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fJ (7UHfINUUM MECHANICS THkUa1kB lfl <br />~~ <br />u ~ <br />a u <br />~w <br />~ u <br />o l a a~ a a r <br />T~ I~til 17t1. f.l.a Time lreWr for aoa arrW (1)). <br />was definitely time-dependent in both longwall and room-and-pillar min- <br />ing. However, the conclusions reached in recent studies differ considera- <br />bly. For example, Gentry (S) tends to agree with the critical area concept, <br />while Dahl and Choi ('n found that surface subsidence is time• <br />independent. <br />9J CONTINUUM MkCIIANICS THFANIFS <br />Continuum mechanics theories have been developed for almost every <br />kind of material behavior. including clustic (14-16), viscoelastic (17, IB) <br />and elastic-elastoplastic (fY). In the following discussions, only represen- <br />tative theories that have been field-tested will be presented, mainly be- <br />cause of the complexity of mathematical derivation involval. <br />9.3.1 FJastk Theory <br />Berry (18) analyzed elastic ground movement for three kinds of under• <br />ground excavation closure development: (I) nonclosurc (floor end root <br />never met), (2) partial closure, and (3) complete closure (Fig. 9.3.q. <br />Excavations were treated as displacement discontinuities by using a wm- <br />plex potential method for partial closure. The results were then extended <br />to nonclosurc and complete closure as the limits. It was found that surface <br />subsidence calculated for isotropic material was smaller than actual mea- <br />aurements. Therefore, the theory for trensverscly isotropic ground was <br />developed in two dimensions such that the vertical surface subsidence S. <br />is given by the equation <br />-S _ 2~nAha(K, + K,)1Q (9.3.1) <br />A is the cross-sectional arcs of the closure, the maximum of which is wm, <br />where w is the width of the opening and m is the thickness of the coal <br /> <br />