A History of the Theory of Elasticity and of the Strength of Materials; From Galilei to the Present Time Volume 2, PT. 2
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 edition. Excerpt: ...same title as the latter: see our Arts. 1571-99. (6) Sur les lois de la distribution plane des pressions a I'interieur des corps isotropes dans Vital cFequilibre limite. T. Lxxviii., 757-9 and 786-9. Paris, 1874. In the first part of this paper Boussinesq supposes a conservative system of body-forces applied to a mass under uniplanar strain. The first two body-stress equations then become: dxx dxy dj) _ Q dxy dyy df dx dy dx ' dx dy dy tf being the potential of the body-forces. Hence we find, as in our Arts. 1576-7, for an elastic body: dx? dy dxdy and for a pulverulent or plastic mass: ( - ) (+4 (--)-0 ( dx' dyjxx + yyj dxdy xx + yy/' The remainder of the first part of the memoir is devoted to the condition of limiting equilibrium discussed in our Art. 1585. (c) The second part of the memoir is devoted to discussing the integration of equations (i) for the case of limiting equilibrium. Boussinesq, as in our Art. 1568, takes T, and T% for the principal tractions, and puts p----(Tt + 7'2), q = (T1-T ), then for limiting equilibrium q will be a function of p and generally a linear one. We have, if a be the angle the greater traction T1 makes with the axis of x: xx =-p + q COS 2a, y?=-p-q cos 2a, 7y = q sin 2a. Hence from (i), if P = p--$: Boussinesq remarks that (iv) can be solved when 1 q is a constant, i.e. 2K: see our Art. 1863, and 2 when f = 0, or the weight of the material is negligible as compared with the pressures to which it is subjected. In both these cases q is a given function (very approximately linear) of the sole variable p--p or P, and (iv) contains therefore only the variables P and a. Boussinesq now proceeds to take P and a as the independent variables, or solves the equations (iv) for x and y in terms of P...
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