symmetry of the stess tensor应力张量的对称性.pptVIP

* SYMMETRY OF THE STRESS TENSOR The stress tensor ?ij satisfies the symmetry condition This condition is a consequence of the conservation of moment of momentum. Consider a volume of moving fluid (rather than a fixed volume through which fluid flows in and out) containing mass m, accelerating at rate and subjected to surface force and body (gravitational) force . Newton’s second law requires that Conservation of momentum requires the following. Where denotes an arbitrarily chosen moment arm, SYMMETRY OF THE STRESS TENSOR A complete proof that the stress tensor ?ij is symmetric is rather tedious. Here we simplify the problem by a) considering only the surface force and b) demonstrating that ?12 = ?21. At the end of the lecture we show how the result generalizes to the other shear stresses (?23 = ?32 and ?13 = ?31) and the case for which the body force (gravity) is included. We demonstrate the desired result (?12 = ?21) by taking moments about the x3 axis of the illustrated control volume, which is moving with the fluid. Since we have dropped the body force, the relevant balance equation is SYMMETRY OF THE STRESS TENSOR x1 x2 x3 ?x1 ?x3 ?x2 a1 a2 a3 The control volume has dimensions ?x1, ?x2 and ?x3. The acceleration vectors a1, a2 and a3 are located at the center of the control volume. We wish to compute the moment of the acceleration vector about the x3 axis, as a prelude to computing . The acceleration a3 contributes nothing to this moment, as it is parallel to x3. The arm for computing the moment of a1 about x3 is (1/2)?x2. (1/2)?x2 (1/2)?x2 a1 The contribution to the moment from a1 is thus SYMMETRY OF THE STRESS TENSOR x1 x2 x3 ?x1 ?x3 ?x2 a1 a2 a3 The contribution to the moment about the x3 axis from a2 is computed as follows. The arm for computing the moment of a2 about x3 is (1/2)?x1. (1/2)?x1 (1/2)?x1 a2 The contribution to the moment from a2 is thus The x3 component of is thu