence of a vector
anddenoted by;
If we take the divergence of the gradient vector, we should
have;
The Laplacian operator
In Cartesian systems
The curl of a vector is defined as the del operator operating on
a vector by means of the cross product;
The quantity n grad of a function is called the normal
derivative of , and is givenby;
and in Catersian system
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Vector expressions and their Cartesian component forms ( A,
B,
and C ) are vector functions, and U isa scalar function; ( 1, 2,
3 ) are the
Cartesian unit vectors
The examples presented illustratethe convenience of index
notation
in establishing vector identities andsimplifying vector
expressions. The
difficult step in these proofs isrecognizing vector operations
from
index notation.
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Cylindrical and Spherical Coordinate Systems
Two commonly used orthogonal curvilinear coordinate systems are
cylindrical coordinate system and spherical coordinate system
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Cylindrical and Spherical Coordinate Systems The matrix of
direction cosines between the orthogonal rectangular Cartesian
system( x, y, z ) and the orthogonal curvilinear systems ( r, , z )
and ( R, , ), respectively, are as
given by the equations below;
Table 2.4.2 summarizes base vectors, del and Laplace operators
in cylindrical andspherical coordinates
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Tensors Dyads and polyads Specification of stress at a point
requires two vectors, one perpendicular to the planeon which the
force is acting and the other in the direction of the force. Such
an object is
known as a dyad , or a second-order tensor . For example, the
surface force acting on a small element of area in a
continuousmedium depends not only on the magnitude of the area but
also on the orientation of thearea. The stress, which is force per
unit area, not only depends on the magnitude of theforce and
orientation of the plane but also on the direction of the
force.
A dyad is defined as two vectors standing side by side and
acting as a unit. A linear combination of dyads is called a dyadic
.
Let A1,A2, . . . , An and B1,B 2, . . . , B n be arbitrary
vectors. Then we can represent adyadic as;
Transpose of a dyadic
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Properties of dyadics
Dot product with a vector V :
The dot operation with a vector produces another vector. In the
first case, the dyad actsas a prefactor and in the second case as a
postfactor . The two operations in generalproduce different
vectors.
The dot product of a dyadic with itself is a dyadic
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Nonion form of a dyadic Let each of the vectors in the dyadic be
represented in a given basis system and usethe Cartesian system
Cartesian
system
We can display all of the components of a dyadic by letting the
k index run to the rightand the j index run downward:
This form is called the nonion form of a dyadic.
A dyad in 3D space has nine independent components in general,
each componentassociated with a certain dyad pair. The components
are thus said to be ordered.
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Explicit form of dyads
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The unit dyad The unit dyad is defined as;
Using the Kronecker delta symbol i j , the unit dyadic in an
orthogonal Cartesiancoordinate system can be written as
follows;
The double dot product The double-dot product between a dyad (
AB) and another dyad ( CD) is defined as thescalar;
The double-dot product is commutative.
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The double-dot product between two dyads in a rectangular
Cartesian system is givenby;
The double dot product
The trace of a dyad is defined as the double-dot product of the
dyad with the unit dyad;
The trace of a dyad
The trace of a tensor is invariant , called the first principal
invariant , and it is denoted byI 1;
The first, second, and third principal invariants of a dyadic
are defined to be;
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The trace of a dyad
In terms of the rectangular Cartesian components, the three
invariants have the form;
In the general scheme that is developed, scalars are the
zeroth-order tensors , vectorsare first-order tensors , and dyads
are second-order tensors . The third-order tensorscan be viewed as
those derived from triads , or three vectors standing side by
side.
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Transformations of components of a dyadic
A second-order Cartesian tensor may be represented in barred and
unbarredcoordinate systems as follows;
The unit base vectors in the barred and unbarred systems are
related by;
Direction cosines between barred and unbarred systems
The components of a second-order tensor transform according
to;
In orthogonal coordinate systems, the determinant of the matrix
of direction cosines isunity and its inverse is equal to the
transpose;
