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CONTINUUM MECHANICS ( STRESS DISTRIBUTION ). Stress vector. State of stress. Stress distribution. Surface traction (loading). Volume V Surface S. Stress ve c tor. x 3. Volume V 0 Surface S 0. Volumetric force. x 2. x 1. GGO theorem.
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Stress vector State of stress Stress distribution
Surface traction(loading) Volume V SurfaceS Stress vector x3 Volume V0 SurfaceS0 Volumetric force x2 x1 GGO theorem
On the body surface stress vector has to be balanced by the traction vector Stress on the body surface Coordinates of vector normal to the surface This equation states statics boundary conditions to comply with the solution of the equation: This equation (Navier equation) reflects internal equilibrium and has to be fulfilled in any point of the body (structure).
Navier equation in coordintes reads: We have to deal with the set of 3 linear partial differential equations. There are 6 unknown functions which have to fulfil static boundary conditions (SBC): We need more equations to determine all 6 functions of stress distribution. To attain it we have to consider deformation of the body.
Comments • Equation is derived from one of two equilibrium equations, i.e. that the sum of forces acting over the body has to vanish. • The other equilibrium equation – sum of the moments equals zero – yield already assumed symmetry of stress matrix, σij= σji • Navier equation is the special case of the motion equation i.e. uniform motion (no inertia forces involved). The inertia effects can be included by adding d’Alambert forces to the right hand side of Navier equation.
