By Jirasek M.
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Extra info for Modeling of Damage and Fracture in Quasibrittle Materials. Lecture Notes
______, )h 2 - a 2• Hint. Usc a fractional linear transformation l ~~z-c X z+c FIGURE to map the given region into a circular ring. to 56. Find the stationary tempe;ature distribution in a wall of thickness a near the corner of a building (see Figure 11), assuming that the temperature of the inside surface of the wall is T0 , while the temperature of the outside surface is zero. Ans. T ~T 0 Re (~;In~). )"' , ~-1 and In and arc tan denote the branches which go to zero as t --+ 0. 38 SOME SPECIAL METHODS I PROB.
However, as a rule, the mixed problem requires the use of the methods considered in C haps. 4- 7, among which the method of the Laplace t ransform (see Chap. 6, Sec. 3) is particularly effective. PROB. JTj p, T is the tension and p the linear density of the string, and the region of integration D is shown m Figure 5. Hint. Introduce new variables x - vt = a, x + vt = 21 Tl - ~ OL----x--~vt~~~~~x~+~v7 t--~'( ~· 30. Study the oscillations of an infiFIGURE 5 nite string produced by a concentrated load Q = Q(t) moving along the string in the direction of the positive x-axis with velocity v0 < v (where v is the velocity of propagation of oscillations along the string).
47 STEADY-STATE HARMONIC OSCILLATIONS 81 78. Find the forced longitudinal oscillations of a rod of length /, if the end 0 is clamped while the end x = I is acted upon by a force A si n (wt rp). + "= Ans. wx u(x, t) = sm Av _ _v_ sin ( wt ESw coswl + rp ), v v -J where = E/ p, E is Young's modulus, p is the density and S the crossNcctional area of the rod. 79. Find the forced oscillations of a beam simply supported at the ends ,. = 0 and x = I, under the action of a uniformly distributed pulsating load rt sin wt.