53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS
SCALAR FIELDS
Ø Gradient of a scalar field f(x), a vector field: ¶f Ñf( x ) = ei ¶xi
Ø Directional derivative of f(x) in the direction s: Let u be a
unit vector that defines the direction s: ¶x u = i e ¶s i
Ø The directional derivative of f in the direction u is given as df = u · Ñf ds
Ø The scalar component of Ñf in any direction gives the rate
of change of df/ds in that direction.
Ø Let q be the angle between u and Ñf. Then df = u · Ñf = u Ñf cos q = Ñf cos q ds
Ø Therefore df/ds will be maximum when cosq = 1; i.e., q = 0.
This means that u is in the direction of f(x).
Ø Ñf points in the direction of the maximum rate of increase
of the function f(x).
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Ø The magnitude of Ñf equals the maximum rate of increase
of f(x) per unit distance.
Ø If direction u is taken as tangent to a f(x) = constant
curve, then df/ds = 0 (because f(x) = c). df = 0 Þ u· Ñf = 0 Þ Ñf is normal to f(x) = c curve or surface. ds
THEORY OF CONSERVATIVE FIELDS
Vector Field
Ø Rule that associates each point (xi) in the domain D (i.e.,
open and connected) with a vector F = Fxi(xj)ei; where xj =
x1, x2, x3. The vector field F determines at each point in
the region a direction.
Flow Lines of F
Ø A curve C passing through a region where F is defined and
is non-zero at all points in the region is called a flow line of
F provided (1) at every point on the curve a unique tangent
vector T is defined whose direction varies continuously
with C, and (2) T is parallel to F.
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Ø If the space curve C is traced by a particle, i.e., its
direction coincides with the direction of the vector field F
at that point, it is called a streamline or characteristic
curve.
Ø Since the direction of a flow line is uniquely determined by
the field F, it is impossible to have two different
directions at the same point, and therefore it is impossible
for two flow lines to intersect.
Ø Define R = position vector of an arbitrary point on the flow
line, and s = the arc length along the curve. The unit
tangent vector to the curve at that point, T is given as dR dx T = = i e ds ds i
Ø Since the point is on the flow line, T has same direction as
F. This can be written as dx T = bF Þ bF = i ; i = 1, 2, 3 i ds dx dx dx dx dy dz Ø Leads to the relationship: 1 = 2 = 3 ; or = = F1 F2 F3 Fx Fy Fz
Line Integral
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Ø Line integral of the tangential component of vector F along
the smooth curve C is defined as
Ø ò F · dR = ò Fidxi = ò(Fxdx + Fydy + Fzdz) C C C dR Ø Let T = Unit tangent vector to the path C. Then, T = ds
Ø Line Integral = Work done by force F in a displacement
along C, if F = Force Field dR ò F · dR = ò F · ds = ò F ·Tds = ò Fsds C C ds C C
where Fs is the tangential component of F
Conservative Fields
Ø Vector Field, F is conservative in domain D if there is a
scalar field in D such that, F = ÑF , F =Potential Function
or Potential of F.
Ø Theorem: Vector Field F, continuous in the domain D is
conservative if and only if the line integral of the
tangential component of F along every regular curve
depends only on the end points of the curve.
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Ø In that case the line integral of the tangential component
is simply the difference in potential of the end points; i.e.,
ò F · dR = Difference in the potential value at the end
points.
ò F · dR = ò Fidxi = ò(Fxdx + Fydy + Fzdz) = F( Q ) - F( P ) C C C
Ø Also, F = ÑF so that ¶F ò F · dR = ò ÑF · dR = ò dxi = ò dF ¶xi
\F · dR = dF
RESTATE
Ø A vector field F continuous in the domain D (i.e., open and
connected) is conservative if and only if it possesses any
one (and hence all) of the following properties:
1. It is the gradient of a scalar function; F = ÑF
2. Its line integral of the tangential component around any
regular closed curve C in D is zero, i.e., ò F · dR = 0
3. Its line integral of the tangential component along any
regular curve in D extending from, a point P to a point Q
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is independent of the path; i.e.,
ò F · dR = ò dF = F( Q ) - F( P ). C C
Ø If the domain D in which F is defined is simply connected,
then there is a fourth property equivalent to any one of
other three: F is curl free, Ñ ´ F = Ñ ´ ÑF = 0; i.e., Curl of F
¶F ¶Fy ¶Fy ¶F ¶F ¶F = 0 which means that x = ; = z ; z = x ¶y ¶x ¶z ¶y ¶x ¶z
Lecture #5 6