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Unformatted text preview: Lecture 3 Marine Hydrodynamics Lecture 3 1.7 Stress Tensor 1.7.1 Stress Tensor ij The stress (force per unit area) at a point in a uid needs nine components to be completely specified, since each component of the stress must be defined not only by the direction in which it acts but also the orientation of the surface upon which it is acting. The first index i specifies the direction in which the stress component acts, and the second index j identifies the orientation of the surface upon which it is acting. Therefore, the i th component of the force acting on a surface whose outward normal points in the j th direction is ij . X 1 X 2 X 3 31 11 21 22 12 32 13 23 33 Figure 1: Shear stresses on an infinitesimal cube whose surfaces are parallel to the coordinate system. 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20 X 2 2 A 1 X 3 3 A 2 1 A 3 area A X 1 n P Q R Figure 2: Infinitesimal body with surface PQR that is not perpendicular to any of the Cartesian axis. Consider an infinitesimal body at rest with a surface PQR that is not perpendicular to any of the Cartesian axis. The unit normal vector to the surface PQR is n = n 1 x 1 + n 2 x 2 + n 3 x 3 . The area of the surface = A , and the area of each surface perpendicular to X i is A i = A n i , for i = 1 , 2 , 3. Newtons law: F i = (volume force) i for i = 1, 2, 3 on all 4 faces Note: If is the typical dimension of the body : surface forces 2 : volume forces 3 An example of surface forces is the shear force and an example of volumetric forces is the gravity force. At equilibrium, the surface forces and volumetric forces are in balance. As the body gets smaller, the mass of the body goes to zero, which makes the volumetric forces equal to zero and leaving the sum of the surface forces equal zero. So, as , all 4 faces F i = for i = 1 , 2 , 3 and i A = i 1 A 1 + i 2 A 2 + i 3 A 3 = ij A j . But the area of each surface to X i is A i = A n i . Therefore i A = ij A j = ij ( A n j ), where ij A j is the notation (represents the sum of all components). Thus i = ij n j for i = 1, 2, 3, where i is the component of stress in the i th direction on a surface with a normal n . We call...
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- Spring '05