By R.F. Streater
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Extra info for Mathematics of contemporary physics; proceedings of an instructional conference organized by the London Mathematical Society
2 Moments of components of the stress tensor The equilibrium equations in the volume , eq. 6), enable us to write the following 3N relationships if) 81 82 3t) d + + 8t2t 8 + 8t 8 T X2 X3 (8t lt 8 83 Xl X2 X3 Xl V if) 81 82 83 X l X2 X3 PK t d T 0. 15) tktNk = Ft is used . 3) Given the volume and surface forces on the entire surface 0 bounding volume V , the right hand sides of these equations are known. e. 3n (n + 1).
The Coriolis acceleration is not included in the right hand side of eq. 3) because the continuum does not move relative to the moving axes. 4) is a mass force. Here a particle is assumed to move along a circle of radius hand e denotes a unit vector directed along this radius from the centre of the circle. The origin of vector R lies on the rotation axis. 1 Field of stresses in a continuum 33 where IT denotes the potential energy of a mass force field. 6) External surface forces are forces distributed over surface 0 of volume V .
4 Examples of the states of stress First example. In the state of stress referred to as pure shear , the stresses on the surfaces which are orthogonal to i 3 as well as stresses tll, t22 are absent. 5) takes the form -t t21 o 0 t12 -t 0 0 -t =-t(t 2-tI2) =0. 6) determining the principal axis of stresses ~ is as follows 1 -t12 e1 +h2 1 1 1 1 12 12 12 e2= 0, t21 e1 -t12 e2= 0, - t12 e3= 0, e1 + e2 + e3= 1. 6. One of these equations should be a consequence of t he ot hers. One can see t hat the second equation repeats t he first equat ion.