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). Lecture #5 1 53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS 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. Lecture #5 2 53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS 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 Lecture #5 3 53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS 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. Lecture #5 4 53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS 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 Lecture #5 5 53:244 (58:254) ENERGY PRINCIPLES IN STRUCTURAL MECHANICS 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.